Discrete Geometry for Computer Imagery: 10th International Conference, DGCI 2002, Bordeaux, France, April 3-5, 2002. Proceedings

Lecture Notes in Computer Science Edited by G. Goos, J. Hartmanis, and J. van Leeuwen 2301 3 Berlin Heidelberg New Y...

Author: Achille Braquelaire | Jacques-Olivier Lauchaud | Anne Vialard

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Lecture Notes in Computer Science Edited by G. Goos, J. Hartmanis, and J. van Leeuwen

2301

3

Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Tokyo

Achille Braquelaire Jacques-Olivier Lachaud Anne Vialard (Eds.)

Discrete Geometry for Computer Imagery 10th International Conference, DGCI 2002 Bordeaux, France, April 3-5, 2002 Proceedings

13

Series Editors Gerhard Goos, Karlsruhe University, Germany Juris Hartmanis, Cornell University, NY, USA Jan van Leeuwen, Utrecht University, The Netherlands Volume Editors Achille Braquelaire Jacques-Olivier Lachaud Anne Vialard LaBRI, Universit´e Bordeaux 1 351 cours de la Lib´eration, 33405 Talence cedex, France E-mail: {achille.braquelaire,lachaud,vialard}@labri.fr

Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Discrete geometry for computer imagery : 10th international conference ; proceedings / DGCI 2002, Bordeaux, France, April 3 - 5, 2002. Achille Braquelaire ... (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Tokyo : Springer, 2002 (Lecture notes in computer science ; Vol. 2301) ISBN 3-540-43380-5

CR Subject Classiﬁcation (1998): I.4, I.3.5, I.5, G.2, I.6.8, F.2.1 ISSN 0302-9743 ISBN 3-540-43380-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Printed in Germany Typesetting: Camera-ready by author, data conversion by PTP-Berlin, Stefan Sossna Printed on acid-free paper SPIN: 10846474 06/3142 543210

Preface

DGCI 2002, the tenth in a series of international conferences on Discrete Geometry for Computer Imagery, was held in Bordeaux, France April 3–5, 2002. The aim of the conference was to present recent advances in both theoretical aspects and applications of discrete geometry. It was organized by the Laboratory of Computer Science of Bordeaux (Bordeaux 1 University) and sponsored by the International Association for Pattern Recognition (IAPR) and the French National Center of Scientiﬁc Research (CNRS). This DGCI conference conﬁrmed the increasing interest of the computer imagery community for discrete geometry, with 67 papers submitted from 23 countries all around the world. After reviewing, 35 contributions were accepted from which 22 were selected for oral presentation and 13 for poster presentation. These contributions focus mainly on the following topics: Models for Discrete Geometry, Topology, Combinatorial Image Analysis, Morphological Analysis, Segmentation, Shape Representation and Recovery, and Applications of Discrete Geometry in Image Processing and Computer Graphics. This program was completed by invited lectures from three internationally known speakers: Alfred M. Bruckstein (Haifa Computer Science Dept, Israel), Gabor Herman (City University of New York, USA), and Walter Kropatsch (Technical University of Vienna, Austria). Many people have contributed to the organization of the conference. In particular we would like to thank all the authors who submitted papers and the invited speakers for their contribution. Also we would like to thank the program committee and the reviewer board for their careful review and the members of the Steering Committee and of the Local Committee for their help. We are grateful to the following institutions for their ﬁnancial support: the Bordeaux 1 University, R´egion Aquitaine, the CNRS, the LaBRI (Laboratory of Computer Science of Bordeaux), the ENSEIRB (National School of Engineers in Electronics, Computer Science, and Telecommunication of Bordeaux), and the City of Bordeaux. Finally we thank all the participants and we hope that they found interest in the scientiﬁc program and that they enjoyed their stay in the capital of Gasconha.

January 2002

Achille Braquelaire Jacques-Olivier Lachaud Anne Vialard

Organization

Conference Co-chairs A. Braquelaire J.P. Domenger J.O. Lachaud

LaBRI, Bordeaux, France LaBRI, Bordeaux, France LaBRI, Bordeaux, France

Steering Committee E. Ahronovitz G. Bertrand G. Borgefors J.M. Chassery A. Montanvert M. Nivat

France France Sweden France France France

Program Committee E. Andres A. Del Lungo U. Eckhardt C. Fiorio R.W. Hall T.Y. Kong W. Kropatsch A. Kuba J.O. Lachaud R. Malgouyres S. Miguet I. Ragnemalm P. Soille G. Szekely

IRCOM-SIC, Poitiers, France University of Sienna, Italy Universit¨ at Hamburg, Germany LIRMM, Montpellier, France Dept. of EE, University of Pittsburgh, USA CUNY, New York, USA TU, Vienna, Austria University of Szeged, Hungary LaBRI, Universit´e Bordeaux 1, France LLAIC, Universit´e Clermont 1, France ERIC, Universit´e Lyon 2, France ISY, Dept. of EE, Link¨ oping University, Sweden Joint Research Center, Ispra, Italy ETH-Z¨ urich, Switzerland

Local Organizing Committee S. Alayrangues A. Ali-Mhammad G. de Dietrich P. Desbarats B. Kerautret B. Taton A. Vialard

LaBRI, LaBRI, LaBRI, LaBRI, LaBRI, LaBRI, LaBRI,

Bordeaux, Bordeaux, Bordeaux, Bordeaux, Bordeaux, Bordeaux, Bordeaux,

France France France France France France France

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Organization

Referees E. Ahronovitz S. Alayrangues E. Andres E. Balogh G. Bertrand G. Borgefors A. Braquelaire R. Breton S. Brunetti J. Burguet L. Buzer J.M. Chassery D. Coeurjolly P. Costantini A. Daurat

A. Del Lungo M. Dudasne U. Eckhardt C. Fiorio S. Fourey A. Frosini S. Gueorguieva R.W. Hall E. Katona T.Y. Kong W. Kropatsch A. Kuba J.O. Lachaud R. Malgouyres E. Mate

Sponsoring Institutions CNRS R´egion Aquitaine Universit´e Bordeaux 1 LaBRI ENSEIRB Communaut´e Urbaine de Bordeaux Ville de Bordeaux

S. Miguet A. Montanvert A. Nagy I. Nystr¨ om K. Palagyi A. Pasini I. Ragnemalm S. Rinaldi G. Sanniti di Baja P. Soille G. Subrenat G. Szekely L. Tougne A. Vialard

Table of Contents

Topology Invited Paper: Abstraction Pyramids on Discrete Representations . . . . . . . . W.G. Kropatsch

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XPMaps and Topological Segmentation – A Uniﬁed Approach to Finite Topologies in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 U. K¨ othe Curves in ZZ n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 G. Th¨ urmer Separation Theorems for Simplicity 26-Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 45 J.C. Ciria, E. Dom´ınguez, A.R. Franc´es Topological Quadrangulations of Closed Triangulated Surfaces Using the Reeb Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 F. H´etroy, D. Attali Non-manifold Decomposition in Arbitrary Dimensions . . . . . . . . . . . . . . . . . . 69 L. De Floriani, M.M. Mesmoudi, F. Morando, E. Puppo

Combinatorial Image Analysis 4D Minimal Non-simple Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 C.J. Gau, T. Yung Kong Receptive Fields within the Combinatorial Pyramid Framework . . . . . . . . . . 92 L. Brun, W.G. Kropatsch A New 3D 6-Subiteration Thinning Algorithm Based on P -Simple Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 C. Lohou, G. Bertrand Monotonic Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Y. Song, A. Zhang Displaying Image Neighborhood Hypergraphs Line-Graphs . . . . . . . . . . . . . . 124 S. Chastel, P. Colantoni, A. Bretto The Reconstruction of a Bicolored Domino Tiling from Two Projections . . 136 A. Frosini, G. Simi

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Table of Contents

Morphological Analysis Invited Paper: Digital Geometry for Image-Based Metrology . . . . . . . . . . . . . 145 A.M. Bruckstein Topological Reconstruction of Occluded Objects in Video Sequences . . . . . . 155 V. Agnus, C. Ronse On the Strong Property of Connected Open-Close and Close-Open Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 J. Crespo, V. Maojo, J.A. Sanandr´es, H. Billhardt, A. Mu˜ noz Advances in the Analysis of Topographic Features on Discrete Images . . . . 175 P. Soille Morphological Operations on Recursive Neighbourhoods . . . . . . . . . . . . . . . . 187 P.P. Jonker

Shape Representation Computing the Diameter of a Point Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 G. Malandain, J.-D. Boissonnat Shape Representation Using Trihedral Mesh Projections . . . . . . . . . . . . . . . . 209 L. Ros, K. Sugihara, F. Thomas Topological Map Based Algorithms for 3D Image Segmentation . . . . . . . . . . 220 G. Damiand, P. Resch On Characterization of Discrete Triangles by Discrete Moments . . . . . . . . . . 232 ˇ c J. Zuni´ Weighted Distance Transforms for Images Using Elongated Voxel Grids . . . 244 I.-M. Sintorn, G. Borgefors Robust Normalization of Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 J. Cortadellas, J. Amat, M. Frigola Surface Area Estimation of Digitized 3D Objects Using Local Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 J. Lindblad, I. Nystr¨ om

Models for Discrete Geometry Invited Paper: An Abstract Theoretical Foundation of the Geometry of Digital Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 G.T. Herman Concurrency of Line Segments in Uncertain Geometry . . . . . . . . . . . . . . . . . . 289 P. Veelaert

Table of Contents

XI

Discretization in 2D and 3D Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 M. Couprie, G. Bertrand, Y. Kenmochi Deﬁning Discrete Objects for Polygonalization: The Standard Model . . . . . 313 E. Andres Visibility in Discrete Geometry: An Application to Discrete Geodesic Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 D. Coeurjolly Multi-scale Discrete Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 J. Burguet, R. Malgouyres Invertible Minkowski Sum of Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 K. Sugihara

Segmentation and Shape Recognition Thinning Grayscale Well-Composed Images: A New Approach for Topological Coherent Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . 360 J. Marchadier, D. Arqu`es, S. Michelin An Incremental Linear Time Algorithm for Digital Line and Plane Recognition Using a Linear Incremental Feasibility Problem . . . . . . . 372 L. Buzer Reconstruction of Animated Models from Images Using Constrained Deformable Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 J. Starck, A. Hilton, J. Illingworth Reconstruction of Binary Matrices from Absorbed Projections . . . . . . . . . . . 392 E. Balogh, A. Kuba, A. Del Lungo, M. Nivat A Simpliﬁed Recognition Algorithm of Digital Planes Pieces . . . . . . . . . . . . . 404 M.M. Mesmoudi

Applications Ridgelet Transform Based on Reveill`es’ Discrete Lines . . . . . . . . . . . . . . . . . . 417 P. Carr´e, E. Andres A Discrete Radiosity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 R. Malgouyres

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Abstraction Pyramids on Discrete Representations Walter G. Kropatsch Institute for Computer-Aided Automation Pattern Recognition and Image Processing Group Vienna Univ. of Technology – Austria

Abstract. We review multilevel hierarchies under two special aspects: their potential for abstraction and for storing discrete representations. Motivated by claims to ‘bridge the representational gap between image and model features’ and by the growing importance of topological properties we discuss several extensions to dual graph pyramids and to topological maps: structural simpliﬁcation should preserve important topological properties and content abstraction could be guided by an external knowledge base.

1

Introduction

At a panel of the last International Workshop on Visual Form (IWVF4) Sven Dickinson asked the following question referring to several research issues in the past and also in the future: “How do we bridge the representational gap between image features and coarse model features?” He identiﬁes the one-to-one correspondence between – salient image features (pixels, edges, corners,...) and – salient model features (generalized cylinders, polyhedrons, invariant models,...) as limiting assumption that makes prototypical or generic object recognition impossible. He suggested to bridge and not to eliminate the representational gap, and to focus eﬀorts on: – region segmentation – perceptual grouping – image abstraction Let us take these goals as a guideline to re-consider research eﬀorts in the area of multiresolution discrete representations under the special viewpoint of abstraction and of representations that are discrete in nature. Regions as aggregations of primitive pixels play an extremely important role in nearly every image analysis

This work was supported by the Austrian Science Foundation under grants P14445MAT and P14662-INF.

A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 1–21, 2002. c Springer-Verlag Berlin Heidelberg 2002

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W.G. Kropatsch

task. Their internal properties (color, texture, shape, ...) help to identify them and their external relations (adjacency, inclusion, similarity of properties) are used to build groups of regions having a particular meaning in a more abstract context. The union of regions forming the group is again a region with both internal and external properties and relations. A representational concept that supports processes working at multiple levels of abstraction with the possibility to access semantic knowledge from the external world are extremely complex. We would like to highlight a few approaches that may have the potential to be extended into a future complex vision system bridging the representational gap identiﬁed by Dickinson. At the last DGCI Udupa [35] considered surfaces as basic descriptive elements for representing boundaries between volumetric regions in 3D. He identiﬁes three ‘important’ properties of such surfaces: 1. connected (topology) 2. oriented (combinatorial maps) 3. closed (Jordan boundary) We would like to address some of these issues in the context of gradually generalizing our discrete image data across levels where geometry dominates up to levels of the hierarchy where topological properties become important. Based on experiences with multiresolution pyramids we present several conceptual extensions with the aim of stimulating further research and collaboration which is necessary to overcome the intrinsic complexity of the proposed system architecture by joint eﬀorts leaving enough room for original contributions. The paper is organized as follows: After considering the formal deﬁnition of abstraction (section 2) and the consequences for representations we review discrete representations including a ‘natural’ example of vision based on an irregular sampling (section 3). Image pyramids are the main focus of section 4 where we present the basic ideas and properties of dual graph pyramids and of multilevel topological maps. Abstraction in such multilevel structures can be done either by modifying the contents of a representational cell (section 5) or by ‘simplifying’ the structural arrangement of the cells while major topological properties are preserved (section 6). In this last section we present two simple 3D conﬁgurations which turned out to be hard to distinguish by current representations.

2

Visual Abstraction

By deﬁnition abstraction extracts essential features and properties while it neglects unnecessary details. Two types of unnecessary details can be distinguished: – redundancies – data of minor importance Details may not be necessary in diﬀerent contexts and under diﬀerent objectives which reﬂect in diﬀerent types of abstraction. In general, three diﬀerent types of abstractionare distinguished:

Abstraction Pyramids on Discrete Representations

3

Isolating abstraction: important aspects of one or more objects are extracted from their original context. Generalizing abstraction: typical properties of a collection of objects are emphasized and summarized. Idealizing abstraction: data are classiﬁed into a (ﬁnite) set of ideal models, with parameters approximating the data and with (symbolic) names/notions determining their semantic meaning. These three types of abstraction have strong associations with well known tasks in computer vision: recognition and object detection tries to isolate the object from the background; perceptual grouping needs a high degree of generalization; and classiﬁcation assigns data to ‘ideal’ classes disregarding noise and measurement inaccuracies. In all three cases abstraction drops certain data items which are considered less relevant. Hence the importance of the data needs to be computed to decide which items to drop during abstraction. The importance or the relevance of an entity of a (discrete) description must be evaluated with respect to the purpose or the goal of processing. The system may also change its focus according to changing goals after knowing certain facts about the actual environment, other aspects that were not relevant at the ﬁrst glance may gain importance. Representational schemes must be ﬂexible enough to accommodate such attentional shifts in the objectives.

3

Discrete Representations

A discrete representation is associated with a countable variable which can be mapped into Zn [12]. A digital image is the result of sampling a continuous image at discrete locations the sampling points. Usually this is a ﬁnite subset of ‘pixels’ of the discrete grid Z2 . This discretization process maps any object of the continuous image into a discrete version if it is suﬃciently large to be captured by the sensors at the sampling points. Resolution relates the unit distance of the sampling grid with a distance in reality. There exist diﬀerent concepts to model the conversion between continuous and discrete representations. Recently Brimkov etal [5] have introduced new schemes for object discretizations in higher dimensions: k-neighbors in Zn , minimal cover and super-cover. Besides their relevance for visualization purposes such concepts allow also the continuous interpretation of discrete measurements. The properties of the continuous object, i.e. color, texture, shape, as well as its relations to other (nearby) objects are mapped into the discrete space, too. The most primitive discrete representation assigns to each sampling point a measurement, be it a gray or color value from a ﬁnite set or a binary value. Hence a digital image is a ﬁnite set of integer triples (ix, iy, ig) ∈ Z3 . In order to express the connectivity or other geometric or topological properties this set must be enhanced by a neighborhood relation. In the regular square grid arrangement of sampling points it is implicitly encoded as 4- or 8-neighborhood with the well known problems in conjunction with Jordan’s curve theorem. Note

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W.G. Kropatsch

that ALL the information about the image’s content is stored at the sampling points! The neighborhood of sampling points can be represented explicitly, too: in this case the sampling grid is represented by a graph consisting of vertices corresponding to the sampling points and of edges connecting neighboring vertices. Although this data structure consumes more memory space it has several advantages, among which we ﬁnd the following: – The sampling points need not be arranged in a regular grid. – The edges can receive additional attributes too. – The edges may be determined either automatically or depending on the data. Since sub-sampling of a discrete representation is also a discrete representation, the arrangement of a human retina can be considered as discrete, too. Fig. 1(a) shows a small portion of the sampling points of a monkey’s retina, which is similar to the one of a human eye. In Fig. 1(b-f) we opposed a simple sampling concept both with the natural but irregular grid and an artiﬁcial but regular grid of similar resolution. Sampling points being within a certain distance to the line have been ﬁlled and the radius has been varied. Let this distance be determined by a circle intersecting the continuous line. Optically similar eﬀects can be observed: with a small radius the sequence of bold points has many gaps (Fig. 1b,c,e), the line is not connected. Increasing the radius, the density of black points increases also and the gaps are closed. With a large radius the line becomes thick since also sampling points not directly along the line reach the line (Fig. 1d,f). These eﬀects are typical for discrete lines and they are overcome in many diﬀerent ways, but most approaches consider the regular grid. The problem arising with irregular grids is that there is no implicit neighbor definition! Usually Voronoi neighbors determine the neighborhood graph. It would be interesting whether concepts for discrete straight lines, discrete planes in 3D and hyper-planes in nD, discrete circles, spheres and hyper-spheres could be recovered also from irregular grids and if the involved computational processes are feasible. The retina example demonstrated that the neighborhood in irregular grids needs to be represented explicitly. This creates a new representational entity: the binary relation of an edge in the neighborhood graph. Together with the fact that a 2D image is embedded in the continuous image plane the line segments connecting the end points of edges partition the image plane into connected faces which are part of the dual graph. In n dimensions n + 1 basic entities are suﬃcient to describe a discrete conﬁguration embedded in the space spanned by the n coordinate axes, e.g. the cells of abstract cellular complexes [24]. Let us shortly remind some of the most frequent names used by diﬀerent authors: dimension geometry spel [35] graph 0 point pixel,pointel vertex 1 line linel edge 2 face, region surfel face 3 volume voxel

Abstraction Pyramids on Discrete Representations

Fig. 1. Sampling a line with irregular and regular grids

5

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The reason to review these basic representational entities is to point out that in nearly all discretization concepts only one of these entities (be it the point or the ‘dual’ correspondent, e.g. the face in 2D or the voxel in 3D) carries the measured information. Only few representational schemes allow the other entities to store information appropriate for the particular dimension of the manifold.

4

Pyramids

In this section we summarize the concepts developed for building and using multiresolution pyramids [33,21,28] and put the existing approaches into a general framework. The focus of the presentation is the representational framework, its components and the processes that transfer data within the framework. A pyra1 ✻ λ λ2 λ3

λn size

✡❏ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ✡ ❏ (a) Pyramid concept

(b) Discrete levels

Fig. 2. Multiresolution pyramid

mid (Fig. 2) describes the contents of an image at multiple levels of resolution. The base level is a high resolution input image. Successive levels reduce the size of the data by a constant reduction factor λ > 1.0 while constant size local reduction windows relate one cell at the reduced level with a set of cells in the level directly below. Thus local independent (and parallel) processes propagate information up and down in the pyramid. The contents of a lower resolution cell is computed by means of a reduction function the input of which are the descriptions of the cells in the reduction window. Sometimes the description of the lower resolution needs to be extrapolated to the higher resolution. This function is called the reﬁnement or expansion function. It is used in Laplacian pyramids [11] and wavelets [30] to identify redundant information in the higher resolution and to reconstruct the original data. The number of levels n is limited by the reduction factor λ: n ≤ log(image size)/ log(λ). The main computational advantage of image pyramids is due to this logarithmic complexity. We intent to extend the expressive power of these eﬃcient structures by several generalizations. The reduction window and the reduction factor relate

Abstraction Pyramids on Discrete Representations

7

two successive levels of a pyramid. In order to interpret a derived description at a higher level this description should be related to the original input data in the base of the pyramid. This can be done by means of the receptive ﬁeld (RF) of a given pyramidal cell ci : RF (ci collects all cells (pixels) in the base level of which ci is the ancestor. Since our goal is to bring up the ‘relevant data’ for solving a particular task let us give the term ‘resolution’ a more general meaning beyond the pure geometric deﬁnition. This is the base of several pyramidal approaches, two of which are chosen as representatives: irregular graph pyramids and topological maps.

4.1

Irregular Graph Pyramids

A graph pyramid is a pyramid where each level is a graph G(V, E) consisting of vertices V and of edges E relating two vertices. In the base level pixels are the vertices and two vertices are related by an edge if the two corresponding pixels are neighbors. This graph is also called the neighborhood graph. The content of the graph is stored in attributes attached to both vertices and edges. Initially only the attributes of the vertices receive the gray values of the pixels. In order to correctly represent the embedding of the graph in the image plane [19] we additionally store the dual graph G( V , E) at each level. Let us denote the original graph as the primal graph. In general a graph pyramid can be generated bottom-up as follows: while further abstraction is possible do 1. determine contraction kernels 2. perform dual graph contraction and simpliﬁcation of dual graph 3. apply reduction functions to compute content of new reduced level. The complete formalism of dual graph contraction is described in [28]. Let us explain it here by means a small window of our line example (Fig. 1). ✄❜ ✂ ✄❜

❜✁ ✄❜ ✄❜ ✂ ✄❜

✄❜

✂❜✁ ✂❜✁ ✄❜ ✄❜ ✂ ✁

✂❜✁ ✄❜ ✂ ✂r✁ ❜✁ ✂❜✁ ✄❜ ❜✁ ✄✂❜ ❜✁ ✂

✄❜

✄❜

✄❜ ✂ ✄❜

❜✁ ✄✂❜ ✄r r✁ ✂❜✁ ✂❜✁ ✂ ✄r r✁ ✂❜✁ ✄✂❜ ❜✁ ✂ ✄❜ ❜✁ ✄✂❜ ❜✁ ✄❜ ✂ ✄ ❜✁ ✄✂❜ ✂❜✁ ✂❜✁ ✂❜

❜✁ ✄❜ ✂❜✁ ✄❜ ✂❜✁

Fig. 3. Neighborhood graph G0 and contraction kernel N01

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The ﬁrst step determines what information in the current top level is important and what can be dropped. A contraction kernel is a (small) sub-tree of the top level the root of which is chosen to survive. Fig. 3 shows the window and the selected contraction kernels each surrounded by an oval. Selection criteria in this case contract only edges inside connected components except for isolated black vertices which are allowed to merge with their background. All the edges of the contraction trees are dually contracted during step 2. Dual contraction of an edge e (formally denoted by G/{e}) consists of contracting e and removing the corresponding dual edge e from the dual graph (formally denoted by G \ { e}). This preserves duality and the dual graph needs not be constructed from the contracted primal graph G at the next level. Since the contraction of an edge may yield multi-edges and self-loops there is a second phase of step 2 which removes all redundant multi-edges and self-loops. Note that not all such edges can be removed without destroying the topology of the graph: if the cycle formed by the multi-edge or the self-loop surrounds another part of the data its removal would corrupt the connectivity! Fortunately this can be decided locally by the dual graph since faces of degree two (having the double-edge as boundary) and faces of degree one (boundary = self-loop) cannot contain any further elements in its interior. Since removal and contraction are dual operations, the removal of a self-loop or of one of the double edges can be done by contracting the corresponding dual edges in the dual graph. The dual ✞❜ ✛❜ ✘ ✞❜ ✞❜ ❜ ❍❍ ✝ ✆ ❅ ❅ ❅ ❍❍ ✞❜ ✛ ✘ ❜ ❅ ❜ ❅ ❅r ✟ ✚ ✙ ✝ ✆ ❅ ❅ ❅ ✟✟ ✁❅ ✛ ✘ ✟ ✁ ❅❜ ❜ ❅❜ ❅r ❅❜ ✝❅ ✆ ✁ ✝✆ ✟✚ ✙ ❆ ❅ ✟ ✟ ✛ ✘ ✛ ✘ ✁ ❆ ❅✟ r❍ ❅❜ ❜❍ ✚ ✙ ❍❍ ❅ ❍❍❆ ❆✞❜ ❍❜ ❅❜ ❜ ❍ ❜ ✚✙✚ ✙ ✝ ✆ Fig. 4. Dually contracted graph G1 and contraction kernel N12

contraction of our example remains a simple graph G1 without self-loops and multi-edges (Fig. 4). Step 2 generates a reduced pair of dual graphs. Their contents is derived in step 3 from the level below using the reduction function. In our example reduction is very simple: the surviving vertex inherits the color of its son. In the only case where the contraction kernel contains two diﬀerent colors, the isolated vertex is always chosen as surviving vertex. The result of another dual contraction is shown in Fig. 5. The selection rules and the reduction function are the same as in the ﬁrst iteration. The result shows that the bridges between the gaps of the original sampling has been closed and the three surviving black vertices are connected after two iterations. This fact

Abstraction Pyramids on Discrete Representations

❜ ❇

9

❜ ❜P ❜ PP ✡❅ PP ✡ ❅ P P ❇ ✏r ✏✏ ✡ ❅ ✏ ❇ ✏ ✡ ❅ r ✏ ❇ ✘✏ ✘ ✘ ✘ ❇ ✡ ✘✘✘ ❇✘ r ✘ ✡ ❜ ❜ P  PP  P  P P ❜ P

Fig. 5. Graph G2 after two steps of dual graph contraction

could be used in a top-down veriﬁcation step which checks the reliability of closing the gap in the more general context. There are lots of useful properties of the resulting graph pyramids. If the plane graph is transformed into a combinatorial map the transcribed operations form the combinatorial pyramid [6,7]. This framework allowed us to prove several of the above mentioned properties and links dual graph pyramids with topological maps which extend the scope to three dimensions. The following table summarizes dual graph contraction in terms of the control parameters used for abstraction and the conditions to preserve topology: Level 0

1

4.2

representation contract / remove conditions (G0 , G0 ) ↓ contraction kernel N0,1 forest, depth 1 (G0 /N0,1 , G0 \ N0,1 ) ↓ redundant multi-edges, self-loops deg v ≤ 2 (G1 , G1 ) ↓ contraction kernel N1,2 forest, depth 1 .. .

The Topological Map

Fiorio, Bertrand and Damiand developed a method that derives the topologically correct region adjacencies both in 2D and in 3D [1,13]. Their base representations are combinatorial and generalized maps. The derivation proceeds in several levels similar to a pyramid with the diﬀerence that the lower levels are not used any more after the higher level has been created. In 2D following levels are identiﬁed: Level representation merge conditions 0 complete inter-pixel map ↓ adjacent faces same label 1 line map ↓ lines (l1 , v, l2 ) deg v = 2 and colinear(l1 , l2 ) 2 border map ↓ lines (l1 , v, l2 ) deg v = 2 3 topological map

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In addition to the topological map an inclusion tree is generated and maintained to cope with holes in regions. The resulting representation is computed by a sequential scan line procedure comparing 2 × 2 windows with prestored ’precodes’ each requiring a special treatment. The current approach has no limit on the size of the reduction window in the sense that any number of elements (’darts’) of level i may be merged into a single element at level i + 1. However a large set of fused elements could be decomposed into a hierarchy of locally independent fusions to allow parallel implementation. The extension into three dimensions is straight forward and contains six different levels [13]: Level representation merge conditions 0 complete inter-voxel map ↓ adjacent volumes same label 1 line map ↓ faces (f1 , l, f2 ) deg l ≤ 2, coplanar(f1 , f2 ) 2 level 2 map ↓ lines (l1 , v, l2 ) deg l = 2, colinear(l1 , l2 ) 3 border map ↓ faces (f1 , l, f2 ) deg v ≤ 2 4 level 4 map ↓ lines (l1 , v, l2 ) deg v = 2, l1 = l2 5 topological map Besides the inclusion tree that captures the 3D holes, ﬁctive edges are maintained to prevent disconnection of faces or lines. Additional conditions are implicitly expressed to prevent disconnections and suppression of basic elements. The process proceeds similar to the 2D case: 2×2×2 windows are compared with a set of ’pre-codes’ and the corresponding procedure is executed for each code. The levels and several categorizations permit to reduce the number of cases drastically from the ones that would be needed for the exhaustive enumeration.

5

Abstraction in Pyramids

In order to discuss the role of abstraction in general multiresolution hierarchies. Let us consider the structure of the representation and the content stored in the representational units separately. In our generalization we allow the resolution cell to take other simply connected shapes and to describe the content by a more complex ‘language’. The ﬁrst generalization is a consequent continuation of the observations of Bister etal [4] to overcome the limited representational capabilities of rigid regular pyramids. It necessitates to consider in more detail the structure of a resolution level which was implicitly coded as a matrix in regular sampling grids. A similar conclusion was expressed by DeFloriani etal [14] where a ‘multi-complex’ was presented as a unifying framework for many multiresolution regular cell complexes and extensions to cope with non-regular shapes are envisioned. Since irregular structures reduce the importance of explicitly representing geometry,

Abstraction Pyramids on Discrete Representations

11

topological aspects become relevant. These aspects will be discussed in more detail in section 6. The second generalization started with the works of Hartley [20] who allowed a resolution cell to contain more than one (gray) value and interpreted them as the parameters of a globally deﬁned model. 5.1

Content Models and Reduction Functions

In the topological map approach each cell contains a label identifying the membership of the cell to the class of all those cells having the same label. In this case the contents of the cells merged during the reduction process can be propagated by simple inheritance: the fused cell ‘inherits’ its label from its children and it does not matter from which since all have the same label. In classical gray level pyramids the contents of a cell is a gray value which is summarized by the mean or a weighted mean of the values in the reduction window. Such reduction functions have been eﬃciently used in Gaussian pyramids. Laplacian pyramids, Ratio pyramids and wavelet pyramids identiﬁed the loss of information that occurs in the reduced level and stored the missing information in the hierarchical structure where it could be retrieved when the original base level is reconstructed. All these approaches use one single globally deﬁned model which must be ﬂexible enough to adapt its parameters to approximate the data. In our generalization we would like to go one step further and allow diﬀerent models to be used in diﬀerent resolution cells as there are usually diﬀerent objects at diﬀerent locations of an image. The models could be identiﬁed by a name or a symbol and may be interrelated by semantic constraints. Simple experiments have been done with images of line drawings. This research used the experiences gained with a regular 2 × 2/2 curve pyramid [25] and the chain pyramid [31] in the more ﬂexible framework of graph pyramids. The model describes symbolically the way how a curve intersects the discrete segments of the boundary of a cell and the reduction function consists in the transitive closure of the symbols collected in the reduction window. The concept works well in areas where the density of curves was low, through the rigidity of the regular pyramid ambiguities arise when more curves appeared within the same receptive ﬁeld. This limitation could be eﬀectively overcome with irregular pyramids in which we could limit the receptive ﬁeld of a cell to a single curve. Fig. 6 gives an impression of the complexity of the data that have been processed in the minimum line property preserving (MLPP) pyramid in order to ﬁnd and identify components in a technical drawing. More details can be found in [8]. The content abstraction in this representation has following features: – models are identiﬁed by (discrete) names (empty cell, line end, line crosses edge, junction), no parameters were used; – adjacent models have to be consistent (‘good continuation’); – contraction kernels were selected such that only one consistent curve is covered in one receptive ﬁeld;

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Fig. 6. Technical drawing used to build an MLPP-pyramid

– a few symbolic rules of the form: new model → local generic sub-graph with model assignments

(1)

governed this selection process; – the reduced content appears in the left hand side of the rule (1). One may notice a certain similarity between the selection rule (1) and the use of pre-codes in the eﬃcient computation of the topological maps [2,1,13]. In both cases the knowledge about the models and in what conﬁgurations they are allowed to occur needs to be stored in a knowledge base. In order to determine which are the best possible abstractions the local conﬁgurations at a given level of the pyramid must be compared with the possibilities of reduction given in the knowledge base. This would typically involve matching the local conﬁguration with the right-hand sides of rules stored in the knowledge base. Such a match may not always be perfect, one may allow a number of outliers. The match results in a goodness of match, which can be determined for all local conﬁgurations. The selection can then choose the locally best candidates as contraction kernels and reduce the contents according to the generic models which matched the local conﬁguration. The goodness of match may also depend on a global objective function to allow the overall purpose, task or intention to inﬂuence the selection process. 5.2

The Knowledge Base

The knowledge to be used in the pyramid can be organized in many diﬀerent ways. It is needed in selecting surviving cells and as the model for the reduction

Abstraction Pyramids on Discrete Representations

13

function. Since the aim is to achieve abstraction it should provide also, besides a goodness of ﬁt, a measure of relevance or importance which may depend on varying goals of the system. These variations in the goals further need to be communicated to the system. Hence the language of interaction must be part of the knowledge base. It is certainly impossible to exhaustively enumerate all the possibilities to organize the knowledge. We list a few exemplary ways (1., 2.) one can ﬁnd in existing approaches and propose a structure (3.) which ﬁts to the concept sketched above. 1. The most common way to enter semantic knowledge into a system is by implicit coding. In the pyramid building process it appears as the parameters that control the selection function or the frequently used ﬁlters in the reduction function. Although often computationally eﬃcient implicit coding has no ﬂexibility to adapt to the data and any change requires a modiﬁcation of the program code. This type of knowledge representation can be found in most regular pyramid approaches which are based on linear ﬁlters. 2. In order to separate the knowledge from the process working on the data, globally coded rules or states are associated with speciﬁc procedures to treat the data. This knowledge is pre-compiled and is accessed through indices based on locally computed features. As examples can be mentioned the precodes for building the topological map and the syntactical rules used for the line drawing application (Fig. 6). These types of knowledge representations are more ﬂexible than the implicit coding since the knowledge base can be extended or adapted to the special needs of an application. However the knowledge is used mostly in a deterministic way without assigning special priorities to items important for the current task. Furthermore the knowledge is in all cases compiled manually which limits the scope of the application. A few ‘rules’ have been used for the line drawing example, application dependent rules would need an enormous amount of eﬀort. Similarly for the deﬁnition of pre-codes of the topological map: the 2D construction needs only 12 pre-codes which grows to more than 4000 cases in 3D. Although this number could be substantially reduced using the six levels and further categorization the extension to higher dimensions seems to be prohibitive. 3. The concept presented in the previous section suggests that the knowledge base can provide pre-stored local conﬁgurations that can be used to identify potential local contraction kernels. There are several approaches pointing in this direction, e.g. the one presented by Kittler etal [22] where three interrelated levels express the knowledge of the system: the measurements, the image features, and the object cues. In addition each ‘conﬁguration’ must be associated with the following: – A function computing the importance based on the goodness of match between the data conﬁguration and the pre-stored conﬁguration. – The more abstract description needs to be identiﬁed, e.g. by a name or a symbol.

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– The reduction function associated with the new identity calculates the speciﬁc parameters (attributes of the survivors) from the attributes of the data conﬁguration. Such a knowledge base could be realized as a formal (graph) grammar. However, as the examples already demonstrated, in order to be eﬀective the knowledge needs a non-negligible degree of complexity. This has consequences for both the knowledge retrieval and updating: Knowledge retrieval requires a high degree of internal organization to quickly access all those conﬁgurations that must be checked in a particular case. Exhaustive search may not be feasible. One could imagine a structure similar to the data pyramid since also the abstract terms have neighbors, e.g. associations in an abstract sense, more speciﬁc terms they are derived from and more abstract terms they are part of. Let us call this the abstraction pyramid in contrast to the data pyramid having the image data in the base. In this case the possible conﬁgurations to be checked in a particular part of the data pyramid could be local neighbors in the abstraction pyramid of the abstract term associated with the data cell. Note that a simpliﬁed version has been proposed for regular pyramids by P. Burt: the pattern tree [10]. The second consequence of the high complexity of the knowledge base concerns the updating: both interactive user interfaces and learning strategies could be integrated with the concept of the abstraction pyramid. However there is still a wide ﬁeld of research necessary before such systems could be used in ‘real’ applications.

6

Preserving Topology

Objects mapped into images remain connected if they are not occluded by other objects nor disturbed by noise. Neither the projection nor the discretization separate the two corresponding adjacent regions in the image. A similar property holds for the adjacency of objects. Hence the connectivity and the adjacency of regions and of boundary segments is a very important property which should not be lost by abstraction. Several authors studied operations that allow the modiﬁcation of the data structure, e.g. its reduction, while the topological properties of the represented objects and their background is preserved (e.g. [23, 32,3,26,15,9]). In the following we ﬁrst look at the simpler cases in two dimensions and refer to the dual graph pyramid. Then some considerations about the reduction operations in 3D based on the recent results of Damiand are discussed. 6.1

Preserving Topology in 2D

Table 1 summarizes the necessary primitive operations: The Euler number characterizes the topology of a description given in terms of points (#P ), lines (#L) and faces (#F ). Since we aim at preserving its value the sum of the changes must be zero: ∆#P − ∆#L + ∆#F = 0.

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15

Table 1. Topology Preserving Operations in 2D Euler Incr. Contract(l, p0 ) Remove(l, f0 ) Any Incr. by a contr. by c remov.

Points Lines Faces Conﬁg. #P −#L +#F ∆#P −∆#L +∆#F -1 -1 0 (p1 , l, p0 ) 0 -1 -1 (fx , l, f0 ) (−a −b −c) (−1 −1) ×a (−1 −1) ×c

PRE-CONDITION = const. =0 p1 = p0 fx = f0 b = a + c;

CCL Euler same label deg(f0 ) ≤ 2

First we observe the changes introduced by contracting an edge l bounded by the two points p0 , p1 . This eliminates one of the points (i.e. p0 ) and the edge l, hence it does not change the Euler characteristic. The only pre-condition is to avoid contracting a self-loop. If we remove an edge l, the number of points remains the same, but two faces f0 , fx are merged into one (fx ). That reduces the number of faces by one. If we would have the same face on both sides of the edge, i.e. fx = f0 , the edge would be a bridge in G the removal of which would disconnect G. If one of the end points of l would have degree 1, the removal of its only connection to the remaining structure would isolate it. Both cases are excluded from removal by the pre-condition fx = f0 . The second pre-condition deg(f0 ) ≤ 2 identiﬁes a redundant self-loop or a redundant multi-edge: in the later case f0 is bounded by two parallel edges connecting the same end-points. This conﬁguration is simpliﬁed in the second phase of dual graph contraction. What about other operations? It is clear that the elimination of an edge must be accompanied by the removal of either a point or a face to preserve the Euler number. So we cannot have less elements involved in a topology preserving modiﬁcation. But we can also show the following: Contraction and removal are the ONLY operations needed to reduce the structure while preserving the topology. Any other topologypreserving operation can be achieved by appropriate combinations of contraction and removals. If we want to remove a number a of points and a number c of faces we have to remove also a number b = a+c of edges to preserve the Euler number. This can be achieved by a contractions and c removals. Pre-conditions for individual operations can be extended to sets of operations to allow a diﬀerent order of execution or even parallelism: The requirement for contraction kernel to form a FOREST is such an extension. If the edges of a cycle would be contracted the last one need to be a self-loop which cannot be contracted. Hence sets of edges to be contracted must be acyclic. 6.2

What Remains after Repetitions?

We can repeat contracting edges the end point of which carry the same label and remove all unnecessary self-loops and multi-edges until no further contraction

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nor removal is possible. Note that a very similar strategy is used to create the border map in [2,1] and the topological map in [13]. At convergence we have the following conditions: 1. All edges (p1 , l, p2 ) with diﬀerent end points have diﬀerent labels: lab(p1 ) = lab(p2 ). 2. A surviving self-loop (p, l, p) separates two diﬀerent faces, (f1 , l, f2 ) and the inner face has degree deg(f1 ) > 2. Since any tree of the dual graph would have been eliminated by the rule deg(f1 ) ≤ 2 starting from the leafs up to the root, there must be a cycle C ∈ G and inside this cycle there exists a point p3 ∈ C : lab(p3 ) = lab(p). 3. All faces have three or more sides: deg(f ) ≥ 3. 4. Pseudo or ﬁctive edges are self-loops (p0 = p1 ) which cannot be contracted in the primal graph and which separate two faces with deg(f0 ) > 2 and deg(f1 ) > 2. Such edges were ﬁrst observed in [29] as an artifact of topology preserving contraction. They connect the boundary of a hole to the surrounding ‘main land’. Holes can be equivalently represented by an inclusion tree as in [13]. 5. Fictive edges appear arbitrarily placed and depend only on the order of contractions and removals. Similar observations can be found in the topological 3D-map [13] where ﬁctive edges appear as the last ones before disconnecting a face or a boundary during the merging of faces and lines (for the process of successive region merging see [1,2]). 6. For each hole there remains exactly one ﬁctive edge (as indicated by the Betti number [17]). 7. Fictive edges are not delineated between two regions as all other edges. Hence they can be continuously deformed and their end points can be moved along the boundary as long as the edge remains fully embedded inside f . Other ﬁctive edges are not excluded from being traversed by the end point! We conjecture that an arrangement of ﬁctive edges can be transformed into any other legal arrangement of ﬁctive edges. Algorithms for continuous deformation [34] or [18] may ﬁnd a new application for re-arranging ﬁctive edges. 6.3

Preserving Topology in 3D

The primitive operations to build the 3D topological map merge voxels (VFusion), faces (F-Fusion) and linels (L-Fusion). In analogy to the 2D table Table 2 summarizes the necessary primitive operations in 3D: 1. V-Fusion, F-Fusion and L-Fusion are the ONLY operations needed, the reasoning is the same as in 2D. Any other topology-preserving operation can be achieved by appropriate combinations. 2. Pre-conditions in 3D are non-trivial except for volumes: a line may delimit more than 2 faces, and a point may be the intersection of more than 2 lines. Damiand [13] lists additional constraints: no disconnection and no suppression of any face or line should be possible.

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Table 2. Topology Preserving Operations in 3D Pts. Lin. Fac. Vol. Conﬁg. Euler #P −#L +#F −#V Incr. ∆#P −∆#L +∆#F −∆V V-Fusion -1 -1 (v1 , f, v2 ) F-Fusion -1 -1 (f1 , l, f2 ) L-Fusion -1 -1 (l1 , p, l2 )

PRE-CONDITION = const. =0 v1 = v2 f1 = f2 l1 = l2

CCL Euler same label deg l ≤ 2 deg p ≤ 2

3. It remains to be checked whether the pre-conditions for individual operations can be extended to sets of operations. 4. Semantic control (i.e. checking the same label as for CCL) occurs only in the initial fusions, all other operations are automatic simpliﬁcations. 6.4

What Should Remain in 3D after Repetition?

Fig. 7. Are the two 3D conﬁguration the same?

It is not surprising that the complexity of minimal topological cases to be considered grows with dimension. In 2D only holes have to be considered and correctly represented by either an inclusion tree or additional ﬁctive edges. In 3D, tunnels may be present additionally both in the foreground and in the background. Conﬁgurations like the two tori as in Fig. 7 need to be distinguished. The answer ‘there are two tori’ is not wrong but it is unsatisfactory since it does not distinguish the two depicted conﬁgurations. Most representations do not even have the necessary relation to express the interlacing between the two tori in the second case. In addition they are not connected. So the description most involve the topology of the surrounding background. Let us sketch a possible solution using ﬁctive elements. A ﬁctive surface intersecting the tunnel could be used to make a torus (genus 1) homotopic to

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Fig. 8. Two tori with two ﬁctive surfaces and one ﬁctive edge

a sphere (genus 0). This ﬁctive surface is not ﬁxed geometrically in space but another object, like the second torus, would intersect it and create a hole. The boundary of the hole would be connected by a ﬁctive edge to the outer boundary of the ﬁctive surface added to the torus. An identical reasoning applies to the second torus creating in total two ﬁctive surfaces and one ﬁctive edge as depicted in Figure 8. The consequences for abstract representations open a wide range for further research and may address deep mathematical problems in diﬀerent ﬁelds: knot theory, Morse complexes, algebraic topology, . . . (see [17,16]).

7

Conclusion

We motivated our discussion by Dickinson’s and Udupa’s claims to ‘bridge the representational gap’, to ‘focus on image abstraction’, and to study ‘topological properties’ in the introduction. We ﬁrst discussed the basic ﬁelds, abstraction and discrete representation, in more detail. It seems that there are much less concepts working on discrete irregular grids than on their regular counterparts. We then recalled two pyramidal approaches having the potential to cope also with irregular grids. These pyramids have some useful properties, e.g. 1. they show the necessity to use multi-edge and self-loop to preserve the topology; 2. they allow to combine primitive operations at one level (i.e. collected by the contraction kernel) and across several levels of the pyramid (i.e. equivalent contraction kernels [27]); 3. repeated contraction converges to speciﬁc properties which are preserved during contraction; 4. termination criteria allow to stop abstraction before a certain property is lost. 5. pseudo/ﬁctive elements characterize topological relations, a ﬁctive edge characterizes a hole, a ﬁctive face characterizes a tunnel, . . . ; 7.1

Open Problems

There are numerous open problems partly addressed in the paper. Let us just enumerate a few important issues:

Abstraction Pyramids on Discrete Representations

19

✎ ✎ ✎ ✞ ✞ ✞ ✎ ✎ ✞ ✞ ✝ ✝ ✆✌ ✝ ✆✌ ✍ ✆✌ ✍ ✍ ✝ ✆ ✝ ✍ ✌ ✍ ✆✌ Fig. 9. The Olympic problem

1. extensions to 3D, 4D, 5D (see [35] for current data sources) 2. how to represent multiple interlaced tori, chains, the Olympic rings (Fig. 9) in a topologically correct representation? 3. Re-insertion of removed edges/darts (like reconstruction with wavelets): after reducing a level of the pyramid the data remaining in the level below could be checked for redundancies and store only the diﬀerences needed for loss-less reconstruction; 4. repeated contraction has several control parameters which allow adaptation to speciﬁc applications: – diﬀerent selection criteria – diﬀerent termination criteria – diﬀerent attributes – diﬀerent reduction functions 5. these control parameters could be organized in a knowledge base to allow further user interaction and automation through learning.

References [1] Y. Bertrand, G. Damiand, and C. Fiorio. Topological Encoding of 3D Segmented Images. In G. Borgefors, I. Nystr¨ om, and G. Sanniti di Baja, editors, Proceedings DGCI’00, Discrete Geometry for Computer Imagery, volume Vol. 1953 of Lecture Notes in Computer Science, pages 311–324, Uppsala, Sweden, 2000. Springer, Berlin Heidelberg, New York. [2] Y. Bertrand, C. Fiorio, and Y. Pennaneach. Border Map: A Topological Representation for nD Image Analysis. In G. Bertrand, M. Couprie, and L. Perroton, editors, Discrete Geometry for Computer Imagery, DGCI’99, volume Vol. 1568 of Lecture Notes in Computer Science, pages 242–257, Marne-la-Vall´ee, France, 1999. Springer, Berlin Heidelberg, New York. [3] J. C. Bezdek and N. R. Pal. An index of topological preservation for feature extraction. Pattern Recognition, 28(3):381–391, March 1995. [4] M. Bister, J. Cornelis, and A. Rosenfeld. A critical view of pyramid segmentation algorithms. Pattern Recognition Letters, Vol. 11(No. 9):pp. 605–617, September 1990. [5] V. E. Brimkov, E. Andres, and R. P. Barneva. Object Discretization in Higher Dimensions. In G. Borgefors, I. Nystr¨ om, and G. Sanniti di Baja, editors, Proceedings DGCI’00, Discrete Geometry for Computer Imagery, volume Vol. 1953 of Lecture Notes in Computer Science, pages 210–221, Uppsala, Sweden, 2000. Springer, Berlin Heidelberg, New York.

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[6] L. Brun and W. G. Kropatsch. Contraction Kernels and Combinatorial Maps. In J.-M. Jolion, W. G. Kropatsch, and M. Vento, editors, Graph-based Representations in Pattern Recognition, GbR 2001, pages 12–21. CUEN, 2001. ISBN 88 7146 579-2. [7] L. Brun and W. G. Kropatsch. Introduction to Combinatorial Pyramids. in print, 2001. Winterschool ”Digital and Image Geometry”, 17.12.2000 - 22.12.2000. [8] M. Burge and W. G. Kropatsch. A Minimal Line Property Preserving Representation of Line Images. Computing, Devoted Issue on Image Processing, 62:pp. 355– 368, 1999. [9] J. Burguet and R. Malgoures. Strong Thinning and Polyhedrization of the Surface of a Voxel Object. In G. Borgefors, I. Nystr¨ om, and G. Sanniti di Baja, editors, Proceedings DGCI’00, Discrete Geometry for Computer Imagery, volume Vol. 1953 of Lecture Notes in Computer Science, pages 222–234, Uppsala, Sweden, 2000. Springer, Berlin Heidelberg, New York. [10] P. J. Burt. Attention mechanisms for vision in a dynamic world. In Proc. 9th International Conference on Pattern Recognition, pages 977–987, Rome, Italy, November 1988. IEEE Comp.Soc. [11] P. J. Burt and E. H. Adelson. The Laplacian pyramid as a compact image code. IEEE Transactions on Communications, Vol. COM-31(No.4):pp.532–540, April 1983. [12] J.-M. Chassery and A. Montanvert. G´eom´etrie discr`ete en analyse d’images. Trait´e des Nouvelles Technologies, s´erie – Images. HERMES, Paris, France, 1991. [13] G. Damiand. D´eﬁnition et ´etude d’un mod` ele topologique minimal de repr´esentation d’images 2d et 3d. PhD thesis, LIRMM, Universit´e de Montpellier, 2001. [14] L. De Floriani, P. Magillo, and E. Puppo. Multiresolution Representation of Shapes Based on Cell Complexes. In G. Bertrand, M. Couprie, and L. Perroton, editors, Discrete Geometry for Computer Imagery, DGCI’99, volume Vol. 1568 of Lecture Notes in Computer Science, pages 3–18, Marne-la-Vall´ee, France, 1999. Springer, Berlin Heidelberg, New York. [15] T. K. Dey, H. Edelsbrunner, S. Guha, and D. V. Nekhayev. Topology Preserving Edge Contraction. in print, 1999. [16] H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse Complexes for Piecewise Linear 2-Manifolds. In SCG’01, June 3–5, 2001, Medford, Mass. USA, ACM 1-58113-357-X/01/0006, 2001. [17] R. Forman. Combinatorial Diﬀerential Topology and Geometry. in print, 1999. [18] S. Fourey and R. Malgouyres. Intersection Number of Paths Lying on a Digital Surface and a New Jordan Theorem. In G. Bertrand, M. Couprie, and L. Perroton, editors, Discrete Geometry for Computer Imagery, DGCI’99, volume Vol. 1568 of Lecture Notes in Computer Science, pages 104–117, Marne-la-Vall´ee, France, 1999. Springer, Berlin Heidelberg, New York. [19] R. Glantz and W. G. Kropatsch. Plane Embedding of Dually Contracted Graphs. In G. Borgefors, I. Nystr¨ om, and G. Sanniti di Baja, editors, Proceedings DGCI’00, Discrete Geometry for Computer Imagery, volume Vol. 1953 of Lecture Notes in Computer Science, pages 348–357, Uppsala, Sweden, 2000. Springer, Berlin Heidelberg, New York. [20] R. L. Hartley. Multi–Scale Models in Image Analysis. PhD thesis, University of Maryland, Computer Science Center, 1984. [21] J.-M. Jolion and A. Rosenfeld. A Pyramid Framework for Early Vision. Kluwer Academic Publishers, 1994.

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21

[22] J. Kittler, K. Messer, W. Christmas, B. Levienaise-Obadia, and D. Koubaroulis. Generation of semantic cues for sports video annotation. In Proceedings IEEE International Conference on Image Processing, ICIP2001, pages pp.26–29, Thessaloniki, Gr., 2001. [23] T. Kong and A. Rosenfeld. Digital topolgy: a comparison of the graph-based and topological approaches. In G. Reed, A. Roscoe, and R. Wachter, editors, Topology and Category Theory in Computer Science, pages 273–289, Oxford, 1991. Oxford University Press. [24] V. A. Kovalevsky. Digital Geometry Based on the Topology of Abstract Cellular Complexes. In J.-M. Chassery, J. Francon, A. Montanvert, and J.-P. R´eveill`es, editors, G´eometrie Discr`ete en Imagery, Fondements et Applications, pages 259– 284, Strasbourg, France, September 1993. [25] W. G. Kropatsch. Preserving contours in dual pyramids. In Proc. 9th International Conference on Pattern Recognition, pages 563–565, Rome, Italy, November 1988. IEEE Comp.Soc. [26] W. G. Kropatsch. Property Preserving Hierarchical Graph Transformations. In C. Arcelli, L. P. Cordella, and G. Sanniti di Baja, editors, Advances in Visual Form Analysis, pages 340–349. World Scientiﬁc Publishing Company, 1997. [27] W. G. Kropatsch. From equivalent weighting functions to equivalent contraction kernels. In E. Wenger and L. I. Dimitrov, editors, Digital Image Processing and Computer Graphics (DIP-97): Applications in Humanities and Natural Sciences, volume 3346, pages 310–320. SPIE, 1998. [28] W. G. Kropatsch, A. Leonardis, and H. Bischof. Hierarchical, adaptive and robust methods for image understanding. Surveys on Mathematics for Industry, No. 9:pp. 1–47, 1999. [29] W. G. Kropatsch and H. Macho. Finding the structure of connected components using dual irregular pyramids. In Cinqui`eme Colloque DGCI, pages 147–158. LLAIC1, Universit´e d’Auvergne, ISBN 2-87663-040-0, September 1995. [30] S. G. Mallat. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI–11(No. 7):pp. 674–693, July 1989. [31] P. Meer, C. A. Sher, and A. Rosenfeld. The chain pyramid: Hierarchical contour processing. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-12(No.4):pp.363–376, April 1990. [32] P. F. Nacken. Image segmentation by connectivity preserving relinking in hierarchical graph structures. Pattern Recognition, 28(6):907–920, June 1995. [33] A. Rosenfeld, editor. Multiresolution Image Processing and Analysis. Springer, Berlin, 1984. [34] A. Rosenfeld and A. Nakamura. Local deformations of digital curves. Pattern Recognition Letters, 18:613–620, 1997. [35] J. K. Udupa. Go Digital, Go Fuzzy. In G. Borgefors, I. Nystr¨ om, and G. Sanniti di Baja, editors, Proceedings DGCI’00, Discrete Geometry for Computer Imagery, volume Vol. 1953 of Lecture Notes in Computer Science, pages 284–295, Uppsala, Sweden, 2000. Springer, Berlin Heidelberg, New York.

XPMaps and Topological Segmentation – A Unified Approach to Finite Topologies in the Plane Ullrich Köthe Cognitive Systems Group, University of Hamburg [email protected]

Abstract. Finite topological spaces are now widely recognized as a valuable tool of image analysis. However, their practical application is complicated because there are so many different approaches. We show that there are close relationships between those approaches which motivate the introduction of XPMaps as a concept that subsumes the important characteristics of the other approaches. The notion of topological segmentations then extends this concept to a particular class of labelings of XPMaps. We show that the new notions lead to significant simplifications from both a theoretical and practical viewpoint.

1

Introduction

Many computer vision researchers now agree that the notion of finite topological spaces is very useful in image analysis and segmentation, as it allows for consistent descriptions of neighborhood relations, boundaries, and so on. Meanwhile, a number of different methods for the representation of finite topologies has been proposed, including cellular complexes [9], block complexes [10], the star topology [1], the Khalimsky grid [6, 3], combinatorial maps [15, 5] and border maps [2]. All methods approach the problem somewhat differently, but they have a lot in common as well. Unfortunately, the commonalties are not immediately apparent from the literature because most authors present their method in isolation or emphasize the differences to other methods. This makes it unnecessarily difficult to understand and compare the different approaches and to provide reusable implementations in a general image analysis framework. In this contribution I’m going to propose a unified approach to finite topologies in the plane. I’ll show that, under fairly general assumptions, different models can be transformed into each other. This will motivate the introduction of two generalized concepts (the XPMap and the topological segmentation) which subsume most aspects of the existing concepts. The concept of a XPMap is essentially a formalization of the border map introduced in [2]. This paper complements a previous paper [7] where solutions for some software design and implementation problems of the new approach were presented. Here, we look at the problem from a theoretical viewpoint in order to show formally why the underlying unification is possible and which properties it has. Due to space constraints, most proofs had to be skipped. They can be found in the long version [8] of this paper which is available from the author’s WWW site. A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 22–33, 2002. © Springer-Verlag Berlin Heidelberg 2002

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Fig. 1. The open stars of a vertex, arc and region (from left to right) in a square tessellation of the plane

2

Models for Finite Topological Spaces in the Plane

In this section, we will introduce the topological models we are going to analyze later. Here we only give definitions but do not yet discuss any relationships between them. The most fundamental definition of a finite topological space is obtained on the basis of a complete division of the Euclidean plane. A plane division is a finite set of disjoint cells whose union completely covers the plane. There are three types of cells:

Vertices: V = {v1, ... , vn} is a set of distinct points of the plane Arcs: A = {a1, ... , am} is a set of disjoint open arcs whose end points are vertices from V. (An arc is a homeomorphic image of the interval [0, 1]. An open arc is an arc less its end points. The two end points need not be distinct.) Regions: R = {r1, ..., rk} are the maximally connected components of the complement of the union of V and A. Since V∪A is a closed set, all regions are open. The Euclidean topology of the plane induces a neighborhood relation between the cells, which is used to define the smallest open neighborhood (the open star) of each cell. The set of open stars forms a basis for a finite topological space:

Definition 1: Let T be a division of the Euclidean plane into cells (regions, arcs, vertices). Then the open star (smallest open neighborhood) of a cell z is the set of cells that intersect with every Euclidean neighborhood of the points in z: open − star ( z ∈ T ) = {z ’∈ T | ∀q ∈ z, ∀ε > 0 : Bε (q) ∩ z ’≠ ∅}

( Bε (q) denotes an open ball of radius ε around the point q). The set of open stars is a basis of a finite topological space over the cells of the plane division. Two divisions of the plane are topologically equivalent if there exists a 1-to-1 mapping between the cells of the division so that the open stars are preserved. Some authors, e.g. Ahronovitz et al. [1], further require all regions to be convex polytopes. We will not do this in this paper. The most common example for a star topology is obtained by tessellating the plane into equal squares. Then the open stars of the vertices, arcs, and regions look as depicted in figure 1. The biggest drawback of the plane division definition is its reliance on geometric concepts which prevents separation between topology and geometry. Therefore, a number of definitions not relying on the Euclidean space exist. The best known is the abstract cellular complex introduced to image analysis by Kovalevsky [9]:

Definition 2: A cell complex is a triple (Z, dim, B) where Z is a set of cells, dim is a function that associates a non-negative integer dimension to each cell, and B ⊂ Z × Z is the bounding relation that describes which cells bound other cells. A cell may bound only cells of larger dimension, and the bounding relation must be transitive. If the largest dimension is k, we speak of a k-complex.

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A cell complex becomes a topological space by additionally defining the open sets as follows: a set of cells is called open if, whenever cell z belongs to the set, all cells bounded by z do also belong to the set. Kovalevsky proved that every finite topological space can be represented as a cell complex. In image analysis, we want to represent the topological structure of 2-dimensional images, so we are naturally interested in 2-complexes whose bounding relation is consistent with a division of the plane as defined above. We will call those planar cell complexes. A special case of a cell complex is the block complex [10]. It is defined on a given cell complex by grouping cells into sets called block cells, or blocks for short. Blocks are defined by the property that they either contain a single 0-cell, or are homeomorphic to an open k-sphere. (A block is homeomorphic to an open k-sphere if it is isomorphic to a simply connected open set in some k-complex – see [10] for details.) A block’s dimension equals the highest dimension of any of its cells.

Definition 3: A block complex is a complete partition of a given cell complex into blocks. The bounding relation of the blocks is induced by the bounding relation of the contained cells: a block bounds another one if and only if it contains a cell that bounds a cell in the other block. Note that the induced bounding relation must meet the requirements of definition 2, i.e. blocks may only bound blocks of smaller dimension, and the relation must be transitive. This means that all 1-blocks must be sequences of adjacent 0- and 1-cells, and all junctions between 1-blocks must be 0-blocks. Another approach to finite topological spaces originates from the field of graph theory – the combinatorial map [15, 5]:

Definition 4: A combinatorial map is a triple (D, σ, α) where D is a set of darts (also known as half-edges), σ is a permutation of the darts, and α is an involution of the darts. In this context, a permutation is a mapping that associates to each dart a unique predecessor and a unique successor. By following successor chains, we can identify the cycles or orbits of the permutation. An involution is a permutation where each -1 orbit contains exactly two elements. It can be shown that the mapping ϕ = σ α is also a permutation. 1 The cycles of the σ permutation are called the nodes of the map, the cycles of α are the edges, and the cycles of ϕ the faces. A combinatorial map is planar (i.e. can be embedded in the plane) if its Euler characteristic equals two (cf. [15]): n−e+ f = 2

(1)

(with n, e, and f denoting the number of nodes, edges, and faces). Each planar combinatorial map has a unique dual map that is obtained by reversing the roles of the σ and ϕ orbits. It should be noted that the graph of a combinatorial map (i.e. the remains of the map when all faces are removed) must be connected, since each face is associated with exactly one orbit. Thus, faces in a combinatorial map may not have holes. 1

Combinatorial maps are sometimes defined by using four darts per edge (“quad-edges” or “crosses”), e.g. [15]. This allows the realization of non-orientable manifolds and the definition of generalized maps (G-maps) that can be generalized to arbitrary dimensions [12]. However, the half-edge definition suffices in the present context.

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The final approach we are going to deal with is Khalimsky’s grid [6, 3]. It is defined as the product topology of two 1-dimensional topological spaces. In particular, one defines a connected ordered topological space (COTS) as a finite ordered set of points which alternate being open and closed. The smallest open neighborhood of an open point is the point itself, of a closed point it is the point and its predecessor and successor (in case of a closed endpoint, the neighborhood consists of two points only). Now Khalimsky’s grid is defined like this:

Definition 5: A Khalimsky grid is defined as the space X×Y with the product topology, where X and Y are COTS with at least three points each. The product topology defines the neighborhood of a grid point as N((x,y))=N(x)×N(y). When x and y are both open, the resulting point (x, y) is also open. When they are closed, (x, y) is closed as well, and its neighborhood consists of the 8 incident points. When an open point is combined with a closed point, a mixed point results, whose neighborhood contains the two incident closed points. These neighborhoods are analogous to the open stars in figure 1, with obvious modifications at the gird’s border.

3

Relationships between Finite Planar Topological Spaces

In this section we show how the concepts defined above relate to each other by describing how a topological space represented in one concept can be translated into another. All missing proofs to the theorems can be found in [8].

Theorem 1: Any finite topology induced by a division of the plane can be represented by a planar cell complex. The transformation is achieved by the following algorithm: Associate a 0-, 1- or 2-cell with every vertex, arc, and region respectively. Define the bounding relation so that cell z bounds another cell z’ whenever z’ is part of the open star of z. In [8], we prove that the algorithm actually creates a valid cell complex. The transformation in the opposite direction is not so simple because not every abstract cell complex corresponds to a division of the plane. We postpone this discussion until after the treatment of combinatorial maps because we will need a map as an intermediate representation during this transformation. The transformation of a division of the plane into a combinatorial map is easy as long as the union of the vertices and arcs (the boundary set of the division) is a connected set. Otherwise, at least one region’s boundary consists of several connected components, which is inconsistent with the requirement that every orbit of the ϕ permutation corresponds to a distinct face. We will first discuss the transformation in the restricted case, and then extend the map definition for the general case.

Theorem 2: A finite topology induced by a division of the plane can be represented by a planar combinatorial map if the union of the arcs and vertices is connected. To create a map from a division of the plane, first associate two darts with each arc, one for either direction. If the arc is defined by the function a(t), 0
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an orbit in the α involution. Then, for each vertex take the darts who start at that vertex, and sort them in the order induced by circling the vertex in mathematically positive direction. This gives the orbits of the σ permutation. Under this definition the -1 orbits of ϕ = σ α indeed correspond to the regions of the plane division (see [8]). Two degenerate cases (not covered by the algorithm) require explicit definitions:

• •

the trivial map: it contains a single empty orbit in the ϕ permutation and corresponds to the trivial division of the plane that contains only a single region. the vertex map: it contains a single empty orbit in both the σ and ϕ permutations and corresponds to one region that contains an isolated vertex.

In order to handle arbitrary divisions of the plane, including holes in regions, we have to extend the notion of planar combinatorial map. In particular, we must allow the graph of the extended map to be non-connected. Such a map is defined as follows:

Definition 6: An extended planar map (XPMap) is a tuple (C, c0, exterior, contains) where C is a set of non-trivial planar combinatorial maps (the components of the XPMap), c0 is a trivial map that represents the infinite face of the XPMap, exterior is a relation that labels one face of each component in C as the exterior face, and contains is a relation that assigns each exterior face to exactly one non-exterior face or the infinite face. Thus, if exterior face f1 is contained in face f2, the two faces are actually identical, and the component of f1 becomes a hole in f2. Euler’s equation must be slightly modified for a XPMap with k non-trivial components (that is, k=|C|): n−e+ f −k =1

(2)

Proof: This is easy to prove by induction. The trivial map has f=1 and n=e=k=0, the vertex map has n=f=k=1, e=0, and a connected map has k=1 and n–e+f=2, so the formula is valid for these cases. Suppose we have proved the formula for some k, i.e. (k) (k) (k) (1) (1) (1) n –e +f –k=1. Now add a new component with n –e +f –1=1. We may add the (k) (1) (k) (1) (k) (1) (k) (1) (k+1) two equations together: n + n –e –e +f +f –k–1=2. It holds that n + n = n , (k) (1) (k+1) (k) (1) (k+1) e + e = e , but f + f = f +1, because the exterior face of the newly added component has already been present in the existing XPMap and must not be counted (k+1) (k+1) (k+1) twice. Thus, n –e +f –(k+1)=1 as claimed. < It should be noted that XPMaps are essentially equivalent to the border maps proposed by Bertrand et al. [2] (but their paper lacks a formal definition). Their approach is based on G-maps because they are interested in 2- and 3-dimensional problems. Since this is not needed here, our definition is sufficient. It is now possible to construct an XPMap from an arbitrary division of the plane: First take each connected component of the division’s boundary set and create the corresponding combinatorial map. These maps become the components of the XPMap. Each component has exactly one ϕ orbit that is traversed in mathematically negative orientation which becomes the exterior face. Now construct the containment relation according to the inclusion of components within regions of the plane division. To go in the opposite direction, we must find a plane imbedding of a given XPMap. This problem is part of the standard problem of drawing a planar graph, suitable algorithms can be found in [4]. It is now easy to see that any XPMap can be transformed into a cell complex:

Theorem 3: Any XPMap defines a corresponding cell complex.

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Proof: For every node, edge, and face of the map create a corresponding 0-, 1-, or 2-cell, respectively. Let a 1-cell bound a 2-cell whenever a dart of the corresponding edge is part of any ϕ orbit associated with the corresponding face. Let a 0-cell bound a 1-cell whenever a dart of the edge is part of the corresponding node’s σ orbit. Let a 0-cell bound a 2-cell whenever its σ orbit contains a dart whose corresponding 1-cell bounds that 2-cell. The bounding relation obviously conforms to definition 2. < The bounding relation of the cells induces an equivalent bounding relation of the nodes, edges, and faces. By extension of terminology, and we will say that a node (edge) bounds an edge (face) if the corresponding 0-cell (1-cell) bounds the corresponding 1-cell (2-cell). The bounding relation defines the topology of an XPMap:

Definition 7: A set of nodes, edges, and faces in a XPMap is called open if the corresponding cells form an open set in the corresponding cell complex. The open star of a node, edge, or face (the smallest open set containing this entity) is the set containing the entity itself and all entities bound by it. Transformations of cell complexes into XPMaps are more difficult. A 2-complex is not necessarily embeddable into the plane – it might correspond to a non orientable surface or a surface of higher genus. So, some necessary conditions must be met. First, one 2-cell must be denoted the infinite face (it can be chosen arbitrarily). Second, Euler’s formula (2) must be fulfilled (again, k is the number of components of the cell complex’ graph, i.e. the remains after removing all 2-cells). Third, all 1-cells must be bounded by one or two 0-cells since all darts must be incident to a node. Unfortunately, its not clear to me whether these requirements are sufficient. Even when these requirements are fulfilled, the transformation of a 2-complex to a XPMap is not necessarily unique. If the cell complex’ graph is 1-connected2 or disconnected, the neighborhood relation does not change when certain sub-graphs are replaced by their mirror images (see figure 2 for examples). In these cases, the order of the darts within each σ or ϕ orbit is not uniquely determined. An algorithm that produces one possible XPMap out of a given planar cell complex is given in [8]. This algorithm terminates with no free darts remaining, if the cell complex is planar. The algorithm implies a method for generating a planar imbedding of a cell complex: first transform the cell complex into an XPMap, and then imbed the map into the plane by a standard imbedding algorithm (see [4]). However, although the algorithm may be interesting from a theoretical point of view, its practical value is probably limited: in practice, a cell complex will usually be equipped with additional geometric information that uniquely determines the correct order of the darts in the orbits and the imbedding into the plane. To allow the establishment of relationships between Khalimsky’s grid and the other topological spaces, we must modify the definition of the grid somewhat so that it gets an infinite point that can be mapped to the infinite face of the map:

Definition 8: A Khalimsky grid with infinite point is defined as follows: the COTS X and Y constituting the grid must be chosen so that their first and last points are all closed. Then the border of the grid consists only of closed and mixed points. Add an additional point – the infinite point – that by definition belongs to the neighborhood of all border points.

2

A graph is k-connected if at least k vertices must be removed to make the graph disconnected.

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Fig. 2. Three imbeddings of the same cell complex that differ in the ordering of the darts in the σ and ϕ orbits. Note that the cell complex’ graph is only 1-connected.

N6

This extension allows formulation of the following theorem:

Theorem 4: A Khalimsky grid with infinite point defines a corresponding XPMap. The transformation is obtained as follows: First, we associate a node with every closed point, an edge with every mixed one, and a face with every open one. Then we create two darts for each mixed point, starting at either incident closed point. Each pair forms an orbit of the α involution. It is convenient to identify each dart by the topological coordinates of its start point and the relative orientation between the start and mixed point (east, north, west, and south). A σ permutation is now defined by taking the darts associated with the same start point and sorting them according to their orientation. This choice also implies the expected ϕ orbits, as is shown in [8]. In contrast, a combinatorial map can only be represented as a Khalimsky grid if it has the required regular structure: all nodes and faces must be adjacent to exactly four darts (with obvious modifications at the exterior border). Since this is a rather strong limitation, Khalimsky’s grid itself is not useful for the representation of arbitrarily shaped topological spaces. Instead, one builds block complexes on top of the grid. That is, one creates irregularly shaped cells by grouping the grid points into appropriate blocks. We will discuss this in detail in section 5.

4

Euler Operators

Before we can turn our attention to block complexes, we must define and discuss Euler operators [13]. Euler operators turn a given planar topological structure into a different one. The term stems from the restriction that Euler’s equation must remain valid in the new structure, so that planarity is preserved (some authors call such operators superficial). The proposed Euler operators are elementary in the sense that they change the number of nodes, edges, faces, and components by at most one. They are complete in the sense that any desired transformation can be expressed as a sequence of Euler operations. Completeness follows from theorem 5 below: every XPMap is an open set and can be reduced to a single face by a sequence of Euler operations. By reversing the sequence, any XPMap can be built out of the trivial map. The operators will be described in the context of XPMaps only. The proofs which check that Euler’s equation indeed remains valid are again found in [8]. Analogous operators in other topological models are implied by the transformations already discussed. For example, figure 3 shows how the operators to be defined act on a division of the plane. Of course, application of Euler operators to the Khalimsky grid

XPMaps and Topological Segmentation

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Fig. 3. Examples for Euler operators. first row: edge contraction and merge edges, second row: merge faces and move component, third row: remove isolated node and remove bridge

is forbidden because the grid’s structure is fixed. We will consider two kinds of operators: first, those that do not change the number of components of the map’s graph (so that we can restrict the discussion to standard combinatorial maps), and second the ones that do (and thus require a XPMap). Node Split and Edge Contraction. A node split creates two nodes from one. Some existing edges are moved to the new node, and a new edge connecting the nodes is added. The split must be arranged so that the order of the edges in the orbits is not changed (formal definition below). The inverse operation is called edge contraction, because the two ends of an edge are contracted into a single node.

Definition 9 (node split): Suppose the node to be split is associated with the orbit σ1=(d0,...,dn–1). Remove σ1 from the set of orbits and chose darts di and dj (0≤i≤j
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Edge Split and Edge Merge. Edge split creates two edges by placing a new node into an existing edge. Edge merge reverses this operation. Merging may only be applied to a pair of distinct edges whose common node has degree 2 (otherwise, forbidden configurations would arise). Edge split is a special case of node split (so we omit the formal definition here). Nonetheless, it is useful to define edge split as an operator of its own because of its special semantic. Face Merge and Face Split. The face merge operator removes an edge, so that its adjacent faces are merged into one. Face split is the inverse operation. The edge to be removed must bound two distinct faces (i.e. it must not be a bridge) as otherwise the number of components would be increased (bridges are discussed below).

Definition 10 (face split): Create a new pair of darts d and d’ and insert the pair as a new orbit into α. Select two darts d1 and d2 which belong to the same ϕ orbit. Insert d after d1 and d’ after d2 into the respective σ orbits. Face merge is executed be reversing those steps, provided that the edge (d, d’) is not a bridge. Face split is the dual operation of node split: it is equivalent to node split in the dual map. Move Component. This operation is very simple and does not involve any orbits:

Definition 11 (move component): Take an entry of an XPMap’s containment relation and replace the enclosing face by another one (which must not itself be an exterior face). This operation is especially useful in the context of splitting faces with holes: if the face to be split contains some components of the XPMap, the definition of the face split operator doesn’t tell which of the newly created faces will contain those components afterwards. Suitable adjustments can be made by means of “move component”. Insert/Remove Isolated Node. This operation is the simplest possibility to create a new component within an XPMap:

Definition 12 (insert isolated node): Create a new vertex map. Make its single face the exterior one and associate it with an arbitrary non-exterior face by adding appropriate entries to the XPMap’s exterior and contains relations. Remove/Insert Bridge. A bridge is an edge that bounds only one face. Removing a bridge from the α and σ orbits is simple, but modification of the ϕ orbits is rather involved. This stems form the fact that “remove bridge” creates a new component, and it is not trivial to decide which ϕ orbit becomes the exterior face of the new component. Due to space constraints, the algorithm must be skipped, see again [8].

5

Block Complexes and Topological Segmentation

In definition 3, block complexes are defined on top of cell complexes. Although we could extend this definition to other topological structures, we will not do so because the restriction “k-blocks must be homeomorphic to open k-spheres” is not acceptable in the context of image analysis: it requires all regions to be simply connected (i.e. holes in regions are not allowed) and each 0-block to consist of a single 0-cell (and thus the maximum degree of 0-blocks is bounded by the 0-cell’s degree – so junctions

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cords

Fig. 4. Left: examples for 2-, 1-, and 0-segments (left to right), right: topological segmentation of a Khalimsky grid with infinite point

of degree > 4 cannot be represented on top of a Khalimsky grid). Therefore, we will develop a more general concept for grouping the cells of a digital topological structure which we will call a topological segmentation. A topological segmentation is a complete partition of a digital topological space into k-segments (k=0, 1, 2) such that every k-segment can be transformed into a single k-cell by a suitable sequence of Euler operations. Before we can give a formal definition we need some preliminaries:

Theorem 5: Any connected open set in a XPMap can be reduced to a single face by a sequence of Euler operators, without changing the structure of the set’s complement. Any simply connected closed set can be reduced to a single node. The proof can be found in [8]. Furthermore, we need the notion of hammocks:

Definition 13: A hammock is the union of a simply connected closed set and two edges emanating from the set such that removal of these edges would turn the closed set into a new component of the XPMap. An edge is said to emanate from a set if one of its end nodes belongs to the set but not the other. The darts starting at these external nodes are called the cords of the hammock. Hammocks are useful because they can be reduced to single edges by a sequence of Euler operations: reduce the closed set in the hammock to a single node which is bound by the two remaining edges of the hammock, but no other edge. Then apply the operator “merge edges” to turn the remaining cells into a single edge.

Definition 14: A topological segmentation is a complete partition of an underlying XPMap into k-segments (k=0, 1, 2) such that every 0-segment is a simply connected closed set, every 1-segment is a hammock, and every 2-segment is a connected open set. Figure 4 shows examples of k-segments and a topological segmentation. According to the discussion above, any 0-segment can be reduced to a single node, any 1-segment to a single edge, and any 2-segment to a single face. By reducing all segments we arrive at a new XPMap which we will call the reduced XPMap of the topological segmentation. Definition 14 replaces the restriction “every k-block must be homeomorphic to an open k-cell” by “every k-segment must be reducible to a single k-cell”. Therefore, a topological segmentation may contain 2-segments with holes and 0-segments that have higher degree than any node of the underlying XPMap. Although, in principle every topological segmentation can be reduced to a XPMap, it is not necessary to actually perform the reduction. Instead, we can just label the segments in the underlying XPMap and work with the segments directly. In fact, this

32

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is the only possibility to represent irregular divisions of the plane by means of a Khalimsky grid: since the grid’s structure is rigid, one cannot perform Euler operations on it, but one can still label points into segments. In order to make a labeled XPMap or Khalimsky grid behave like a reduced XPMap, we must specify the orbits of the topological segmentation. To avoid confusion, we will denote these orbits by Greek capitals Α, Φ, Σ, in contrast to the lower case α, ϕ, and σ orbits of the underlying XPMap. The cords of the hammocks are the basis of the orbit definitions. These definitions amount to contour following algorithms that go around segments in order to find all incident cords. Efficient and elegant implementations of these algorithms can be obtained by means of generic programming, as described in [7]:

Α orbits of a topological segmentation: The Α orbits associate the two cords of a hammock: given one cord, we want to find the other. To do so, observe that the left face of the first cord is the right face of the second. Thus, we can follow the ϕ orbit of the first cord until we arrive at a dart that doesn’t belong to the present hammock. Its σ-successor is the sought for Α successor of the first cord. Σ orbits of a topological segmentation: Σ orbits associate cords with 0-segments. This is easy if the 0-segment consists of a single node: then the Σ orbit is obtained from the node’s σ orbit by simply dropping the darts that are not cords. In case of a more complicated 0-segment, the principle is the same, but instead of just considering a single σ orbit we must now follow the entire contour of the segment. The following algorithm finds the Σ successor of a given cord: 1. Find the σ successor of the present dart. If the successor is a cord, stop. If it belongs to an edge that is part of the 0-segment, go to 2. Otherwise go to 1. 2. Find the α successor of the present dart, and find the σ successor of the resulting dart. If this dart is a cord, stop. If it belongs to an edge that is part of the 0-segment, go to 2. Otherwise go to 1.

Φ orbits of a topological segmentation: The Φ orbits associate cords with 2-segments. The Φ successor of a given cord is found as follows: go to the Α successor of the present cord, and find its Σ predecessor. Thus, Φ = Σ-1Α. It can be shown that the orbits of a topological segmentation are equivalent to the corresponding orbits in the reduced XPMap. Furthermore, we can define Euler operators for a topological segmentation. They perform the same logical operation as their XPMap equivalent, but do not merge cells. Instead, they re-label the segments in order to reflect the modified structure. The only difference to XPMaps is that split operations can only be applied as long as the segments contain more than one cell.

6

Conclusions

In this paper, we have presented a novel, unified approach to the representation of finite topologies in the plane. XPMaps incorporate the full range of functionality from cell complexes, combinatorial maps and plane divisions. The concept of a topological segmentation allows to define labelings of XPMaps that again have the properties of a

XPMaps and Topological Segmentation

33

topological space. This provides a smooth transition from the pixel plane to an abstract topological space as it allows for the definition of irregular topologies on top of regular grids. Furthermore, by applying labeling and reduction repeatedly, a wide variety of irregular pyramids can be realized. This idea leads to a generalization of concepts form [11, 14] and will be explored in an upcoming paper. Of course, it would be desirable to carry the analysis to 3-dimensional spaces. However, given the wealth of unsolved problems in 2D image analysis and segmentation, its appears equally worthwhile to utilize the power of the new approach here. The generalized formal definitions presented provide a firm basis for studying the relationships between topology and the imaging process. By using XPMaps as a common representation for segmentation results, we will be able to develop more robust segmentation algorithms and achieve a unification of approaches in general.

References [1] E. Ahronovitz, J.P. Aubert, C. Fiorio: “The Star Topology: a Topology for Image Analth sysis”, Proc. 5 Intl. Conf. Discrete Geometry for Computer Imagery, DGCI’95, 1995 [2] Y. Bertrand, C. Fiorio, Y. Pennaneach: “Border Map: a Topological Representation for th nD Image Analysis”, G. Bertrand, M. Couprie, L. Perroton (eds.): Proc. 8 Intl. Conf. Discrete Geometry for Computer Imagery, DGCI’99, Springer LNCS 1568, 1999 [3] J.-P. Braquelaire, J.-P. Domenger: “Representation of Region Segmented Images with Discrete Maps”, Université Bordeaux, Laboratoire Bordelais de Recherche en Informatique, Technical Report 1127-96, 1996 [4] G. Di Battista, P. Eades, R. Tamassia, I.G. Tollis: “Graph Drawing”, Prentice Hall, 1999 [5] J.-F. Dufourd, F. Puitg: “Functional specification and prototyping with oriented combinatorial maps”, Computational Geometry 16 (2000) 129-156 [6] E. Khalimsky, R. Kopperman, P. Meyer: “Computer Graphics and Connected Topologies on Finite Ordered Sets”, J. Topology and its Applications, vol. 36, pp. 1-27, 1990 [7] U. Köthe: “Generic Programming Techniques that Make Planar Cell Complexes Easy to Use”, G.Bertrand, A.Imiya, R.Klette (eds.): Proc. of Dagstuhl WS on Digital and Image Geometry 2000, Springer LNCS 2243, 2001 [8] U. Köthe: “XPMaps and Topological Segmentation – a Unified Approach to Finite Topologies in the Plane (long version)”, Uni Hamburg, Preprint FBI-HH-M-308/0, 2001 http://kogs-www.informatik.uni-hamburg.de/~koethe/papers/XPMapsLong.ps.gz [9] V. Kovalevsky: “Finite Topology as Applied to Image Analysis”, Computer Vision, Graphics, and Image Processing, 46(2), pp. 141-161, 1989 [10] V. Kovalevsky: “Computergestützte Untersuchung topologischer Eigenschaften mehrdimensionaler Räume”, Preprints CS-03-00, Computer Science Department, University of Rostock, 2000 (in German) [11] W. Kropatsch: “Building Irregular Pyramids by Dual Graph Contraction”, IEE Proceedings – Vision, Image and Signal Processing, 142(6), 366-374, 1995 [12] P. Lienhardt: “Topological models for boundary representation: a comparison with ndimensional generalized maps”, computer aided design, 23(1), 59-82, 1991 [13] M. Mäntylä: “An Introduction to Solid Modeling”, Computer Science Press, 1988 [14] A. Montanvert, P. Meer, A. Rosenfeld: “Hierarchical Image Analysis Using Irregular Tessellations”, IEEE Trans. Pattern Anal. and Machine Intelligence, 13(4), 307-316, 1991 [15] W.T. Tutte: “Graph Theory”, Cambridge University Press, 1984

Curves in ZZ n Grit Th¨ urmer Computer Graphics, Visualization, Man-Machine Communication Group Faculty of Media Bauhaus-University Weimar 99421 Weimar, Germany [email protected]

Abstract. A new deﬁnition of closed curves in n-dimensional discrete space is proposed. This deﬁnition can be viewed as a generalization of closed quasi curves and is intended to overcome the limitations of known deﬁnitions for practical purposes. Following the proposed deﬁnition, a set of points forms a closed curve in discrete space if the set admits a parameterization, i.e. there exists a Hamiltonian cycle in the set. Adjacencies that do not indicate the parameterization are allowed only between points that are “close to each other” along the parameterization. Additionally, it is proven that discrete curves satisfying the new deﬁnition in two-dimensional discrete space have the Jordan property.

1

Introduction

Curves are basic objects in geometry and play an important role in modeling. New application ﬁelds of volume data have been emerging during the last years, which require modeling of synthetic discrete objects, i.e. geometrically deﬁned objects in ZZ 3 or data sets generated by rastering [1,16] geometric descriptions of objects given in IR3 . Such application ﬁelds are, e.g., virtual reality in medicine [17], volume-based interactive design and sculpturing [15,14]. Curves in two-dimensional discrete space, i.e. in ZZ 2 , are a well explored subject [12,7,4]. The certainly best known deﬁnition is the one of simple closed curves [10,11]. This deﬁnition is very restrictive for many practical purposes. To overcome this problem, closed quasi curves have been introduced [7]. Although, the deﬁnition of closed quasi curves works well in many cases in ZZ 2 , it is still rather limited in higher dimensional space. This point is further discussed in Sect. 4. In this paper, a new deﬁnition of closed quasi curves is proposed. This deﬁnition can be viewed as a generalization of quasi curves as introduced in [7] and overcomes the limitations of them. Moreover, the deﬁnition is given for curves in n-dimensional discrete space. The paper is organized as follows: Section 2 gives some deﬁnitions and notations used throughout the paper. In Sect. 3 simple closed curves are explored,

This work has been supported by a postdoctoral grant of the DFG (Deutsche Forschungsgemeinschaft).

A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 34–44, 2002. c Springer-Verlag Berlin Heidelberg 2002

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35

and Sect. 4 deals with closed quasi curves. Afterwards, the new deﬁnition of closed quasi curves is suggested and discussed in Sect. 5. Finally, Sect. 6 summarizes the paper.

2

Deﬁnitions and Notations

The n-dimensional discrete space ZZ n is constituted by the n-dimensional array of points with integer coordinates in the Cartesian coordinate system. A few deﬁnitions are given below particularly regarding ZZ 2 and ZZ 3 . These deﬁnitions can be easily modiﬁed and extended to a discrete space of higher dimension. Adjacency relations. Two points p(xp , yp ) and q(xq , yq ) of ZZ 2 , where (| xp − xq |≤ 1) ∧ (| yp − yq |≤ 1), are said to be 4-adjacent if | xp − xq | + | yp − yq |= 1, and 8-adjacent if 0 <| xp − xq | + | yp − yq |≤ 2. Two points p(xp , yp , zp ) and q(xq , yq , zq ) of ZZ 3 , where (| xp − xq |≤ 1) ∧ (| yp − yq |≤ 1) ∧ (| zp − zq |≤ 1), are said to be 6-adjacent if | xp − xq | + | yp − yq | + | zp − zq |= 1, 18-adjacent if 0 <| xp − xq | + | yp − yq | + | zp − zq |≤ 2, and 26-adjacent if 0 <| xp − xq | + | yp − yq | + | zp − zq |≤ 3. Points k-adjacent to p, where k ∈ {4, 8, 6, 18, 26}, are called k-neighbours of p. The points in the k-neighbourhood of a point p are the points k-adjacent to p and p itself, and are denoted with Nk (p). In ZZ n , n diﬀerent types of adjacency relations can be deﬁned, similarly as stated above for n = 2, 3. Following the deﬁnitions of adjacency relations as introduced for ZZ 2 and ZZ 3 , one type of adjacency relation is a subset of the other. For example, in ZZ 3 , the set of 6-adjacencies is a subset of the set of 18-adjacencies, which in turn is a subset of the set of 26-adjacencies. This correlation of the types of adjacency relation can be used as criterion for an ordering, such that for a point p in ZZ n the adjacency relation of the sth -order with 0 < s ≤ n covers all neighbouring points of ZZ n that vary in exactly s coordinates from p. Paths. A k-path P in ZZ n is a sequence of distinct points P = {p0 , p1 , ..., ph } of ZZ n such that any two consecutive points pi and pi+1 with 0 ≤ i < h along the sequence are k-adjacent. The k-path P is said to be a closed path if p0 is k-adjacent to ph . Connectivity. Two points p, q are said to be k-connected in a subset A of ZZ n if there exists a k-path from p to q consisting entirely of points of A. A set A of ZZ n is said to be k-connected if every two points of A are k-connected in A. A component A is k-adjacent to a point p, and equivalently p is k-adjacent to A, if A contains some k-neighbour of p. The tth -order of connectivity is deﬁned by the types of adjacency of sth -order with 0 < s ≤ t. For example, a set in ZZ 3 connected in the 2nd -order is 18-connected.

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Graph-theoretical approach. Considering ZZ n as the set of vertices V of a non-oriented graph G(V, A) and the adjacencies between the points of ZZ n as set of edges A is commonly deﬁned as graph-theoretical approach [2] and the graph G itself is referred to as adjacency graph. An object O ⊂ ZZ n then becomes a subgraph G (V , A ) of G with O = V ⊂ V and A ⊂ A where A is the set of edges representing the adjacencies between the points of O.

3

Simple Closed Curves

A curve in continuous space IRn is denoted with γ and, in contrast, a curve in ZZ n is denoted with γd throughout the paper. In continuous space, the Jordan Curve Theorem considers a closed curve γ homeomorphic to a circle in IR2 [9] and states that such a curve γ separates IR2 into two disjoint connected sets. This feature of γ is known as Jordan property, which is an important property of closed curves in IR2 and shall be kept for closed curves in discrete space. The Jordan curve theorem was originally proposed for discrete curves in ZZ 2 by Rosenfeld [10,11] for pairs of adjacency relations (k, l) where k represents the connectedness of a curve γd and l represents the connectedness of ZZ 2 − γd . The Jordan curve theorem holds for (8, 4) and (4, 8). Theorem 1 (Discrete Jordan Curve Theorem [2]). Let γd be an kconnected set of ZZ 2 containing at least ﬁve points and any point p of γd is k-adjacent to exactly two other points of γd . Then ZZ 2 − γd has exactly two l-components, each of which is l-connected to every point of γd . A curve for which Theorem 1 holds is called a simple curve. Since a Jordan curve in IR2 is a one-manifold, it is possible to derive an analogous deﬁnition of “one-manifold” discrete closed curves from the theorem. These are ﬁnite curves in ZZ 2 for which the theorem holds: Deﬁnition 1 (Simple Closed Curve in ZZ 2 [13]). A simple closed curve γd is a ﬁnite set of points in ZZ 2 for which the following conditions hold. SCC1 γd is k-connected. SCC2 γd consists of at least ﬁve points. SCC3 Each point of γd is k-adjacent to exactly two other points of γd . Condition SCC2 was intended to rule out cases for which ZZ 2 − γd is not separated into two non-empty subsets as illustrated in Fig. 1(a) for a 4-connected curve, and in (b) for an 8-connected curve. Note, that this condition is actually too restrictive for 8-connected curves. An example is given in Fig. 1(c) for which an 8-connected curve separates ZZ 2 into two components and consists itself of four points. Using the terminology of the graph-theoretical approach, a simple closed curve γd is a connected graph in which any vertex has degree two. An arc is deﬁned as an k-connected set of points, denoted with α, with each of its points satisfying the conditions SCC1-3, except two points which are kadjacent to just one other point of α. In [13], Rosenfeld has shown that no simple closed curve in ZZ 2 is both a 4-connected curve and an 8-connected curve.

Curves in ZZ n

(a)

(b)

37

(c)

Fig. 1. Simple closed curves: (a) a 4-connected set to be ruled out, (b) an 8-connected set to be ruled out, (c) a valid 8-connected curve.

The generalization of Def. 1 to simple closed curves in ZZ n is straightforward. Subsequently, we talk about simple curves if we do not care about closeness of γd , i.e. a simple curve is a closed curve or it is an inﬁnite set of points, for which SCC1-3 are true.

4

Closed Quasi Curves

Often, a discrete curve γd is the result of a rasterization of a continuous curve γ or it is the border of a two-dimensional object rastered in ZZ 2 . The generation of a simple curve γd as a discrete representation of a continuous curve γ, that possesses high curvature, is a well-known problem. It has been shown in [5], that a simple curve cannot be obtained generally even in areas of γ with almost no curvature, unless γ is a straight line. This may necessitate an elimination of “additional” points of the rasterization to obtain a simple curve. That problem in discrete space has been already reported, e.g., in [7]. An example is given in Fig. 2(a) for a rastered circle. The points in light grey would have been eliminated to obtain a simple closed 8-connected curve in ZZ 2 .

(a)

(b)

Fig. 2. (a) 8-connected closed quasi curve, (b) adjacency graph.

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As shown above, the deﬁnition of simple curves is too restrictive for many cases. To overcome the problem, a generalization of simple closed curves has been proposed by Malgouyres in [7], which were named closed quasi curves. Closed quasi curves admit – like simple closed curves – a unique parameterization, that has been identiﬁed as one of the fundamental features of curves to be kept in discrete space. Loosely speaking, a closed quasi curve is obtained by adding “triangles” to the adjacency graph of simple closed curves following certain rules. This is illustrated for an example in Fig. 2(b). The notation of closed quasi curves are generally stated for graphs in [8] and for 18-connected closed quasi curves in [7]. The latter one is given below and can be easily adapted to 8-connected curves in ZZ 2 . Deﬁnition 2 (18-connected Closed Quasi Curve). A set γd of ZZ 3 is said to be an 18-connected closed quasi curve iﬀ the cardinality of γd is greater or equal to four and γd satisﬁes the following properties. CQC1 γd is 18-connected. CQC2 Any point of γd is 18-adjacent to either two, three or four points of γd . CQC3 For any subset α of γd consisting of three points such that the elements of α are pairwise 18-adjacent, there is a point p ∈ α which is 18-adjacent to no point in γd − α. If q and r are the two points of α − p, then q and r satisfy the following conditions. CQC3a q and r are 18-adjacent to a point in γd − α. CQC3b No point in γd − α is 18-adjacent to both q and r. CQC4 For each point p in γd which is 18-adjacent to at least three points of γd , there exists a non-trivial partition of N18 (p) ∩ γd into two non-empty subsets α and β such that the following conditions hold. CQC4a The points of α are pairwise 18-adjacent. CQC4b The points of β are pairwise 18-adjacent. CQC4c No point of α is 8-adjacent to a point of β. It has been shown [6], that an 8-connected closed quasi curve γd in ZZ 2 satisﬁes the Jordan property, i.e. ZZ 2 − γd consists of exactly two 4-connected components. The deﬁnition of closed quasi curves as given above works well for many cases in ZZ 2 but is still rather limited in higher dimensional space, i.e. ZZ 3 as shown in Fig. 3. This example does not satisfy the Def. 2 as can be seen from the 18adjacency graph in Fig. 3(b). Either the point q or p must be deleted such that Def. 2 holds. Even more, considering the same curve as a 26-connected curve, the points q and p and the two points adjacent to them are mutually 26-adjacent as shown in the 26-adjacency graph in Fig. 3(c). The points p and q can be viewed as “additional” points of a simple closed 26-connected curve, which can arise only in dimension three or higher. The point r is covered by the Def. 2 as some “additional” point to simple closed curves. Consequently, we need some further generalization of closed curves.

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q

q

p

39

q

p

p

r r

(a)

(b)

r

(c)

Fig. 3. (a) Closed curve in ZZ 3 , (b) 18-adjacency graph, (b) 26-adjacency graph.

5 5.1

A New Deﬁnition of Closed Quasi Curves in ZZ n Properties of Closed Quasi Curves

Before we come to the new deﬁnition of closed quasi curves in ZZ n , basic properties of closed curves are identiﬁed, which we wish to ensure by our deﬁnition of a discrete curve γd in ZZ n . The parameterization is a natural and fundamental property of continuous manifold curves and, of course, must be kept for the representation of such curves in ZZ n . The necessity of a unique Hamiltonian cycle in the graph of a curve – and with it a unique parameterization of the curve – is very restrictive, particularly in three or higher dimensional discrete space. This is illustrated in the example of Fig. 3. On the other hand, a curve connected in the 1st -order meets our understanding of a maximal connected curve, i.e. whenever a point p is added to a simple closed curve connected in the 1st -order, e.g. a 4-connected curve γd in ZZ 2 , the set γd ∪ p does not corresponds to the common understanding of a curve anymore. Hence, the Hamiltonian cycle in the graph of a curve should contain all adjacency relations of the 1st -order. Fig. 4 illustrates a further problem. In both cases, (a) and (b), a Hamiltonian cycle exists (thick lines) which is not unique. The point r can be viewed as some “additional” point with respect to a simple closed curve. However, in (a) the adjacency between the points r and q violates the understanding of a curve since the neighbours p and q of r have a common neighbour which is not adjacent to r. Hence, in Fig. 4(a) r is not a valid “additional” point of a curve, but in (b) it is. Consequently, a speciﬁcation of the adjacencies, that do not indicate the parameterization, is required. 5.2

Deﬁnition of Closed Quasi Curves

Assume the graph G(γd , B) of a k-connected curve γd where B are the edges representing the k-adjacencies between the points of γd . The properties of closed curves as discussed in the previous section are kept using the following deﬁnition.

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p r q (a)

(b)

Fig. 4. Hamiltonian cycle in a closed curve (a) invalid (b) valid.

Deﬁnition 3 (Closed Quasi Curve in ZZ n ). A k-connected subset γd of ZZ n is a closed quasi curve in ZZ n iﬀ γd satisﬁes the following properties. CQC1’ There exists a Hamiltonian cycle H(γd , A) in the graph G(γd , B), where A ⊂ B and A contains all edges representing the adjacencies of the 1st -order between points of γd . CQC2’ For each point p of γd the subgraph H (Nk (p) ∩ γd , A ) of H(γd , A) induced by the points of γd in the k-neighbourhood of p, constitutes a k-path. CQC3’ If for H(γd , A) A represents exclusively adjacencies of the 1st -order, the cardinality of γd is greater or equal to ﬁve and it is greater or equal to four, otherwise. Condition CQC1’ can be veriﬁed as follows. Associate each element of B of the graph G with the order of the adjacency represented. Then the problem of ﬁnding any Hamiltonian cycle in G is replaced by ﬁnding a Hamiltonian cycle H of minimal length in the weighted graph G. This is a special case of the traveling salesman problem [3]: each pair of vertices with an edge between them has a distance depending on the weight, while pairs of vertices without an edge between them are separated by a distance of inﬁnite length. When a Hamiltonian cycle of minimal length has been obtained in G, the cycle must contain all adjacencies of the 1st -order. Otherwise, the set is not a quasi curve. Since each point of γd has degree two in H(γd , A) and A contains all adjacencies of the 1st -order, no point of a closed quasi curve has more than two neighbours adjacent in the 1st -order. Remark 1. If any point in the set γd has more than two neighbours of the 1st order of adjacency, the set is not a quasi curve. Condition CQC2’ of Def. 3 ensures that any additional adjacency with respect to a simple curve arise only between points that are “close to each other” along the parameterization. Such additional adjacencies may arise in areas where curves perform high curvature. Finally, condition CQC3’ rules out cases, e.g. as shown in Fig. 1(a), in which ZZ 2 − γd is not separated. In many cases a closed quasi curve connected in the 1st -order is also a closed quasi curve connected in some higher order. This fact is illustrated by examples in Fig. 5(a) and (c) for a curve in ZZ 2 and ZZ 3 , respectively. The thick lines in the adjacency graphs represent the adjacencies of the 1st -order. Note that the

Curves in ZZ n

41

example in Fig. 5(a) is not an 8-connected closed quasi curve according to the deﬁnition by Malgouyres.

(a) 4 1 1 2 3 5 18 18 6 7 17 16 8 17 15 9 15 14 10 13 13 12 11

(c)

(b) 4

2

3 5 16 6 7 14 9 8 11 12 10

(d)

(e)

Fig. 5. (a) Closed quasi curve in ZZ 2 with (b) the adjacency graph for the 8connectedness, and (c) closed quasi curve in ZZ 3 with adjacency graph for the (d) 18-connectedness and (e) 26-connectedness.

The examples in Fig. 5 suggest that any closed quasi curve connected in the 1st -order is also a closed quasi curve connected in some higher order and, inversely, for each closed quasi curve γd connected in an order greater than one there exists a closed quasi γd connected in the 1st -order such that γd ⊂ γd . This reasoning is not true, as can be seen from the examples in Fig. 6: in (a) and (c) closed quasi curves connected in the 1st -order are shown that are not quasi curves for any higher order of connectivity, because condition CQC2’ of Def. 3 would not be satisﬁed in these cases. For the example in Fig. 6(b), there does not exist any 4-connected closed quasi curve γd , of which the shown curve γd is a subset. Indeed, it is impossible to identify two 4-neighbours of p - call them q and r - which connect p with the set γd − p in the 1st -order such that each point q and r has exactly two 4-neighbours in γd . For a closed quasi curve that satisﬁes Def. 3 and is connected in the 1st -order, the following is true: Theorem 2. A closed quasi curve γd in ZZ n connected in the 1st -order is a simple closed curve. Proof. Assume the adjacency graph G(γd , A1 ) of a closed quasi curve γd connected in the 1st -order. Then A1 covers all adjacencies of the 1st -order between

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p

(a)

(b)

(c)

Fig. 6. (a) 4-connected closed quasi curve, (b) 8-connected closed quasi curve, and (c) 6-connected closed quasi curve.

the points of γd . Since the Hamiltonian cycle H(γd , A) contains all adjacencies of the 1st -order H(γd , A) = G(γd , A1 ). Consequently, each point of γd is adjacent in the 1st -order to exactly two other points of γd . Moreover, following Def. 3 the cardinality of G is equal or greater to ﬁve. From Theorem 2 follows that a 4-connected closed quasi curves in ZZ 2 that satisﬁes Def. 3 is a 4-connected simple closed curve. The Jordan Curve Theorem 1 is true for a simple closed curve. Consequently, the Jordan Curve Theorem is also true for a 4-connected closed quasi curves in ZZ 2 : Corollary 1. A 4-connected closed quasi curve γd in ZZ 2 separates ZZ 2 − γd in exactly two 8-components each of which is 8-connected to every point of γd . Finally, it is left to show that an 8-connected closed quasi curve in ZZ 2 satisfying the new deﬁnition has the Jordan property. For this, the terms maximal and minimal connected closed quasi curve are used and introduced below. A maximal connected closed quasi curve is a closed quasi curve that satisﬁes Def. 3 and to which no point can be added such that the result still fulﬁlls the conditions of Def. 3. For example, a 4-connected closed quasi curve in ZZ 2 is a maximal connected curve. As stated in Theorem 2, a 4-connected closed quasi curve γd in ZZ 2 is a simple closed curve. Thus, each point of γd has exactly two 4-neighbours. Adding a point p to γd such that p is 4-connected to γd leads to at least one point of γd that has more than two 4-neighbours in γd ∪ p. As already stated in Remark 1, then the set γd ∪ p is not a closed quasi curve. In contrast, a minimal connected closed quasi curve is a closed quasi curve that satisﬁes Def. 3 and from which no point can be deleted such that the result still fulﬁlls the conditions of Def. 3. A minimal 8-connected closed quasi curve in ZZ 2 , which satisﬁes Def. 3 is a curve γd for which the Hamiltonian cycle H(γd , A) = G(γd , B), i.e. all 8-adjacencies between the points of γd belong to the Hamiltonian cycle. In this case, each point of γd has exactly two 8-neighbours in γd . Thus, a minimal 8-connected closed quasi curve in ZZ 2 is an 8-connected simple closed curve.

Curves in ZZ n

43

Theorem 3. An 8-connected closed quasi curve γd in ZZ 2 that satisﬁes Def. 3 separates ZZ 2 − γd in exactly two 4-components. Proof. Assume γd is a minimal connected closed quasi curve, i.e. γd is an 8connected simple closed curve. Then γd separates ZZ 2 −γd into two 4-components. One can view any 8-connected closed quasi curve as a minimal 8-connected closed quasi curve with added points. It will be shown below, that any point added to γd neither removes one of the 4-components separated by the minimal curve nor increases the number of 4-components by subdividing an existing 4component. Assume an added point p removes one of the 4-components of ZZ 2 − γd and call this component I. Then I must consist of exactly one point q and all points in N4 (q)−q belong to γd . Consequently, p would have four 4-neighbours in γd and this contradicts Def. 3, because in this case there does not exist a Hamiltonian cycle in γd ∪ p containing all 4-adjacency relations between the points of γd ∪ p. Assuming a point p ∈ ZZ 2 − γd , that is added to a minimal connected closed quasi curve γd , introduces at least one additional 4-component by subdividing an existing one such that ZZ 2 − γd − p consists of at least one more 4-component than ZZ 2 − γd . To achieve this, N8 (p) − γd must consist of m 4-components and N8 (p) − γd − p of at least m + 1 4-components. Hence, there are at least two 8-components in N8 (p) ∩ γd . Assume there exist a Hamiltonian cycle for the set γd ∪ p which contains all edges representing 4-adjacencies between the points of γd ∪ p. To satisfy condition CQC2’ of Def. 3, for any point p added to a minimal 8-connected closed quasi curve γd , (N8 (p) − p) ∩ γd must form an 8-path. However, this contradicts the conditions of a point p that subdivides an existing 4-component of ZZ 2 − γd .

6

Summary

It has been shown that simple closed curves and closed quasi curves proposed in [7] might be too restrictive for practical purposes. This has motivated the new deﬁnition of closed quasi curves introduced in this paper. The new deﬁnition can be viewed as some generalization of quasi curves as introduced in [7] and overcomes the limitations of them. Moreover, the deﬁnition is given for curves in n-dimensional discrete space and is, therefore, not restricted to any certain dimension of space. It has been proven that curves in ZZ 2 , which satisfy the new deﬁnition, have the Jordan property. It is a drawback of the new deﬁnition - in contrast to the known deﬁnition of quasi curves - that it cannot be decided locally for a point p if this point belongs to a quasi curve considering only the neighbourhood conﬁguration of the point. This is a point of future work. At the moment it can be stated only, that the existence of a path in the subgraph of the Hamiltonian cycle induced by the points of a quasi curve in the neighbourhood of each point of the quasi curve is necessary. It has to be shown, if this condition is already suﬃcient or if further local properties have to be identiﬁed.

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G. Th¨ urmer

Consider the Jordan curve theorem for an 8-connected closed quasi curve γd in ZZ 2 . It seems to be clear that each point of γd is 8-connected to the two separated sets of ZZ 2 − γd . However, at the moment this is just a conjecture and needs further investigations to provide a formal proof. This is also a point of future work.

References 1. Cohen-Or, D., and Kaufman, A. Fundamentals of surface voxelization. Graphical Models and Image Processing 57, 6 (1995), 453–461. 2. Kong, T. Y., and Rosenfeld, A. Digital topology: a comparison of the graphbased and topological approaches. In Topology and category theory in computer science, G. M. Reed, A. W. Roscoe, and R. F. Wachter, Eds. Oxford University Press, 1991, pp. 273–289. 3. Lawler (ed.), E. L., and Rinnooy-Kan, A. H. The Traveling Salesman problem: A guided tour of combinatorial optimization. Wiley-Interscience Series in Discrete Mathematics. John Wiley and Son, 1985. 4. Lebied´ z, J. Discrete arcs and curves. Machine Graphics and Vision 9, 1/2 (2000), 25–30. ¨thrich, C. A. Morphologically closed surfaces and their 5. Lincke, C., and Wu digitization. submitted for publication. 6. Malgouyres, R. Phd thesis, Universit´e d’Auvergne, Clermont-Ferrand, France, 1994. 7. Malgouyres, R. Graphs generalizing closed curves with linear construction of the Hamiltonian cyle - parameterization of discretized curves. Theoretical Computer Science 143 (1995), 189–249. 8. Malgouyres, R. A deﬁnition of surfaces of Z 3 : A new 3D discrete Jordan theorem. Theoretical Computer Science 186 (1997), 1–41. 9. Moise, E. E. Geometric topology in dimensions 2 and 3. Springer–Verlag, New York, 1977. 10. Rosenfeld, A. Connectivity in digital pictures. Journal of the ACM 17, 1 (1970), 146–160. 11. Rosenfeld, A. Arcs and curves in digital pictures. Journal of the ACM 20, 1 (1973), 81–87. 12. Rosenfeld, A. A converse to the Jordan curve theorem for digital curves. Information and Control 29 (1975), 292–293. 13. Rosenfeld, A. Three-dimensional digital topology. Information and Control 50 (1981), 119–127. 14. Shareef, N., and Yagel, R. Rapid previewing via volume-based solid modeling. In Proc. of Solid Modeling ’95 (1995), pp. 281–292. 15. Wang, S. W., and Kaufman, A. E. Volume-sampled 3D modeling. IEEE Computer Graphics and Applications 14, 5 (1994), 26–32. ¨thrich, C. A. A model for curve rasterization in n-dimensional space. Com16. Wu puters and Graphics 22, 2–3 (1998), 153–160. 17. Yagel, R., Stredney, D., Wiet, G. J., Schmalbrock, P., Rosenberg, L., Sessanna, D. J., and Kurzion, Y. Building a virtual environment for endoscopic sinus surgery simulation. Computers and Graphics 20, 6 (1996), 813–823.

Separation Theorems for Simplicity 26-Surfaces J.C. Ciria, E. Dom´ınguez, and A.R. Franc´es Dpt. de Inform´ atica e Ingenier´ıa de Sistemas. Facultad de Ciencias. Universidad de Zaragoza. E-50009 – Zaragoza. Spain. {jcciria,afrances}@posta.unizar.es

Abstract. The main goal of this paper is to prove a Digital Jordan– Brouwer Theorem and an Index Theorem for simplicity 26-surfaces. For this, we follow the approach to Digital Topology introduced in [2], and ﬁnd a digital space such that the continuous analogue of each simplicity 26-surface is a combinatorial 2-manifold. Thus, the separation theorems quoted above turn out to be an immediate consequence of the general results obtained in [2] and [3] for arbitrary digital n-manifolds. Keywords: Digital surface, simplicity 26-surface, digital separation theorems.

Introduction Several diﬀerent notions of digital surface have been introduced on the grid ZZ 3 , within the graph theoretical approach to Digital Topology, since Morgenthaler and Rosenfeld [9] deﬁned the (26, 6)- and (6, 26)-surfaces in 1981. Without going into all the existing notions, let us recall the (α, β)-surfaces, for α, β ∈ {6, 18, 26} and (α, β) = (6, 6), deﬁned by Kong and Roscoe [7], which extend the original notion to other couples of adjacencies in ZZ 3 , and the strong n-surfaces, n ∈ {18, 26}, of Bertrand and Malgouyres [6], where the set of strong 26-surfaces strictly contains all the (26, 6)-surfaces (see [4]). More recently, Couprie and Bertrand [5] have introduced the notion of simplicity n-surface, for n ∈ {6, 26}; and they have also shown that any strong 26-surface is a simplicity 26-surface. Usually, some kind of separation theorem is obtained in order to show that these discrete objects are suitable digital counterparts to the notion of topological surface in the Euclidean space IR3 . For instance, in [7] and [6] it is shown that a digital version of the Jordan–Brouwer Theorem holds for (α, β)-surfaces and strong 26-surfaces, respectively. Moreover, Morgenthaler and Rosenfeld show in [9] an Index Theorem for (26, 6)- and (6, 26)-surfaces. However, similar results for simplicity 26-surfaces cannot yet be found in the literature. Our main goal in this paper is to prove a digital Jordan–Brouwer Theorem and an Index Theorem for these surfaces. These results have been found within the framework for Digital Topology introduced in [1] and [2].

This work has been partially supported by the project DGES TIC2000-1368-C03-01.

A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 45–56, 2002. c Springer-Verlag Berlin Heidelberg 2002

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A digital space, as deﬁned in [2], is a pair (K, f ) consisting of a polyhedral complex K, which represents the spatial layout of pixels, and a lighting function f providing a “continuous interpretation” for each digital image in K. In this approach, a quite natural notion of digital n-dimensional manifold arises for which general digital versions of the Jordan–Brouwer and Index Theorems are stated and proved in arbitrary dimension (see [2,3]). Moreover, (α, β)-surfaces and strong 26-surfaces turn out to be digital 2-manifolds (digital surfaces) for suitable digital spaces (R3 , fαβ ) and (R3 , f BM ), respectively, deﬁned on the standard cubical decomposition of the Euclidean space R3 (see [1,2]). This means, in particular, that the proof of such separation theorems can be directly applied to these kinds of surfaces. However, there does not exist a lighting function f on the device model R3 such that every simplicity 26-surface is a digital surface in the digital space (R3 , f ). For this reason, in this paper, we consider a weaker version for one of the axioms deﬁning lighting functions, in order to deal with a larger class of digital spaces for which the proofs of the general separation theorems quoted above still work. From these results, we derive in Section 3 a Jordan–Brouwer Theorem and an Index Theorem for simplicity 26-surfaces as follows. Firstly, we translate the notion of simplicity 26-surface into terms of the device model R3 (see Section 2). Then we ﬁnd, also in Section 3, a lighting function f ss on R3 satisfying the new weaker axiom, and we show that any simplicity 26-surface is a digital surface in the digital space (R3 , f ss ). The long proof of this result is outlined in Section 4, while Section 1 introduces the basic notions of the approach to Digital Topology in [1,2] that are needed in this paper. It is worth pointing out that simplicity 26-surfaces are not the only digital surfaces in the space (R3 , f ss ). At the present time, we do not know about the existence of a digital space whose digital surfaces coincide with the class of simplicity 26-surfaces. This will be the subject of a future work. On the other hand, the class of simplicity 6-surfaces is the class of digital surfaces in the digital space (R3 , f6,26 ) given in [1]. This is a consequence of the fact that the former class coincides with Morgenthaler and Rosenfeld’s (6, 26)-surfaces ([5]), and these are exactly the digital surfaces in (R3 , f6,26 ) (see [1]).

1

Our Approach to Digital Topology

The main purpose of Digital Topology is to study the topological properties of digital images. Digital images are discrete objects in nature, but they usually represent continuous objects or, at least, they are perceived as continuous objects. In our approach to Digital Topology [1], [2] we propose a multilevel architecture to represent, using diﬀerent levels, each digital object together with the continuous perception that an observer may have of it, that we call its continuous analogue. In addition, several other intermediate levels allow us to relate the properties of digital objects with the corresponding properties of their continuous analogues. However, for simplicity, we introduce in this section only the levels of this architecture that are explicitly used in this paper.

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The ﬁrst level, called device model , is a homogeneously n-dimensional locally ﬁnite polyhedral complex K, which is used to represent the spatial layout of space elements (xels for short). Each n-cell in a device model K represents a xel, and so the digital object displayed in an image is a subset of the set celln (K) of n-cells in K; while the other lower dimensional cells in K are used to describe how the xels could be linked to each other. In addition, the selection of a continuous analogue for each digital object in a device model K is determined by a lighting function f deﬁned on K. So, a digital space is a pair (K, f ); that is, a device model K together with a “continuous interpretation” for each digital object in K. Next we describe these two notions in detail. By a homegeneously n-dimensional locally ﬁnite polyhedral complex we mean a set K of polytopes, in some Euclidean space IRd , provided with the natural ordering given by the relationship “to be face of”, that in addition satisﬁes the four following properties: 1. If σ ∈ K and τ is a face of σ then τ ∈ K. 2. If σ, τ ∈ K then σ ∩ τ ∈ K is a face of both σ and τ . 3. For each point x in the underlying polyhedron | K | = ∪{σ; σ ∈ K} of K, there exists a neighbourhood of x which intersects only a ﬁnite number of polytopes in K; in particular, each polytope of K is a face of a ﬁnite number of other polytopes in K. 4. Each polytope σ ∈ K is a face of some n-dimensional polytope in K. These complexes are particular cases of cellular complexes, as they are usually deﬁned in polyhedral topology. So, in this paper, polytopes in K will be simply referred to as cells, and K itself will be called a polyhedral complex. The next paragraph recalls some elementary notions from polyhedral topology used in this paper. We refer to [10] for further notions on this subject. Given a polyhedral complex K and two cells γ, σ ∈ K, we shall write γ ≤ σ if γ is a face of σ, and γ < σ if in addition γ = σ. A centroid-map on K is a map ◦ c : K → | K | such that c(σ) belongs to the interior of σ; that is, c(σ) ∈σ= σ −∂σ, where ∂σ = ∪{γ; γ < σ} stands for the boundary of σ. Remark 1. The device model Rn , called the standard cubical decomposition of the Euclidean n-space IRn , is the complex determined by the collection of unit n-cubes in IRn whose edges are parallel to the coordinate axes and whose centers are in the set ZZ n . The centroid-map we will consider in Rn associates each cube σ with its barycentre c(σ), which is a point in the set Z n . Here, Z = 12 ZZ stands for the set of points {z ∈ IR; z = y/2, y ∈ ZZ}. In particular, if dim σ = n then c(σ) ∈ ZZ n , where dim σ denotes the dimension of σ; and, thus, every digital object O in Rn can be identiﬁed with a subset of points in ZZ n . Before proceeding with the deﬁnition of lighting function, we need to introduce the following notions. The ﬁrst two notions formalize two types of “digital neighbourhoods” of a cell α ∈ K in a given digital object O ⊆ celln (K). Indeed, the star of α in O is the set stn (α; O) = {σ ∈ O; α ≤ σ} of n-cells (xels) in O having α as a face.

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Similarly, the extended star of α in O is the set st∗n (α; O) = {σ ∈ O; α∩σ = ∅} of n-cells (xels) in O intersecting α. Finally, the support of a digital object O is the set supp(O) of cells of K (not necessarily xels) that are the intersection of n-cells (xels) in O. Namely, α ∈ supp(O) if and only if α = ∩{σ; σ ∈ stn (α; O)}. Notice that O ⊆ supp(O) for any digital object O, and the elements in supp(O) − O are the lower dimensional cells of K that directly connect at least two xels in O. For ease of writing, we use the following notation: supp(K) = supp(celln (K)), stn (α; K) = stn (α; celln (K)) and st∗n (α; K) = st∗n (α; celln (K)). Finally, we shall write P(A) for the family of all subsets of a given set A. Deﬁnition 1. Given a device model K, a lighting function on K is a map f : P(celln (K)) × K → {0, 1} satisfying the following ﬁve axioms for all O ∈ P(celln (K)) and α ∈ K: 1. 2. 3. 4. 5.

object axiom: if dim α = dim K, then f (O, α) = 1 if and only if α ∈ O; support axiom: if α ∈ / supp(O) then f (O, α) = 0; monotone axiom: f (O, α) ≤ f (celln (K), α); local axiom: f (O, α) = f (st∗n (α; O), α); and, complement connectivity axiom: if O ⊆ O ⊆ celln (K) and α ∈ K are such that stn (α; O) = stn (α; O ), f (O , α) = 0 and f (O, α) = 1, then: (a) the set of cells α(O ; O) = {ω < α; f (O , ω) = 0, f (O, ω) = 1} is not empty; and, ◦ (b) the set ∪{ω; ω ∈ α(O ; O)} is connected in ∂α.

If f (O, α) = 1 we say that f lights the cell α for the object O. As quoted above, the lighting function f provides a continuous interpretation for each digital object in a digital space (K, f ), which is represented by an Euclidean polyhedron in the ﬁfth level of our architecture. To introduce these polyhedra we use an arbitrary but ﬁxed centroid–map c : K → | K | on the device model K. Deﬁnition 2. Let O ⊆ celln (K) be a digital object in the digital space (K, f ). The continuous analogue of O is the underlying polyhedron | AO | of the simplicial complex AO , whose k-simplexes are c(α0 ), c(α1 ), . . . , c(αk ) where α0 < α1 < · · · < αk are cells in K such that f (O, αi ) = 1, for 0 ≤ i ≤ k. The complex AO is called the simplicial analogue of O. Since continuous analogues represent the continuous interpretation that an observer makes of digital objects, our architecture allows us to introduce digital notions in terms of the corresponding continuous ones. For example, we will say that an object O is connected if its continuous analogue | AO | is a connected polyhedron. Similarly, if we consider the set of xels celln (K) − O as the complement of the digital object O, we say that celln (K) − O is connected if the complement of the object’s continuous analogue | Acelln (K) |−| AO | is a connected topological space. And, an object O is also naturally called a m-dimensional digital manifold if | AO | is a combinatorial m-manifold without boundary; that is, we call O a digital surface (2-manifold) if it looks like a surface. However, doing

Separation Theorems for Simplicity 26-Surfaces

49

this, the problem of characterizing these notions in digital terms arises (see [2] for connectivity, and [1],[2] for digital surfaces). But, even more interesting than the deﬁnition of new digital notions is the possibility of translating results from polyhedral topology, through the levels of this architecture, in order to obtain general results in Digital Topology. Two examples of this powerful technique are the Digital Jordan–Brouwer and Index Theorems for arbitrary m-manifolds proved in [2] and [3], respectively, which generalize the well–known result of Morgenthaler and Rosenfeld ([9]) to all types of (α, β)-surfaces ([7]) and to the strong 26-surfaces ([6]). At this point, it is worth pointing out that, sometimes, the continuous result can be translated into digital terms directly, as was the case of the Digital Jordan–Brouwer Theorem; but, in general, some previous work must be done at the continuous level to adapt the continuous result before the translation can be carried out. In this paper we consider, however, a class of digital spaces larger than the class for which these separation theorems were proved in [2] and [3]. The lighting functions deﬁning this restricted class of spaces satisfy, in addition to the ﬁve axioms in Deﬁnition 1, the following condition Subspace condition: Let O ⊆ O ⊆ celln (K) be two digital objects and α ∈ K a cell such that stn (α; O) = stn (α; O ), f (O , α) = 0 and f (O, α) = 1. Then, if O ⊆ O ⊆ celln (K), f (O, β) = 1 for every β ∈ α(O ; O). This is a suﬃcient condition to ensure that each digital object O in a digital space (K, f ) induces a digital subspace (K(O), fO ), where K(O) = {α ∈ K; α ≤ σ ∈ O} is the subcomplex of K induced by the cells in O and the lighting function fO is deﬁned by fO (O , α) = f (O, α)f (O , α) for every O ⊆ O and α ∈ K(O). It is not diﬃcult to observe that the proofs given in [2] and [3] do not make use of this additional condition anywhere; and, thus, the Jordan–Brouwer and Index Theorems hold for the extended class of digital spaces considered in this paper.

2

About Simplicity 26-Surfaces

Couprie and Bertrand [5] deﬁne a simplicity 26-surface as a “thin” subset of ZZ 3 such that a certain neighbourhood of each one of its points constitutes a simple closed curve under the structure of a particular graph, which is deﬁned using the notions of homotopy and simple point. In the same paper [5], two local characterizations of simplicity 26-surfaces are provided. One of them states that any simplicity 26-surface can be obtained by concatenating a certain family of patterns according to a number of rules; in particular, the patterns around each point must deﬁne a cycle. In this paper we will use this characterization as the deﬁnition of simplicity 26-surfaces. Before introducing this deﬁnition of simplicity 26-surfaces in terms of digital objects in the device model R3 , we ﬁrstly recall, in this language, some basic notions of the graph–theoretical approach to Digital Topology. For this, we use the identiﬁcation between the sets ZZ 3 and cell3 (R3 ) given in Remark 1.

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Given a xel σ ∈ cell3 (R3 ) we consider the following three neighbourhoods ([8]): N26 (σ) = {τ ∈ cell3 (R3 ); σ ∩ τ = ∅}, N18 (σ) = {τ ∈ N26 (σ); dim σ ∩ τ ≥ 1} ∪ {σ}, and N6 (σ) = {τ ∈ N26 (σ); dim σ ∩ τ = 2} ∪ {σ}. Notice that N26 (σ) = st∗3 (σ; R3 ). Two xels σ, τ ∈ cell3 (R3 ) are said to be n-adjacent if τ ∈ Nn∗ (σ) = Nn (σ) \ {σ}, where n ∈ {6, 18, 26}. Given a digital object O ⊆ cell3 (R3 ) and two xels σ, τ ∈ O, a n-path in O from σ to τ is a sequence (σi )ki=0 ⊆ O such that σ0 = σ, σk = τ , and σi−1 is n-adjacent to σi , for 1 ≤ i ≤ k. Notice that n-paths deﬁne an equivalence relation in the object O, whose classes are called the n-components of O. Finally, a digital object O ⊆ cell3 (R3 ) is said to be n-connected if it consists of only one n-component. Next, we introduce the family of patterns deﬁning simplicity 26-surfaces. For this we need the following Deﬁnition 3. Let O ⊆ cell3 (R3 ) be a digital object. A vertex α ∈ R3 is said to be maximal with respect to O if st3 (α; O) = ∅ and, in addition, st3 (α; O) = st3 (β; O) for any other vertex β ∈ R3 such that st3 (α; O) ⊆ st3 (β; O). If α is maximal with respect to O, we say that st3 (α; O) is a maximal trace (or simply a trace) of O. Given a digital object O ⊆ cell3 (R3 ), the set of traces of O will be denoted by Tr(O), while Tr(σ, O) will stand for the traces of O containing a given xel σ ∈ O. Couprie and Bertrand deﬁne in [5] a graph on the set of traces of a given digital object as follows. Deﬁnition 4 (Def. 16 in [5]). Let O ⊆ cell3 (R3 ) be a digital object. The vertices of the graph GTr(O) are the traces of O, and two distinct traces t1 , t2 ∈ Tr(O) are adjacent in GTr(O) if the intersection t1 ∩ t2 consists of exactly two elements. Remark 2 (Lemma 15 in [5]). The traces of an arbitrary digital object O may correspond (up to rotations and symmetries) to 21 diﬀerent patterns. However, only the eight patterns from the set A26 = {a1 , . . . , a8 } depicted in Fig. 1 may appear in a simplicity 26-surface. Deﬁnition 5 (Th. 17 in [5]). A 26-connected digital object S ⊆ cell3 (R3 ) is said to be a simplicity 26-surface if the following conditions are satisﬁed for any xel σ ∈ S: 1. T26 (σ, S) = 1 and T6 (σ, cell3 (R3 ) − S) = 2. 2. Every trace of S corresponds (up to rotations and symmetries) to an element of the set A26 . 3. σ is the only xel in S belonging to all of the traces of Tr(σ, S); that is, σ = ∩{t; t ∈ Tr(σ, S)}. 4. For any two distinct traces t1 , t2 ∈ Tr(σ, S), t1 ∩t2 has at most two elements. 5. The set of traces Tr(σ, S) constitutes a simple closed curve in the graph GTr(O); that is, for each trace t ∈ Tr(σ, O) there exist exactly two other traces t1 , t2 ∈ Tr(σ, O) which are adjacent to t in the graph GTr(σ, O).

Separation Theorems for Simplicity 26-Surfaces

a1

a2

a3

a4

a5

a6

a7

a8

51

Fig. 1. The eight patterns deﬁning simplicity 26-surfaces.

3

Main Results

As was quoted above, our main goal in this paper is to state and prove a Jordan– Brouwer Theorem and an Index Theorem for simplicity 26-surfaces. For this purpose, we ﬁrstly ﬁnd a digital space in which the continuous analogue of each simplicity 26-surface is a combinatorial surface. This space is deﬁned on the device model R3 , the standard cubical decomposition of the Euclidean 3-space IR3 , by the lighting function f ss , which is given by f ss (O, α) = 1 if and only if: (a) dim α = 3 and α ∈ O; (b) dim α = 2 and α ∈ supp(O); (c) dim α = 0, α ∈ supp(O) but st3 (α; O) does not correspond to the pattern a7 in Fig. 1; and (d) dim α = 1 and one of the two following conditions holds: (d1) st3 (α; R3 ) ⊆ O (d2) st3 (α; O) = {σ1 , σ2 }, with α = σ1 ∩ σ2 , and the two vertices β1 and β2 of α are both maximal with respect to O. Next we give several immediate properties of the digital space (R3 , f ss ) needed in the sequel. Remark 3. 1) For any cell α ∈ R3 , f ss (cell3 (R3 ), α) = 1. So that, the continuous analogue of the digital object cell3 (R3 ) consisting of all the xels in (R3 , f ss ) is | Acell3 (R3 ) | = IR3 the Euclidean 3-space. 2) The digital space (R3 , f ss ) is solid; that is, f ss (O, α) = 1 for any cell α ∈ R3 and any digital object O ⊆ cell3 (R3 ) such that st3 (α; R3 ) ⊆ O. 3) The digital space (R3 , f ss ) is strongly local at dimensions 0, 2 and 3; that is, if dim α ∈ {0, 2, 3} then f ss (O, α) = f ss (st3 (α; O), α) for any object O. Now we are ready to state our basic result, whose long and quite technical proof is outlined in Section 4. Theorem 1. Every simplicity 26-surface is a digital surface in the digital space (R3 , f ss ); that is, the continuous analogue | AS | of any simplicity 26-surface S is a combinatorial surface.

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J.C. Ciria, E. Dom´ınguez, and A.R. Franc´es

Fig. 2. A digital surface in (R3 , f ss ) which is not a simplicity 26-surface.

Remark 4. Simplicity 26-surfaces are not the only digital surfaces in the digital space (R3 , f ss ). For example, it is easy to check that the digital object S depicted in Fig. 2 is a digital surface. However, it is not a simplicity 26-surface since any two traces in S corresponding to the pattern a7 meet in three xels. Both the Jordan–Brouwer Theorem and the Index Theorem are related to the connectivity of the complement of digital objects; so that, we need to determine the connectivity associated to the digital space (R3 , f ss ). This is done in the next result, whose proof is straightforward from the deﬁnition of the lighting function f ss and the characterization of the connectivity of an object and its complement given in [2, Theorem 4.2] (see also [1]). Proposition 1. Let O be a digital object in (R3 , f ss ). Then 1. a susbset C ⊆ O is a 26-component of O if and only if C = {σ ∈ O; c(σ) ∈ X} for some component X of the continuous analogue | AO | of O; and 2. a susbset D ⊆ cell3 (R3 ) − O is a 6-component of cell3 (R3 ) − O if and only if D = {σ ∈ O; c(σ) ∈ Y } for some component Y of the complement IR3 − | AO | = | Acell3 (R3 ) | − | AO | of the continuous analogue of O. As quoted in Section 1, the Digital Jordan–Brouwer and Index Theorems were originally proved in [2, Theorem 5.3] and [3, Theorem 3.16], respectively, for the class of digital spaces satisfying the subspace condition. However, these proofs do not make use of such condition. Thus, to reach our goal, it will suﬃce to check that the hypotheses of these general separation theorems are satisﬁed in the digital space (R3 , f ss ), which is actually done in Remarks 3(1) and (2). In this way, the Jordan–Brouwer Theorem for simplicity 26-surfaces follows from the next result, which is an immediate consequence of Proposition 1 and Theorem 5.3 in [2]. Theorem 2 (Jordan–Brouwer Theorem). Let S be a 26-connected digital surface in (R3 , f ss ). Then, its complement cell3 (R3 ) − S consists of two 6components. Moreover, if S is ﬁnite then one of the 6-components of cell3 (R3 )−S is also ﬁnite.

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To state the Index Theorem for simplicity 26-surfaces we need the following additional notions (see [3] for details). Let σ ∈ cell3 (R3 ) be a xel with centroid c(σ) = (x1 , x2 , x3 ) ∈ ZZ 3 . We call the object Hσ = {τ ∈ cell3 (R3 ); c(τ ) = (x1 , x2 , λ) ∈ ZZ 3 , λ ≥ x3 } a digital half–line starting at σ. Given a ﬁnite digital surface S in (R3 , f ss ) such that σ ∈ / S, the 26-components of Hσ ∩ S turn out to be ﬁnite digital segments; and thus, we can consider the predecessor πC and the successor σC of C in Hσ . Then, we say that the half-line Hσ meets S transversally at C if πC and σC lie in the same 6-component of the object ∪σ∈C N26 (σ) − S. Now, Proposition 1 and Theorem 3.16 in [3] yield the following result. Theorem 3 (Index Theorem). Let S be a ﬁnite digital surface in (R3 , f ss ). A xel σ ∈ / S belongs to the ﬁnite 6-component of the complement of S if and only if the half-line Hσ meets S transversally an odd number of times.

4

An Outline of the Proof of Theorem 1

Our goal in this Section is to prove that, for any simplicity 26-surface S, the continuous analogue | AS | in the digital space (R3 , f ss ) is a combinatorial surface. For this we will ﬁnd a suitable cellular decomposition of the polyhedron | AS |. By a (principal) cellular decomposition of a polyhedron P we mean a triple C = (F, E, P) of families of faces (2-balls), edges (1-balls) and points (0-balls), respectively, satisfying the following properties: 1. for any two distinct faces f1 , f2 ∈ F, their intersection f1 ∩ f2 is a ball in E ∪ P or it is the empty set; 2. for any two distinct edges e1 , e2 ∈ E, the intersection e1 ∩ e2 is a point in P or it is empty; 3. the polyhedron P is the union of all balls in F ∪ E ∪ P; and, 4. each point p ∈ P is contained in some edge e ∈ E; and, similarly, each edge e ∈ E is contained in some face f ∈ F. A polyhedron P is said to be a combinatorial surface if it has a cellular decomposition C = (F, E, P) for which the two following additional conditions holds: 5. every edge e ∈ E is included in exactly two faces; and, 6. for every point p ∈ P, the union ∪{f ∈ F; p ⊆ f } is a 2-ball. We claim that, for any simplicity 26-surface, there exists a cellular decomposition CS = (FS , ES , PS ) of its continuous analogue | AS | satisfying the ﬁve properties above. The triple CS is deﬁned as follows: – the faces in FS are the polyhedra fα = | AS | ∩ | Ast3 (α;R3 ) |, where α ranges over the set of vertices of R3 which are maximal with respect to S; – the edges in ES are the intersections eαβ = fα ∩ fβ , where fα , fβ are two distinct faces in FS such that the traces st3 (α; S) and st3 (β; S) share exactly two xels of S; and, ﬁnally, – the family PS of points is the set of centroids c(σ) where σ ∈ S.

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f1

f2

f3

f4

f5

f6

f7

f8

Fig. 3. The eight faces of the cellular decomposition of | AS |. The bold segments are the edges in ES contained in these faces.

In the rest of this Section we give an outline of the long and rather technical proof of our claim. The ﬁrst step is to determine the shape of the faces in the set FS , which is done in the following result. Proposition 2. Let S be a simplicity 26-surface in (R3 , f ss ) and α ∈ R3 a maximal vertex with respect to S. If the trace st3 (α; S) corresponds to the pattern ai (1 ≤ i ≤ 8) in Fig. 1, then the face fα = | AS |∩| Ast3 (α;R3 ) | is (up to rotations and symmetries) the 2-ball fi depicted in Fig. 3. The proof of Proposition 2 makes use of the fact that the simplicial analogues AS and Ast3 (α;R3 ) are both full subcomplexes of the ﬁrst derived subdivision of the device model R3 , which is determined by the centroid–map c : R3 → | R3 | that associates each cell γ ∈ R3 with its barycentre (see Remark 1). Moreover, the vertices of Ast3 (α;R3 ) are the centroids c(γ) of all cells γ ∈ R3 having α as a face. Thus, the 2-ball fα is determined by the set of cells {γ ∈ R3 ; α ≤ γ and f ss (S, γ) = 1}. To characterize this set we distinguish two cases: dim γ ∈ {0, 2, 3} and dim γ = 1. The following result, which is almost immediate from Remark 3(3), states that, in the ﬁrst case, the cell γ is lighted for the simplicity 26-surface S if and only if it is lighted for the trace determined by α. Proposition 3. Let S be a simplicity 26-surfaces and α ∈ R3 a maximal vertex with respect to S. If α ≤ γ and dim γ ∈ {0, 2, 3}, then f ss (S, γ) = f ss (st3 (α; S), γ). If dim γ = 1, it suﬃces, by the deﬁnition of the lighting function f ss , to analyze the case st3 (γ; S) = {τ1 , τ2 } with γ = τ1 ∩ τ2 . For this, we use the pattern corresponding to the trace st3 (α; S) to consider two subcases, which are summarized in the next result.

Separation Theorems for Simplicity 26-Surfaces

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Proposition 4. Let S be a simplicity 26-surfaces and α ∈ R3 a maximal vertex with respect to S such that the trace st3 (α; S) does not correspond to the pattern a8 in Fig. 1. Then, f ss (S, γ) = 1 for any 1-cell α ≤ γ such that st3 (γ; S) = {τ1 , τ2 } with γ = σ ∩ τ . On the other hand, if the trace st3 (α; S) = {σ, σ1 , σ2 , σ3 } corresponds to the pattern a8 , then exactly one of the three 1-cells γiσ = σ ∩ σi , i = 1, 2, 3, is not lighted for S. Now, it is not diﬃcult to show Proposition 2 from Propositions 3 and 4. Moreover, we derive from the deﬁnition of the family ES that the edges eαβ are the bold segments in Fig. 3. Here, the only tricky point occurs when the trace t = st3 (α; S) corresponds to the pattern a7 . But, in this case is possible to show that t is the only trace of S containing the pair {σ, τ }, where the points c(σ) and c(τ ) are the extremities of the segment shared by the two squares in the 2-ball f7 ; and, thus, such a segment is not an edge in ES by deﬁnition. This shows that FS , ES and PS are families of 2-balls, 1-balls and 0-balls, respectively. To prove that CS = (FS , ES , PS ) is a cellular decomposition we start with Property (4). This property follows from the next technical result since a xel σ ∈ cell3 (R3 ) belongs to S if and only if σ ∈ supp(S). Proposition 5. Let O be a digital object in the device model R3 and γ ∈ R3 a cell in supp(O). Then, at least one of the vertices α ≤ γ is maximal with respect to O. By Property (4) it suﬃces to check that | AS | ⊆ ∪{fα ; fα ∈ FS } to obtain Property (3); and this can be also straightforwardly proved from Proposition 5. For the proof of Property (1), let fα , fβ ∈ FS be two distinct faces such that fα ∩ fβ = ∅. By condition (4) in Deﬁnition 5 the traces st3 (α; S) and st3 (β; S) share at most two xels of S; and, thus, Property (1) follows by the deﬁnition of the family of edges ES . And, if eα1 β1 = fα1 ∩ fβ1 and eα2 β2 = fα2 ∩ fβ2 are two distinct edges, the set {fα1 , fβ1 , fα2 , fβ2 } contains at least three diﬀerent elements; and, then, condition (3) in Deﬁnition 5 yields Property (2). Finally, from conditions (4) and (5) in Deﬁnition 5 is not diﬃcult to show that the cellular decomposition CS satisfy also Properties (5) and (6), and thus | AS | is a combinatorial manifold.

References 1. R. Ayala, E. Dom´ınguez, A.R. Franc´es, A. Quintero. Digital Lighting Functions. Lecture Notes in Computer Science. 1347 (1997) 139–150. 2. R. Ayala, E. Dom´ınguez, A.R. Franc´es, A. Quintero. Weak Lighting Functions and Strong 26-surfaces. To appear in Theoretical Computer Science. 3. R. Ayala, E. Dom´ıguez, A. R. Franc´es and A. Quintero. A Digital Index Theorem. Int. J. Patter Recog. Art. Intell. 15(7) (2001) 1–22. 4. G. Bertrand, R. Malgouyres. Topological Properties of Discrete Surfaces. Lect. Notes in Comp. Sciences. 1176 (1996) 325–336. 5. M. Couprie, G. Bertrand. Simplicity Surfaces: a new deﬁnition of surfaces in ZZ 3 . SPIE Vision Geometry V. 3454 (1998) 40–51.

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6. R. Malgouyres, G. Bertrand. Complete Local Characterization of Strong 26Surfaces: Continuous Analog for Strong 26-Surfaces. Int. J. Pattern Recog. Art. Intell. 13(4) (1999) 465-484. 7. T.Y. Kong, A.W. Roscoe. Continuous Analogs of Axiomatized Digital Surfaces. Comput. Vision Graph. Image Process. 29 (1985) 60–86. 8. T.Y. Kong, A. Rosenfeld. Digital Topology: Introduction and Survey. Comput. Vision Graph. Image Process. 48 (1989) 357–393. 9. D.G. Morgenthaler, A. Rosenfeld. Surfaces in three–dimensional Digital Images. Inform. Control. 51 (1981) 227-247. 10. C.P. Rourke, and B.J. Sanderson. Introduction to Piecewise-Linear Topology. Ergebnisse der Math. 69, Springer 1972.

Topological Quadrangulations of Closed Triangulated Surfaces Using the Reeb Graph Franck H´etroy and Dominique Attali LIS Laboratory, Domaine Universitaire, BP 46, 38402 Saint Martin d’H`eres cedex, France {Franck.Hetroy,Dominique.Attali}@lis.inpg.fr

Abstract. Although surfaces are more and more often represented by dense triangulations, it can be useful to convert them to B-spline surface patches, lying on quadrangles. This paper presents a method to construct coarse topological quadrangulations of closed triangulated surfaces, based on theoretical results about topological classiﬁcation of surfaces and Morse theory. In order to compute a canonical set of generators, a Reeb graph is constructed on the surface using Dijkstra’s algorithm. Some results are shown on diﬀerent surfaces.

1

Introduction

Surfaces of arbitrary shape and topology are often represented in computer graphics by large ﬁne triangle meshes, obtained with complex data acquisition hardware. Such discrete surfaces are not easy to store and handle because of the huge amount of data. Many works trying to reduce the number of triangles have been carried, see e.g. [9] for a survey. Other approaches convert these dense triangle meshes representation to other suitable models, such as parametric surfaces [5,12]. Our goal is to create a new discrete representation of the surface by large non planar quadrangles, in order to later set B-spline surface patches on them (one patch per quadrangle). Constructing a quadrangulation can be a very diﬀerent task depending on whether we try to optimize the shape of the quadrangles (i.e. we try to maximize the minimum angle among quadrangles), the number of quadrangles, or the number of extraordinary vertices. Minimizing the number of extraordinary vertices can be an important point since problems generally occur at their vicinity, when analyzing continuity between B-spline surface patches or the limit surface of a subdivision scheme for example. On one hand, most of existing techniques try to construct good shaped quadrangles with few extraordinary vertices (see e.g. [15] for a survey), but, as a result, the constructed quadrangulations are often ﬁne. On the other hand, even if some theoretical results exist concerning the minimal number of quadrangles in a quadrangulation of a surface [8], there exists no algorithm able to construct such a minimal quadrangulation for a surface of arbitrary topology. A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 57–68, 2002. c Springer-Verlag Berlin Heidelberg 2002

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Unlike most of the previous works, we focus on the number of quadrangles, in order to reduce the amount of data used to describe the surface. This paper presents an algorithm to construct a coarse topological quadrangulation of a closed triangulated surface with strictly positive genus. First, we recall some theoretical deﬁnitions and give a theoretical topological quadrangulation of the surface based on its canonical polygonal schema (Section 2). Then we describe an algorithm to construct ﬁrst a Reeb graph, then a canonical set of generators and ﬁnally a coarse topological quadrangulation of a closed triangulated surface (Section 3). Some results are shown in Section 4. Finally we conclude and discuss future work in Section 5.

2 2.1

Theoretical Background Quadrangulations

We call surface a connected compact 2-manifold of IR3 . A quadrangle designates any subset of IR3 homeomorphic to a closed square of IR2 . In this paper, we consider special decompositions of surfaces into vertices, edges and quadrangles that we call pre-quadrangulations. Deﬁnition 1 (Pre-quadrangulation). A pre-quadrangulation Q of a surface M is a decomposition of M into a ﬁnite number of vertices, edges and quadrangles such that: 1. Q is a cell complex; 2. any pair of vertices is joined by at most two edges; 3. any two quadrangles of Q are either disjoint or meet in a common vertex or intersect along one common edge or intersect along two non-consecutive common edges; Our deﬁnition includes quadrangulations as deﬁned in [8]. But, unlike usual quadrangulations, two vertices in a pre-quadrangulation can be joined by two edges and two quadrangles can intersect along two edges. Figure 1 shows a prequadrangulation of the torus: we have four distinct quadrangles, all having the same four vertices A, B, C and D. A vertex in a pre-quadrangulation is said to be ordinary if its valence (or degree) is 4. Vertices with a valence diﬀerent from 4 are said to be extraordinary. We have the following remarkable relation: Property 2. Let Q be a pre-quadrangulation of M , Ve the number of extraordinary vertices in Q, and v1 , . . . , vVe the valences of these extraordinary vertices. Ve We have: i=1 (vi − 4) = 8(g − 1), where g is the genus of M . Property 2 shows that the sum of the orders (that is to say, the valences minus four) of all extraordinary vertices is a surface invariant. This also shows that, except for surfaces with genus equal to 1, any pre-quadrangulation of a surface contains extraordinary vertices. Since a pre-quadrangulation is a cell complex, we have the Euler relation χM = V − E + F = 2 − 2g.

Topological Quadrangulations of Closed Triangulated Surfaces

D

A

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B

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Fig. 1. Pre-quadrangulation of a torus.

Proposition 3. Let Q be a pre-quadrangulation of a surface M . Let V be the number of vertices in Q, E the number of edges, F the number of faces (quadrangles) and χM the Euler characteristic of M : χM = 2 − 2g = V − E + F where g is the genus of M . A lower √ bound for the number of√quadrangles needed to construct Q is 12 ( 9 − 8χM + 3) − χM = 12 ( 16g − 7 − 1) + 2g. Example 4. For a torus (g = 1), we have F ≥ 3. For a 2-torus (g = 2), we have F ≥ 6. Proof. According to rule nr. 2 of the deﬁnition, E is lower or equal to twice the number of pairs of vertices in Q, which is V (V2−1) : E ≤ V (V − 1). Since each edge belongs to two faces, and each face has four distinct edges, 2E = 4F . We thus have F = V − χM and F ≤ V (V2−1) . Consequently, 2V − 2χ√ M ≤ V (V − 1). This quadratic inequality has√the solution: V ≥ 12 ( 9 − 8χM + 3). Finally, F = V − χM ≥ 12 ( 9 − 8χM + 3) − χM . We do not know if a general formula gives the minimum number of quadrangles in a pre-quadrangulation of a surface. We now construct a prequadrangulation of a surface with only 4g quadrangles, based on the canonical polygonal schema of the surface. 2.2

Canonical Polygonal Schema of a 2-Manifold

We assume here that the reader knows some basic notions of combinatorial topology. We recall here the classiﬁcation theorem for surfaces [6]. Theorem 5. An orientable 2-manifold M with genus g > 0 can be represented canonically using a 4g-gon G, where all 4g vertices represent the same vertex on M and edges are oriented. The labels of the edges around the polygon are of −1 −1 −1 the form: a1 b1 a−1 1 b1 ...ag bg ag bg . Curves formed by partnered edges (e.g. a1 −1 and a1 ) are called canonical generators of M , and G is called the canonical polygonal schema (or canonical fundamental polygon) of M . A canonical set of generators is composed of two kinds of generators: g generators surround the g holes (we will call them longitudinal generators) and g other turn around each handle associated with a hole (we will call them latitudinal generators). See Fig. 2 for an example and [4] for more details.

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b2

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Fig. 2. Canonical set of generators, canonical polygonal schema, and prequadrangulation of a 2-torus. a1 and a2 are the longitudinal generators, and b1 and b2 are the latitudinal ones.

2.3

Quadrangulation of the Canonical Polygonal Schema

We present here a simple pre-quadrangulation of a surface with genus g > 0, with 4g quadrangles and 2g + 2 vertices. This pre-quadrangulation is based on the canonical polygonal schema of the surface: half the edges of the prequadrangulation follow the canonical generators. First, we choose an arbitrary vertex on the surface, O, to be the base-point of the canonical polygonal schema. We then choose 2g other vertices A1 , B1 , . . . , Ag , Bg , and join twice O and each Ai (resp. each Bi ) together, along a longitudinal (resp. latitudinal) canonical generator. We eventually choose a ﬁnal vertex C (which is not on an edge) and join twice C and each Ai (resp. each Bi ) together. We end up with 2g + 2 vertices, 8g edges and 4g quadrangles. An example is shown on Fig. 2 (right). The number of quadrangles, 4g, is close to the lower bound of Prop. 3. The only extraordinary vertices of the pre-quadrangulation we propose are O and C, which valences are equal to 4g (if g = 1, our pre-quadrangulation does not contain any extraordinary vertex). We have chosen to minimize the number of extraordinary vertices (with respect to the number of vertices), but it implies the valence of these extraordinary vertices to be high, according to Prop. 2. We will now call this pre-quadrangulation of the surface based on a quadrangulation of its canonical polygonal schema a topological quadrangulation. 2.4

The Reeb Graph

To construct a canonical set of surface generators on a triangulated surface, we will use a topological structure for compact manifolds named the Reeb graph. The Reeb graph, named after Reeb [16], and Morse theory are powerful tools to describe topological features on surfaces. Both are increasingly used in computer graphics, mainly for terrain analysis and shape modeling [17,11,18,1]. See [6] for an introduction about Morse theory and the Reeb graph. A Reeb graph is deﬁned as follows:

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61

Deﬁnition 6. Let f : M → IR be a real-valued continuous function on a compact manifold M . The Reeb graph of f is the quotient space of M by the equivalence relation ∼ deﬁned by: x1 ∼ x2 ⇔ f (x1 ) = f (x2 ) and x1 and x2 are in the same connected component of f −1 (f (x1 )). Figure 3 shows a Reeb graph of a height function on a torus (ﬁgure inspired by [6]). E

F D C

f

B A

Fig. 3. A torus and the critical points of its height function f (left), connected components of some level sets for f (middle), and the Reeb graph of f on this torus.

There are two kinds of nodes in a Reeb graph: – Extremal nodes are nodes incident to only one edge (e.g. nodes A, E and F on Fig. 3). They represent local extrema of the Morse function. – Internal nodes are nodes incident to at least three edges (e.g. nodes B, C and D on Fig. 3). They represent saddle points of the Morse function. To ﬁnd a canonical polygonal schema of a surface, we just need to ﬁnd a cycle basis in the Reeb graph; then each cycle will give us two canonical generators of the surface (one longitudinal generator and the associate latitudinal one).

3 3.1

Constructing Quadrangulations of a Triangulated Surface Construction of a Reeb Graph on a Triangulated Surface

To deﬁne our Reeb graph, we must choose a function f . As said before, the Reeb graph is mainly used in terrain analysis. In this case f is often a height function, because we have a given orientation where the height is meaningful. In our case (surfaces) we do not have such a particular orientation, so we choose, as Lazarus and Verroust did before us [13], the shortest path distance to a given source point on the surface.

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Very recently Hilaga et al. [10] have chosen to integrate this function over the surface, in order to avoid the choice of the source point. Since we are only interested in detecting a cycle basis on the Reeb graph, that is to say on the topology of the surface and not the exact location of the critical points of f , we do not need to choose a very subtle (but computationally more expensive) function f . We approach f in our discrete case by the shortest edge path to the source point on the triangulated surface. The source point S is chosen as the furthest vertex from an arbitrary vertex: in practice S will be at the end of some long branch of the surface; consequently the direction induced by f is geometrically meaningful. If f is a Morse function and if all its singularities are simple, we can prove that the Reeb graph we construct using this function determines the genus of the surface [20]. If singularities are not simple, we can use Carr et al. algorithm [2] to break them into multiple simple singularities. This method turns f into a Morse function. Proposed Algorithm. To avoid confusion between the nodes (resp. edges) of the Reeb graph and the vertices (resp. edges) of the mesh, the ﬁrst will be called Reeb nodes (resp. Reeb edges) and the last just vertices (resp. edges). To construct the Reeb graph of f on the surface, we must ﬁrst compute the value of f at all the vertices of the mesh, then detect the critical points of f and ﬁnally link them to create the Reeb edges. Actually, we will do the three steps at the same time, using Dijkstra’s algorithm. With each vertex X of the triangulation we associate a value v(X) (at the end of the algorithm v(X) will be equal to the value of f at X), and a list of attributes. Attributes will be used to identify Reeb edges: two vertices X and Y will have the same attribute if and only if the segment [f (X), f (Y )] contains non critical value of f and X and Y belong to the same connected component of f −1 ([f (X), f (Y )]). At the beginning, each vertex value is +∞ and vertices have no attribute, except the source point which value is 0 and which has one attribute. At each step the current processed vertex is the vertex X with minimal value: we know that f (X) = v(X). Values of all neighbouring vertices of X are updated using Dijkstra’s algorithm. We then compute the number of sign change n(X) for v − v(X) when we cover all neighbouring vertices Y . Since we use Dijkstra’s algorithm, if v(Y ) < v(X) then v(Y ) = f (Y ) and thus f (Y ) < v(X), otherwise v(X) < f (Y ). Thus n(X) is also the number of sign change for f − f (X). Furthermore, note that n(X) is always even (our triangulated surface is closed, so the neighbourhood of X is a ring). If n(X) = 0, X is a local extremum for f , that is to say an extremal Reeb node. His only attribute will be equal to the (ﬁrst) attribute of its predecessor (for Dijkstra’s algorithm). If n(X) = 2, X is not a Reeb node. His only attribute will also be equal to the (ﬁrst) attribute of its predecessor. Finally if n(X) ≥ 4,

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63

X is an internal Reeb node. We compute its local level set and its connected components (see below). Each vertex of each upper connected component will be given a new attribute (the same for all vertices in the same connected component). X will be given all attributes of its upper and lower connected components. A Reeb edge between vertices A and B with f (A) < f (B) will be computed as a path made from B to its successive predecessors through Dijkstra’s algorithm until A or a vertex with the corresponding attribute is found, and the path made from this vertex to A following a connected component of A (if A is not the source point, A is an internal node of the graph, so its local level set has been computed). Local Level Sets. The local level set LS(X) of X is a set of pairs of vertices (Y, Z) where f (Y ) > f (X) and f (Z) < f (X) (see Fig. 4). An upper (resp. lower) connected component of X will be a connected component of the set of all Y (resp. Z) in LS(X) such that f (Y ) > f (X) (resp. f (Z) < f (X)). Note that a connected component of X will be a set of vertices, and not a set of pairs of vertices.

Y3 Y18 Y17 Y4 Y3 Y2 Y1 Z17 Y31Y30 Y29 Y28 X Z4 Z31 Z3 Z2 Z Z1 Z35 Z34Z33 32 Y2

1

Fig. 4. Local level set of an internal node X and its connected components. Here we have two upper connected components (Y1 ...Y17 Y1 Y2 and Y18 ...Y31 Y3 ) and one lower connected component (Z1 ...Z17 ...Z35 ).

To compute LS(X) and the connected components of X, we use a very simple algorithm: LS:=0 Repeat Choose two adjacent vertices Y and Z in the neighbourhood of X such as f(Y)>f(X) and f(Z)
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Add (Y,Z) to LS avoid_vertex:=X new_vertex:=Y While new_vertex is different from X do: new_vertex:=the vertex adjacent to Y and Z which is not avoid_vertex If f(new_vertex)f(X) then Add (new_vertex,Z) to LS avoid_vertex:=Y Y:=new_vertex End if. End while. We now have two new connected component of X (not complete) Until it is no more possible to choose such two adjacent vertices.

To complete our connected components and possibly merge them, we compute the vertex N V adjacent to X and Y (where Y is the last vertex of a connected component) which is not the previous vertex of the connected component. We add it to the connected component, replace Y by this new vertex and do the same until N V is in LS(X). If the last vertex N V (which is in LS(X)) is also in the connected component, this connected component is complete. If not, we must merge it with the connected component where is N V . 3.2

Construction of a Canonical Set of Generators

Only a few algorithms have been proposed to construct a canonical set of generators [19,3,14]. All three only use mesh properties and not the Reeb graph. Providing that the genus of the surface is strictly positive, ﬁnding a cycle basis B on our constructed Reeb graph will give us information about the location of surface holes, and thus about surface generators. To ﬁnd B, we use a traditional graph theory algorithm: we ﬁnd a maximal forest1 on the Reeb graph, then for each edge which is not in this forest we ﬁnd a minimal cycle containing this edge. The set of all minimal cycles found is B. Such algorithms are detailed in graph theory books, e.g. [7]. Once we have found B we can construct a canonical set of surface generators: for each cycle C in B there exists one longitudinal generator and one latitudinal generator. The longitudinal generator is simply C. To construct the latitudinal generator we ﬁnd the cycle vertex V with the minimal value, and then follow an upper connected component of V (we have computed the local level set of V when we have constructed the Reeb graph). 1

A forest is a subgraph with no cycle.

Topological Quadrangulations of Closed Triangulated Surfaces

3.3

65

Topological Quadrangulations of a Triangulated Surface

To construct our pre-quadrangulation on the triangulated surface, we must ﬁrst choose which vertices will be O, the Ak , the Bk and C (cf. Section 2.3 for the notations). We take O equals to the source point and C equals to the vertex for which f is maximum. Then for each k we deﬁne Bk as the internal node of the Reeb graph with the lowest value and belonging to the k th latitudinal cycle of the basis, and Ak as the internal node with the greatest value and belonging to the k th longitudinal cycle of the basis. To construct the edges OAk and OBk of the quadrangulation, we continuously distort each generator of the constructed canonical set of generators so that O belongs to it (this can be done in time O(g.n), n being the number of vertices in the initial triangulation, since Dijkstra’s algorithm computes for each vertex a path to the source point). Then we follow the distorted generators: the longitudinal ones for the edges OAk and the latitudinal ones for the edges OBk . We can do the same to construct the edges CAk and CBk , since O and C are dual points on the pre-quadrangulation (see Fig. 5 for an example): edges CAk correspond to the latitudinal generators and edges CBk correspond to the longitudinal generators. C

C

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Fig. 5. Generators of a torus (top left), edges OAi and OBi of the topological quadrangulation (top middle), edges CAi and CBi of the topological quadrangulation (top right). The bottom ﬁgure shows the quadrangulation “centered” either at C or at O (quadrangles are numbered from 1 to 4).

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Results

Some results are shown on Fig. 6. Top left ﬁgure shows the Reeb graph computed on a topological sphere (no hole, 1071 vertices, 2138 triangles). Note the local (non global) extremum on the middle of the helix. The two other top ﬁgures show the Reeb graph and a canonical set of generators computed on a rocker arm with one hole (10044 vertices, 20088 triangles). This model is courtesy of Cyberware (www.cyberware.com). The two bottom left ﬁgures shows canonical sets of generators of a knot-shaped torus (4245 vertices, 8490 triangles; the latitudinal generator is bottom left of the ﬁgure) and a topological 3-torus (1572 vertices, 3152 triangles). Finally bottom right ﬁgure shows the coarse topological quadrangulation of a simple torus (400 vertices, 800 triangles) constructed with our algorithm. The computation time of our algorithm is O(n2 ). Time used by Dijkstra’s algorithm to construct the Reeb nodes is O(n. log(n)), but then to construct the Reeb edges we need to compute the local level sets of Reeb nodes. This takes O(n2 ) time.

(a)

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Fig. 6. Reeb graph (a,b), canonical set of generators (c,d,e), and topological quadrangulation (f) of some closed surfaces.

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Conclusion and Future Work

We have presented theoretical results and a new algorithm to construct a new discrete representation of a closed triangulated surface with strictly positive genus by a coarse topological quadrangulation. Our method ﬁrst construct a Reeb graph on the surface using Dijkstra’s algorithm and then the generators of this surface. Further work includes quadrangulation reﬁnement to take surface geometrical singularities into account, and optimization (in terms of angles in quadrangles, extraordinary vertices, . . . ). The choice of the function used to deﬁne the Reeb graph can also be improved to fulﬁll these goals. Acknowledgment. The authors would like to thank Francis Lazarus for fruitful discussions and helpful comments, especially concerning the number of cycles in the Reeb graph.

References 1. S. Biasotti, B. Falcidieno, M. Spagnuolo. Extended Reeb Graphs for Surface Understanding and Description. Proceedings of DGCI’00, Lecture Notes in Computer Science, Vol. 1953, pp.185-197, 2000. 2. H. Carr, J. Snoeyink, U. Axen. Computing Contour Trees in All Dimensions. Proceedings of ACM 11th Symposium on Discrete Algorithms, pp. 918-926. San Francisco, California, USA, Jan. 2000. 3. T. Dey, H. Schipper. A new Technique to Compute Polygonal Schema for 2Manifolds with Application to Null-Homotopy Detection. Discrete and Computational Geometry,14:93-110, 1995. 4. T. Dey, S. Guha. Computing Homology Groups of Simplicial Complexes in IR3 . Journal of ACM, Vol. 45, No. 2, pp. 266-287, 1998. 5. M. Eck, H. Hoppe. Automatic Reconstruction of B-Spline Surfaces Of Arbitrary Topological Type. Proceedings of SIGGRAPH’96, pp. 325-334, August 1996. 6. A.T. Fomenko, T.L. Kunii. Topological Modeling for Visualization. Springer, 1997. 7. M. Gondran, M. Minoux. Graphs and Algorithms. Wiley, 1995. 8. N. Hartsﬁeld, G. Ringel. Minimal Quadrangulations of Orientable Surfaces. Journal of Combinatorial Theory, Series B, Vol. 46, No. 1, pp. 84-95, 1989. 9. P.S. Heckbert, M. Garland. Survey of Polygonal Surface Simpliﬁcation Algorithms. Multiresolution Surface Modeling Course, SIGGRAPH’97, 1997. 10. M. Hilaga, Y. Shinigawa, T. Kohmura, T.L. Kunii. Topology Matching for Fully Automatic Similarity Estimation of 3D Shapes. Proceedings of SIGGRAPH’01, August 2001. 11. M. van Kreveld, R. van Ostrum, C. Bajaj, V. Pascucci, D. Schikore. Contour Trees and Small Seed Sets for Isosurface Traversal. Proceedings of ACM 13th Symposium on Computational Geometry, pp. 212-220. Nice, France, June 1997. 12. V. Krishnamurthy, M. Levoy. Fitting Smooth Surfaces to Dense Polygon Meshes. Proceedings of SIGGRAPH’96, pp. 313-324, August 1996. 13. F. Lazarus, A. Verroust. Level Set Diagrams of Polyhedral Objects. Proceedings of ACM 5th Symposium on Solid Modeling and Applications, pp. 130-140. Ann Arbor, Michigan, USA, June 1999.

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14. F. Lazarus, M. Pocchiola, G. Vegter, A. Verroust. Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface. Proceedings of ACM 17th Symposium on Computational Geometry, pp. 80-89. Tufts University, Medford, USA, June 2001. 15. S. Owen. A Survey of Unstructured Mesh Generation Technology. Proceedings of the 7th International Meshing Roundtable, Sandia National labs, pp. 239-267. Dearborn, Michigan, U.S.A., October 1998. 16. G. Reeb. Sur les Points Singuliers d’une Forme de Pfaﬀ Compl`etement Int´egrable ou d’une Fonction Num´erique. Comptes Rendus Acad. Sciences, Paris, France, 222:847-849, 1946. 17. Y. Shinagawa, T.L. Kunii. Surface Coding based on Morse Theory. IEEE Computer Graphics and Applications, pp. 66-78, September 1991. 18. S. Takahashi, Y. Shinagawa, T.L. Kunii. A Feature-based Approach for Smooth Surfaces. Proceedings of ACM 4th Symposium on Solid Modeling and Applications, pp. 97-110. Atlanta, Georgia, USA, May 1997. 19. G. Vegter, C.K. Yap. Computational Complexity of Combinatorial Surfaces. Proceedings of ACM 6th Symposium on Computational Geometry, pp. 102-111. Berkeley, California, USA, June 1990. 20. Z. Wood. Semi-Regular Mesh Extraction from Volumes. Master’s thesis, Caltech, Pasadena, California, USA, 2000.

Non-manifold Decomposition in Arbitrary Dimensions Leila De Floriani, Mostefa Mohammed Mesmoudi, Franco Morando, and Enrico Puppo Department of Computer and Information Sciences, Universit` a di Genova, Via Dodecaneso 35, 16146 Genova - Italy [deflo, mesmoudi, morando, puppo]@disi.unige.it

Abstract. In this paper we consider the problem of decomposing a nonmanifold n-dimensional object described by an abstract simplicial complex into an assembly of ‘more-regular’ components. Manifolds, which would be natural candidates for components, cannot be used to this aim in high dimensions because they are not decidable sets. Therefore, we deﬁne d-quasi-manifolds, a decidable superset of the class of combinatorial d-manifolds that coincides with d-manifolds in dimension less or equal than two. We ﬁrst introduce the notion of d-quasi-manifold complexes, then we sketch an algorithm to decompose an arbitrary complex into an assembly of quasi-manifold components abutting at non-manifold joints. This result provides a rigorous starting point for our future work, which includes designing eﬃcient data structures for non-manifold modeling, as well as deﬁning a notion of measure of shape complexity of such models.

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Introduction

A d-manifold M is any subset of the Euclidean space such that the neighborhood of any point of M is locally equivalent to a d-dimensional open ball. Geometric objects that do not fulﬁll such property at all points are said to be non-manifold. Non-manifold objects are very common in practical applications, because of their high expressive power [5,10,20,21,22,25,26,27]. However, most objects considered in the applications have a relatively small number of non-manifold joints (geometric singularities), while most parts of such objects fulﬁll the manifold condition. Examples are objects containing parts of diﬀerent dimensionality. Valid representation of non-manifolds is a central issue in the geometric modeling of solids. Non-manifold objects are hard to control and to understand, both from a theoretical and from a practical point of view. Non-manifolds are not well understood and classiﬁed mathematically, especially in high dimensions. Non-manifold cell complexes, which can be used to represent such objects, are hard to encode and manipulate with eﬃcient data structures. Models and data structures for representing such objects in computers have been proposed in the literature in two dimensions [13,16,22,25,26,27]. Moreover, most such models and data structures do not scale well with the degree of “non-manifoldness” (i.e., the number of geometric singularities) of the model, thus resulting often verbose and ineﬃcient. A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 69–80, 2002. c Springer-Verlag Berlin Heidelberg 2002

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A possible approach in addressing the representation of non-manifold objects consists of decomposing them into simpler parts. So far, this approach has been pursued in the literature in two dimensions for non-manifold objects by using pseudo-manifold components [10,21]. A decomposition into manifold components, which seems a natural choice, is actually possible only in two dimensions, but it is not feasible in the general case, since the class of manifolds is not decidable in higher dimensions. The purpose of our paper is to deﬁne a mathematically founded decomposition of non-manifold objects into components that belong to a well-understood class, that we call d-quasi-manifolds, which is a decidable proper superset of d-manifolds for d ≥ 3, while it coincides with the class of d-manifolds for d ≤ 2. Our ﬁnal goal is to use such a decomposition as a basis for designing eﬃcient data structures to represent non-manifold objects in any dimension, as well as deﬁning a measure of structural complexity of singularities. Our approach allows also a better understanding of the combinatorial structure of non-manifolds and represents a contribution in the direction of topology-based geometric modeling [17]. The remaining of this paper is organized as follows. In Section 2, we review related work, from geometric modeling and from combinatorial topology. In Section 3, we introduce basic deﬁnitions from combinatorial topology. In Section 4, we develop the notion of quasi-manifold, recall the classic deﬁnition of combinatorial manifolds, and compare these two class of objects. In Section 5, we deﬁne a decomposition procedure that will split a non-manifold complex into an assembly of quasi-manifold objects. Finally, in Section 6, we draw some conclusions, and discuss future work.

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Related Work

Work related to the results presented in this paper can be found in the literature on geometric modeling and on combinatorial topology. In geometric modeling, the main interest about non-manifold objects is to develop modeling schemes and related data structures for representing and manipulating such objects. Since CAD applications has motivated most such research, most contributions are concerned with the description of the boundary of solid objects. A common assumption is to consider only those objects that can be described by a general geometric cell complex formed by cells homeomorphic to closed balls. Some proposals also exist that consider objects with open boundaries [16,22]. In the context of this paper we will stick to the above assumption. A possible approach to the representation of such objects is to encode general cell complexes directly. Existing data structures developed under this approach (e.g., the incidence graphs [7,28], the Radial Edge [27] and the Tri-ciclic Cusps [13]) may be quite verbose and do not scale well with the degree of “nonmanifoldness” of the modeled object. In other words, such data structures have been designed by assuming to have a relatively high number of non-manifold situations, and, thus, they are less computationally eﬃcient and compact, when

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applied to manifolds, then data structures developed for boundary representation of manifold solids [2,12,18]. The Facet-Edge [6] data structure, even if developed for subdivided manifolds, can model some special cases of non-manifold objects, but it is not clear how to extend it to deal with more general situations. However, the representation domain of the Facet-Edge suggested the notion of regularly adjacent complexes that we introduce in Section 3. The n-G-maps proposed in [16] are deﬁned in arbitrary dimensions, and they can describe a subclass of pseudo-manifolds (hence, of non-manifolds). The quasi-manifolds, that we will introduce in Section 5, can be modeled with nG-maps, while the reverse is not true, since n-G-maps can model open cells as well. A representation based on chains of n-G-Maps [8] can be used to represent arbitrary cell complexes with a mix of open and closed cells. An alternative approach to modeling consists of decomposing a non-manifold object into simpler and more manageable parts. There are a few proposals that follow this approach, which are limited to the two-dimensional case. In [10, 21], decompositions of two-dimensional r-sets (i.e., uniformly two-dimensional objects) into pseudomanifolds are presented. In [11], the idea of cutting a twodimensional non-manifold complex into manifold pieces, and to use a two-level representation to stitch them together is exploited in order to develop a geometric compression algorithm. In order to pursue an approach based on decomposition, a crucial point is to understand the possible structure of the components. Combinatorial topology establishes limits on what can be done, and motivates the fact that d-manifolds cannot be used as building blocks, in general. In order to use objects of a given class as building blocks, we must be able to decide whether an object belongs to this class or not. Following the classical main stream from point set to combinatorial topology (see, e.g., [15] for a survey), we restrict our attention to combinatorial manifolds (see Section 5). The set of combinatorial d-manifolds is decidable if and only if the (d − 1)-spheres can be recognized. However, while a decision procedure for the recognition of the 3-sphere is known [24], this problem is open for d = 4, and d-spheres are not recognizable for d ≥ 5 (see [14,19] and [23] Chapter 9). Therefore, manifolds are not decidable for d ≥ 6. For this reason, we will use building blocks from a superclass of combinatorial manifolds, which we will deﬁne in Section 4. Such class of objects proves to be decidable in all dimensions, and to coincide with the class of manifolds for d ≤ 2.

3

Background Notions

In this section, we introduce abstract simplicial complexes and their basic related notions [9]. We will use geometric complexes just as examples and completely avoid all deﬁnitions from point set topology. Abstract Simplicial Complexes Let V be a ﬁnite set of points that we call vertices. An abstract simplicial complex on V is a subset Ω of the set of (non empty) parts of V such that {v} ∈ Ω for

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every point v ∈ V , and, if γ ⊂ V is an element of Ω, then every subset of γ is also an element of Ω. Each element of Ω is called an abstract simplex. We associate an integer dim(γ) with each abstract simplex γ ∈ Ω , called dimension of γ, deﬁned by dim(γ) = |γ|−1, where |γ| is the number of vertices in γ. Whenever no ambiguity arises, we will use the term cell or simplex to denote an abstract simplex. We will use also the term complex or abstract complex to denote an abstract simplicial complex. A cell of dimension s is called an s-cell. A complex Ω is called d-dimensional or a d-complex if maxγ∈Ω (dim(γ)) = d; each d-cell of a d-complex Ω is called a maximal cell of Ω. Boundary, Star, Link, Subcomplexes, and Closures The boundary ∂γ of a cell γ is deﬁned to be the set of all proper parts of γ. Cells ξ in ∂γ are called faces of γ. Similarly, the co-boundary, or star, of a cell γ is deﬁned as γ = {ξ ∈ Ω | γ ⊂ ξ}. Cells ξ in γ are called co-faces of γ. Any cell γ such that γ = {γ} is called a top cell of Ω. Two distinct cells are said to be incident if and only if one of them is a face of the other. Otherwise, they are called disjoint cells. Both star and boundary are deﬁned by referring to the set of cells in Ω. We can emphasize this by using ∂Ω , instead of ∂ and using Ω , instead of . The above deﬁnition can be easily extended to a set of cells. Thus if Θ is a set of cells, we will have that Θ = γ∈Θ γ. The link of a set of cells Θ (denoted by lk(Θ)) is the set of all faces of co-faces of cells in Θ, that are not incident to cells in Θ. A subset of cells Ω ⊂ Ω which has the property to be a complex is called a sub-complex Ω of Ω. A subset Θ of Ω is open (resp. closed) (in Ω) if, for every cell γ of Θ, all cells of the co-boundary (resp. boundary) of γ in Ω are also cells of Θ. Note that a closed subset of Ω is a sub-complex of Ω. Given a set of cells Θ, the closure Θ of Θ will be the smallest sub-complex whose cells include those in Θ. Note that the star of an arbitrary set of cells might not be a sub-complex while the link of a set of cells is always a sub-complex. With this deﬁnitions, we can write that, for a generic set of cells Θ, lk(Θ) = Θ − Θ. Similarly we can emphasize the dependence of the link on the complex Ω by writing lkΩ (Θ) = Ω Θ − Ω Θ. Carrier and Embedding If cells of a complex Ω are subsets of a the same topological space E, we deﬁne the carrier ∆(Ω) as the topological subspace deﬁned by the union γ∈Γ γ. We will say that ∆(Ω) is an embedding of Ω into the topological space E. The carrier of an abstract simplicial complex is usually called a polyhedron. Adjacency, Paths, and Connected Complexes Two cells are called s-adjacent if they share a s-face. In particular, two p-cells are said to be adjacent if they are (p − 1)-adjacent. A sequence of elements (γi )ki=0 in a subset Ω of a complex Ω is called a path in Ω from γ0 to γk if for every two consecutive cells γi−1 , γi , i = 1, . . . , k we have γi−1 ∈ ∂γi or γi−1 ∈ γi . A subset Ω of a complex Ω is called connected if it contains at least a path for any pair

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of its points. Equivalence classes of this connectivity relation are sub-complexes called connected components of Ω. Combinatorial Equivalence Given a d-cell γ in Ω and a point w ∈ / γ, we deﬁne the cone from a vertex w to γ as the cell {w} ∪ γ. Since w ∈ / γ, the cone is a (d + 1)-cell. An abstract simplicial complex Ω is a starring subdivision of another abstract simplicial complex Ω if Ω is obtained from Ω by deleting a cell γ (which is not a vertex) and by adding a vertex w plus all the cones from w to all the faces, that are diﬀerent from γ, of cells in γ. An abstract simplicial complex Ω is a subdivision of another abstract simplicial complex Ω if Ω is obtained from Ω by a sequence of starring subdivisions. Two abstract simplicial complexes will be combinatorial equivalent, or stellar equivalent, if and only if they have two isomorphic subdivision.

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Quasi-Manifolds

In this section, we introduce the new concept of quasi-manifolds, and discuss the relation between quasi-manifolds and combinatorial manifolds. To this aim, we need to introduce some other deﬁnitions. Deﬁnition 1. A d-complex in which all top cells are maximal is called a regular, or a uniformly d-dimensional complex. Let γ be a (d − 1)-cell in a regular d-complex Ω. We say that γ is a manifold cell if and only if there exist at most two d-cells incident at γ. Such incidence is called manifold adjacency. Similarly, we say that γ is a non-manifold cell in Ω if and only if there exist at least three d-cells incident at γ. Let Ωbd−1 be the collection of all (d − 1)-cells having only one incident d-cell. The

sub-complex ∂Ω = Ωbd−1 is called the boundary of Ω, while the collection of the cells in Ω − ∂Ω is called the set of internal cells of Ω. Deﬁnition 2. A regular complex which has only manifold cells is called a combinatorial pseudo-manifold (with possible boundary).

For a given regular complex Ω, we call the equivalence classes induced by the reﬂexive and transitive closure of the manifold adjacency relation the pseudomanifold components in Ω . Note that in the literature, d-pseudo-manifolds are required to be connected via manifold adjacency [1]. In deﬁnition 2 we do not need this fact. We will mention the connectivity separately when it is necessary. Figure 1(a) shows an example of a two dimensional pseudo-manifold since it is uniformly two-dimensional and for every 1-simplex there are at most two 2-simplexes incident in it. The complex in Figure 1(b) is the closure of a pseudomanifold component but it is not a pseudo-manifold. This is due to the fact that segment nm is the face of three triangles.

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o p

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Fig. 1. (a) An example of a 2-pseudo-manifold; (b) the closure of a 2-pseudo-manifold component that is not a pseudo-manifold.

Using the fact that, for any vertex v, the map that associate with each cell in v − {v} its face in lk(v) is bijective, the following properties can be easy proven: Proposition 1 – A d-complex, with d > 0, is a regular complex if and only if the link of each vertex is a regular (d − 1)-complex. All 0-complexes are regular. – A d-complex, with d > 1, is a pseudo-manifold if and only if the link of each vertex is a (d − 1)-pseudo-manifold. For d = 1, only regular 1-complexes (i.e. graphs) with at most two 1-simplexes incident at every vertex are pseudomanifolds. We deﬁne regularly adjacent complexes inductively as follows. Deﬁnition 3. A regular abstract simplicial 1-complex is a regularly adjacent complex. A regular abstract simplicial d-complex is regularly adjacent if and only if the link of each vertex is a connected regularly adjacent (d − 1)-complex. Vertices for which the link condition in the deﬁnition above is not satisﬁed are called singular vertices. Complex in Figure 1(a) is a pseudo-manifold which is not regularly adjacent. The link of vertex l is not connected since it is represented by the two thick disconnected 1-simplexes {o, p}, {q, r}. For this reason, we can say that vertex l is a singular vertex. Complex in (b) is regularly adjacent but not a pseudo-manifold. The link of each point is a connected graph. Vertices n and m in Figure 1(b), are singular vertices because they violate link condition for pseudo-manifoldness. The complex in ﬁgure 2(a) is a 3-pseudo-manifold but not regularly adjacent. This is due to the fact that the link of vertex p is represented by the strip of triangles pinched at q, which is not regularly adjacent since it has the same property of the complex in Figure 1(a). Complex in Figure 2(b) is a regularly adjacent 3-pseudo-manifold. Therefore, pseudo-manifoldness and regular adjacency are two independent requirements for the regularity of a complex. Deﬁnition 4. We say that a complex is a quasi-manifold if and only if it is both a pseudo-manifold and a regularly adjacent complex.

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Fig. 2. (a) An example of a 3-pseudo-manifold that is not a regularly adjacent; (b) an example of a 3-pseudo-manifold which is a regularly adjacent

The complex in Figure 2(b) is an example of quasi-manifolds. It is easy to reformulate this deﬁnition of quasi-manifolds in term of link conditions. This yields to the following inductive characterization of quasi-manifolds. Proposition 2 A d-complex is a quasi-manifold if and only if, for d > 1, the link of each vertex is a connected (d − 1)-quasi-manifold. Only regular 1-pseudomanifolds are quasi-manifolds. A similar characterization can be given for d-quasi-manifolds with boundary by just considering pseudo-manifolds with boundary. We now introduce the deﬁnition of combinatorial manifolds and we discuss their relation with quasimanifolds. Deﬁnition 5. A combinatorial d-pseudo-manifold Ω (with possible boundary) is called a combinatorial d-manifold if and only if the link of every vertex v is a (d − 1)-complex that is combinatorially equivalent either to the boundary of the standard d-dimensional simplex, if v is an internal vertex, or to the standard (d − 1)-simplex if v is a boundary vertex. The following property, that can be easily proven by induction on the manifold dimension d, gives the relation between manifolds and quasi-manifolds. Proposition 3 For any d ≥ 0, every combinatorial d-manifold is a d-quasimanifold. Conversely, every 2-quasi-manifold is a 2-manifold; however, for d > 2, there exist d-quasi-manifolds that are not d-manifolds. Figure 2(b) shows an example of a three dimensional quasi-manifold which is not a combinatorial manifold, because the link of vertex p is a equivalent to a 2-cylinder (and, hence, is not equivalent to the boundary of a sphere or to a disc).

5

Decomposition of a Non-manifold Complex

In this section we brieﬂy describe an algorithmic procedure to decompose a dcomplex into a natural assembly of quasi-manifolds of dimension h ≤ d. Details

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and proofs about correctness and time complexity of the procedure are given in [4]. The decomposition procedure consists of two sub-procedures, that we will call SplitVertices and CheckTies. Such two procedures are applied iteratively to the complex until no more singularities remain. It can be shown that a single iteration of procedure SplitVertices alone is suﬃcient to decompose a complex for d ≤ 2. For d ≥ 3 procedure CheckTies is necessary to remove situations such as those depicted in Figure 4, which are not detected by SplitVertices. Procedure SplitVertices works recursively on the dimension d of the complex to decompose, on the basis of the following inductive process. A 1-complex is (isomorphic to) a graph possibly with isolated vertices. We deﬁne the decomposition of this graph as follows. – For every vertex v of degree o > 2 (i.e., with a number of incident 1-simplexes > 2) we introduce o distinct copies of v, one for each edge incident into v. This will split Ω into a set of disconnected points, cycles and chains. – If a chain of this expanded graph starts and ends on two copies of the same vertex, we convert it into a cycle by identifying such two copies. The result of this decomposition is a graph whose connected components are either isolated points, or single cycles, or linear chains of adjacent edges. In this decomposition scheme, connected components will be no further decomposed. For d ≥ 2, we consider each vertex v of a d-complex Ω, and, we recursively decompose its link lk(v) (which is a (d − 1)-complex) into an assembly of k connected (d − 1)-complexes. If k > 1, we call vertex v a splitting vertex. In this case, given the collection {li (v), 1 ≤ i ≤ k} of connected components of lk(v) (decomposed recursively), we will modify the input complex Ω by replacing v with k copies of it. More precisely, for every i ∈ {1, . . . k}, vi replaces v in all cells of the star of v that are incident at the cells of component li (v). We iterate this local decomposition process until no more splitting vertices remain. It can be shown that this procedure works in time linear in the number t of top simplexes in the input complex. The resulting decomposition is dependent on the order in which we examine vertices of the input complex. In Figure 3 we give an example of two local decompositions of the 2-complex in (a). The ﬁrst decomposition (b) starts by considering vertex v, while the second decomposition (c) starts by considering vertex u. The components resulting from the two choices are diﬀerent.

Fig. 3. An example of decomposition depending on the order of choice of vertices.

For d-complexes, with d > 2, some singularities may remain after running procedure SplitVertices. Such singularities are due to vertices having a link

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which remains made of a single connected component, but needs to be decomposed at some of its vertices (without disconnecting it) in order to fulﬁll the link conditions for quasi-manifold complexes. We call such vertices ties. Vertices p and q in Figure 4 are examples of ties. Note that vertices that are potential ties are detected by procedure SplitVertices, depending on the structure of their link components. However, such vertices are left unchanged at that stage, because some such situations can be resolved while splitting some of their neighboring vertices afterwards. Procedure CheckTies tests only such potential ties. If a tie v is found, then we introduce a new copy of v, as above, for the closure of each top simplex in the star of v (see Figure 4(b), for an example). Note that it is suﬃcient to split just vertex p to remove the tie also at q. Tie removal causes a sort of “explosion” of the complex in the neighborhood of a removed tie. This explosion may generate other singularities. Therefore, procedure SplitVertices is repeated only on those vertices that might be aﬀected by tie removal, and the process is iterated until no more ties are found. It can be shown that a single application of procedure CheckTies has a complexity linear in the number of cells it visits. Moreover, the total number of iterations of the two procedures is bounded from above by the number t of top simplexes in the input complex. Therefore, the total decomposition process for d ≥ 3 is completed in time O(t2 ). However, we believe that this estimate is quite pessimistic. For instance, under the (reasonable) assumption that the number of cells incident in a vertex is bounded from above by a constant, it can be shown that the complexity of the whole decomposition is always Θ(dt).

q p2

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Fig. 4. An example of decomposition of a tie: Complex in (a) contains two ties p and q. Tetrahedra in (b) correspond to the decomposition of tie p.

We can obtain the original complex Ω from the set of quasi-manifold components by identifying all vertex copies of each non-manifold vertex of Ω. Therefore, such a set of quasi-manifold components, together with a simplicial map that identiﬁes all vertex copies of every non-manifold vertex, can describe completely the original complex Ω.

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Fig. 5. A non-manifold 2-complex Ω and its decomposition with hyper-graph

An example of the results of the decomposition for a 2-non-manifold is depicted in Figure 5. Consider the complex Ω in Figure 5(a). Points b,i and f are the non-manifold points. The decomposed link of vertex b has ﬁve connected components: cycles (c, e, f ), (c, d, e), (h, i, m) and (p, q, r), arc (i, f ) and isolated point a. The link of vertex i has two connected components: cycle (b, h, j, m) and arc (b, f, g). The link of f has two connected components: cycle (b, c, e) and arc (b, i, g). Therefore (b, i) and (i, g) are non-manifold edges. Figure 5(b) shows the manifold decomposition of the complex in Figure 5(a).

6

Concluding Remarks

Results in this paper show that any abstract simplicial complex can be decomposed into an assembly of subcomplexes. Each subcomplex belongs to the class of quasi-manifolds complexes. The resulting decomposition is natural (i.e., not arbitrary) in the sense that it is obtained by induction and depends only on the order in which the vertices are selected. The decomposition described can be used to deﬁne a hierarchical two-level representation for non-manifolds. In a upper level we can describe, through a hypergraph, the way in which quasi-manifold components are stitched together to form the original complex. In the lower level, we describe each quasi-manifold component. This idea is already well understood and developed for 2-dimensional complexes in [3]. For d-dimensional complexes for d > 2 we expect to ﬁnd a compact data structure to encode d-quasi-manifolds. Simplex connectivity in a d-quasi-manifold can be completely described by giving the pairs of (d − 1)adjacent d-simplexes. Finally, we note that our decomposition operates on a local basis and, therefore, the structure of the decomposition can be understood by local analysis. This notion will be used to deﬁne a measure of the amount of shape

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complexity at each non-manifold point. This will be used to guide shape simpliﬁcation process which preserves the iconic content of an object (see [3] for details).

Acknowledgments. This work has been supported by the European Research Training Network “MINGLE - Multiresolution in Geometric Modelling”, contract number HPRN-CT-1999-00117, and by the National Project founded by the Italian Ministry of Instruction, University and Research (MIUR) “SPADA - Representation and processing of spatial data in Geographic Information Systems”.

References [1] M.K. Agoston. Algebraic Topology, A First Course,. Pure and Applied Mathematics, Marcel Dekker, 1976. [2] Bruce G. Baumgart. Winged edge polyhedron representation. Technical Report CS-TR-72-320, Stanford University, Department of Computer Science, October 1972. [3] L. De Floriani, P. Magillo, F. Morando, and E. Puppo. Non-manifold multitessellation: from meshes to iconic representations of 3d objects. In C. Arcelli, L.P. Cordella, and G. Sanniti di Baja, editors, Proceedings of the 4th International Workshop on Visual Form (IWVF4), volume 2059 of Springer-Verlag (LNCS), page 654 ﬀ, Berlin, 2001. Springer-Verlag. [4] L. De Floriani, M. M. Mesmoudi, F. Morando, and E. Puppo. Decomposition of n-dimensional complex into quasi-manifold components. Technical Report DISITR-01-11, Department of Computer and Information Sciences of Genova (DISI), Genova-Italy, 2001. [5] H. Desaulnier and N. Stewart. An extension of manifold boundary representation to r-sets. ACM Trans. on Graphics, 11(1):40–60, 1992. [6] D. Dobkin and M. Laszlo. Primitives for the manipulation of three-dimensional subdivisions. Algorithmica, 5(4):3–32, 1989. [7] H. Edelsbrunner. Algorithms in combinatorial geometry. In Brauer, W., Rozenberg, G., and Salomaa, A., editors, EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1987. [8] H. Elter and P. Lienhardt. Diﬀerent combinatorial models based on the map concept for the representation of sunsets of cellular complexes. In Proc. IFIP TC 5/WG 5.10 Working Conference on Geometric Modeling in Computer Graphics, pages 193–212, 1993. [9] R. Engelking and K.Svekhicki. Topology: A Geometric Approach. Heldermann Verlag-Berlin, 1992. [10] B. Falcidieno and O. Ratto. Two-manifold cell-decomposition of r-sets. In A. Kilgour and L. Kjelldahl, editors, Computer Graphics Forum (EUROGRAPHICS ’92 Proceedings), volume 11 n 3, pages 391–404, sep 1992. [11] A. Gueziec, G. Taubin, F. Lazarus, and W. Horn. Converting sets of polygons to manifold surfaces by cutting and stitching. In Scott Grisson, Janet McAndless, Omar Ahmad, Christopher Stapleton, Adele Newton, Celia Pearce, Ryan Ulyate, and Rick Parent, editors, Conference abstracts and applications: SIGGRAPH 98, July 14–21, 1998, Orlando, FL, Computer Graphics, pages 245–245, New York, NY 10036, USA, 1998. ACM Press.

80

L. De Floriani et al.

[12] Leonidas Guibas and Jorge Stolﬁ. Primitives for the manipulation of general subdivisions and the computation of voronoi diagrams. ACM Transaction on Graphics, 4(2):74–123, April 1985. [13] E. L. Gursoz, Y. Choi, and F. B. Prinz. Vertex-based representation of nonmanifold boundaries. In M. J. Wozny, J. U. Turner, and K. Preiss, editors, Geometric Modeling for Product Engineering, pages 107–130. Elsevier Science Publishers B.V., North Holland, 1990. [14] V. E. Kuznetsov I. A. Volodin and A. T. Fomenko. The problem of discriminating algorithmically the standard three-dimensional sphere. Russisan Math. Surveys, 29(5):71–172, 1974. Original Russian article in Uspekhi Mat. Nauk, 29(1), (1974), pp. 72–168. [15] W. R. B. Lickorish. Simplicial moves on the complexes and manifolds. Geometry and Topology Monographs: Proceedings of the Kirbyfest, 2:299–320, 1999. [16] P. Lienhardt. Topological models for boundary representation: a comparison with n-dimensional generalized maps. CAD, 23(1):59–82, 1991. [17] P. Lienhardt. Aspects in Topology-Based Geometric Modeling : Possible Tools for Discrete Geometry ? In Proceedings of Discrete Geometry in Computer Science LNCS 1347, pages 33–48, 1997. [18] M. Mantyla. An introduction to solid modeling. Computer Science Press, 1983. [19] A. A. Markov. Unsolvability of the problem of homeomorphy. In International Congress of Mathematics, pages 300–306, 1958. In Russian. [20] J. Popovic and H. Hoppe. Progressive simplicial complexes. In ACM Computer Graphics Proc., Annual Conference Series, (Siggraph ’97), 1997. (to appear). [21] J. Rossignac and D. Cardoze. Matchmaker: Manifold BReps for non-manifold R-Sets. In Willem F. Bronsvoort and David C. Anderson, editors, Proceedings of the Fifth Symposium on Solid Modeling and Applications (SSMA-99), pages 31–41, New York, June 9–11 1999. ACM Press. [22] J.R. Rossignac and M.A. O’Connor. SGC: A dimension-indipendent model for pointsets with internal structures and incomplete boundaries. In J.U. Turner M. J. Wozny and K. Preiss, editors, Geometric Modeling for Product Engineering, pages 145–180. Elsevier Science Publishers B.V. (North–Holland), Amsterdam, 1990. [23] John Stillwell. Classical Topology and Combinatorial Group Theory. Number 72 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1993. [24] A. Thompson. Thin position and the recognition problem for s3 . Mat. Res. Lett., 1:613–630, 1994. [25] K. Weiler. Boundary graph operators for non-manifold geometric modeling topology representations. In J.L. Encarnacao M.J. Wozny, H.W. McLaughlin, editor, Geometric Modeling for CAD Applications, pages 37–66, North-Holland, 1988. Elsevier Science. [26] K. Weiler. The radial edge data structure: A topological representation for non-manifold geometric boundary modeling. In J.L. Encarnacao M.J. Wozny, H.W. McLaughlin, editor, Geometric Modeling for CAD Applications, pages 3– 36, North-Holland, 1988. Elsevier Science. [27] Kevin Weiler. Topological Structures for Geometric Modeling. Ph.D. thesis, Computer and Systems Engineering, Rennselaer Polytechnic Institute, Troy, NY, August 1986. [28] Tony C. Woo. A combinatorial analysis of boundary data structure schemata. IEEE Computer Graphics and Applications, 5(3):19–27, March 1985.

4D Minimal Non-simple Sets C.J. Gau1 and T. Yung Kong2 1

Department of Computer Science Graduate School and University Center City University of New York New York, NY 10036, U.S.A. [email protected] 2 Department of Computer Science Queens College City University of New York Flushing, NY 11367, U.S.A. [email protected]

Abstract. One method of verifying that a given parallel thinning algorithm “preserves topology” is to show that no iteration ever deletes a minimal non-simple (“MNS”) set of 1’s. The practicality of this method depends on the fact that few types of set can be MNS without being a component. The problem of ﬁnding all such types of set has been solved (by Ronse, Hall, Ma, and the authors) for 2D and 3D Cartesian grids, and for 2D hexagonal and 3D face-centered cubic grids. Here we solve this problem for a 4D Cartesian grid, in the case where 80-adjacency is used on 1’s and 8-adjacency on 0’s. Keywords. 4D xel, attachment, minimal non-simple, MNS, parallel thinning.

1

Introduction

Loosely speaking, a set of 1’s in a binary image is said to be simple if parallel deletion of those 1’s “preserves topology”. A non-simple set of 1’s must contain a minimal non-simple (MNS) set. So one way to prove that a given parallel thinning algorithm always “preserves topology” is to show that no iteration of that algorithm can ever delete an MNS set of 1’s. The practicality of this method depends on the fact that there are few types of possible MNS sets, and even fewer types of sets that can be MNS without being components of the 1’s. (Here we consider two sets of grid points to be of the same type if one set is a translate of the other.) The problem of identifying all such types of sets has been solved for 2D and 3D Cartesian grids, and for 2D hexagonal and 3D face-centered cubic grids. Ronse [1] introduced the concept of an MNS set (called an MND set in [1]), and solved the problem for a 2D Cartesian grid with (8,4) or (4,8) connectedness. Hall [2] solved the problem for a 2D hexagonal grid with (6,6) connectedness. Ma [3] solved the problem for a 3D Cartesian grid with (26,6), (6,26), (18,6), or A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 81–91, 2002. c Springer-Verlag Berlin Heidelberg 2002

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(6,18) connectedness. The authors [4] solved the problem for a 3D face-centered cubic grid with (18,12), (12,12), or (12,18) connectedness. In this paper we solve the problem for a 4D Cartesian grid with (80,8) connectedness: We determine which sets of grid points can be MNS, and also determine which of those sets can be MNS without being a component of the 1’s. This gives a method of verifying that a given 4D parallel thinning algorithm always “preserves topology”. Such algorithms might be of use in temporal image analysis, in which a time sequence of 3D images is represented as a 4D image. As in [4] and [5], we do not work directly with grid points, but prefer to think in terms of their Voronoi neighborhoods. For a 4D Cartesian grid the Voronoi neighborhoods of grid points are “upright” 4-dimensional hypercubes. We will call such a hypercube a 4-xel. Assuming without loss of generality that the grid points have coordinates of the form (i1 + 0.5, i2 + 0.5, i3 + 0.5, i4 + 0.5) where the i’s are integers, the vertices of a 4-xel have integer coordinates and the edges of a 4-xel have length 1. We identify each binary image on a 4D Cartesian grid with the set of all 4-xels that are centered at the grid points which have value 1 in the image. This allows us to deﬁne a binary image simply as a ﬁnite set of 4-xels. The concepts of a simple 1 and a simple set of 1’s will be made precise by the concepts of a simple 4-xel and a simple set of 4-xels (in a binary image). We will use the same general approach to MNS sets as was used in [4] and [5]. Our work will be based on a characterization of simple 4-xels in terms of their attachment sets that was given in [6]. The attachment set of a 4-xel in a binary image is the subset of the boundary of the 4-xel that is shared with other 4-xels in the image.

2

n-Xels, 4D Binary Images, Xel-Complexes

An elementary 0-cell is a singleton set {i}, where i is an integer. An elementary 1-cell is a closed unit interval [i, i + 1] of the real line, where i is an integer. A 4xel is a Cartesian product of 4 elementary 1-cells. More generally, for 0 ≤ n ≤ 4 an n-xel is a Cartesian product of n elementary 1-cells and 4 − n elementary 0-cells, in some order. If q is an n-xel for some n, then we say q is a xel. A 0-xel will also be called a vertex and a 1-xel will also be called an edge. A 4D binary image is a ﬁnite set of 4-xels. An n-xel is the trajectory of an (n−1)-xel as it moves one unit in the positive or negative direction of the coordinate axis that is perpendicular to the (n − 1)xel. This is illustrated in Figure 1. The reader may sometimes ﬁnd it helpful to think of 4-xels as trajectories of 3-xels that move through a unit of time. If a xel y is a proper subset of a xel x, then we say y is a proper face of x, and write y < x. A xel x is a Cartesian product of elementary 1-cells and elementary 0-cells, and each proper face of x can be obtained from the product by replacing one or more of its elementary 1-cells [i, i + 1] with {i} or {i + 1}. A set K of xels is called a xel-complex if K is ﬁnite and, for every xel x ∈ K, every proper face of x is also in K. If y < x is a k-xel then we say y is a k-face of x.

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Fig. 1. An n-xel can be viewed as the trajectory of a moving (n − 1)-xel.

Each 4-xel intersects just 80 other 4-xels, and shares a 3-face with just 8 other 4-xels. Accordingly, two 4-xels are said to be 80-adjacent if they are distinct but intersect, and are said to be 8-adjacent if their intersection is a 3-xel. For any set S of 4-xels, and κ = 80 or 8, two 4-xels q1 , q2 ∈ S are said to be κ-connected in S if they are related by the reﬂexive transitive closure of the κadjacency relation on S. This is an equivalence relation on S, and its equivalence classes are called the κ-components of S.

3

Simple 4-Xels and Attachment Sets of 4-Xels

A 4-xel in a 4D binary image I is said to be simple in I if (loosely speaking) its deletion “preserves topology”. More precisely,if I is a 4D binary image we say that a 4-xel q ∈ I is simple if the polyhedron (I \ {q}) is a strong deformation retract of the polyhedron I. In other words, q is simple if and only if the union of the 4-xels in I can be continuously deformed over itself onto the union of the 4-xels in I \ {q}, in such a way that all points in (I \ {q}) at the start remain ﬁxed throughout the deformation process. This deﬁnition of simpleness in terms of the polyhedra I and (I \ {q}) is consistent with the use of 80-connectedness as the deﬁnition of connectedness on the 4-xels in I and the use of 8-connectedness as the deﬁnition of connectedness on the 4-xels in the complement of I. Indeed, two 4-xels belong to the same 80-component of I if and only if they lie in the same connected component of I, and they belong to the same 8-component of the 4-xels in the complement of I if and only if their interiors lie in the same connected component of R4 \ I. Our deﬁnition of simpleness has the advantage of being independent of dimensionality and of the shapes of xels. (For example, the same deﬁnition of simpleness could be used in 3D binary images on a face-centered cubic grid, except that instead of 4-xels we would have dodecahedral rhomboids belonging to a tessellation of 3-space by congruent dodecahedral rhomboids. In fact our deﬁnition of simpleness would be appropriate even in 5- and higher-dimensional binary images.) The deﬁnition involves continuous deformation, but Theorems 1 and 2 below give essentially discrete sets of necessary and suﬃcient conditions for q to be simple in I.

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These two theorems depend on the concept of the attachment complex of a 4-xel q in I, which is denoted by Attach(q, I) and is deﬁned by Attach(q, I) = {f | f < q and f < y for some y ∈ I − {q}} The closed polyhedral set Attach(q, I) is called the attachment set of q in I. This set is also given by Attach(q, I) = q ∩ (I \ {q}). The boundary complex of a 4-xel q, denoted by Boundary(q), is the set of all the proper faces of q. Evidently, Attach(q, I) ⊆ Boundary(q). For any xel-complex K, the Euler characteristic of K is the integer χ(K) deﬁned by χ(K) = c0 (K) − c1 (K) + c2 (K) − c3 (K) + c4 (K), where cn (K) is the number of n-xels in K. If P is the union of the xels of a xel-complex K, then we deﬁne χ(P) to be χ(K). It can be shown that if x is any xel then χ(x) = 1. We are now ready to state the above-mentioned essentially discrete sets of necessary and suﬃcient conditions for a 4-xel q to be simple. Theorem 1. Let q be a 4-xel in a 4D binary image I. Then q is simple in I if and only if the following all hold: 1. Attach(q, I) is connected and nonempty. 2. Boundary(q) − Attach(q, I) is connected and nonempty. 3. Attach(q, I) is simply connected. Theorem 2. Let q be a 4-xel in a 4D binary image I. Then q is simple in I if and only if the following all hold: 1. Attach(q, I) is connected. 2. Boundary(q) − Attach(q, I) is connected. 3. χ(Attach(q, I)) = 1. Theorems 1 and 2 are Theorems 7 and 9 in [6], except that the deﬁnition of a simple 4-xel used in that paper might seem to be more stringent than the deﬁnition given above: In [6], a 4-xel q is said to be simple in a 4D binary image I if Attach(q, I) is a strong deformation retract of q. As is explained in [6], it follows from the main result of [7] that the three conditions of Theorem 1 are equivalent to the three conditions of Theorem 2. So the two theorems are equivalent. An elementary proof of the “if” parts of these theorems is given in [6]. A shorter proof can be given using results of algebraic topology—notably Fact 3.4 in [8] (which follows from Corollaries 3.2.5, 1.3.11, and 1.4.10, and Theorem 1.4.11, in [9]) and Corollary 8.3.11 in [10]. Although the “only if” parts of Theorems 1 and 2 are easy to prove if simple 4-xels are deﬁned as in [6], we have to work a little harder to give a proof for the deﬁnition of simpleness used in this paper. However, standard techniques of algebraic topology suﬃce. Indeed, assuming one can q is simple in I use the exact homology sequences of the pairs (q, Attach(q, I)) and ( I, (I \ {q})) together with the excision theorem to deduce that the reduced homology groups

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of Attach(q, I) are all trivial. The three conditions of Theorem 2 follow from this and the Alexander duality theorem. The deﬁnition of simpleness used in [6] is in fact equivalent to the deﬁnition used in this paper, because both deﬁnitions are equivalent to the three conditions of Theorem 1 or 2. An advantage of the deﬁnition we are now using is that it involves only 4-xels (and does not involve their attachment sets). It may be worth mentioning here that, while Theorem 2 provides an easy way of determining whether or not a 4-xel in a 4D binary image is simple, there seems to be no analog of Theorem 2 for 5- and higher-dimensional images. Indeed, for n ≥ 5 the authors do not currently have an easy way to determine if the attachment set of an n-xel in an n-dimensional binary image is simply connected.

4

MNS Sets and the Main Theorem

A set D of 4-xels in a 4D binary image I is said to be simple in I if the elements of D can be arranged in a sequence in which each element is simple after all of its predecessors in the sequence have been removed from the image. In particular, the empty set is simple in I, and a singleton set {q} is simple in I if and only if q is a simple 4-xel of I. Since the deletion of a single simple 4-xel “preserves topology”, so does the parallel deletion of a simple set of 4-xels. More precisely, if D is a simple set of 4-xels of a 4D binary image I, then it follows from the deﬁnition of a simple 4-xel (and the transitivity of the relation “is a strong deformation retract of”) that (I \ D) is a strong deformation retract of I. We are interested in ways of proving that the set of 4-xels deleted at each iteration of a given parallel thinning algorithm for 4D binary images is always a simple set. (This would imply that the algorithm “preserves topology”.) Since a non-simple set must evidently contain a minimal non-simple (MNS) set, one method of proof would be to show that at each iteration no set of 4-xels that all satisfy the algorithm’s deletion condition can be an MNS set of the image. (A set D of 4-xels of I is an MNS set of I if and only if D is a non-simple set of I but every proper subset of D is a simple set of I.) In fact, this would show that the set of 4-xels deleted at each iteration is not only simple but also hereditarily simple—i.e., all of its subsets are simple sets of the image. One example of an MNS set in a 4D binary image is shown in Figure 2. The next four theorems state fundamental properties of MNS sets. These results were established for 3D binary images in [5]—see Propositions 4.3, 4.5, 4.6, and 4.7 in that paper—and can be proved for 4D images in the same way. Theorem 3. Let D be a nonempty set of 4-xels in a 4D binary image I. Then D is MNS in I if and only if the following conditions both hold: 1. Each element q ∈ D is non-simple in I \ (D \ {q}). 2. Each element q ∈ D is simple in I \ D whenever D D \ {q}.

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We say that a set D of 4-xels can be MNS if there is some 4D binary image I such that D is an MNS set of I. We say that a set D of 4-xels can be MNS without being a component if there is some 4D binary image I such that D is an MNS set of I and D is not an 80-component of I. Theorem 4. A set of 4-xels can be MNS only if it is a subset of some 2×2×2×2 block of sixteen 4-xels. Theorem 5. Let D be an MNS set of a 4D binary image I, and suppose D is not an 80-component of I. Then every element of D is 80-adjacent to a 4-xel of I that is not in D. Theorem 6. If a set D of 4-xels can be MNS without being a component, then every subset D of D can be MNS without being a component. We now state our Main Theorem, which identiﬁes all sets of 4-xels that can be MNS, and all such sets that can be MNS without being a component: Theorem 7 (Main Theorem). Let D be a set of 4-xels. Then: 1. D can be MNS if and only if D is contained in some 2 × 2 × 2 × 2 block of sixteen 4-xels. 2. D can be MNS without being a component if and only if D is a subset of some 2 × 2 × 2 block of eight 4-xels. Note that there are four types of 2 × 2 × 2 block: Such a block could be a 1 × 2 × 2 × 2, a 2 × 1 × 2 × 2, a 2 × 2 × 1 × 2, or a 2 × 2 × 2 × 1 block.

5 5.1

Proof of the Main Theorem Useful Results

The purpose of this subsection is to present three results that will be used in our proof of the Main Theorem. The ﬁrst result is the Inclusion-Exclusion Principle for Euler characteristics, which is the following identity: χ(

n i=1

Ki ) =

T ⊆{1,2,...,n},T =∅

(−1)|T |−1 χ(

Ki )

(1)

i∈T

This holds for arbitrary xel-complexes K1 , K2 , . . . , Kn . The identity follows from the Inclusion-Exclusion Principle for ﬁnite sets and the deﬁnition of χ(K). The second result is the next proposition, which is related to the following lemma: Lemma 1. Let P be a union of xels and let x be an edge or a 2-xel such that x ⊆ Pand χ(x ∩ P) = 1. Then one of the following is true:

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1. 2. 3. 4.

x∩P x is a x is a x is a

87

consists of a single vertex of x. 2-xel and x ∩ P is one of the four edges of x. 2-xel and x ∩ P is a union of two edges of x that share a vertex. 2-xel and x ∩ P is a union of three of the four edges of x.

This lemma is easily veriﬁed by considering all possible forms of x ∩ P. From the lemma it is not hard to deduce Proposition 1 below, which will be used in section 5.3. We omit the proof of the lemma, but expect most readers will ﬁnd it intuitively clear that all parts of the proposition are valid in each of the four cases of the lemma. Proposition 1. Let q be a 4-xel. Let P be a union of xels in Boundary(q) and let x be an edge or a 2-xel in Boundary(q) such that χ(x ∩ P) = 1. Then: 1. P is connected if and only if P ∪ {x} is connected. 2. Boundary(q) \ P is connected if and only if Boundary(q) \ (P ∪ {x}) is connected. 3. χ(P) = χ(P ∪ {x}). The following proposition is the third result. This will save us a lot of casechecking in section 5.2: Proposition 2. Let q be a 4-xel, and let X be any nonempty set of xels in Boundary(q) that satisﬁes one of the following two conditions: A. There is some vertex that belongs to all of the xels in X. B. X = Y ∪ Z, where Y ∩ Z = ∅, there is some vertex that belongs to all of the xels in Y , there is some vertex that belongs to all of the xels in Z, and no xel in Y \ Z intersects a xel in Z \ Y . Then X satisﬁes the following conditions: 1. X is connected. 2. Boundary(q) \ X is connected. 3. χ( X) = 1. In fact condition A in this proposition is a special case of condition B (since we may take Y = Z = X in B). The proposition follows from Theorem 4.1 in [8]: Condition B implies that X is SN in the sense of [8], and so the theorem implies X is contractible. By standard results ofalgebraic topology (including the Alexander duality theorem) [10], this implies X satisﬁes conditions 1 – 3. 5.2

The “If ” Parts of the Main Theorem

For any 4-xel q in a 4Dbinary image I, let A(q, I) = {q ∩ x | x ∈ I \ {q}} \ {∅}. Note that A(q, I) = Attach(q, I). To show that the “if” part of assertion 1 of the Main Theorem is valid, let D be a subset of a 2 × 2 × 2 × 2 block of 4-xels such that D is an 80-component of a 4D binary image I. We claim D is MNS in I. Evidently, D satisﬁes condition 1

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Fig. 2. Let I be the 4D binary image consisting of ten 4-xels shown in this ﬁgure. Then the 2×2×2 block of eight 4-xels in the center constitute an MNS set of I. (In the ﬁgure a larger scale is used in the direction of the 4th coordinate axis than in the directions of the other three, so the 4-xels appear to have been stretched in that direction.) Notice that there is an edge, indicated by the thick gray line, which connects the two “end” 4-xels; this edge is shared by the eight central 4-xels. Let q be any one of those eight 4-xels. Then it is easy to deduce from Theorem 2 and Proposition 2 that q is simple in the image that remains after any proper subset of the other seven central 4-xels is deleted from I. But if we delete all seven of the other central 4-xels from I then q is non-simple in the remaining image (which consists just of q and the two end 4-xels). Hence, by Theorem 3, the central block of eight 4-xels is MNS in I as we claimed.

of Theorem 3. It remains to show that D also satisﬁes condition 2 of Theorem 3. Let q ∈ D and let I be obtained from I by deleting any proper subset of the other elements of D. We need to show that q is simple in I . Let X = A(q, I ). As D is contained in a 2 × 2 × 2 × 2 block of 4-xels, the central vertex of that block belongs to all the 4-xels in D and hence to all the xels in X. So, since X = Attach(q, I ), it follows from Proposition 2 that the three conditions of Theorem 2 hold with I in place of I. Thus q is simple in I , as required. To show that the “if” part of assertion 2 is also valid, let I be a 2 × 2 × 2 × 3 block of 4-xels and let D be its central 2 × 2 × 2 × 1 block (which is clearly not an 80-component of I). We claim D is MNS in I. If we can prove this then, by Theorem 6, the “if” part of assertion 2 is valid. By symmetry we may assume that I = {i1 × i2 × i3 × i4 | i1 , i2 , i3 ∈ {[0, 1], [1, 2]}, i4 ∈ {[0, 1], [1, 2], [2, 3]}} so that D = {i1 × i2 × i3 × [1, 2] | i1 , i2 , i3 ∈ {[0, 1], [1, 2]}}. Then D clearly satisﬁes condition 1 of Theorem 3. To show that D also satisﬁes condition 2 of Theorem 3, let q ∈ D and let I be obtained from I by deleting any proper subset of the other seven 4-xels in D. We need to show that q is simple in I . Let D− = {i1 × i2 × i3 × [0, 1] | i1 , i2 , i3 ∈ {[0, 1], [1, 2]}} and let D+ = {i1 × i2 × i3 × [2, 3] | i1 , i2 , i3 ∈ {[0, 1], [1, 2]}}. Let X = A(q, I ). Since I ⊆ D− ∪ D ∪ D+ , we have X = Y ∪ Z, where Y = A(q, I ∩ (D ∪ D− )) and Z = A(q, I ∩ (D ∪ D+ )). Since D ∪ D− = {i1 × i2 × i3 × i4 | i1 , i2 , i3 , i4 ∈ {[0, 1], [1, 2]}}, the vertex (1,1,1,1) belongs to all the 4-xels in D ∪D− and hence to all the xels in Y . Similarly, the vertex (1,1,1,2)

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belongs to all the xels in Z. Moreover, Y ∩ Z = A(q, I ∩ D) = ∅ because at least one element of D \ {q} is in I . Also, no xel in Y\ Z = A(q, I ∩ D− ) intersects a xel in Z \ Y = A(q, I ∩ D+ ). Since X = Attach(q, I ), it follows from Proposition 2 that the three conditions of Theorem 2 hold with I in place of I, which implies q is simple in I , as required. 5.3

The “Only If ” Parts of the Main Theorem

The “only if” part of assertion 1 is just Theorem 4. To prove the “only if” part of assertion 2, let S be any MNS set of a 4D binary image I. By Theorem 4, S is contained in some 2 × 2 × 2 × 2 block. A set T of 4-xels that is contained in some 2 × 2 × 2 × 2 block will be called a spanning set if there is no 2 × 2 × 2 block that contains T . We now suppose that our MNS set S is a minimal spanning set—i.e., we suppose S is a spanning set but no proper subset of S is a spanning set—and deduce that S must be an 80-component of I. This will show that no minimal spanning set can be MNS without being a component, which (by Theorem 6) is enough to establish the “only if” part of assertion 2 of the Main Theorem, since every spanning set contains a minimal spanning set. For any two 4-xels p and q let p − q denote the vector from the centroid of q to the centroid of p. We deﬁne the l1 -diameter of S to be maxp,q∈S p − q1 , where v1 is the l1 -norm of the vector v (i.e., the sum of the absolute values of the four components of v). Since S is a spanning set, the l1 -diameter of S is at least 2, and is therefore equal to 2, 3, or 4. Case 1: The l1 -Diameter of S Is 4 In this case S = {q, a} for some 4-xels q and a such that q − a1 = 4. Note that q∩ a consists of just one vertex, v say. Let P = Attach(q, I \ {a}), so that Attach(q, I) = P ∪ {v}. Since S is MNS in I, it follows from Theorem 3 that q is non-simple in I \ {a} but q is simple in I. The latter and Theorem 2 imply P ∪ {v} = Attach(q, I) is connected, and so either v ∈ P or P = ∅. But v ∈ P would imply Attach(q, I) = P ∪ {v} = P = Attach(q, I \ {a}), which (by Theorem 2) would make it impossible for q to be simple in I but non-simple in I \ {a}. Hence P = ∅ and so, by Theorem 5, S is an 80-component of I. Case 2: The l1 -Diameter of S Is 3 In this case it is readily conﬁrmed that S = {q, a, b} for some 4-xels q, a, and b such that q − a1 = q − b1 = 3 and a − b1 = 2. Let q ∩ a = ea and q ∩ b = eb . Then ea andeb are edges and ea ∩ ebconsists of just a vertex. Let P = Attach(q, I \{a, b}), so Attach(q, I) = P ∪ea ∪eb , Attach(q, I \ {a}) = P ∪ eb , and Attach(q, I \ {b}) = P ∪ ea . Since S is MNS in I, it follows from Theorem 3 that q is non-simple in I \ {a, b} but q is simple in I, in I \ {a}, and in I \ {b}. So it follows from Theorem 2 and Proposition 1 that neither χ(P ∩ ea ) nor χ(P ∩ eb ) is equal to 1.

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As Attach(q, I) = P ∪ ea ∪ eb , Attach(q, I \ {a}) = P ∪ eb , Attach(q, I \ {b}) = P ∪ ea , and χ(ea ) = χ(eb ) = χ(ea ∩ eb ) = 1, it follows from Theorem 2 and the Inclusion-Exclusion Principle for Euler characteristics that 1 = χ(P ∪ ea ) = χ(P) + 1 − χ(P ∩ ea ) 1 = χ(P ∪ eb ) = χ(P) + 1 − χ(P ∩ eb ) 1 = χ(P ∪ ea ∪ eb ) = χ(P) + 1 + 1 − χ(P ∩ ea ) − χ(P ∩ eb ) − 1 + χ(P ∩ ea ∩ eb ) and therefore χ(P) = χ(P ∩ ea ) = χ(P ∩ eb ) = χ(P ∩ ea ∩ eb ). So, since neither χ(P ∩ea ) nor χ(P ∩eb ) is equal to 1, χ(P ∩ea ∩eb ) = 1. Thus P ∩ ea ∩ eb = ∅ (since ea ∩ eb consists of just a vertex) and χ(P ∩ ea ∩ eb ) = 0. Therefore χ(P) = χ(P ∩ ea ) = χ(P ∩ eb ) = 0, so P ∩ ea = ∅. Now if P =

∅ then P ∪ e is disconnected, which contradicts Theorem 2 because P ∪ e = a a Attach(q, I \{b}) and q is simple in I \{b}. Hence P = ∅ and so, by Theorem 5, S is an 80-component of I. Case 3: The l1 -Diameter of S Is 2 In this case it is quite easy to verify that S = {q, a, b, c}, where x − y1 = 2 for all distinct x and y in S. Let q ∩ a = fa , q ∩ b = fb , and q ∩ c = fc . Then fa , fb , and fc are 2-xels, every pair of them share an edge, and fa ∩ fb ∩ fc consists of just a vertex. Let P = Attach(q, I \ {a, b, c}), so that Attach(q, I \ {a, b}) = P ∪ fc . Since S is MNS in I, it follows from Theorem 3 that q is non-simple in I \{a, b, c} but q is simple in I, I \ {c}, I \ {b, c}, and I \ {a, b}. So it follows from Theorem 2 and Proposition 1 that χ(P ∩ fc ) = 1. Since Attach(q, I \ {b, c}) = P ∪ fa , it follows from Theorem 2, the fact that χ(x) = 1 for any xel x, and the Inclusion-Exclusion Principle for Euler characteristics that 1 = χ(P∪fa ) = χ(P)+1−χ(P∩fa ). Hence χ(P) = χ(P∩fa ). By symmetrical arguments we must have χ(P) = χ(P ∩ fa ) = χ(P ∩ fb ) = χ(P ∩ fc )

(2)

Similarly, since Attach(q, I \ {c}) = P ∪ fa ∪ fb , we have 1 = χ(P ∪ fa ∪ fb ) = χ(P) + 1 + 1 − χ(P ∩ fa ) − χ(P ∩ fb ) − 1 + χ(P ∩ fa ∩ fb ) and so by eqn. (2) we have χ(P) = χ(P ∩ fa ∩ fb ). By symmetrical arguments we must have χ(P) = χ(P ∩ fa ∩ fb ) = χ(P ∩ fb ∩ fc ) = χ(P ∩ fa ∩ fc ) (3) Again, since Attach(q, I) = P ∪ fa ∪ fb ∪ fc , we have 1 = χ(P ∪ fa ∪ fb ∪ fc ) = χ(P) + 1 + 1 + 1 − χ(P ∩ fa ) − χ(P ∩ fb ) − χ(P ∩ fc ) − 1 − 1 − 1 + χ(P ∩ fa ∩ fb ) + χ(P ∩ fb ∩ fc ) + χ(P ∩ fa ∩ fc ) + 1 − χ(P ∩ fa ∩ fb ∩ fc ) and so by equations (2) and (3) we have: χ(P) = χ(P ∩ fa ∩ fb ∩ fc ) (4) Recalling that χ(P ∩ fc ) = 1, we see that equations (2) and (4) imply χ(P ∩ fa ∩ fb ∩ fc ) = 1. Thus P ∩ fa ∩ fb ∩ fc = ∅ (since fa ∩ fb ∩ fc consists of just

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a vertex) and χ(P ∩ fa ∩ fb ∩ fc ) = 0. Therefore, by equations (2), (3), and (4), the 2-xel fa satisﬁes χ(P ∩ fa ) = χ(P ∩ fa ∩ fb ) = 0. Since χ(P ∩ fa ) = 0, P ∩ fa either is empty or is the union of the four edges of fa . But the latter is impossible because fa ∩ fb is an edge of fa that is not contained in P (since χ(P ∩ fa ∩ fb ) = 0). Hence P ∩ fa = ∅. Now if P =

∅ then P ∪ fa is disconnected, which contradicts Theorem 2 because P ∪ fa = Attach(q, I \ {b, c}) and q is simple in I \ {b, c}. So P = ∅ and, by Theorem 5, S is an 80-component of I.

6

Concluding Remarks

We have identiﬁed all types of sets of 4-xels that can be minimal non-simple (MNS) in a 4D binary image, and all types that can be MNS without being an 80-component of the image, when 4D 80-connectedness is used on 4-xels in the image and 4D 8-connectedness on 4-xels in its complement. This work is based on a characterization of simple 4-xels that was given in [6], and the InclusionExclusion Principle for Euler characteristics.

References 1. Ronse, C.: Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Applied Mathematics 21 (1988) 67–79 2. Hall, R.W.: Tests for connectivity preservation for parallel reduction operators. Topology and Its Applications 46 (1992) 199–217 3. Ma, C.M.: On topology preservation in 3D thinning. CVGIP: Image Understanding 59 (1994) 328–339 4. Gau, C.J., Kong, T.Y.: Minimal nonsimple sets of voxels in binary images on a facecentered cubic grid. International Journal of Pattern Recognition and Artiﬁcial Intelligence 13 (1999) 485–502 5. Kong, T.Y.: On topology preservation in 2D and 3D thinning. International Journal of Pattern Recognition and Artiﬁcial Intelligence 9 (1995) 813–844 6. Kong, T.Y.: Topology preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In Ahronovitz, E., Fiorio, C., eds.: Discrete Geometry for Computer Imagery: 7th International Workshop (DGCI ’97, Montpellier, France, December 1997), Proceedings. Springer (1997) 3–18 7. Kong, T.Y., Roscoe, A.W.: Characterizations of simply-connected ﬁnite polyhedra in 3-space. Bulletin of the London Mathematical Society 17 (1985) 575–578 8. Saha, P.K., Kong, T.Y., Rosenfeld, A.: Strongly normal sets of tiles in N dimensions. Electronic Notes in Theoretical Computer Science 46 (2001) URL: http://www.elsevier.nl/locate/entcs/volume46.html 9. Spanier, E.H.: Algebraic Topology. Springer (1989) 10. Maunder, C.R.F.: Algebraic Topology. Dover (1996)

Receptive Fields within the Combinatorial Pyramid Framework Luc Brun1 and Walter G. Kropatsch2 1

´ Laboratoire d’Etudes et de Recherche en Informatique (EA 2618) Universit´e de Reims - France [email protected] 2 Institute for Computer-aided Automation Pattern Recognition and Image Processing Group Vienna Univ. of Technology- Austria [email protected]

Abstract. A hierarchical structure is a stack of successively reduced image representations. Each basic element of a hierarchical structure is the father of a set of elements in the level below. The transitive closure of this father-child relationship associates to each element of the hierarchy a set of basic elements in the base level image representation. Such a set, called a receptive ﬁeld, deﬁnes the embedding of one element of the hierarchy on the original image. Using the father-child relationship, global properties of a receptive ﬁeld may be computed in O(log(m)) parallel processing steps where m is the diameter of the receptive ﬁeld. Combinatorial pyramids are deﬁned as a stack of successively reduced combinatorial maps, each combinatorial map being deﬁned by two permutations acting on a set of half edges named darts. The basic element of a combinatorial pyramid is thus the dart. This paper deﬁnes the receptive ﬁeld of each dart within a combinatorial pyramid and study the main properties of these sets.

1

Introduction

Regular image pyramids have been introduced 1981/82 [11] as a stack of images with exponentially reduced resolution. Each image of this sequence is called a level. Such Pyramids present several interesting properties within the image processing and analysis framework such as [4]: The reduction of noise, the processing of local and global features within the same frame and the eﬃciency of many computations on this structure. Using the neighborhood relationships deﬁned on each image the Reduction window relates each pixel of the pyramid with a set of pixels deﬁned in the level below. The pixels belonging to one reduction window are the children of the pixel which deﬁnes it. This father-child relationship maybe extended by transitivity down to the base level image. The set of children of one pixel in the base level is named its receptive ﬁeld (RF) and

The authors wish to thank the anonymous reviewers for their useful comments. This Work was supported by the Austrian Science Foundation under P14445-MAT.

A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 92–101, 2002. c Springer-Verlag Berlin Heidelberg 2002

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deﬁnes the embedding of this pixel on the original image. Using the father-child relationship global properties of a receptive ﬁeld RF (v) with a diameter m may be computed in O(log(m)) parallel processing steps thanks to local calculus. However, receptive ﬁelds deﬁned within the regular pyramid framework are not necessarily connected [4]. Furthermore, the adjacency of two pixels v and w deﬁned at level k may not be easily interpreted on the base level image. Indeed, the boundary between the receptive ﬁelds RF (v) and RF (w) associated to this adjacency at level k may be disconnected and even incomplete. Irregular pyramids, ﬁrst introduced by Meer [16], Montanvert [17] and Jolion [13] are deﬁned as a stack of successively reduced simple graphs (i.e. graphs without double edges nor self-loops). The base level graph may be built from a sampling grid using one pixel adjacency such as the 4−neighborhood. Each graph of the hierarchy is built from the graph below by selecting a set of vertices named surviving vertices and mapping each non surviving vertex to a surviving one [17]. This mapping induces a father-child relationship between a surviving vertex and the set of non surviving vertices mapped to it. The reduction window of one surviving vertex is then deﬁned as its set of children. The receptive ﬁeld of one surviving vertex is deﬁned by the transitive closure of the father-child relationship. Using this reduction scheme, the receptive ﬁeld of each vertex in the hierarchy is a connected set of vertices in the base level graph. However, using simple graphs, the adjacency between two vertices is encoded by only one edge while the receptive ﬁelds of two vertices may share several boundaries. Edges in the hierarchy may thus encode non connected set of boundaries between the associated receptive ﬁelds. Moreover, the lack of self-loops in simple graphs does not allow to diﬀerentiate an adjacency relationship between two receptive ﬁelds from an inclusion relationship. The last two drawbacks may be overcome by using the Dual graph pyramids introduced by Kropatsch [14]. Using Kropatsch’s reduction scheme, the reduction operation is encoded by edge contractions [14]. This operation contracts one edge and its two end points into a single vertex. The contraction of a graph reduces the number of vertices while maintaining the connections to other vertices. As a consequence some redundant edges such as self-loops or double edges may occur. These redundant edges may be characterized in the dual of the contracted graph. The removal of such edges is called a dual decimation step. Since the reduction scheme requires both features of the initial graph and its dual such pyramids are called Dual graph pyramids. Within such hierarchies, each receptive ﬁeld is a connected set of vertices in the base level. Moreover, each edge between two vertices in the hierarchy encodes an unique connected boundary between the associated receptive ﬁelds. Finally, the use of self-loops within the hierarchy allows to diﬀerentiate adjacency relationships between receptive ﬁelds from inclusions relations. The basic entity of all the above pyramids is the vertex/pixel. Combinatorial maps are based on darts. Hence receptive ﬁelds of Combinatorial Pyramids are expressed in terms of darts. Combinatorial pyramids are equivalent to dual graph pyramids with the exeption that they represent the orientation ex-

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plicitely. The expected advantages of such hierarchies within the image analysis framework are presented in [9]. The remaining of this paper is as follows: In Section 2 we present the combinatorial map model together with the expected advantages of this model within the Pyramid framework. In Section 3 we present the construction scheme of a combinatorial pyramid. Finally Section 4 deﬁnes the notion of receptive ﬁeld within the combinatorial pyramid framework and states its major properties.

2

Combinatorial Maps

Combinatorial maps and generalized combinatorial maps deﬁne a general framework which allows to encode any subdivision of nD topological spaces orientable or non-orientable with or without boundaries. The concept of maps has been ﬁrst introduced by Edmonds [12] in 1960 and later extended by several authors [15]. This model has been applied to several ﬁelds of computer imagery such as geometrical modeling [3] and 2D segmentation [1,5]. An exhaustive comparison of combinatorial maps with other boundary representations such as cell-tuples and quad-edges is presented in [15]. Recent trends in combinatorial maps apply this framework to the segmentation of 3D images [6,2] and the encoding of hierarchies [8,9]. The remaining of this paper will be based on 2D combinatorial maps which will be just called combinatorial maps. A combinatorial map may be seen as a planar graph encoding explicitly the orientation of edges around a given vertex. Fig. 1a) demonstrates the derivation of a combinatorial map from a plane graph. First edges are split into two half edges called darts, each dart having its origin at the vertex it is attached to. The fact that two half-edges (darts) stem from the same edge is recorded in the reverse permutation α. A second permutation σ encodes the set of darts encountered when turning counterclockwise around a vertex (see e.g. the σ-orbit (−8, −3, 11, 4) encoding the central vertex in Fig. 1a)). ✞ ✞ ✞ -1 ✆ -2 ✆ ✝7 ✆ ✝ ✝ σ ✞ ✞ σ ✞ ✞ σ 8 9 7 8 ✆✝ 2 ✆✝ 9 ✝1 ✆ ϕ✝ ϕ✞ ✆ -8 ✞ ✞ ✞ -7 -9 3 -3 4 -4 -7 ✆✝ -3 ✆✝ -8 ✆✝ -4 ✆✞ ✞ ✝ ✇ ✇ ✇ ✝ σ ✞ ✞ σ ✞ ✞ σ✝ 10 ✆ -9 ✆ 11 ✆✝ 4 ✆✝ 12 ✆ ✝3 ✆ ✝ 10 11 12 ϕ✞ ✞ ϕ✞ ✞ -11 -10 -5 ✆-11 -6 ✆ ✝ ✆ ✝ ✝ ✆ ✝ -10 5 -5 σ σ σ✞ 6 -6 -12 ✞ ✞ ✇ ✇ ✇ -12 ✝5 ✆ ✝6 ✆ ✝✆ ✇

1 -1

2 -2

✇

✇

a)

b)

✞✛ ✞ ✞ ✔-1 ✛ ✗ -2 ✆ ✝7 ✆ ✝✆ ✝ σ✞ ✞✁ σ❆ σ ✻ ❑✞ ✞❄ ☛ ✲ 1✲ 8✲ 2✲ 9 ✕ ✖ ✝✆ ✆✝ ✆ ✆ ϕ✝ ϕ✝ ✞✻ ✞❄ ✞✻ ✞❄ ✛ ✛ ✛ -7 ✆✝ -3 ✆✝ -8 ✆✝ -4❍ ✞❄ ✝ ✆✞ ✙ ✟  σ ✞❄ σ ✞✻ σ✝ 10 ✆ -9 ✆ ✞❄ ✝ ❍ ❥✞3✻ ✯ ✲11✲ ✲ ✟ 4 ✆✝ 12 ✆ ✻ ✝✆ ✝ ✆ ✝ ϕ✞❄ ϕ✞❄ ✞✻ ✞✻ ✛ ✛ ✗ ✔ -10 -5 ✛ -11 -6 ✛ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ σ σ✁✕ σ✞ ✞❄ ✞ ✻ ❆  ✲6 ✲-12 ✕ ✖ ✝5 ✆ ✝✆ ✝✆ c)

Fig. 1. A 3 × 3 grid encoded by a combinatorial map

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The symbols α∗ (d) and σ ∗ (d) stand, respectively, for the α and σ orbits of the dart d. More generally, if d is a dart and π a permutation we will denote the π-orbit of d by π ∗ (d). A combinatorial map G is the triplet G = (D, σ, α), where D is the set of darts and σ, α are two permutations deﬁned on D such that α is an involution: ∀d ∈ D

α2 (d) = d

(1)

Note that, if the darts are encoded by positive and negative integers, the involution α may be implicitly encoded by sign (Fig. 1a)). This convention is often used for practical implementations [5] where the combinatorial map is simply implemented by an array of integers encoding the permutation σ. Given a combinatorial map G = (D, σ, α), its dual is deﬁned by G = (D, ϕ, α) with ϕ = σ ◦ α. The orbits of the permutation ϕ encode the set of darts encountered when turning around a face (see e.g. the ϕ-orbit (1, 8, −3, −7) in Fig. 1a)). Note that, using a counter-clockwise orientation for permutation σ, each dart of a ϕ-orbit has its associated face on its right. Figures 1b) and 1c) illustrate an alternative representation of the combinatorial map encoding. Within such a representation, each dart is represented by one vertex and one edge connects a dart d1 to d2 iﬀ either d2 = σ(d1 ) or d2 = ϕ(d1 ). Using this representation, the σ and ϕ orbits of the combinatorial map are represented by the faces of the oriented graph.

3

Combinatorial Pyramids

The aim of combinatorial pyramids is to combine the advantages of combinatorial maps with the reduction scheme deﬁned by Kropatsch [14] (see also Section 1). A combinatorial pyramid is thus deﬁned by an initial combinatorial map successively reduced by a sequence of contraction or removal operations. In order to preserve the number of connected components of the initial combinatorial map, we forbid the removal of bridges and the contraction of self-loops. A self-loop in the initial combinatorial map becomes a bridge in its dual and vice-versa [10]. In the same way, a contraction operation in the initial combinatorial map is equivalent to a removal operation performed in its dual. Therefore, the exclusion of bridges and self-loops from respectively removal and contraction operations corresponds to a same constraint applied alternatively on the dual combinatorial map and the original one. In order to avoid the contraction of self-loops, the set of edges to be contracted must form a forest of the initial combinatorial map. The graph of a combinatorial map is a forest if it does not contain a cycle. A more formal deﬁnition may be found in [7][Def. 4]. A set of edges to be contracted satisfying the above requirement is called a contraction kernel: Deﬁnition 1. Contraction Kernel Given a connected combinatorial map G = (D, σ, α) the set K ⊂ D will be called a contraction kernel iﬀ K is a forest of G. The set SD = D − K is called the set of surviving darts.

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In the same way, a removal kernel is deﬁned as a forest of the dual combinatorial map. This constraint insures that no self-loop will be contracted in the dual combinatorial map and thus that no bridge may be removed in the initial one. Within our framework, a removal kernel is used to remove redundant edges created by contraction operations. These edges are characterized as self-loops or double edges [14]. A removal kernel is thus deﬁned as a forest of the dual combinatorial map removing any redundant double edge and self-loop. Since one contraction encodes a merge between two regions, the set of darts encoding the adjacency relationships between the image and its background must be excluded from contraction operations. However, after a contraction step, some edges of the image boundary may become double edges. The removal of such edges corresponds to the concatenation of the associated boundaries in the base level image. Such operation preserves the image boundary and is thus allowed. Contraction and removal kernels specify the set of edges which must be contracted or removed. The creation of the reduced combinatorial map from a contraction or a removal kernel is performed in parallel by using connecting walks [9]. Given a combinatorial map G = (D, σ, α), a kernel K and a surviving dart d ∈ SD = D − K, the connecting walk associated to d is either equal to: CW (d) = d, ϕ(d), . . . , ϕn−1 (d) with n = M in{p ∈ IN∗ | ϕp (d) ∈ SD} if K is a contraction kernel and CW (d) = d, σ(d), . . . , σ n−1 (d) with n = M in{p ∈ IN∗ | σ p (d) ∈ SD} If K is a removal kernel. Given a kernel K and a surviving dart d ∈ SD, such that CW (d) = d.d1 . . . dp , the successor of d within the reduced combinatorial map G = G/K = (SD, σ , α) is retrieved from CW (d) by the following equations [9]: ϕ (d) = ϕ(dp ) if K is a contraction kernel σ (d) = σ(dp ) if K is a removal kernel

(2)

Note that, if K is a contraction kernel, the connecting walk CW (d) allows to compute ϕ (d). The σ-successor of d within the contracted combinatorial maps may be retrieved from CW (α(d)) = α(d).d1 , . . . , dp . Indeed, we obtain by using equations 1 and 2: ϕ (α(d)) = σ (α(α(d))) = σ (d) = ϕ(dp ). Fig. 2a) shows the three connection walks deﬁned by kernel α∗ (1, 7, 10) corresponding to the contracted maps in Fig. 2b). On this example, we have ϕ (−2) = 5, ϕ (−5) = 3 and ϕ (−3) = 8. The permutation ϕ remains unchanged for the other surviving darts. We have thus: σ (2) = 5, σ (5) = 3 and σ (3) = 8 while the permutation σ is unchanged for the other surviving darts. Based on the above property we designed two algorithms to traverse connecting walks respectively deﬁned by contraction and removal kernels [9]. One straightforward application of these algorithms consists to compute the reduced combinatorial maps. However, connecting walks may also be used to relate one combinatorial map of the pyramid with the one below. Indeed, we showed [8] that

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each non surviving dart belongs to one and only one connecting walk. Therefore, the connecting walk CW (d) of each surviving dart d contains the dart d and all non surviving darts mapped to it. Such a sequence of darts corresponds to the reduction window associated to the surviving dart d.

4

Receptive Fields of Darts

The notion of connecting walks introduced in Section 3 allows us to build one reduced combinatorial map from an initial one and a contraction or removal kernel. Therefore, given a sequence of kernels K1 , . . . , Kn and an initial combinatorial map G0 = (D, σ, α) deﬁned from a planar sampling grid (e.g. the 4−neighborhood one) one can deﬁne the sequence of reduced combinatorial maps G0 , G1 , . . . , Gn where Gi = Gi−1 /Ki = (SDi , σi , α) for each i ∈ {1, . . . , n}. Note that according to the deﬁnition of surviving darts (Deﬁnition 1) we have SDi = SDi−1 − Ki = D − ∪ij=1 Kj . Intuitively, one connecting walk CWi (d) deﬁnes the set of darts that we have to traverse in the level below in order to connect d to ϕi (d) if Ki is a contraction kernel and d to σi (d) if Ki is a removal kernel (see equation 2). Let us consider the sequence of darts CDSi (d) that we have to traverse in the base level graph G0 to connect d to ϕi (d) if Ki is a contraction kernel and d to σi (d) if Ki is a removal kernel. Such a sequence of darts is called a connecting dart sequence(CDS). Moreover, using the construction scheme described below, we showed [8] that the ﬁrst dart of CDSi (d) is d. A connecting dart sequence CDSi (d) without its ﬁrst dart will be denoted CDSi∗ (d). To construct a CDS, let us ﬁrst suppose that Ki is a contraction kernel. If Ki+1 is a removal kernel we have to traverse the sequence of darts CWi+1 (d) = d.d1 , . . . , dp in Gi to connect d to σi+1 (d). Each dart of CWi+1 (d) is related to the following one by: d1 = σi (d), ∀j ∈ {1, . . . , p − 1} dj+1 = σi (dj ) Since Ki is a contraction kernel, CDSi (dj ) connects dj to ϕi (dj ) for any j in {1, . . . , p − 1}. However, CDSi (α(dj )) connects α(dj ) to ϕi (α(dj )) = σi (α ◦ α(dj )) = σi (dj ) (see equation 1). The connection between dj and σi (dj ) is thus performed by dj CDSi∗ (α(dj )) and the connecting dart sequence of d at level i + 1 is equal to CDSi+1 (d) = d1 · CDSi∗ (α(d1 )) · · · dp · CDSi∗ (α(dp )). Fig. 2c) illustrates the connecting dart sequences deﬁned by the applications of the contraction kernel K1 = α∗ (1, 7, 10) followed by the removal kernel K2 = α∗ (3). Connecting walks deﬁned by K1 are illustrated in Fig. 2a) while the contracted combinatorial map G1 = G0 /K1 is represented in Fig. 2b) together with the connecting walks deﬁned by K2 . According to Fig. 2b) we have to traverse the dart 3 in G1 to connect 5 to σ2 (5) = 8. Moreover, using Fig. 2a), the connection between 5 and 3 requires to traverse the dart −10 in G0 while the connection between 3 and 8 requires to traverse the darts −7 and 1. The connecting dart sequence associated to 5 is thus equal to CDS2 (5) = 5CDS1∗ (−5)3.CDS1∗ (−3) = 5. − 10.3. − 7.1 (Fig. 2c)).

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Conversely, if both kernels Ki and Ki+1 are contraction kernels, the connecting dart sequence of d at level i + 1 is equal to [8]: CDSi (d).CDSi (d1 ) · · · CDSi (dp ). The same construction scheme holds if Ki is a removal kernel. Both cases are resumed in the following deﬁnition: Deﬁnition 2. Given a combinatorial map G0 = (D, σ, α) and a sequence of contraction or removal kernels K1 , K2 . . . , Kn . The connecting dart sequences are deﬁned by the following recursive construction: ∀d ∈ D

CDS0 (d) = d

For each level i in {1, . . . , n} and for each dart d in SDi – If Ki and Ki−1 have the same type: CDSi (d) = CDSi−1 (d1 ) · · · CDSi−1 (dp ) – If Ki and Ki−1 have diﬀerent types: ∗ ∗ CDSi (d) = d1 · CDSi−1 (α(d1 )) · · · dp · CDSi−1 (α(dp ))

Where (d1 . . . dp ) is equal to CWi (d). The kernels K0 = ∅ and K1 have the same type by convention. ✞✛ ✞ ✞ ✞ ✞✛ ✞ ✔-1 ✛ ✗ -2 ✆ -2 ✆ ✝ 7✆ -1 ✆ ✎ ✗ ✝7 ✆ ✝✆ ✝ ✝ ✝ ϕ✞❄ σ✞ ✞✁ σ❆ σ ✻ σ✞ σ ✻ ❑✞ ✞❄ ❆❑✞ ✞ ✞ ☛ ✲ 1✲ 8✲ 2✲ 9 ✕ ✲ 8✲ 2✲ 9 ✕ ✖ ✎ 1✆ ✝✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✝2 ✆ ϕ✞❄ ϕ✞❄ ϕ✞✞❄ ϕ✞❄ ✞✞ ✞ ✞✻ ✞✻ ✞✻ ✞✻ ✛ ✛ ✛ ✛ ✛ ✛ ✍ -7 ✆✝ -3 ✆✝ -8 ✆✝ -4❍ -3 ✆✝ -8 ✆ -4❍ -7 ✆ -3 ✆✝ -8 ✆ ✞❄ ✝ ✆✞ ✆✞ ✞❄✝ ✝✝ ✆✝ ✝✝ ✆ ✙ ✟   σ ✞❄ σ ✞✻ σ✝ σ ✞✻ σ✝ 10 ✆ -9 ✆ -9 ✆ ✝ 10 ✆✞✻ ✞❄ ✞✲ ✞❄ ✞❄ ✝ ❍ ❥✞3✻ ✯ ✯ ✲11✲ ✲ ✟ ✲ ✲ ✟ 4 12 3 11 4 12 3 ✝✆ ✆✝ ✆ ✆ ✻ ✝✆ ✝ ✆✝ ✆ ✆ ✻ ✝✆ ϕ✝ ϕ✝ ϕ✝ ✞✞✻ ✞❄ ✞✻ ✞❄ ✞✻ ✞❄ ✞✻ ✻ϕ✞❄ ✛ ✛ ✛ ✛ ✛ ✛ ✛ ✗ ✔ ✖ ✔ -10 -10 -5 -11 -6 -5 -11 -6 ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝ ✆ ✝✆ ✝ ✆ σ σ✁✕ σ✞ ✞❄ σ σ✁✕ σ✞ ✞ ✞❄ ✻ ✲❆✞ ❆✞ ✻ ✲ ✲ ✲ ✕ ✖ ✕ ✖ ✕ -12 -12 ✝5 ✆ ✝6 ✆ ✝✆ ❅ ✝5 ✆ ✝6 ✆ ✝✆ ❅ ✝5 ✆ ❅ ❅ a)G0 , CW1 (−2) ,

CW1 (−3) , CW1 (−5)

b)G1 , CW2 (−8) , CW2 (5)

c) CDS2 (2) ,

CDS2 (5) , CDS2 (−8)

Fig. 2. Receptive ﬁelds CDS2 are based on connecting walks CW1 and CW2 .

The set of connecting dart sequences deﬁned at each level deﬁnes a partition of the initial set of darts D. Moreover, each connecting dart sequence CDSi (d) deﬁned at level i by d ∈ SDi satisﬁes [8] CDSi∗ (d) ⊂ ∪ij=1 Ki . Therefore, d is the only dart of CDSi (d) surviving up to level i and CDSi∗ (d) encodes the set of non surviving darts mapped to it at level i. Connecting dart sequences encode thus the notion of receptive ﬁeld within the combinatorial map framework.

Receptive Fields within the Combinatorial Pyramid Framework

99

Using Deﬁnition 2 connecting dart sequences must be computed at each level from the level below. However, such a construction scheme may induce useless calculus if one does not need to compute all connecting dart sequences deﬁned from level 1 to level i. This recursive construction scheme may be avoided by using the following theorem [8]: Theorem 1 Given a combinatorial map G0 = (D, σ, α), a sequence of contraction kernels or removal kernels K1 , K2 . . . , Kn , the relation between the successive darts of a connecting dart sequence CDSi (d) = (d.d1 , . . . , dp−1 ), with i ∈ {1, . . . , n} and d ∈ SDi is as follows: ϕ(d) If Ki is a contraction kernel d1 = σ(d) If Ki is a removal kernel ϕ(dj−1 ) if dj−1 has been contracted ∀j ∈ {2, . . . , p} dj = σ(dj−1 ) if dj−1 has been removed Therefore, connecting dart sequences may be computed directly from the base level graph G0 given the type of the kernel Ki and the type of the reduction operation applied to each dart in ∪ij=1 Kj . This last information may be stored in each dart during the construction of the pyramid by a function state from ∪ij=1 Kj to {Contracted, Removed} such as state(d) is equal to either Contracted or Removed according to the type of the kernel applied to d before level i. Note that the storage of this function only adds on bit per dart. Algorithm 1 uses the function state and the properties established by Theorem 1 to compute one connecting dart sequence at level i from the base level graph G0 . The complexity of this algorithm is linear in the length of the computed connecting dart sequence. Algorithms computing all connecting dart sequences deﬁned at level i in parallel are described in [8]. The complexity of these algorithms is O(log(Mi )) where Mi denotes the longest connecting dart sequence deﬁned at level i. Moreover, by construction connecting dart sequences encode the set of darts in G0 that we have to traverse to connect one surviving dart at level i with its ϕi or σi successor according to Ki . Indeed, given one dart d ∈ SDi , such that CDSi (d) = d.d1 , . . . , dp we showed [8] that: – If Ki is a contraction kernel: ϕ(dp ) if dp has been contracted ϕi (d) = σ(dp ) if dp has been removed – If Ki is a removal kernel: ϕ(dp ) if dp has been contracted σi (d) = σ(dp ) if dp has been removed

(3)

(4)

Note that equations 3 and 4 are similar to equations 2 deﬁned for connecting walks. Given an encoding of the function state and the set of surviving darts SDi , equations 3 and 4 combined with Algorithm 1 allow us to retrieve the reduced

100

L. Brun and W.G. Kropatsch cds compute cds(map G0 ,dart d,level i) { cds CDS=d if (Ki is a contraction kernel) compute cds rec(G0 ,ϕ(d),i,CDS); else compute cds rec(G0 ,σ(d),i,CDS); return CDS } cds compute cds rec(map G0 ,dart d’,level i,cds CDS) { if (d ∈ SDi ) return CDS; CDS=CDS.d’ if (state(d ) = Contracted) compute cds rec(G0 ,ϕ(d ),i,CDS); else compute cds rec(G0 ,σ(d ),i,CDS); return CDS }

Algorithm 1: Computation of the connecting dart sequence of a dart d at level i.

combinatorial map Gi without computing G1 , . . . , Gi−1 . This last property may be useful for the analysis of hierarchies where the ﬁrst levels of the pyramid often corresponds to low level operations.

5

Conclusion

The presented concept of connecting dart sequences uses the principle of receptive ﬁelds for the ﬁrst time within the combinatorial pyramid framework. The darts of a combinatorial map reﬁne the pixel subdivision of a digital image. Hence receptive ﬁelds based on darts can distinguish more conﬁgurations. Moreover, using the order deﬁned on such sequences we showed that any reduced combinatorial map may be retrieved directely from the base level. Higher level concepts are typically associated with higher level structural entities like a dart or a vertex at a high pyramid level. Using the receptive ﬁeld of these structural entities properties and parameters of the corresponding high level concepts may be computed, maybe through complicated calculations, maybe using the hierarchical decomposition, from the gray values, the colors or the coordinates of the discrete (pixel) measurements. In the future we plan to study the interactions between the highly ﬂexible contraction scheme of combinatorial pyramids and the eﬃciency of computing high level descriptions

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References [1] E. Ahronovitz, J. Aubert, and C. Fiorio. The star-topology: a topology for image analysis. In 5th DGCI Proceedings, pages 107–116, 1995. [2] Y. Bertrand, G. Damiand, and C. Fiorio. Topological map: Minimal encoding of 3d segmented images. In J. M. Jolion, W. Kropatsch, and M. Vento, editors, 3rd Workshop on Graph-based Representations in Pattern Recognition, pages 64–73, Ischia(Italy), May 2001. IAPR-TC15, CUEN. [3] Y. Bertrand and J. Dufourd. Algebraic speciﬁcation of a 3D-modeler based on hypermaps. CVGIP: Graphical Models and Image Processing, 56(1):29–60, Jan. 1994. [4] M. Bister, J. Cornelis, and A. Rosenfeld. A critical view of pyramid segmentation algorithms. Pattern Recognit Letter., 11(9):605–617, Sept. 1990. [5] J. P. Braquelaire and L. Brun. Image segmentation with topological maps and inter-pixel representation. Journal of Visual Communication and Image representation, 9(1), 1998. [6] J. P. Braquelaire, P. Desbarats, and J. P. Domenger. 3d split and merge with 3-maps. In J. M. Jolion, W. Kropatsch, and M. Vento, editors, 3rd Workshop on Graph-based Representations in Pattern Recognition, pages 32–43, Ischia(Italy), May 2001. IAPR-TC15, CUEN. [7] L. Brun and W. Kropatsch. Pyramids with combinatorial maps. Technical Report PRIP-TR-057, PRIP, TU Wien, 1999. [8] L. Brun and W. Kropatsch. The construction of pyramids with combinatorial maps. Technical Report 63, Institute of Computer Aided Design, Vienna University of Technology, lstr. 3/1832,A-1040 Vienna AUSTRIA, June 2000. [9] L. Brun and W. Kropatsch. Contraction kernels and combinatorial maps. In J. M. Jolion, W. Kropatsch, and M. Vento, editors, 3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition, pages 12–21, Ischia Italy, May 2001. IAPR-TC15, CUEN. [10] L. Brun and W. G. Kropatsch. Dual contraction of combinatorial maps. In W. G. Kropatsch and J.-M. Jolion, editors, 2nd IAPR-TC-15 Workshop on Graphbased Representations, volume 126, pages 145–154, Haindorf, Austria, May 1999. ¨ Osterreichische Computer Gesellschaft. [11] P. Burt, T.-H. Hong, and A. Rosenfeld. Segmentation and estimation of image region properties through cooperative hierarchial computation. IEEE Transactions on Sustems, Man and Cybernetics, 11(12):802–809, December 1981. [12] J. Edmonds. A combinatorial representation for polyhedral surfaces. Notices American Society, 7, 1960. [13] J. Jolion and A. Montanvert. The adaptative pyramid : A framework for 2d image analysis. Computer Vision, Graphics, and Image Processing, 55(3):339–348, May 1992. [14] W. G. Kropatsch. Building Irregular Pyramids by Dual Graph Contraction. IEEProc. Vision, Image and Signal Processing, Vol. 142(No. 6):pp. 366–374, December 1995. [15] P. Lienhardt. Topological models for boundary representations: a comparison with n-dimensional generalized maps. Computer-Aided Design, 23(1):59–82, 1991. [16] P. Meer. Stochastic image pyramids. Computer Vision Graphics Image Processing, 45:269–294, 1989. [17] A. Montanvert, P. Meer, and A. Rosenfeld. Hierarchical image analysis using irregular tessellations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(4):307–316, APRIL 1991.

A New 3D 6-Subiteration Thinning Algorithm Based on P -Simple Points Christophe Lohou and Gilles Bertrand Laboratoire d’Algorithmique et Architecture des Syst`emes Informatiques (A2 si), ´ ´ ´ Ecole Sup´erieure d’Ing´enieurs en Electrotechnique et Electronique (Esiee), 2, bld Blaise Pascal, Cit´e Descartes, BP 99, F-93162 Noisy-le-Grand Cedex, France {lohouc,bertrang}@esiee.fr

Abstract. In a recent study [1], we proposed a new methodology to build thinning algorithms based on the deletion of P -simple points. This methodology may permit to conceive a thinning algorithm A from an existent thinning algorithm A, such that A deletes at least all the points removed by A, while preserving the same end points. In this paper, by applying this methodology, we propose a new 6-subiteration thinning algorithm which deletes at least all the points removed by the 6-subiteration thinning algorithm proposed by Pal´ agyi and Kuba [2].

1

Introduction

Some graphical applications require to transform objects while preserving their topology [2,3]. That leads to the well-known notion of simple point: a point in a binary image is said to be simple if its deletion from the image “preserves the topology” [4,5,6,7,8,9,10,11,12,13,14]. A process deleting simple points is called a thinning algorithm. During the thinning process, certain simple points are kept in order to preserve some geometrical properties of the object. Such points are called end points. The result obtained by a thinning algorithm is called a skeleton. A process deleting simple points in parallel may not preserve the topology. For example, a two-width ribbon may vanish because all its points are simple. Therefore, a parallel thinning algorithm must use a “certain deletion strategy” in order to preserve the topology. For example, we may consider a deletion strategy based on subiterations, which consists in dividing a deletion iteration into several subiterations. These subiterations may be based on directions [15, 16,17,2,18] or on subgrids [19,20]. Another example of deletion strategy consists in using an extended neighborhood; such a strategy may lead to fully parallel thinning algorithms [21,3,22]. One of the authors has proposed the notion of P -simple point [23]. A subset composed solely of P -simple points may be deleted in one time while preserving the topology. Furthermore, a P -simple point may be locally characterized, once P is known. In a recent paper [1], we proposed a set P x , locally deﬁned for each point x and from a set P . That has permitted us to propose a new A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 102–113, 2002. c Springer-Verlag Berlin Heidelberg 2002

A New 3D 6-Subiteration Thinning Algorithm Based on P -Simple Points U x

W

E S D

(a)

(b)

p19 p20 p10 p11 p0 p1 p2 p21 p22 p23 p12 p13 p14 p3 p4 p5 p24 p25 p26 p15 p16 p17 p6 p7 p8 p9

N

103

p18

(c)

Fig. 1. (a) The 6-, 18-, and 26-neighbors of x, (b) the six major directions, (c) the used notations

thinning scheme, based on the deletion of P x -simple points, which needs neither a preliminary step of labelling nor the examination of an extended neighborhood, in contrast to the already proposed thinning algorithms based on P -simple points [23]. In this paper, our purpose is to design a new 3D 6-subiteration thinning algorithm based on the deletion of P x -simple points. We apply our general methodology proposed in [1]: from the 6-subiteration thinning algorithm devised by Pal´agyi and Kuba [2], we conceive a ﬁrst thinning algorithm deleting P x -simple points; then we improve it in such a way that it may delete at least all the points removed by the Pal´ agyi and Kuba’s thinning algorithm, while preserving the same end points.

2

Basic Notions

A point x ∈ ZZ 3 is deﬁned by (x1 , x2 , x3 ) with xi ∈ ZZ. We consider the three neighborhoods: N26 (x) = {x ∈ ZZ 3 : M ax[|x1 − x1 |, |x2 − x2 |, |x3 − x3 |] ≤ 1}, N6 (x) = {x ∈ ZZ 3 : |x1 −x1 |+|x2 −x2 |+|x3 −x3 | ≤ 1}, and N18 (x) = {x ∈ ZZ 3 : |x1 −x1 |+|x2 −x2 |+|x3 −x3 | ≤ 2}∩N26 (x). We deﬁne Nn∗ (x) = Nn (x)\{x}. We ∗ call respectively 6-, 18-, 26-neighbors of x the points of N6∗ (x), N18 (x) \ N6∗ (x), ∗ ∗ N26 (x) \ N18 (x); these points are respectively represented in Fig. 1 (a) by black triangles, black squares, and black circles. The 6-neighbors of x determine six major directions (Fig. 1 (b)): Up, Down, North, South, West, East; respectively ∗ denoted by U , D, N , S, W and E. Each point of N26 (x) may characterize one direction amongst the 26 that we can obtain from the 6 major ones; e.g. SW , ∗ U SW . . . Let Dir denote one of these 26 directions. The point in N26 (x) along the direction Dir is called the Dir-neighbor of x and is denoted by Dir(x). In the following, points in N26 (x) are often denoted by pi , with 0 ≤ i ≤ 26 (Fig. 1 (c)); for example, p0 is the U SW -neighbor of p13 , i.e. p0 = U SW (p13 ). Let X ⊆ ZZ 3 . The points belonging to X (resp. X, the complement of X in ZZ 3 ) are called black points (resp. white points). Two points x and y are said to be n-adjacent if y ∈ Nn∗ (x) (n = 6, 18, 26). An n-path is a sequence of points x0 , . . . , xk , with xi n-adjacent to xi−1 and

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x

x

x

x

(a)

(b)

(c)

(d)

Fig. 2. Points belonging to X and X are respectively represented by black discs and white circles. Only the point x in (d) is 26-simple

1 ≤ i ≤ k. If x0 = xk , the path is closed. Let X ⊆ ZZ 3 . Two points x ∈ X and y ∈ X are n-connected if they are linked by an n-path included in X. The equivalence classes relative to this relation are the n-connected components of X. If X is ﬁnite, the inﬁnite connected component of X is the background, the other connected components of X are the cavities. In order to have a correspondence between the topology of X and the one of X, we have to consider two diﬀerents kinds of adjacency for X and for X [6]: if we use an n-adjacency for X, we have to use another n-adjacency for X. In this paper, we only consider (n, n) = (26, 6). The presence of an n-hole in X is detected whenever there is a closed n-path in X that cannot be deformed, in X, into a single point (see [5], for further details). For example, a hollow ball has one cavity and no hole, a solid torus has one hole and no cavity, and a hollow torus has one cavity and two holes. Let X ⊆ ZZ 3 . A point x ∈ X is said to be n-simple if its removal does not “change the topology” of the image, in the sense that there is a one to one correspondence between the components, the holes of X and X and the components, the holes of X \ {x} and X ∪ {x} (see [5], for a precise deﬁnition). The set composed of all n-connected components of X is denoted by Cn (X). The set of all n-connected components of X and n-adjacent to a point x is denoted by Cnx (X). Let #X denote the number of elements which belong to X. The topological numbers relative to X and x are the two numbers [10]: T6 (x, X) = ∗ ∗ #C6x [N18 (x) ∩ X] and T26 (x, X) = #C26 [N26 (x) ∩ X]. These numbers lead to a very concise characterization of 3D simple points [24]: x ∈ X is 26-simple for X if and only if T26 (x, X) = 1 and T6 (x, X) = 1. Some examples are given in Fig. 2. The topological numbers relative to x and X or X are: (T26 (x, X), T6 (x, X)) = (1, 2), (2, 1), (1, 2), (1, 1) for the conﬁgurations (a), (b), (c) and (d), respectively. Only the conﬁguration in Fig. 2 (d) corresponds to a 26-simple point.

3

P -Simple Points

Let us introduce the notions of P -simple point and P -simple set [23]. In the following, we consider a subset X of ZZ 3 , a subset P of X, and a point x of P . The point x is P -simple (for X) if for each subset S of P \ {x}, x is 26simple for X \ S. Let S(P ) denote the set of all P -simple points. A subset D

A New 3D 6-Subiteration Thinning Algorithm Based on P -Simple Points

r

x

x

x

(a)

(b)

(c)

p

105

q

Fig. 3. Points belonging to R, P and X are respectively represented by black discs, black stars and white circles. Only the points x in (a) and (b) are P -simple

of X is P -simple if D ⊆ S(P ). We have the remarkable property that any algorithm removing only P -simple subsets (i.e. subsets composed solely of P simple points) is guaranteed to keep the topology unchanged [23]. We give a local characterization of a P -simple point [25] (see also [26]): Proposition 1. Let R denote the set X \ P . The point x is P -simple iﬀ:  T (x, R) = 1,  26   T6 (x, X) = 1 , ∗ ∀y ∈ N26 (x) ∩ P, ∃ z ∈ R such that z is 26-adjacent to x and to y ,    ∀y ∈ N6∗ (x) ∩ P, ∃ z ∈ X and ∃ t ∈ X such that {x, y, z, t} is a unit square . Some examples are given in Fig. 3: only the points x in (a) and (b) are P -simple. Let us consider the subset X depicted in Fig. 3 (c). The subset S = {p, q, r} is a subset of P \ {x}; and x is non-simple for X \ S. Therefore, by deﬁnition, the point x cannot be a P -simple point; or directly with the Proposition 1, the ﬁrst P -simplicity condition is not veriﬁed because T26 (x, R) = 2. For each x of ZZ 3 , we consider a ﬁnite family T of pairs of subsets of ZZ 3 k (B (x), W k (x)) with 1 ≤ k ≤ l, such that B k (x) ∩ W k (x) = ∅ and x belongs to B k (x); T is said to be a family of templates. In the following, we consider a subset X of ZZ 3 . Let P (T , X) = {x ∈ ZZ 3 : ∃k with 1 ≤ k ≤ l such that B k (x) ⊆ X and W k (x) ⊆ X}. In fact, P (T , X) corresponds to a Hit or Miss transform of X by T [27,28]. A thinning algorithm, based on the deletion of P -simple points, usually considers subsets P which may be characterized by a certain family T of templates. Such an algorithm must decide whether a point x is P (T , X)-simple or not: it must check if the point x belongs to P (T , X), and in order to check the four ∗ conditions of the Proposition 1, it must check if the points y of N26 (x) belong to P (T , X). Such an algorithm may operate according to diﬀerent ways to detect the points belonging to P (T , X) and the points being P (T , X)-simple: it use either a preliminary step of labelling or the examination of an extended neighborhood [23] (see details in [1]). Note that a general strategy has already been proposed to design diﬀerent thinning schemes and algorithms based on P -simple points [29] (see also [30,31]).

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Let us introduce a subset P x , locally deﬁned for each point x of ZZ 3 and from a set P (described as previously by a family T of templates) [1,32]. From this subset, we will derive the notion of a P x -simple point. For each x of ZZ 3 , we deﬁne a new subset P x (T , X) of ZZ 3 , determined by P x (T , X) = {y ∈ N26 (x) : ∃k with 1 ≤ k ≤ l such that [B k (y) ∩ N26 (x)] ⊆ X and [W k (y) ∩ N26 (x)] ⊆ X}. In fact, P x (T , X) is constituted by the points y of N26 (x) ∩ X which “may belong” to P (T , X), by the only inspection of membership to X or to X of points belonging to [B k (y) ∪ W k (y)] ∩ N26 (x). We have P x (T , X) ⊇ [P (T , X) ∩ N26 (x)]. We have proven that a P x (T , X)-simple point is P (T , X)-simple [32]. This implies that an algorithm deleting in parallel P x (T , X)-simple points is guaranteed to preserve the topology, because it deletes P (T , X)-simple subsets. In addition, since P x (T , X) is completely known in N26 (x) for each point x, that permits us to propose a new thinning scheme, based on the deletion of P x (T , X)simple points x, which needs neither a preliminary step of labelling nor the examination of an extended neighborhood, in contrast to the already proposed thinning algorithms based on P (T , X)-simple points (see [1] for further details). In Sect. 5, we will propose a thinning algorithm deleting P x (T , X)-simple points. Notations: In the following, we write P (resp. P x ) instead of P (T , X) (resp. x P (T , X)) and “x is a P -simple point (resp. P x -simple point) means “x is a P (T , X)-simple point (resp. P x (T , X)-simple point)”.

4

Description of the Used Thinning Algorithms

A thinning scheme consists in the repetition until stability of deletion iterations. In the case of 6-subiteration thinning algorithms, an iteration is divided into 6 subiterations, each of them successively corresponding to one of the 6 following directions: Up, Down, North, South, East and West (see Fig. 1 (b)). Let α denotes such a direction. The stability is obtained when there is no more deletion during 6 successive subiterations. Such a thinning scheme can be described by X i = X i−1 \ DEL(X i−1 , α) for the ith deletion subiteration (i > 0), with X 0 = X, and DEL(Y, α) being the set of points to be deleted from Y , according to the direction α corresponding to the ith subiteration. The stability is obtained when X k = X k+6 . Pal´ agyi and Kuba have proposed a 6-subiteration thinning algorithm [2], denoted by pk in the following. A set of 3 × 3 × 3 matching templates is given for each direction. For a given direction α, a point is deletable by pk if at least one template (or theirs rotations around the axis along the direction α) in the set of templates matches it. The set of templates used by pk along the direction α, is denoted by Tα and is represented in Fig. 5 for the direction α = U ; see notations in Fig. 4. The templates for the other directions can be obtained by appropriate rotations and reﬂections of these templates. Sometimes, we will write that “Tα deletes a point” to mean pk deletes this point during a subiteration along the direction α. We recall the deﬁnition of an end point, adopted in [2], that we will also use in our proposed algorithm. A black point x is an end point if the set ∗ N26 (x) contains exactly one black point. We note that end points are prevented

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A position marked by a matches a black point. A position marked by a matches a white point. At least one position marked by a belongs to X. Every position non marked matches either a black or a white point. A position marked by a matches a black point belonging to a considered set P . Fig. 4. Notations used in the following of the paper

x

α=U

M1

x

M2

x

M3

x

x

x

M4

M5

M6

Fig. 5. The set TU of thinning templates for the direction U , up to rotations around the vertical axis (see notations in Fig. 4)

to be deleted by the templates of Tα . According to the previous general thinning scheme (described in the beginning of this section), DEL(Y, α) is the set of points of Y such that at least one of the templates of Tα matches them, for the direction α corresponding to the deletion subiteration. A 6-subiteration thinning algorithm removing P -simple points, has already been proposed [23]. Now, we give a general scheme for 6-subiteration thinning algorithms deleting P x -simple points. It can be described by the scheme of the beginning of this section with DEL(Y, α) = S(P x ); S(P x ) being the set of P x simple points for Y which are not end points and according to the direction α corresponding to the deletion subiteration. From this scheme, we will propose our algorithm by deﬁning an appropriate P (Sect. 5), in the sense that we investigate P such that our algorithm deletes at least all the points removed by pk. In the following, we write lb to indicate our ﬁnal algorithm which deletes P x -simple points, while preserving end points. See [1] for details concerning the eﬃcient implementation of such algorithms, with the use of Binary Decision Diagrams [33,34,35]; in fact, pk and lb have the same computational complexity.

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p3 p6

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Fig. 6. This conﬁguration (a) is not P1x -simple (b), and is P2x -simple (c)

5

Our Thinning Algorithm (LB)

In this section, we give the entire reasoning which leads us to propose two successive conditions of membership to a set P . The used methodology consists in proposing successive “reﬁnements” of P , until to obtain a set P such that at least all points deleted by pk are P -simple. This is achieved with our second proposal of a set P . We note that the ﬁrst proposal, detailed in Sect. 5.1, is directly deduced from pk. We ﬁrst deal with the direction U until a general comparison of our results. In the following, when we write “a point belongs to P x ” then x is the point p13 for the considered conﬁguration (see Fig. 1 (c)). We write “a conﬁguration is P x -simple” to mean that the central point x(= p13 ) of this conﬁguration is P x -simple. Let y be a point of a conﬁguration, y belongs to {p0 , . . . , p26 }, see Fig. 1 (c); we write “a point y veriﬁes a template T ” to mean that the template T matches the conﬁguration whose central point is y. 5.1

First Membership Condition

We observe that any point of X deleted by TU is such that its U -neighbor belongs to X (see templates in Fig. 5). Thus, we propose to consider P1 = {x ∈ X : the U -neighbor of x belongs to X}. Among all 226 possible conﬁgurations, we obtain 4 423 259 ones corresponding to P1x -simple and non end points, for the direction U. Let us consider the conﬁguration in Fig. 6 (a). The three points p3 , p7 and p13 belong to P1x (Fig. 6 (b)) because they belong to X, and each U -neighbor of these points belongs to X. The ﬁrst and the third P1x -simplicity conditions are not veriﬁed for the central point p13 . Thus, the point p13 is not P1x -simple. Nevertheless, it is matched by a rotation around the vertical axis of M5 of TU . Therefore, it should be deleted by our wanted algorithm. Let us examine the behavior of the other points of this conﬁguration with the templates TU (see Fig. 6 (a)). The point p3 may verify a rotation around the vertical axis of M5 or of M6 . The point p7 cannot be deleted, because p6 (= W (p7 )) belongs to X and p3 (= U (p6 )) belongs to X, and the templates are such

A New 3D 6-Subiteration Thinning Algorithm Based on P -Simple Points

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Fig. 7. A point x belongs to P2 iﬀ it veriﬁes at least one of these templates, according to the direction α

that for any point x deleted by TU and for any y belonging to N6∗ (x) ∩ X, y being neither U (x) nor D(x), then the point U (y) must belong to X. With these remarks, we can propose a new set P2 . 5.2

Second Membership Condition

We ﬁrst introduce some notations. We recall that α denote one of the six deletion directions. Let α denote the opposite direction. Let Nα6 (x) denote the four 6neighbors of x which belong to the 3 × 3 window perpendicular to the direction α and containing x (in fact, Nα6 (x) = N6∗ (x) \ {α(x), α(x)}). We propose to consider P2 = {x ∈ X : the α-neighbor of x belongs to X and for any point y belonging to Nα6 (x), if y belongs to X then α(y) must belong to X}, according to the direction α. With notations used in Sect. 3, the set P2 can be described by the family composed of 16 pairs of subsets of ZZ 3 (B k (x), W k (x)) with 1 ≤ k ≤ 16, depicted in Fig. 7 for the direction α = U ; in fact, there are 6 main templates, up to rotations around the axis (α(x), α(x)). Let us consider the non P1x -simple conﬁguration in Fig. 6 (b) (see notations in Fig. 6 (c)). The point p13 belongs to P2x , as it veriﬁes the template in Fig. 7 (a). The point p3 belongs to P2x , as p3 may verify the templates in Fig. 7 (a), (c), (e), or (g). The point p7 does not belong to P2x because there exists a

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p2 p3

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(c)

Fig. 8. (a) This conﬁguration cannot be deleted by pk whatever the deletion direction, and is P2x -simple (b), in (c) (obtained from (a)) no point is deleted by pk nevertheless x is deleted by lb

point y(= p6 (= W (p7 ))) in NU6 (p7 ) ∩ N26 (x) which belongs to X, and such that p3 (= U (y)) belongs to X; or more directly because p7 veriﬁes no template in Fig. 7. So, this non P1x -simple conﬁguration (Fig. 6 (b)) is now P2x -simple (Fig. 6 (c)). We obtain 6 129 527 conﬁgurations corresponding to P2x -simple and non end points, for the direction U . The 2 124 283 conﬁgurations deleted by TU , are also P2x -simple. The fact that the conﬁgurations deletable by pk are P2x -simple (for each direction and therefore for the whole algorithm) guarantees that the topology is preserved by pk (as pk deletes subsets of P2x -simple points, see Sect. 3). For a better comparison between pk and lb, we generate the conﬁgurations deleted by these algorithms for each direction: pk deletes 9 916 926 conﬁgurations, i.e. there exists at least one deletion direction such that a given conﬁguration among these ones is deleted for this direction by pk; lb deletes 23 721 982 conﬁgurations (139.2% “better”). We recall that there are 25 985 118 simple and non end points amongst the 67 108 864(= 226 ) possible 3 × 3 × 3 conﬁgurations. The conﬁguration depicted in Fig. 8 (a) cannot be deleted by pk, whatever the deletion direction. This conﬁguration is P2x -simple (Fig. 8 (b)), with α = U . Indeed, the point p2 belongs to P2x as p2 may verify the templates in Fig. 7 (a), (b), (c) or (d); p3 belongs to P2x as p3 may verify the templates in Fig. 7 (a), (c), (e) or (g); p13 belongs to P2x as it veriﬁes the template in Fig. 7 (a); p5 does not belong to P2x as p2 (= U (p5 )) belongs to X; and p7 does not belong to P2x as there exists a point y(= p6 (= W (p7 ))) in NU6 (p7 ) ∩ N26 (x) which belongs to X and such that U (y) (= p3 ) belongs to X (or more directly, as p7 veriﬁes no template in Fig. 7). The ﬁgure 8 (c) shows an image built from the conﬁguration in Fig. 8 (a) such that each point is either a non simple point (except x) or an end point, and no point can be deleted by pk, nevertheless the point x is deleted by lb.

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Initial object

pk: 16 − 2 256

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pk: 142 − 170 001

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Fig. 9. Skeletons of a synthetic object and of a vertebra, with pk and lb. Under each ﬁgure are given the number of the last subiteration of deletion and the number of deleted points

5.3

Results

The skeletons of some images, obtained respectively by pk and lb are shown in Fig. 9. We observe that the number of deletion subiterations required by lb is less than or equal to the one of pk. The number of points deleted by lb is less

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than or equal to the one of pk. We recall that it is possible that lb needs more subiterations to obtain a skeleton than pk needs (see Fig. 8 (c)).

6

Conclusion

We have conceived a new 6-subiteration thinning algorithm, based on the deletion of P x -simple points, by applying a recent methodology that we proposed in [1]. As it deletes solely P x -simple points, this algorithm is guaranteed to preserve the topology. Furthermore, we have proposed some various sets P such that our ﬁnal algorithm deletes at least all the points removed by pk, while preserving the same end points. This also implies that pk is guaranteed to preserve the topology. In addition, our ﬁnal algorithm also deletes points removed by the Gong and Bertrand’s algorithm [16] (in the variant proposed by Rolland et al. [17]) while preserving the same end points. In another study [1,32], we succeeded in proposing a new 12-subiteration thinning algorithm for 3D binary images, which produces curve or surface skeletons, and such that it deletes at least the points removed by one other 12-subiteration thinning algorithm [18]. A future work will propose new fully parallel thinning algorithms for 2D and 3D binary images.

References 1. C. Lohou and G. Bertrand. A new 3D 12-subiteration thinning algorithm based on P -simple points. In 8th IWCIA 2001, volume 46 of ENTCS, pages 39–58, 2001. 2. K. Pal´ agyi and A. Kuba. A 3D 6-subiteration thinning algorithm for extracting medial lines. Pattern Recognition Letters, 19:613–627, 1998. 3. C.M. Ma and M. Sonka. A fully parallel 3D thinning algorithm and its applications. Computer Vision and Image Understanding, 64(3):420–433, 1996. 4. D.G. Morgenthaler. Three-dimensional simple points: Serial erosion, parallel thinning, and skeletonization. Technical Report TR-1009, Computer Vision Laboratory, University of Maryland, 1981. 5. T.Y. Kong. A digital fundamental group. Computer and Graphics, 13(2):159–166, 1989. 6. T.Y. Kong and A. Rosenfeld. Digital topology: introduction and survey. Computer Vision, Graphics and Image Processing, 48:357–393, 1989. 7. P.K. Saha, B. Chanda, and D.D. Majumder. Principles and algorithms for 2D and 3D shrinking. Technical Report TR/KBCS/2/91, N.C.K.B.C.S. Library, Indian Statistical Institute, Calcutta, India, 1991. 8. R.W. Hall. Connectivity preserving parallel operators in 2D and 3D images. In Vision Geometry, volume 1832 of SPIE, pages 172–183, 1992. 9. T.Y. Kong. On the problem of determining whether a parallel reduction operator for n-dimensional binary images always preserves topology. In Vision Geometry II, volume 2060 of SPIE, pages 69–77, 1993. 10. G. Bertrand. Simple points, topological numbers and geodesic numbers in cubic grids. Pattern Recognition Letters, 15:1003–1011, 1994. 11. C.M. Ma. On topology preservation in 3D thinning. Computer Vision, Graphics, and Image Processing: Image Understanding, 59(3):328–339, 1994.

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12. T.Y. Kong. On topology preservation in 2-D and 3-D thinning. International Journal of Pattern Recognition and Artiﬁcial Intelligence, 9(5):813–844, 1995. 13. T.Y. Kong. Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In 7th DGCI, volume 1347 of LNCS, pages 3–18, 1997. 14. S. Fourey and R. Malgouyres. A concise characterization of 3D simple points. In 9th DGCI, volume 1953 of LNCS, pages 27–36, 2000. 15. Y.F. Tsao and K.S. Fu. A parallel thinning algorithm for 3D pictures. Computer Graphics and Image Processing, 17:315–331, 1981. 16. W. Gong and G. Bertrand. A simple parallel 3D thinning algorithm. In International Conference on Pattern Recognition, pages 188–190, 1990. 17. F. Rolland, J.-M. Chassery, and A. Montanvert. 3D medial surfaces and 3D skeletons. In Visual Form 1991, pages 443–450, 1991. 18. K. Pal´ agyi and A. Kuba. A parallel 3D 12-subiteration thinning algorithm. Graphical Models and Image Processing, 61:199–221, 1999. 19. G. Bertrand and Z. Aktouf. A three-dimensional thinning algorithm using subﬁelds. In Vision Geometry III, volume 2356 of SPIE, pages 113–124, 1994. 20. K. Pal´ agyi and A. Kuba. A hybrid thinning algorithm for 3D medical images. Journal of Computing and Information Technology, CIT 6, pages 149–164, 1998. 21. C.M. Ma. A 3D fully parallel thinning algorithm for generating medial faces. Pattern Recognition Letters, 16:83–87, 1995. 22. A. Manzanera, T. M. Bernard, F. Prˆeteux, and B. Longuet. A uniﬁed mathematical framework for a compact and fully parallel n-D skeletonization procedure. In Vision Geometry VIII, volume 3811 of SPIE, pages 57–68, 1999. 23. G. Bertrand. On P -simple points. Compte Rendu de l’Acad´ emie des Sciences de Paris, t. 321(S´erie 1):1077–1084, 1995. 24. G. Malandain and G. Bertrand. Fast characterization of 3D simple points. In IEEE International Conference on Pattern Recognition, pages 232–235, 1992. 25. G. Bertrand. Suﬃcient conditions for 3D parallel thinning algorithms. In Vision Geometry IV, volume 2573 of SPIE, pages 52–60, 1995. 26. G. Bertrand and R. Malgouyres. Some topological properties of discrete surfaces. In 6th DGCI, volume 1176 of LNCS, pages 325–336, 1996. 27. J. Serra. Image analysis and mathematical morphology. Academic Press, 1982. 28. P.P. Jonker. Morphological operations on 3D and 4D images: From shape primitive detection to skeletonization. In 9th DGCI, volume 1953 of LNCS, pages 371–391, 2000. 29. G. Bertrand. P -simple points: A solution for parallel thinning. In 5th DGCI, pages 233–242, 1995. 30. R. Malgouyres and S. Fourey. Strong surfaces, surface skeletons and image superimposition. In Vision Geometry VII, volume 3454 of SPIE, pages 16–27, 1998. 31. J. Burguet and R. Malgouyres. Strong thinning and polyhedrization of the surface of a voxel object. In 9th DGCI, volume 1953 of LNCS, pages 222–234, 2000. 32. C. Lohou and G. Bertrand. A 3D 12-subiteration thinning algorithm based on P -simple points. Submitted for publication. 33. R.E. Bryant. Graph-based algorithms for boolean function manipulation. IEEE Transactions on Computer, Vol. C-35(8):677–691, 1986. 34. K.S. Brace, R.L. Rudell, and R.E. Bryant. Eﬃcient implementation of a bdd package. In 27th IEEE Design Automation Conference, pages 40–45, 1990. 35. L. Robert and G. Malandain. Fast binary image processing using binary decision diagrams. Computer Vision and Image Understanding, 72(1):1–9, 1998.

Monotonic Tree Yuqing Song and Aidong Zhang Department of Computer Science & Engineering State University of New York at Buﬀalo Buﬀalo, NY 14260, USA

Abstract. Contour trees have been used in geographic information systems (GIS) and medical imaging to display scalar data. Contours are only deﬁned for continuous functions. For an image represented by discrete data, a continuous function is ﬁrst deﬁned as an interpolation of the data. Then the contour tree is deﬁned on this continuous function. In this paper, we introduce a new concept termed monotonic line, which is directly deﬁned on discrete data. All monotonic lines in an image form a tree, called monotonic tree. As compared with contour trees, monotonic trees avoid the step of interpolation, thus can be computed more eﬃciently. Monotonic tree can also be used as a hierarchical representation of image structures in computer imagery.

1

Introduction

The concepts of contour trees have been developed by Morse [1], Roubal and Peucker [2], and recently by van Kreveld et al. [3]. In geographic information systems (GIS), contour trees are used to display scalar data deﬁned over the plane, or the three-dimensional space. For example, the elevation in the landscape can be modeled by scalar data over the plane, where a contour (also called an isoline) is a line where the elevation function assumes the same value. Contour trees are also used in medical imaging to show the scanned data. Contours are only deﬁned for continuous functions. For an image represented by discrete data, a continuous function is ﬁrst deﬁned as an interpolation of the data. Then the contour tree is deﬁned on this continuous function. In this paper, we introduce a new concept termed monotonic line, which is directly deﬁned on discrete data. We observe that for any 2D Morse function, a curve is a normal contour with value v iﬀ it’s a boundary of the set {x ∈ IR2 |f (x) > v}. This is not true for non-Morse functions. However, the equivalent condition is more general, and can be used to deﬁne contours for discontinuous or discrete functions. Speciﬁcally, an outward-falling/climbing monotonic line of an gray image is a boundary where the image assumes higher/lower values in the pixels adjacent to the boundary from inside than those from outside (see Figure 1(a)). The two kinds of monotonic lines correspond to positive and negative contour lines in [1], respectively. To make the boundary of the image domain a monotonic line, we extend the A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 114–123, 2002. c Springer-Verlag Berlin Heidelberg 2002

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input image to the whole digital plane such that the extended function assumes −∞ out of the domain. It can be proved that monotonic lines don’t cross each other, i.e., if l1 = ∂X, l2 = ∂Y are two monotonic lines, where X, Y are two simply connected regions, then X ⊆ Y , Y ⊆ X or X Y = Ø. Based on this property, we can deﬁne a parent-child relation: monotonic line l1 is the parent of monotonic line l2 , if l2 is directly enclosed by l1 . Under this parent-child relation, all monotonic lines in an image form a rooted tree, called monotonic tree. See Figure 1(b)(c). A monotonic tree can be reduced. A maximal sequence of uniquely enclosing monotonic lines is called a monotonic slope. All monotonic slopes in an image form the topological monotonic tree (TMT). See Figure 1(d). Algorithms for computing traditional contour trees can be easily modiﬁed to compute monotonic trees or topological monotonic trees. Because the monotonic line is directly deﬁned on pixels, the interpolation step is avoided. Thus the monotonic trees and the topological monotonic trees can be computed more eﬃciently.

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Fig. 1. (a) An outward-falling monotonic line (the solid line in the ﬁgure), (b) a set of monotonic lines, (c) the monotonic tree, (d) the topological monotonic tree (TMT).

Monotonic tree can be used as a hierarchical representation of image structures in computer imagery. As compared with other models such as wavelet, the monotonic tree model has following advantages. (1) The monotonic tree retrieves and represents the structures of an image at all scales. In addition, these structures are organized hierarchically as a tree,

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which gives us a better way to analyze the relationship between diﬀerent levels. (2) The monotonic tree retrieves the structures of an image directly and maintains their original shapes. An example is shown in Figure 2. In this example, the tree region and water wave region are characterized by the shapes and permutation of the TMT elements in these two regions, which makes it possible to recognize the trees and water waves by classifying and clustering the TMT elements. Based on our monotonic tree model, we made an online demo for scenery analysis, which is available at “http://monet.cse.buﬀalo.edu:8888”.

(b)

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(c) Fig. 2. (a) Original image, (b) the elements of the TMT at a smaller scale, and (c) the elements of the TMT at a larger scale. In (b) and (c), the TMT elements are shown by black and white regions in a gray background.

2

Preliminary

In our theoretical discussion, we choose hexagonal grid for the digital plane. We use notations in [6] with modiﬁcations. The digital plane is a pair (Vh , Πh ), where Vh is the set of pixels: √ Vh = {h1 (1, 0) + h2 (−0.5, 0.75)|h1 , h2 ∈ ZZ},

(1)

and Πh is the edge-adjacency shown in Figure 3(a). Πh is a symmetric binary relation on Vh such that Vh is Πh -connected. Formally, Πh = {(p, q)|p, q ∈ Vh , and p − q = 1}.

(2)

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Each pixel P has 6 edges, named as ei (P ) for i = 0, 1, ..., 5. See Figure 3(b). Each ei (P ) is an element of Πh . We deﬁne function nextP on {ei (P )}5i=0 by nextP (ei (P )) = e(i+1)%6 (P ).

(3)

Function nextP deﬁnes the counter-clockwise direction in the border of P . e 3 (P)

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Fig. 3. (a) Hexagonal grid, (b) six edges of pixel P , and (c) the connected boundary of a digital region.

For any subset X of Vh , its border is deﬁned as ∂X = {(p, q) ∈ Πh |p ∈ X, and q ∈ X}.

(4)

We deﬁne function nextX on ∂X such that for e = (p, q) ∈ ∂X with p ∈ X and q ∈ X, nextp (e) if nextp (e) ∈ ∂X; nextX (e) = (5) next−1 q (e) otherwise. Function nextX deﬁnes the counter-clockwise direction in the border of X. For example, in Figure 3(c), nextX (α) = β and nextX (β) = γ. The following deﬁnition deﬁnes connected regions. Deﬁnition 1. A region X is called a connected region if (X, Πh |X ) is a connected graph, i.e., for any p, q ∈ X, there exists a Πh -connected path in X connecting p and q, where a Πh -connected path is a sequence of pixels {pi }ni=1 such that (pi , pi+1 ) ∈ Πh for i = 1, 2, ..., n − 1. For a connected region with no holes, its border is a connected boundary, which is an boundless list of pixel edges. See Figure 3(c). Formally, we give two deﬁnitions.

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Deﬁnition 2. A boundless list is a pair (X, next) such that X is a set and (1) next is a bijective function from X to X; and (2) ∀x, y ∈ X, there exists an integer n ≥ 0 such that either x = nextn (y), or y = nextn (x). A circular list is a boundless list (X, next) such that X is a ﬁnite set. It’s easy to see that for a circular list (X, next) and any x, y ∈ X, there exists an integer n ≥ 0 such that x = nextn (y). Deﬁnition 3. For a region X ⊆ Vh , a connected boundary of X is a subset S of its border ∂X such that nextX (S) = S and (S, nextX |S ) is a boundless list. About the boundaries of regions, we have following lemma. Lemma 1. For any X ⊆ Vh , (1) ∂X = ∂(Vh − X); (2) nextX = (nextVh −X )−1 ; (3) the border of X can be decomposed into a set of connected boundaries; and (4) if X is bounded (i.e., it’s ﬁnite), then its connected boundaries are circular lists. This lemma is obvious. Next we give the deﬁnition for simple connection. Deﬁnition 4. A region X ⊆ Vh is called simply connected if both X and Vh − X are connected. The simple connection we deﬁned here is a little bit diﬀerent from the traditional deﬁnition on Euclidean plane. However, if we add an inﬁnite pixel to the hexagonal grid and make it a digital sphere, then our deﬁnition ﬁts the deﬁnition of simple connection on Euclidean sphere. About simple connection, we have following lemma. Lemma 2. For a bounded region X ⊂ Vh , X is simply connected iﬀ (∂X, nextX ) is a circular list. That is to say, for a bounded region X, (∂X, nextX ) is a circular list iﬀ both X and Vh − X are connected. This lemma is an equivalent of Jordan’s curve theorem on the hexagonal grid. Due to the limited space, in this paper, we give our lemmas and theorems with the proof omitted. Next, we model gray images on the hexagonal grid. Deﬁnition 5. A gray image is a pair (f, Ω) such that Ω ⊂ Vh is a bounded and simply connected region, and f is a real valued function deﬁned on Ω. For any gray image (f, Ω), we extend the function f to the whole plane by f (p) if p ∈ Ω; fE (p) = (6) −∞ otherwise. fE is called the extended function of f . In fact, instead of choosing −∞, we can choose any value which is less than all values assumed by f , or greater than all values assumed by f .

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Monotonic Line

In this and following sections, let I = (f, Ω) be a ﬁxed gray image, and fE be the extended function of f . For a region X ⊆ Vh , we denote the immediate interior [6] of ∂X as IntBorderP ixelSet(X), and the immediate exterior as ExtBorderP ixelSet(X), i.e., IntBorderP ixelSet(X) = {x ∈ X|(x, y) ∈ Πh f or some y ∈ X}; (7) ExtBorderP ixelSet(X) = {y ∈ Vh − X|(x, y) ∈ Πh f or some x ∈ X}. (8) Now we can deﬁne monotonic line. Deﬁnition 6. A monotonic line of I is a boundary ∂X such that X ⊆ Ω is simply connected and not empty, and there exists some v ∈ IR with the property that either of the following is true: (1) ∀x ∈ IntBorderP ixelSet(X), fE (x) > v, and ∀y ∈ ExtBorderP ixelSet(X), fE (y) < v; (2) ∀x ∈ IntBorderP ixelSet(X), fE (x) < v, and ∀y ∈ ExtBorderP ixelSet(X), fE (y) > v. If (1) is true, ∂X is called outward falling; if (2) is true, ∂X is called outward climbing. We denote the set of all monotonic lines of I as M onotonicLineSet(I). We can prove that monotonic lines don’t cross each other, i.e., for any ∂X, ∂Y in M onotonicLineSet(I), one of the following is true: X ⊆ Y , Y ⊆ X or X Y = Ø. Theorem 1. ∀∂X, ∂Y ∈ M onotonicLineSet(I), one of following is true: X ⊆ Y , Y ⊆ X or X Y = Ø. The basic idea to prove this theorem is simple. Suppose we have two monotonic lines ∂X and ∂Y crossing each other. Let a, b, c, d be four pixels around a crossing point. See Figure 4. By the deﬁnition of monotonic line, we discuss on four cases: (1) both ∂X and ∂Y are outward falling; (2) ∂X is outward falling, ∂Y is outward climbing; (3) ∂X is outward climbing, ∂Y is outward falling; and (4) Both ∂X and ∂Y are outward climbing. For each case, we can get a contradiction. For example, in case (1), there exists v1 such that fE (a) > v1 , fE (c) > v1 , and fE (b) < v1 , fE (d) < v1 ; and there exists v2 such that fE (c) > v2 , fE (d) > v2 , and fE (a) < v2 , fE (b) < v2 . Then we get both fE (a) > v1 > fE (d) and fE (d) > v2 > fE (a), which is a contradiction.

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X

c

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Fig. 4. ∂X crosses ∂Y .

4

Monotonic Tree and Topological Monotonic Tree

We ﬁrst deﬁne some relations on M onotonicLineSet(I). Deﬁnition 7. For any monotonic lines ∂X, ∂Y ∈ M onotonicLineSet(I), – ∂X encloses ∂Y , denoted as Enclose(∂X, ∂Y ), if X ⊃ Y ; – ∂X directly encloses ∂Y , denoted as DirectEnclose(∂X, ∂Y ), if Enclose(∂X, ∂Y ) and there is no ∂Z ∈ M onotonicLineSet(I) such that X ⊃Z ⊃Y; – ∂X uniquely directly encloses ∂Y , denoted as U niqueDirectEnclose(∂X, ∂Y ), if DirectEnclose(∂X, ∂Y ) and ∀∂Z ∈ M onotonicLineSet(I), DirectEnclose(∂X, ∂Z) ⇒ ∂Y = ∂Z. The relation DirectEnclose is a parent-child relation on the set of monotonic lines in gray image I. Based on Theorem 1, we can easily prove that: Theorem 2. (M onotonicLineSet(I), DirectEnclose) is a rooted tree, and ∂Ω is its root. The tree (M onotonicLineSet(I), DirectEnclose) is called the monotonic tree of image I, and denoted as M onotonicT ree(I). Next we can deﬁne monotonic slope and topological monotonic tree. Deﬁnition 8. A monotonic slope s is a maximal sequence of monotonic lines s = {li }ni=1 with n ≥ 1 such that (1) ∀i = 1, 2, ..., n − 1, U niqueDirectEnclose(li , li+1 ); and (2) either all li are outward-falling, or all li are outward-climbing.

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s is called outward-falling/outward-climbing if all li are outwardfalling/outward-climbing. The ﬁrst monotonic line l1 is called the enclosing line of the slope s. The set of all monotonic slopes is denoted as M onotonicSlopeSet(I). We can also deﬁne some relations on M onotonicSlopeSet(I). Deﬁnition 9. For any sa , sb ∈ M onotonicSlopeSet(I), we deﬁne (1) Enclose(sa , sb ) if ∃la ∈ sa , ∃lb ∈ sb , Enclose(la , lb ); (2) DirectEnclose(sa , sb ) if ∃la ∈ sa , ∃lb ∈ sb , DirectEnclose(la , lb ). The relation DirectEnclose is a parent-child relation on the set of all monotonic slopes. Theorem 3. (M onotonicSlopeSet(I), DirectEnclose) is a rooted tree, and the monotonic slope which contains ∂Ω is the root. The tree (M onotonicSlopeSet(I), DirectEnclose) is called the topological monotonic tree of I, and denoted as T opologicalM onotonicT ree(I).

5

Properties of Monotonic Tree

In this section, we introduce some theorems about monotonic tree. Theorem 4. Inside Intersecting Theorem For any ∂X, ∂Y ∈ M onotonicLineSet(I), if IntBorderP ixelSet(X) IntBorderP ixelSet(Y )

= Ø, then ∂X is outward falling ⇔ ∂Y is outward falling. This theorem is equivalent to the statement that for a sequence of monotonic lines which intersect from inside, either they are all outward falling, or they are all outward climbing. See Figure 5(a).

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Fig. 5. (a) An inside-intersecting sequence, and (b) two inside-intersecting sequences which intersect from outside.

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Theorem 5. Outside Intersecting Theorem For any ∂X, ∂Y ∈ M onotonicLineSet(I), if IntBorderP ixelSet(X) ExtBorderP ixelSet(Y )

= Ø, then ∂X is outward falling ⇔ ∂Y is outward climbing. This theorem is equivalent to the statement that for two inside-intersecting sequences, if they intersect from outside, then the monotonic lines in one sequence are all outward falling, and the monotonic lines in the other sequence are all outward climbing. See Figure 5(b). Theorem 6. Separation Theorem Let x, y be two pixels in Ω. (1) If there is a Πh -connected path P = {w1 = x, w2 , ..., wn = y} in Ω such that f is constant along this path, i.e., f (w1 ) = f (w2 ) = ... = f (wn ), then ∀ ∂X ∈ M onotonicLineSet(I), x ∈ X ⇔ y ∈ X. (2) If f (x)

= f (y), then there is some ∂X ∈ M onotonicLineSet(I) such that x ∈ X ⇔ y ∈ X. This theorem states that for any two pixels in the image domain, (1) if they are connected by a path where the function assumes constant value, then no monotonic line separates them; and (2) if the function assumes diﬀerent values at the two pixels, then there is some monotonic line separating them. Theorem 7. Private Region Theorem For any ∂X ∈ M onotonicLineSet(I), let {∂Yi }ni=1 (n ≥ 0) be the set of children of ∂X in themonotonic tree. Then n (1) (X − i=1 Yi ) IntBorderP ixelSet(X) is not empty; and n (2) ∀x, y ∈ (X − i=1 Yi , ) f (x) = f (y). This theorem states that each monotonic line has a nonempty private region and that the function is constant over this region. Based on this theorem, we give following deﬁnition. Deﬁnition 10. For any ∂X ∈ M onotonicLineSet(I), let {∂Yi }ni=1 (n ≥ 0) be the set of children of ∂X in the monotonic tree. We deﬁne the private region of ∂X to be: n P RegionI (∂X) = X − i=1 Yi . We deﬁne the assumed value of ∂X to be the constant value assumed by f over P RegionI (∂X). The assumed value of ∂X is denoted as V alueI (∂X). Theorem 8. Value Jumping Theorem For any ∂P, ∂C ∈ M onotonicLineSet(I), if ∂P is the parent of ∂C, then (1) V alueI (∂P )

= V alueI (∂C); and (2) ∂C is outward falling ⇔ V alueI (∂P ) < V alueI (∂C). The property (1) in this theorem says that there is a value jump (up or down) from a child to its parent. Value jumping is a natural property for contour trees, but it’s not straight forward for monotonic trees.

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Conclusion and Discussion

Contour trees are only deﬁned on continuous functions. When applying the contour tree model to computer imagery, we have to make an interpolation of the discrete data, which make the computation not eﬃcient. We solved this problem by introducing our monotonic tree model. One main diﬃculty of computer imagery comes from the discrete and noise nature of images. While we have plenty of powerful theories to analyze the structures of smooth functions, we need more for discrete functions. The monotonic tree can be used as a theoretical tool for discrete functions. The capacity of the monotonic tree model can be extended. The topological structure of the monotonic lines in an image is captured by the topological monotonic tree. We may further deﬁne diﬀerential slope and diﬀerential monotonic tree to capture the diﬀerential information. A diﬀerential slope may be deﬁned as a sequence of monotonic lines where the gradient is smooth.

References 1. S. Morse. Concepts of use in computer map processing. Communications of the ACM, 12(3):147-152, March 1969. 2. J. Roubal and T.K. Peucker. Automated contour labeling and the contour tree. In Proc. AUTO-CARTO 7, pages 472-481, 1985. 3. M. van Kreveld, R. van Oostrum, C. Bajaj, V. Pascucci, and D. Schikore. Contour trees and small seed sets for iso-surface traversal. In Proc. 13th Ann. Sympos. Comput. Geom., pages 212-220, 1997. 4. Mark de Berg and Marc J. van Kreveld. Trekking in the alps without freezing or getting tired. In European Symposium on Algorithms, pages 121-132, 1993. 5. M. van Kreveld. Eﬃcient methods for isoline extraction from a tin. International Journal of GIS, 10:523-540, 1996. 6. Gabor T. Herman. Geometry of Digital Spaces. Birkhauser Boston, 1998.

Displaying Image Neighborhood Hypergraphs Line-Graphs S. Chastel, P. Colantoni, and A. Bretto ´ LIGIV – Universit´e de Saint-Etienne 3, rue Javelin Pagnon BP 505 ´ F-42007 SAINT-ETIENNE Cedex 01 [email protected]

Abstract. Graph-based structures are commonplace in image processing. Our contribution in this article consists in giving hints representing a new modeling of digital images: image neighborhood hypergraphs. We give some results on the hyperedges coloring of them. We also describe techniques we used to display image neighborhood hypergraphs line-graphs. These techniques form the basis of a tool that allows the exploration of these structures. In addition, this tool can be used to visualize, explore and describe features of image regions of interest such as object edges or noise.

For nearly ﬁfty years, image processing, image analysis, and computer vision are ﬁelds that have been intensively studied. Many various mathematical theories and techniques have been widely involved in these scientiﬁc ﬁelds: linear algebra, continuous optimization, statistics, combinatorics, to name a few. One of the main problematics in image analysis is the foundation of formal modelings of the digital image. In any scientiﬁc ﬁeld, an eﬃcient modeling should establish structural relationships between the diﬀerent objects of that domain. A digital image may be considered as a union of elementary spatial and colorimetric units. Therefore, working on the organization of these components may be seen as a combinatorial problem, since combinatorics is the science that studies organization of discrete objects in mathematically formalized modelings. This is maybe why graph theory and associated techniques arose in image analysis and had been widely used by many researchers. An attractive approach is to develop a modeling technique based on a generalized graph theory: hypergraph theory. Hypergraph theory arose from the seminal work of C. Berge in the early seventies [2]. In graph theory, one studies the binary relationships between elements. In hypergraph theory, the relationships between basics elements are generalized: elements in relation belong to a same set if they share a common property. We recently proposed a modeling based on hypergraph theory [6]. Our ﬁrst intentions in dealing with this modeling was to try to characterize regions of interest in a picture. That is why we tried to represent the image neighborhood hypergraph structure itself and this paper presents some results on this. Trying A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 124–135, 2002. c Springer-Verlag Berlin Heidelberg 2002

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to represent it in a bidimensional way in simply drawing and coloring the hyperedges, we will show that this might appear as an unappropriate representation of that modeling. A tridimensional representation of the INH line-graph appeared to be a better way to display them and is the objet of our third part.

1

Theoretical Background

Given a set X, we will denote by X (2) the set of unordered pairs of X. A graph G [1,13] is an ordered pair (X, E) such that E is a subset of X (2) . The size of G is the cardinality of X. Elements of X are called vertices, those of E edges. If e = {x, y} is an edge of G, x and y are said to be adjacent. We say that a graph G = (X , E ) is a subgraph of a graph G = (X, E) if X ⊂ X and E ⊂ E. A graph is said to be complete if E = X (2) . A complete graph whose size is n will be denoted by Kn . A complete maximal (in the sense of inclusion) subgraph of a graph G is said to be a clique of G. A clique whose size is maximum is called a maximum clique [5]. To any vertex of a graph may be associated its neighborhood Γ (x) = {y ∈ X, {x, y} ∈ E}. The degree of a vertex x is the cardinality of the set Γ (x). If any vertex in the graph has the same degree, the graph is said to be regular. A hypergraph H [4] on a set X is an ordered pair (X, E) where E is a set of nonempty subsets of X such that e∈E e = X. Elements of X are called vertices, those of E hyperedges. The size of a hypergraph is the cardinality of its set of vertices. The line-graph of a hypergraph is deﬁned as the graph whose set of vertices are the hyperedges of the hypergraph and where there is an edge between two vertices if the associated hyperedges have a nonempty intersection. To any graph may be associated its neighborhood hypergraph deﬁned by: X, ({x} ∪ Γ (x))x∈X . Moreover, we will say that the hyperedge {x} ∪ Γ (x) is generated by the vertex x and x will be called source pixel of {x} ∪ Γ (x). A digital image I [11] is a map from a subset X (generally ﬁnite) from ZZ 2 in a subset C of ZZ n . Elements of X are called points, those of C colors. A couple (x, I(x)) where x belongs to X is called a pixel. However, the confusion between a point x and (x, I(x)) is often made and we will not depart from it since it keeps its meaningfulness. A tiling (or a tessellation) of IR2 [19] is a partition of IR2 . The tilings generally studied are constrained by a limited number of geometric conﬁgurations called tessels. Given a tiling, the choice of an arbitrary point in the tessels, and the fact to link two points if the tessels share a common side allows to build a mesh. For the case of regular tilings where the tessels are regular polygons, the center of gravity of the polygon is often chosen and therefore leads to a regular mesh. In image processing, three types of meshes are used: hexagonal, triangular and square ones. Because of the current technological devices and the natural data structuring, the last type is the most used and we will therefore restrict our study to this kind of mesh. If a distance on ZZ 2 deﬁnes a undirected, simple, loopless and regular graph on a mesh, that distance will be called a grid distance. A grid is then a nonempty set of ZZ 2 with an associated grid distance. On square grids, two distance are mainly used: the city block (or

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square) distance for which a given pixel has four neighbors and the chessboard (or diamond) distance for which a given pixel has eight ones. This paper will only deal with the chessboard distance.

2 2.1

Image Neighborhood Hypergraph Modeling Spatiocolorimetric Neighborhood of a Pixel

Let d be a distance on the set of colors C and d be a distance that deﬁnes a grid on X ⊂ ZZ 2 . Let α and β be two strict positive reals. A unique neighborhood Γα,β (x) for the digital image I may be associated to any pixel x of X by: Γα,β (x) = {y ∈ X, y = x such as d (I(x), I(y)) < α and d (x, y) < β} We will call d grid distance and d colorimetric distance. The deﬁnition of colorimetric distances highly correlated with the perception of the human brain is always nowadays an extremely delicate domain of study of the neuroscientiﬁc ﬁeld. However we will assume that such functions exist even if the ones we will use may not appear appropriate with our sensations at the sight of an image. Looking at the previous deﬁnition, it appears that the ﬁrst part of it deﬁnes a neighborhood in the space C whereas the second one only involves the spatial domain. We will therefore respectively call α and β, colorimetric threshold and spatial threshold. Moreover we will qualify Γα,β (x) of spatiocolorimetric neighborhood. That notion allows to describe some consistency or homogeneity of a pixel with its environment. It is also interesting to see that such a neighborhood has a useful property: it increases in the sense of inclusion both with α and β. For instance, if we chose two colorimetric thresholds α and α such as α < α , for a ﬁxed β, Γα,β (x) ⊆ Γα ,β (x). That is due to the fact that if a pixel y is, from the color point of view, close of x at α, it remains close to x at α . This argument is the same if we chose the spatial thresholds and it is possible to show it also for them. The main interest of that property is that it implies a certain regularity in the modeling that we will now present. 2.2

Image Neighborhood Hypergraph

It is now possible to deﬁne an image neighborhood hypergraph (INH) Hα,β on X by Hα,β = X, ({x} ∪ Γα,β (x))x∈X . It is useful to precise that such a deﬁnition is correct since any hyperedge Γα,β (x) is nonempty: it contains at least the pixel x. Moreover the union of all the hyperedges is X itself. We will call Γα,β (x) hyperedge centered in (or generated by) the pixel x. As it directly inherits from the spatiocolorimetric neighborhood deﬁnition, an INH also increases with α and β in the sense of the hyperedges inclusion. The ﬁgure 1 shows a part of an INH and its associated line-graph for a spatial threshold β of 1 for the chessboard distance. For this ﬁgure we did not precise the colorimetric distance and its associated threshold as that is not relevant. However the ﬁrst noticing is that displaying hypergraphs structures in this way leads to a muddled interlacing of patterns of various shapes and colors.

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3

First Attempts of Representation

In this section we will describe the diﬃculties we encountered with our ﬁrst attempts to represent the image neighborhood hypergraph structure in a simple bidimensional way as in the ﬁgure 1. That simple idea consisted in associating to represent hyperedges by the set of pixels that they contained. 3.1

Coloring Hyperedges

The ﬁrst idea in representing an INH will probably be to associate a color to any hyperedge of it. Moreover it is then natural to color them in such a way that two intersecting hyperedges will have diﬀerent colors. In order to make the representation as simple as possible it should be interesting to have the least number of colors. Generally speaking, this problem is known as the hyperedge coloring of a hypergraph H and that minimum number of colors associated with it is called the chromatic index of H, denoted by q(H). That problem relies on the chromatic number of a graph as lemma 1 precises it. The chromatic number of a graph is the minimum number of colors that are needed to color the vertices of that graph in such a way such that two adjacent vertices have distinct colors [1]. Lemma 1. The chromatic index q(H) of a hypergraph H is also the chromatic number χ(L(H)) of its associated line-graph L(H). q(H) = χ(L(H)) Proof. The proof directly comes from the deﬁnitions. Two intersecting hyperedges will have distinct colors and therefore their associated vertices in the linegraph of the hypergraph will also have diﬀerent colors. That fact allows us to claim that necessarily q(H) ≤ χ(L(H)). Conversely two distinct vertices of the line-graph with diﬀerent colors ensures that their associated hyperedges will also have diﬀerent colors and therefore that χ(L(H)) ≤ q(H). Hence we can conclude to the equality of both quantities χ(L(H)) and q(H).

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Unfortunately computing the chromatic number of a graph is generally not an easy thing... it is NP-hard [15]. Nevertheless we can particularize it to our line-graphs. The worst case that may appear is the one such that any hyperedge of the hypergraph has maximal cardinality, that is, when only the spatial threshold is discriminating. In this case, on the 8-connected grid, for a ﬁxed threshold β, a pixel generates a square hyperedge whose cardinality is (2β + 1)2 (we will assume that there are no border problems if we suppose that the image loops itself). It is easy to see that the resulting line-graph is regular: as the same hyperedges conﬁguration appears on the whole image, the number of intersections is the same for every vertex of the graph. Remark that, in the case of a regular graph, computing the chromatic number is as hard as in the general case [17]. However it is possible to represent the vertices of the line-graph by the pixel that generated the corresponding hyperedge and it is clear that any vertex-pixel of a ﬁxed hyperedge e (that we will call as the “initial” hyperedge) is the center of a hyperedge e that intersects with e. This is the case of the pixel we called “member” in the ﬁgure 2. But these hyperedges are not the only ones that intersects e. This is the case for instance for the pixel that we called “external” on the ﬁgure 2. It is however easy to see that no pixel further than 2β from the “initial” pixel will generate a hyperedge that intersects with the initial hyperedge. Hence we can precisely give the number of intersections that a hyperedge have in the INH: (4β + 1)2 − 1 (if we assume that a hyperedge does not intersect with itself). The degree of a vertex in the line-graph is (4β + 1)2 − 1 (see Fig. 3). external hyperedge pixel generating the external hyperedge or external pixel member pixel initial pixel initial hyperedge Fig. 2. Determination of the worst case for the chromatic index in a image neighborhood hypergraph. We display in this the case for β = 2

Lemma 2. In the line-graph L(H ,β ) of an image neighborhood hypergraph H ,β , the degree of any vertex is (4β + 1)2 − 1.

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Computing the chromatic number of that graph is similar to determining the size of the largest clique in it [1]. As we represent the vertices of the line-graph by pixels on a grid, we can make some claims from the geometrical point of view to determinate the size of this clique. Let us take a vertex o (associated to a pixel O) in the line-graph. There is at least one clique C that lies in the neighborhood of o in the graph. Using the representation of the line-graph that we previously deﬁned, we may see the pixel O as a “central” one (in the sense of geometrical central symmetric) for the clique and we will say that the clique C is centered in o. Therefore if we take another vertex p in the line-graph that belongs to the neighborhood of o and if we consider the pixel P associated with it, we can conclude that the symmetric Q of P is associated to a vertex q that also lies in the neighborhood of o and therefore belongs to the same clique. Moreover we may apply the same arguments to any vertex associated with a pixel which is at the same distance of P from O thanks to isometric transformations of the discrete plane [25,20]. Then we can assert the following result: Lemma 3. If a pixel P such that d (O, P ) = β0 is associated to a vertex p that belongs to a clique C centered in o, then the set of pixels {P ∈ X such that d (O, P ) = β0 } is associated with vertices of the graph that also belong to C. Under the previous assumptions and notations, we may also say that any pixel located in the disk whose center is O and whose radius is β0 is also associated with vertices that also belong to the clique. That fact is simply due to the deﬁnition of our worst case as any hyperedges are of size β and a triangular inequality argument. The idea is now to determinate precisely the maximal size of that disk.

P O Q

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Fig. 3. Illustration of some geometric remarks concerning hyperedges coloration. Example of a partial line-graph built with β = 2

Assume that such a disk D exists. Let us call its center O and its radius β0 . Let us take a pixel P at β0 from its center. It is clear that P belongs to

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D. Its symmetrical Q around O also belong to D. If we want P and Q to be respectively associated with vertices p and q that belong to the clique, necessarily P and Q are linked by an edge in the line-graph and therefore each lies in the neighborhood of the other one if the line-graph. Moreover the distance between P and Q is simply 2β0 and cannot exceed 2β (as we already prove it). Therefore we have to solve the equation: arg maxβ0 ∈IN 2β0 ≤ 2β ... whose solution is simply: β0 = β The maximal radius of the disk we sought is then β and we can give the following lemma: Lemma 4. The upper bound for the size of the maximum clique of a line-graph associated with an image neighborhood hypergraph H ,β is (2β + 1)2 . In the worst case i.e. when the colorimetric threshold is not discriminating, such a size is reached. So, for instance, for an INH built for β = 2, the representation of its hyperedges already leads to a 25-colored muddle. That quadratic increase of the number of colors is unfortunately not the only problems we were confronted to represent the hyperedges of an INH. 3.2

Other Representation Problems

Hyperedges Connectivity. On the 8-connected grid, let us consider two hypergraphs representations built for α = 5 on the subimage of the ﬁgure 4 and the colorimetric distance simply compares the absolute value of the diﬀerence between the colorimetric value of the pixels with α. For β = 1, the hyperedge generated by the “central” pixel is represented in blue, for β = 2, in red. In order not to overload that ﬁgure, the superimposition of the two hyperedges has not been explicitly shown but it must be clear that due to the increasing property of the INH, the blue hyperedge is contained in the red one. What is remarkable in that ﬁgure is that such a representation of hyperedges may lead to the appearance of disconnected areas in it and holes in hyperedges increasing the diﬃculty of the reading for a potential user. Which Curves to Chose? As there are no convexity properties for the hyperedges (see ﬁgure 4), our initial choice of curves of the splines family did not appeared as very relevant. Moreover, after coloration and connectivity diﬃculties, the problem of the choice of the control points of these curves was added. The representation of the hypergraph structure itself appeared us then so tricky that we chose to go towards another representation of the structure: the one of its line-graph.

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4

Line-Graph Representation of an INH

Our main motivation with that representation was not to display a particular graph structure: we had the intention to make it interactively investigational by a potential user. It is certainly of great interest for the one interested in discovering our modeling to chose himself his own point of view on the structure and, better, in allowing him to navigate through the structure itself. We develop such a tool and in investigating the structure we were able to characterize image phenomena such as noise or object edges lying inside an image [10,22]. 4.1

Image Correlation

Our ﬁrst concern was to strongly correlate our representation with the data it came from. That is why we chose to display the original image on the representation. In order not to disturb the visualization of the graph structure, we chose to draw it with a parametric opacity. As any hyperedge is generated by a pixel of the digital image, it is natural to represent it by a point or better by a polyhedron or a sphere located at the vertical of the pixel. The elevation of the center of gravity of that polyhedron from the source pixel is controllable and may moreover be proportional to the intensity of the source pixel giving the representation a tridimensional aspect that makes easier its reading. To show the size of the hyperedge we make its radius proportional to the cardinality of the hyperedge (that proportional factor is also controllable). 4.2

Line-Graph Structure Representation

The line-graph is not a planar graph. For β = 1 for instance, we showed that the degree of its vertices may reach 24 and therefore it should not be rare to

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P N O

Fig. 5. The representation of the line-graph structure may gives bad impression. Some vertices may appear linked in the structure whereas they are not. The local display of a hyperedge of an image neighborhood hypergraph.

Fig. 6. Sample representations of the line-graph of an INH. The image is a RGB-color picture. For a pixel x, color is coded by (xr , xg , xb ). The colorimetric distance used was d(x, y) = (xr − yr )2 . α was set to 25 and β to 1. Note the weak degree of some vertices in that graph: that allowed us to partially characterize edges of objects in an image [10].

see K5 or K3,3 in that graph [1]. However we chose to link the vertices of the graph by simple segment lines even if it could give bad impression on the graph. From the ﬁgure 5 for instance, two hyperedges are explicitly drawn. They are generated by the pixels O and P . They intersect and therefore a link between O and P must be shown. Let us assume that the hyperedge generated by N is reduced to N itself. The user may therefore have the (bad) impression that N is linked with both P and O as N lies between these two points. Firstly it must be clear that such cases may be corrected by the user himself. As the elevation

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Fig. 7. Sample representations of the line-graph of an INH. The image is a RGB-color picture. For a pixel x, color is coded by (xr , xg , xb ). The colorimetric distance used was d(x, y) = max{|xr − yr |, |xg − yg |, |xb − yb |}. α was set to 20 and β to 2. Note that some areas are disconnected from the rest of the graph, some vertices are also completely isolated from it. That allowed us to characterize noise in an image [7,8,9,22].

of the polyhedron representing the hyperedge generated by N is controllable by the user. As its cardinality is 1, its intensity will probably diﬀerent enough from its neighbors and therefore its elevation may also be diﬀerent enough in order not the segment line not to cross the polyhedron. Moreover, as the previous argument may not appear as suﬃcient, we give the possibility to the user to individually select a polyhedron and to display the corresponding hyperedge. Only the pixels belonging to the hyperedge are drawn and moreover in order not to give also bad impression on that representation, parabola arcs are drawn if the pixels are located are located at more than 2 units (in the sense of the grid distance) from the source pixel (see Fig. 5). 4.3

Current Restrictions

The main problem with this kind of representation is that it is not possible, at the moment, to display too large images. For instance the ﬁgures 6 and 7 only involves 33 × 21 and 55 × 46 color images. The main drawback is the memory cost of the structure and the slowness of the current video cards1 . However, at the sight of the progress recently made in that technical ﬁeld, we do not despair to display larger structures in a near future. Moreover as it is possible to deal 1

For the ﬁgures 6 and 7, the construction and display have been made on a 1.2 MHz Athlon GeForce2 MX video card. Navigation through the structure is very ﬂowing but begins to be diﬃcult when the size of the image reaches more than 50000 pixels.

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with sub-images and parallelize processes, it should be possible to build subrepresentations of the image and to deal only with the nearest of them when the display is made. 4.4

Characterizing Regions of Interest in an Image

However, at the sight of pictures resulting of the display, it is interesting to note that regions of interest appear from the graph representation. For instance, weakly connected vertices appear in the left upper part of ﬁgure 6 characterizing edges of objects or of homogeneous areas in the original image [10]. Totally disconnected vertices, small sets of them or very weak connections between a set of vertices and larger size areas of the graph as shown in ﬁgure 7 allowed us to give non-statistically based deﬁnitions of noise in an image [7,8,9,22].

References 1. B. Bollob´ as Modern Graph Theory, Springer–Verlag, (1998). 2. C. Berge, “Introduction ` a la th´eorie des hypergraphes”, Les Presses de l’Universit´e de Montr´eal, (1973) 3. C. Berge, “Graphs”. North Holland, (1987). 4. C. Berge, “Hypergraphs”. North Holland, (1989). 5. I. Bomze, M. Budinich, P. Pardalos, and M. Pelillo, The maximum clique problem, http://www.dsi.unive.it/ pelillo/papers/tr99-1.ps.gz, (1999). 6. A. Bretto, J. Azema, H. Cheriﬁ, and B. Laget, Combinatorics and image processing in “Computer Vision Graphic in Image Processing”, 59, (5), September, (1997) 265–277. 7. A. Bretto, and H. Cheriﬁ, A noise cancellation algorithm based on hypergraph modeling in “IEEE Workshop on Digital Signal Processing” (1996) 1–4. 8. A. Bretto, and H. Cheriﬁ, Noise detection and cleaning by hypergraph model in “IEEE Computer Sciences” International Symposium on Information Technology: Coding and computing, IEEE Computer Sciences, (2000) 416–419. 9. A. Bretto and S. Chastel, A note on valued polyominoes hypergraph image modeling, International Journal of Computational and Numerical Analysis and Applications, to appear. 10. S. Chastel, A. Bretto, and H. Cheriﬁ, Image neighborhood hypergraph model and edge detection in “JCIS Proceedings” 3, Association for Intelligence Machinery, (1998), 272–275. 11. J. P. Cocquerez and S. Philipp, Analyse d’images : ﬁltrage et segmentation, Enseignement de la physique, Masson, (1995). 12. T.K. Dey, H. Edelsbrunner, S. Guha “Computational topology”, Technical report, (1998). 13. R. Diestel, “Graph Theory”, Graduate Text in Mathematics, Springer-Verlag, (1997). 14. M. Gondran, and M. Minoux, “Graphs and Algorithms” Wiley, Chichester, (1984). 15. I. Hoyler, The NP-completeness of edge coloring, SIAM J. Comput., 10: 718–720, (1981) 16. J. Jing-Ying, Multilevel Median Filter Based on Fuzzy Decision in “IEEE Trans. Image Processing”, 4: 680–682, (1995)

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17. D. Leven and Z. Galil, NP-completeness of ﬁnding the chromatic index of regular graphs, J. Algorithms, 4: 35–44 (1983) 18. V.A. Kovalevsky, Finite topology as applied to image processing in “Computer Vision, Graphics, and Image Processing” 46 (1989) 141–161. 19. A. Montanvert, and J.M. Chassery, “G´eometrie discr`ete en analyse d’images” Trait´e des Nouvelles Technologies. S´erie Images. Herm`es, Paris (1991). 20. J.-P. R´eveill`es, G´eom´ etrie discr`ete, calculs en nombre entiers et algorithmique, Th`ese de doctorat, Universit´e L. Pasteur, Strasbourg, (1991). 21. F.P. Preparata, and M.I. Shamos, “Computational Geometry : An Introduction”, Springer, New York, (1985). 22. S. Rital, A. Bretto, H. Cheriﬁ, and D. Aboutajdine, Mod´elisation d’images par hypergraphe. Application au d´ ebruitage, ISIVC Proceedings (ed. IAPR), Rabat (2000), 25–34. 23. A. Rosenfeld, “Digital Image Processing”, Academic Press, San Diego, (1982) 24. J.C. Russ, “The image processing handbook” CRC press, IEEE press, Springer, Berlin (1999) 25. K. Voss, Discrete Images, Object and Functions in ZZ n , Springer-Verlag (1993)

The Reconstruction of a Bicolored Domino Tiling from Two Projections A. Frosini and G. Simi Universit` a degli Studi di Siena Dipartimento di Matematica Via del Capitano 15 53100 Siena, Italy frosini,[email protected]

Abstract. We challenge the problem of the reconstruction of a bicolored domino tiling of a rectangle from its horizontal and vertical projections. We give two N P -completeness results after having deﬁned two non equivalent and very natural notions of projections on a generic bicolored domino tiling. The more general problem of the reconstruction of monochromatic domino tilings is still left open. Keywords. Domino tiling, Reconstruction Problem, N P -completeness.

1

Introduction

The aim of Discrete Tomography is the reconstruction of a discrete ﬁnite set of points in the d-dimensional integer lattice Zd using projections on lower dimensional subspaces. Important applications of Discrete Tomography are in image processing, in reconstructing structures from data get by an electronic microscopy, in data security and data compression (regarding the projections as an encoding process and reconstruction as a decoding process of a given object), and in computer-aided tomography. Interesting results have been achieved in reconstructing planar sets using projections on one, two or more mono-dimensional subspaces. In [2], Ryser studies the problem of how to reconstruct a binary matrix (which models bidimensional sets) from its projections and ﬁnds a P-time algorithm for it. In [1] the authors extend the above result to matrices with a ﬁnite number of diﬀerent entries (which models colored bidimensional sets) and ﬁnd the N P -completeness when n > 3 (n − colors problem). These results become relevant when applied to the algorithms for the reconstruction of polyatomic crystal structures. In this paper we use standard techniques of Discrete Tomography in order to reconstruct tilings of rectangular subsets of the plane with two colored dominoes. The horizontal and vertical projections of the tilings are a priori knowledge. In [5] the author shows that the reconstruction problem of a bicolored domino tiling is at least as hard as the reconstruction problem of a three entries matrix (3−colors problem) and leaves as an open problem its computational complexity. A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 136–144, 2002. c Springer-Verlag Berlin Heidelberg 2002

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Furthermore in [6] a polynomial time algorithm which reconstructs a bicolored domino tiling from one projection is given. The paper is organized as follows: in Section 2 we give some general deﬁnitions about bicolored domino tilings, in Section 3 we ﬁnd the computational complexity of the reconstruction problem on such a class using two orthogonal projections. This problem arose in [5]. In the last section we consider and solve the same reconstruction problem using the diﬀerent and non equivalent notion of horizontal and vertical projections introduced in [6].

2

Deﬁnitions

Let us consider an inﬁnite squared surface S composed by cells and consider a two length horizontal or vertical bar called dominoes. A rectangular subset of S of dimension m×n has a domino tiling if it could be completely covered with non overlapping dominoes. Furthermore, a bicolored domino tiling, is a domino tiling which use two diﬀerent kinds of dominoes: white and black ones. Columns are numbered from 1 to n, starting from the leftmost one, while rows are numbered from 1 to m, starting from the topmost one. A vertical domino covering two cells ci,j and ci+1,j is said to start on line i and end on line i + 1, while an horizontal domino covering two cells ci ,j and ci ,j +1 is said to start on column j and end on column j + 1. V

(4,1) (2,2) (2,3) (4,0) (3,2) (4,0) (3,2) (1,3) (3,2) (2,3) (3,3) (4,4) (6,1)

H

(3,3) (5,4) (6,2)

Fig. 1. A bicolored domino tiling of size 6 × 10 and its horizontal and vertical projections.

Let B be a bicolored domino tiling of dimension m × n, we deﬁne H = ((w1 , b1 ), . . . , (wm , bm )) the vector of the horizontal projections of B, where wi is the number of white dominoes which intersect at least one cell of line i and bi is the number of black dominoes which intersect at least one cell of line i, for each 1 ≤ i ≤ m. V = ((w1 , b1 ), . . . , (wn , bn )) the vector of the vertical projections of B, where wj is the number of white dominoes which intersect at last one cell of column j and bj is the number of black dominoes which intersect at least one cell of column j, for each 1 ≤ j ≤ n.

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The two vectors H and V do not univocally determine the number of horizontal (vertical as well) white dominoes on each row. The same holds for the black dominoes. Through the paper we will indicate with B a bicolored domino tiling of dimension m × n; the vectors of horizontal and vertical projections of B will be denoted with H and V in Section 3 and with R and C in Section 4.

3

The Reconstruction of a Bicolored Domino Tiling Consistent with Two Given Projections

We deﬁne the two problems: Consistency(H,V ) (b.d.t). Instance: two vectors H ∈ (N × N)m and V ∈ (N × N)n . Question: does there exist a bicolored domino tiling such that its horizontal and vertical projections are H and V respectively? Reconstruction(H,V ) (b.d.t). Instance: two vectors H ∈ (N × N)m and V ∈ (N × N)n . Output: a bicolored domino tiling such that its horizontal and vertical projections are H and V . In [5] the author made a step towards the solution of the 3 − colors problem by reducing its instances to instances of Consistency(H,V ) (b.d.t.) so that if this problem allowed a P -time solution, then also 3 − colors would. Unfortunately in this paragraph we prove that Consistency(H,V ) (b.d.t.) is N P complete: this prevent us from making any conjecture about the complexity of the 3 − colors problem. We achieve the above mentioned result by using a reduction which involves the N P -complete problem PARTITION (see [3]). PARTITION Instance: a ﬁnite sequence of integers A = a1 , . . . , ak . Question: let J = {1, . . . , k}. Is there J ⊂ J such that aj = aj ? j∈J

j∈J−J

Lemma 1. Let B be a bicolored domino tiling and H its horizontal projection. m The number of white dominoes in B is lower bounded by w1 +···+w and upper 2 bounded by w1 + · · · + wm . Proof. We get the lower bound when in B there are only vertical white dominoes so that each of them is counted in two diﬀerent entries of H. We get the upper bound when in B there are only horizontal white dominoes, so that each of them is counted only in one entry of H.

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Theorem 1. Consistency(H,V ) (b.d.t.) is N P -complete. Proof. The problem obviously belongs to N P . Let A = a1 , . . . , ak be an instance I of PARTITION with i=J ai = 2s. We want to deﬁne in polynomial time k − 1 instances I1 , . . . , Ik−1 of the problem Consistency(H,V ) (b.d.t.) such that a solution for I exists if and only if a solution for at least one of the I1 , . . . , Ik−1 exists. The index i of the generic instance Ii represents the cardinality (which is obviously less than, but not equal to k) of the set J . Let us construct Ii and let Bi be one of its solutions, if it exists. We deﬁne H ∈ (N × N)4s and V ∈ (N × N)3 as follows: − H is composed by couples which can be arranged in k blocks, each of them encoding a diﬀerent element of A. The height of the j-th block is 2aj : H = ((1, 1), (2, 1), . . . , (2, 1), (1, 1), . . . , (1, 1), (2, 1), . . . , (2, 1), (1, 1)) block 1 of length 2a1 block k of length 2ak − V is composed by 3 couples: V = ((s, s + i), (2s − k, 2k), (s, s + k − i)) (see Figure 2 a)). We make some useful remarks: let us consider the ﬁrst block of horizontal projections of H, and let Bi1 be a bicolored domino tiling consistent with it as shown in Figure 2 b) and c), it holds: i)

by Lemma 1, the minimum number of white dominoes in Bi which are k required by H is j=1 2aj − 1 = 4s − k, which is also the maximum number of white dominoes required by V (we compute it by adding w1 + w2 + w3 of V , that is s + (2s − k) + s). It follows that Bi has exactly 4s − k white dominoes; ii) again by Lemma 1, the minimum number of white dominoes in Bi1 is 1 (1 + 2 + · · · + 2 + 1) = 2a1 − 1. 2 2a1 times Figure 2 c) shows a solution having a higher number of white dominoes; iii) since Bi is composed by k blocks, using i) we get that, for each 1 ≤ j ≤ k, Bij has exactly 2aj − 1 white dominoes, which are all vertical ones; iv) from the previous remarks it follows that each Bij has two horizontal black dominoes which are placed on the ﬁrst and the last row of the block; vi) since the number of black dominoes in the second column of Bi is 2k and remark iii) and iv) hold, we get that each Bij has only two horizontal black dominoes in the ﬁrst and on the last rows.

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(5,6) (1,1) (2,1) (2,1) (2,1) (2,1) (2,1) (2,1) (2,1) (2,1) (1,1)

a)

(1,1)

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(2,1)

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(2,1)

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(2,1) (2,1)

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(2,1)

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(2,1)

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b)

c)

Fig. 2. The tiling which leads to a solution of the instance A = 5, 3, 2 of PARTITION with i = 2.

The above remarks imply that Bi can assume only ﬁxed conﬁgurations: each of them leads us to a solutions of PARTITION. On line 1, Bi has a white vertical domino and a black horizontal domino; if the white domino is placed on column 1, then the tiling of the whole ﬁrst block Bi1 is univocally ﬁxed. We can consider such a tiling as the coding of the belonging or no-belonging of the index 1 to J (see Figure 2 a)). If the white domino is placed on column 3 we get a symmetrical tiling. For the remaining k − 1 blocks a similar behavior holds: by placing the ﬁrst line of each of them we determine the whole tiling of the block. On column 1, Bi has only the white dominoes belonging to the coding of i elements of A whose sum is s. The indexes of such elements forms the required set J . On the other hand let J be a solution for I having cardinality i. We proceed by coding each element having index in J by a block which has white dominoes

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in column 1 and each element having index in J − J by a block which has white dominoes in column 3. The constructed tiling is one of the solutions of the instance Ii of Consistency(H,V ) (b.d.t.).

Corollary 1. The problem Reconstruction(H,V ) (b.d.t.) is N P -hard.

4

A New Reconstruction Problem on the Class of Bicolored Domino Tilings

In this section we give a diﬀerent and very natural deﬁnition of projections for a bicolored domino tiling (as introduced in [6]). Through the section, the new vectors of horizontal and vertical projections will be referred to as R and C in order to distinguish them from H and V previously deﬁned. In [6] the author shows a polynomial algorithm which solves Reconstruction(R) (b.d.t.) and he leaves Reconstruction(R,C) (b.d.t.) as an open problem. In spite of the two notions of projections are not equivalent, we prove the N P -completeness of Consistency(R,C) (b.d.t.) by reducing it to the problem PARTITION in a similar way as in the Theorem 1. This result solves a question proposed in [5] and [6]. However a general question about domino tilings remains still open: the computational complexity of the reconstruction of a domino tiling from two projections. A new notion of projections. Let B be a m × n bicolored domino tiling, we deﬁne R = (r1 , . . . , rm ) and C = (c1 , . . . , cn ) as the vectors of the horizontal and vertical projections respectively, where, for each 1 ≤ i ≤ m and 1 ≤ j ≤ n, ri is the number of cells covered with a white domino on line i and cj is the number of white cells covered with a white domino on column j. Theorem 2. Let H and R be the two horizontal projections of B, there does not exist neither a function f : (N × N)m → Nm which maps H into R nor a function g : Nm → (N × N)m which maps R into H (this means that the two notion of projections are not equivalent). Proof. examples a) and b) of Fig. 3 shows that a function f which maps H into R can not be deﬁned. Examples c) and d) prevent us from deﬁning the function g which maps R into C.

The same result holds if we use the vertical projections V and C. Again a proof of N P -completeness which involves PARTITION.

Theorem 3. Consistency(R,C) (b.d.t) is N P -complete

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H

R

H

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H

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2 2 2 2

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1 1

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1 1

a)

b)

c)

d)

Fig. 3. Examples showing the non equivalence between H and R.

Proof. The problem obviously belongs to N P . Let A = a1 , . . . , ak be an instance I of PARTITION, J = {1, . . . , k} and i∈J ai = 2s. We want to construct in polynomial time an instance I of Consistency(R,C) (b.d.t) such that a solution of I exists if and only if a solution of I exists. We deﬁne R ∈ N4s and C ∈ N3 as follows: – R is composed of k blocks such that, for each 1 ≤ j ≤ k, the j-th block encodes the element aj ∈ A and its length is 2aj : R=(

1, 2, . . . , 2, 1 ,..., 1, 2, . . . , 2, 1 ); block 1 of length 2a1 block k of length 2ak

– C is the following vector: C = (2s, 4s − 2k, 2s). We prove that if B exists, then it can assume only ﬁxed conﬁgurations, each of them leading us to a solution of I. We immediately note that, for each 1 ≤ j ≤ k, the block B j of B has 2aj − 1 white dominoes. The reconstructing procedure of B is the following: – on row 1 we place a vertical white domino on column 1 or 3 and an horizontal black domino covering the two remaining cells. No other tile is possible, otherwise we get stuck with the reconstruction on row 2. We choose to put the white domino on column 1 (a symmetrical behavior if we choose column 3) (see Figure 4 a)); – on row 2 we place a vertical white domino and a vertical black domino in order to satisfy the second entry of R; – on row 3 we have only a free cell where we place a vertical white domino; – in an iterative manner we ﬁll all the positions till row 2a1 where the entry of R is 1 and the cell on column 1 is covered with an ending vertical white domino. So the two free cells should be covered with one or two black dominoes; – the above observations can be extended to each one of the k blocks which has a black cell on column 2 on the ﬁrst and the last row. The total number of black cells on such a column is 2k and so the remaining 4s − 2k cells are covered with white vertical dominoes, as requested from the second entry of C(see Figure 4 b));

The Reconstruction of a Bicolored Domino Tiling from Two Projections

10

14

10

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10

14

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1 2 2 2 2 2 2 2 2 1 1 2 2 1 1 2 2 2 2 1 a)

b)

c)

d)

Fig. 4. The reconstruction of a bicolored domino tiling associated to the instance A = 5, 2, 3 of PARTITION from R and C.

– now we can tile column 1 and 3 of each block of B with white or black vertical dominoes according to the entries of C (see Figure 4 c) and d)). Finally we observe that, for each 1 ≤ j ≤ k, the block j which codes the element aj ∈ A has the whole column 1 or the whole column 3 covered with aj vertical white dominoes. Furthermore in column 1 of B we have s vertical white dominoes which belongs to the blocks j1 . . . jt and which code some elements aj1 , . . . , ajt whose sum is s. The set J = {j1 , . . . , jt } is the desired solution of the instance I. On the other hand, if a solution of I exists, it is easy to construct a solution for the corresponding instance I .

Corollary 2. The problem Reconstruction(R,C) (b.d.t.) is N P -hard.

5

Conclusions

In this paper we have studied and solved the computational complexity of the reconstruction problem on the class of the bicolored domino tilings using two different pairs of orthogonal projections. These two problems arose while searching

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a solution for the complexity of the 3 − colors problem and for the reconstruction of a monochromatic domino tiling from two projections. These two classical challenges are still left unsolved.

References 1. M.Chrobak, C.Durr , Reconstructing Polyatomic Structures from Discrete X-Rays: NP-Completeness Proof for Three Atoms, Theoretical Computer Science, 259:81-98, (2001), 2. H.Ryser , Combinatorial Mathematics, Mathematical Association of America and Quinn & Boden, Rahway, New Jersey, (1963) 3. M.R.Garey , D.S. Johnson , Computers and intractability: A guide to the theory of N P -completeness, Freeman, New York, (1979) 4. C.D¨ urr, E.Goles, I.Rapaport, E.R´emila, Tiling with bars under tomographic constraints Theoretical Computer Science to appear. 5. C.Picouleau, Reconstruction of domino tiling from its two orthogonal projections Theoretical Computer Science, 255, pgg.437-447 (2001). 6. C.Picouleau, Reconstruction of a coloured domino tiling from its projections Theoretical Computer Science submitted. 7. G. J. Woeginger , The reconstruction of polyominoes from their orthogonal projections, Inf. Proc. Letters 77 pagg.225-229 (2001).

Digital Geometry for Image-Based Metrology Alfred M. Bruckstein Ollendorﬀ Professor of Science Computer Science Department Technion, IIT 32000, Haifa, Israel [email protected]

Abstract. This paper surveys several interesting issues arising in digital geometry due to the need to perform accurate automated measurements on objects that are seen through the eyes of various types of imaging devices. These devices are typically regular arrays of (light) sensors and provide us matrices of quantized probings of the objects being looked at. In this setting, the natural questions that may be posed are: how accurately can we locate and recognize these objects from classes of possible objects, and how precisely can we measure various geometric properties of the objects of interest, how accurately can we locate them given the limitations imposed upon us by the geometry of the sensor lattices and the quantization and noise omnipresent in the sensor device output. Yet another exciting area of investigation is the design of (classes of) objects that enable optimal exploitation of the imaging device capabilities, in the sense of yielding the most accurate measurements possible.

1

Introduction

Scanned character recognition systems are working quite well by now, several companies have grown based on the need to do image based inspection for quality control in the semiconductor industry and, in general, automated visual inspection is by now widely used in many areas of manufacturing. In these important applications one needs to perform a series of precise geometric measurements based on images of various types of (planar) objects or shapes. The images of these shapes are provided by sensors with limited capabilities. These sensors are spatially arranged in (regular) planar arrays providing (matrices of) quantized (pixel-)values that need to be processed by automated metrology systems to extract information on the location, identity, size and orientation, texture and color, of the objects being looked at. The geometry of the sensor array is a crucial factor in the measurement performances that are possible. When sensor arrays are regular planar grids, we have to deal with a wealth of issues involving geometry on the integer grid. This is how digital geometry enters the picture in industrial metrology tasks, in very fundamental ways.

2

The Digitization Model and the Metrology Tasks

We shall here assume that the planar shapes, the objects we are interested to locate, measure and recognize are binary (black on a white background) and live A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 145–154, 2002. c Springer-Verlag Berlin Heidelberg 2002

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in the real plane, R. Hence their full description can be given via an indicator function ξ(x, y) which is 1 (black) on the shape and 0 (white) on the background. The digitization process will be by point sampling on the integer grid, Z2 , hence the result of digitization will be a discrete indicator function on the integer grid: a discrete binary image, or matrix of picture elements, or pixels. The generic problem we shall deal with is: given the discretized shape recover as much information as possible on the “pre-image”, i.e. on the original binary shape that lives on the continuous real plane. The necessary information on the pre-image shape might be its location and orientation area, perimeter, etc. In order to solve the particular problem at hand we shall also exploit whatever prior information we may have on the continuous pre-images. This prior information sometimes deﬁnes the objects or shapes we digitize as members of parameterized sets of possible pre-images. For example, we might know that the shapes we are called upon to measure are circular with varying locations and sizes. In this case the parameter deﬁning the particular object instance being analyzed from its digitization is a vector comprising three numbers: two coordinates pointing out the center of the disk and a positive number providing its radius.

3

The Wonders and Uses of Digital Lines

Digital lines, or digital straight segments result from sampling half-plane preimages or planar polygonal shapes (on the integer grid). More is known about this topic than anyone can possibly know, but I would dare to say that the basic facts are both simple and beautiful. Half-planes are not very interesting or practically useful objects, however they already pose the following metrology problem: given the digital image of a half-plane, locate it as precisely as possible. Of course, we must ask ourselves whether and how our location estimation improves as we see more and more of the digitized boundary. Well, it turns out that we can think about the location estimation problem as a problem of determining the halfplane pre-images that satisfy all the constraints that the digitized image provides. Indeed every grid-point pixel that is 0 (white) will tell us that the half-plane does not cover that location while every black (1) pixel will indicate that the halfplane covers its position. It should come as no surprise that the boundary pixels, i.e. the locations where white pixels are neighboring black ones, carry all the information. The constraint that a certain location in the plane belongs, or does not belong to the half-plane that is being probed translates into a condition that the boundary line has a slope and intercept pair in a half-plane deﬁned in the dual representation space (which is called in pattern recognition circles the Hough parameter plane). Therefore, as we collect progressively more data in the “imageplane” we have to intersect more an more half-planes in the Hough plane to get the so called “locale”, or the uncertainty region in parameter space where the boundary line parameters lie, see [8]. Looking at the grid geometry and analyzing the lines that correspond to grid-points in the dual plane one quickly realizes that only the boundary points contribute to setting the limits of the locale of interest, and a careful analysis reveals that, due to the regularity of the sampling grid,

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the locales are always polygons of at most four sides, see [6]. Hence as more and more consecutive boundary points are added to the pool of information on the digitized half plane, we have to perform half-plane intersections with at most four sided polygonal locales to update them. Clearly the locales generally decrease in size as the number of points increases, and we can get exact estimates on the uncertainty behavior as the jagged boundary is progressively revealed. This idea, combining Leo Dorst’s discovery on the geometry of locales for digital straight lines with the process of successively performing the half-plane intersections for each new data point while walking along the jagged digitized boundary, led to the simplest, and only recursive O(length) algorithm for detecting straight edge segments. For a complete description of this algorithm see [13]. The jagged edges that result from discretizing half-planes have a beautiful, self-similar structure, intimately related to the structure of the real number that deﬁnes the slope of their boundary line. One can readily see that at various sampling resolutions the boundary maintains its jaggedness in a fractal manner, but here we mean a diﬀerent type of self-similarity, inherent in the jagged boundaries at any given resolution! The paper [3] summarizes the wealth of interesting and beautiful properties that were described over many years of research on digital straight lines using a very simple unifying principle: invariance under re-encoding with respect to regular grids embedded into the integer lattice. Not only does this principle help in deriving in a very straightforward manner digital straight edge properties that were discovered and proved in sinuous ways, but it also points out all the selfsimilarity type properties that are possible, making nice connections to number-theoretic issues that arise in this context and the general linear group GL(2, Z) that describes all integer lattice isomorphisms. Using the wonderful properties of digital straight lines, we can not only solve the above-mentioned, and somewhat theoretical issue of locating a half-plane object of inﬁnite extent but we can also address some very practical issues like measuring the perimeters of general planar shapes from their versions digitized on regular grids of pixels. Indeed, analyzing the properties of digitized lines made possible the rational design of some very simple and accurate perimeter estimators, based on classiﬁcations of the boundary pixels into diﬀerent classes according to the jaggedness of their neighborhoods. Building upon earlier work of Proﬃt and Rosen [16], Koplowitz and Bruckstein proposed a general methodology for the design of simple and accurate perimeter estimation algorithms that are based on minimizing the maximum error that would be incurred for digitized straight edges over all orientations [11]. This methodology enables predictions of the expected performance for shapes having arbitrary, but bounded curvature, boundaries. Note also that using the recursive O(1)-per boundary pixel algorithm for detecting digital straightness of Lindenbaum and Bruckstein [13] one could parse general, curved object boundaries into digitally straight segments and then estimate the pre-image object’s perimeter as a sum of the lengths of the line-segments so-detected. In terms of the methodology presented in [13] this algorithms yields zero error for digital straight edges of inﬁnite extent at all orientations, and hence should be the best perimeter estimator ever obtainable!

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Digital Disks, Convex and Star-Shaped Objects

From the realm of half-plane objects we can move to either inﬁnite extent regions that have more complex boundaries (say parabolas, hyperbolae or some periodic functions along a principal direction) or to the analysis of ﬁnite extent objects like polygons, disks and other interesting shapes. Some work has indeed been done on detecting polygonal preimages from their digitized versions, and in fact a good algorithm for parsing a jagged boundary into digital straight segments turns out to be a crucial ingredient in solving various issues regarding the metrology of such objects. Suppose next that we have the prior information that the objects discretized are disks of various locations and sizes. Then the metrology question arising naturally is: how precisely can we determine the location of a disk and its radius. Considering the digitization model by point sampling, as discussed above, given a digitized image of black and white pixels, we know that if a certain point in the plane is the center of a disk of unknown radius, this point will necessarily be closer to all black grid points than to any white grid point. Hence the locus of all possible points in the plane closer to all black points than to any white points is the locale of possible disk centers, and its size will quantify our uncertainty in locating the object in the preimage plane. It is interesting to note that this locale can be found without knowledge on the radius, which will still need to be estimated. It turns out that the locale as deﬁned above is a well-known concept in computational geometry, and it is known that it is a convex region in the plane. Efrat and Gotsman have done a careful analysis of the problem and produced an O(R log R) algorithm to determine the locale, where R is the radius of the disk. We refer the interested reader to the paper [7] for details. Note again that the locale we are talking about is independent of the radius parameter. Had we prior knowledge on the exact radius, the location of the disk center could be determined by intersecting all disks of radius R around the black grid points with all the complements of disks or radius R around the white (uncovered) grid points. The resulting intersection locale is generally not a convex shape, due to the precise knowledge of the radius. For general convex shapes the question of determining the location and area and perimeter cannot be addressed in any generality. The digitized version of a convex shape is a set of black grid points on a background of white ones. As a union of square pixels the digitized shape will not be convex. Hence much work was done addressing the question whether there is a good deﬁnition of convexity for discrete objects [18]. A variety of proposals were made and can be found in the literature. The metrology questions however, in all cases remain: determine with best precision the location (ﬁrst order moments), orientation (second order moments) and other metric properties, like area (zeroth order moment) and perimeter of the shape. These questions, too have received some attention. It turns out that computing the moments of the black grid points yields good estimates for the corresponding continuous quantities, and more reﬁned, boundary estimation procedures (say, based on polygonalization of the jagged boundary via an eﬃcient digital straight segment detection, as discussed

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above) do indeed provide improved estimates but the improvement needs to be carefully weighed against the increased complexity involved. Among the many procedures that propose polygonal approximations to preimages based on the discrete grid points that were covered by the shape, and also based on the ones that were not covered, one stands out in elegance and usefulness: the minimum perimeter polygon that is enclosing all black (covered) points and excludes all white (uncovered) ones. This minimum perimeter polygon turns out to be the relative convex hull of the black points with respect to the white ones. It can be computed easily and may serve as a good approximation for preimages for all metrology purposes. So far we talked about disks and convex objects. The next level of complexity in planar shapes are the so called star-shaped objects. These are deﬁned as the shapes that have a “kernel region” inside them so that from any point in the kernel the entire boundary of the shape can be “seen”, i.e. a line from the chosen point to any boundary point will lie entirely inside the shape. It is easy to see that this deﬁnition generalizes convexity an a rather natural way and that the kernels must be convex regions. Determining star-shapedness of a planar shape is not a too diﬃcult task for polygons and for spline-gons and the algorithms for doing this rely on locating and using the inﬂection points on the boundary, and intersecting the regions in the plane from where the convex boundary regions are seen, see [1]. As with the notion of convexity, determining digital star-shapedness posed a series of special problems that needed careful analysis. This was the topic of a paper by Shaked, Koplowitz and Bruckstein, and there is was shown that the relative convex hull, or minimal perimeter polygon of the grid points covered by the shape wuth respect to the ones that remained uncovered, provides a convenient computational way to determine digital star-shapedness, see [17].

5

Shape Designs for Good Metrology

Up to this point we have discussed ways to analyze and measure planar shapes when seen through the looking glass of grid probing, or point-sampling discretization. The classes of shapes were assumed given in some perhaps parameterized form, and we dealt with questions about recovering their various features and parameters, or about measuring their size and perimeter and determining their location with the highest precision possible. When considering such issues, a further question that can be posed is the following: design planar shapes or collections of shapes that will interact with the discretization process in such a way that the quantities we need to measure will be very easily read out in the discretized images we get. Could we design an object in the plane (that can be a union of continuous binary shapes), so that digitization of this object translated to various locations, will yield black and white patterns on the (discretization) grid that clearly exhibit, say in a binary representation, the X and Y translation values up to a certain desired precision? Interestingly, recently a new pen-like device was invented and advertised, that has the following feature: it automatically computes with very high precision the

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location of its tip on any of the pages of a paper pad by looking at a faint pattern of dots that is printed on these sheets of paper. The pattern of these dots is so designed that the image obtained on any small region as seen by the pen near it’s tip (with the help of a tiny light detector array) uniquely and easily locates the pen-tip’s position on any of the pages of the pad, see [19]. This example shows that it is good engineering to think about designing shapes to have such “self-advertising” properties and this approach could provide us surprisingly eﬃcient and precise metrology devices. This problem was posed by Bruckstein O’Gorman and Orlitsky, at Bell Laboratories, already in 1989, with the aim of designing planar patterns that will serve as location marks, or ﬁducials on printed circuit boards. The need for location or registration ﬁducials in printing circuit boards and in processing VLSI devices is quite obvious. When layers of printing and processing are needed in the manufacturing operation, the precision in performing the desired processes in perfect registration with previously processed layers is indeed imperative. The work of [2] showed that there exists an information theoretic bound that limits the location precision for any shape that has an spatial extent of say AxA in pixel-size. Such a shape, when digitized will provide for us about A2 meaningful bits of information, via the pattern of black and white pixels in the digitized image. This number of bits 2 can only eﬀectively encode 2A −1 diﬀerent locales, and hence the precision to which we can reﬁne a region one pixel-square in size has a maximal area that 2 must exceed 1/(2A −1 ). If we want balanced X and Y axis precision, we can 2 only locate the pattern to a subpixel precision of 1/ 2(A −1)/2 . This is the best precision possible assuming optimal exploitation of the real estate of area AxA, assigned to the location mark. The important issue that was further settled in [2] is the existence of a ﬁducial pattern that indeed achieves this precision. The pattern is so cute that we exhibit it in Fig. 1. Looking at this ﬁducial pattern it becomes obvious what it does. It is indeed a continuous 2D (analog) input that employs the point sampling discretization process to compute its X and Y displacement by providing a binary read-out of the subpixel location of the ﬁducial within the one pixel to which it can readily be located using the left lowest grid-point (the “rough location” mark) covered by the shape. This leftmost bit of information is also the reason we can only use A2 − 1 bits for subpixel precision, i.e. for cutting the one pixel precision (provided by the “rough location” bit) into locale slices. This process turns the ﬁducial and the discretization process into a nice analog computer that yields the displacements in the X and Y direction easily, and achieves the highest precision in this task that is possible based on the available data. The analysis provided in [2] goes even further. The optimal ﬁducials turn out to require highly precise etchings on the VLSI or circuit board devices and hence might be diﬃcult to realize in practice. Hence there is a need to analyze other types of ﬁducial shapes that achieve suboptimal exploitation of the area, however can provide good location accuracies. For rotational invariance, circularly symmetric shapes turn out to be necessary, and therefore bull-eye ﬁducials were also proposed in [2] and further analyzed by [7].

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Fig. 1. An optimal 2D ﬁducial of area 3x3

The most interesting question that remains to be addressed here is the following: can we invent shapes that provide other metrological measures as easily as the above discussed example advertised its location?

6

The Importance of Being Gray

So far we have discussed the case of binary continuous images being pointsampled into matrices of zeros and ones, or Black and White pixels. However the real world is far richer in possibilities and complications. First of all, point sampling is not a good model of the imaging process as performed by real life cameras. Those carry out, at each sensor level, a weighted integration of the incoming light from the continuous input pattern. This integration happens around each grid point, and the pixel inﬂuence region may be assumed circular. The integration yields, at each grid point, values that continuously vary from a lowest value for white (no object) input over the pixel inﬂuence region to to highest value that corresponds to having the input object cover the entire area of integration. The result of this integration is then transformed into a discrete value encoded by several bits, via quantization. Therefore even for binary preimages, we get at each grid point a pixel value that is the quantization of a continuous variable proportional to the fraction of the pixel inﬂuence region that is covered by the input object. Furthermore we may also consider the advantages of using non-binary, grayscale of color pre-images. The combination or more realistic sampling and quantization processes with the use or grey levels in preimages open for us a great

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variety of further possibilities. As an example, Kiryati and Bruckstein have analyzed, following a question of Professor Pavlidis, the trade-oﬀ between spatial resolution and number of grey levels when the aim is to get as much information as possible on a class of binary pre-images that comprise polygonal shapes. The conclusion of this research was that “Gray Levels Can Improve the Performance of Binary Image Digitizers”, see [10]. The paper introduces a measure of digitization-induced ambiguity in recovering the the binary preimage, hence it is quite relevant to metrology under such sampling conditions. It is then shown that, if the sampling grid is suﬃciently dense (i.e. the sampling rate is high!) and if the pixels would provide us exact grey-levels rather than quantized values, then error-free reconstruction of the binary pre-image becomes possible. This is not too surprising, however, when the total bit budget for the digitized image representation is limited (i.e. the sampling rate and the quantization depth are related, both being ﬁnite) the bit allocation problem that arises shows that the best resource allocation policy is to increase the grey level quantization accuracy as much as possible, once a suﬃciently dense spatial sampling resolution has been reached. Therefore once we have a grid dense enough to ensure that all linear borders of the binary input image polygonal shapes can adequately be “seen” in the sampled image, all the remaining bit resources should go towards ﬁner gray level quantization. Professor Pavlidis’ question, which prompted this research asked to explain why grey-level fax machines at low resolution yield nicer images than fax machines at higher resolution, even for binary document images. It was clear that some sort of anti-aliasing eﬀect is in place, however [10] showed quantitatively that even in terms of a well-deﬁned metrology error measure, the grey-levels help considerably more than increased spatial resolution. Imagine next that we allow grey level input images too. In this case we shall certainly have, in conjunction with multilevel quantizations at each pixel much more information for location and various other measurements. A gradual boundary in the input image, or equivalently an area integration sensor providing a quantized multilevel pixel value at each grid-point, will transform the issue of locating a half plane into a problem of locating precisely several parallel digital straight edges, when they are simultaneously sampled. Such richness of detail will certainly dramatically reduce the size of the uncertainty locales, and enable us to design a wealth of improved location and orientation ﬁducials in the future. For a beginning of work in this direction see the recently completed thesis of Barak Hermesh, titled “Fiducials for Precise Location Estimation”. The conclusion therefore is that gray levels matter, they are good for us! And the last word on these issues certainly has not been said yet.

7

Concluding Remarks

Somebody said that every paper and book are autobiographical. This is certainly true of this short paper. It surveys my own research that dealt with digital geometry and metrology issues. As is clear from the list of papers below, my excitement with the interesting topics of research that arise at the interface

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between continuous and discrete geometry is permanent. And I am sure that the attendees of the Discrete Geometry for Computer Imaginary International Conferences all share my excitement with these topics. More on the vast subject of discrete geometry can be found in several books [4,5,12,14,15]. Acknowledgements. Thanks to my past and present students who over the years have taught me so many things on digital geometry and other topics. Thanks to Professors Theo Pavlidis and Azriel Rosenfeld for their leadership in digital geometry and for their friendship. Thanks to Yana Katz who made the speedy preparation of this paper possible.

References 1. R. Bornstein and A.M. Bruckstein, “Finding The Kernel Of Planar Shapes”, Pattern Recognition, Vol. 24, No. 11, 1019-1035, 1991. 2. A. M. Bruckstein, L. O’Gorman, and A. Orlitsky, “Design Of Shapes For Precise Image Registration”, IEEE Transactions on Information Theory, Vol IT-44/7, 3156-3162, 1998, ( AT&T Bell Laboratories Technical Memorandum ). 3. A. M. Bruckstein, “Self-Similarity Properties of Digitized Straight Lines”, Contemporary Mathematics, Vol. 119, 1–20, 1991. 4. J.-M. Chassery and A. Montanvert, G´eom´etrie Discr´ete en Analyse d’Images, Hermes, Paris, 1991. 5. L. S. Davis, editor, The Azriel Rosenfeld Book:“Foundations of Image Understanding”, University of Maryland, U.S.A., Kluwer, 2001. 6. L. Dorst, “Discrete Straight Line Segments: Parameters, Primitives and Properties”, Ph.D. Thesis, Technological University Delft, 1986. 7. A. Efrat and C. Gotsman, “Subpixel Image Registration Using Circular Fiducials”,International Journal of Computational Geometry and Applications, Vol.4, 403–422, 1994. 8. D. I. Havelock, “The topology of Locales and Its Eﬀects on Position Uncertainty” IEEE Transaction on Pattern Analysis and Machine Intelligence, Vol. 13, No. 4 380–386, 1991. 9. B. Hermesh, “Fiducials for Precise Location Estimation”, Research Thesis, Technion, IIT, Haifa Israel, 2001 10. N. Kiryati and A. M. Bruckstein, “Gray Levels Can Improve the Performance of Binary Image Digitizers” CVGIP: Graphical Models and Image Processing, Vol. 53, No. 1 31–39, 1991. 11. J. Koplowitz, and A. M. Bruckstein,“Design Of Perimeter Estimators For Digitized Planar Shapes”, IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. PAMI-11/6, 611-622, 1989. 12. L. J. Latecki, Discrete Representation of Spatial Objects in Computer Vision, Vol. 11, Kluwer, 1998. 13. M. Lindenbaum and A. M. Bruckstein , “On Recursive, O(N ) Partitioning of a Digitized Curve into Digital Straight Segments”, IEEE Transaction on Pattern Analysis and Machine Intelligence, Vol. 15, No. 9 949–953, 1993. 14. M. D. McIlroy, “Number Theory in Computer Graphics”, The Unreasonable Effectiveness of Number Theory, Proceedings of Symposia in Applied Mathematics, Vol. 46, 105–121, 1992

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15. R. A. Melter, A. Rosenfeld, P. Bhattacharya, editors, “Vision Geometry”, Contemporary Mathematics, Vol. 119, AMS, Rhode Island, 1991. 16. D. Proﬃt and D. Rosen, “Metrication Errors and Coding Eﬃciency of Chain Coding Schemes for the Representation of Lines and Edges”, Computer Graphics Image Processing, Vol. 10, 318–332, 1979. 17. D. Shaked, J. Koplowitz, and A. M. Bruckstein, “Star-Shapedness of Digitized Planar Shapes”, Contemporary Mathematics, Vol. 119, 137–158, 1991. 18. K. Voss, Discrete Images, Objects, and Functions in Z n , Springer-Verlag, 1991. 19. http://www.anoto.com/technology/anotopen/

Topological Reconstruction of Occluded Objects in Video Sequences Vincent Agnus and Christian Ronse Laboratoire des Sciences de l’Image, de l’Informatique et de la T´el´ed´etection (UMR 7005 CNRS–ULP), 67400 Illkirch, France {agnus,ronse}@dpt-info.u-strasbg.fr http://lsiit.u-strasbg.fr/

Abstract. In [1,2] we have introduced a new approach for the spatiotemporal segmentation of image sequences. Here a 2D+t sequence is considered as a 3D image, and 2D objects moving in time (or following a given motion model) are segmented as 3D objects with the use of connected morphological ﬁlters, and are represented as spatio-temporal ﬂat zones. However when an object undergoes occlusion by another in the sequence, their 3D trajectories intersect, and the spatio-temporal segmentation will fuse the two objects into a single ﬂat zone. In this paper we introduce a method for separating occluded objects in spatio-temporal segmentation. It is based on a study of the changes of topology of the temporal sections of a ﬂat zone. A topologically constrained watershed algorithm allows to separate the objects involved in the occlusion.

1

Introduction

The goal of segmentation is to partition an image into regions having the same attributes (color, contrast, texture, etc.). In the case of 2D+t image sequences, attributes for segmentation are generally contrast and/or motion. In [1,2] we introduced a new morphological approach enabling the segmentation of 2D+t regions following a given (local) spatio-temporal translation model. The method also provides a natural framework for tracking objects and, using an extension of the method, for classifying motions. Other morphological approaches [4,5] produce a 2D spatial segmentation and try to preserve space-time consistency by propagating markers in the future and applying a new spatial segmentation. (see [1,2] for a detailed discussion of other methods, morphological or based on optical ﬂow). Our 2D+t morphological approach avoids the computation of the optical ﬂow and the control of marker propagation in the future. It naturally handles the case of new object emerging in the sequence or objects leaving the scene. However our method encounters a problem: the occlusion between objects produces a unique ﬂat zone covering many diﬀerent objects. In this paper we present a new method of topological reconstruction in order to split these ﬂat zones and seperate the diﬀerent objects. A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 155–164, 2002. c Springer-Verlag Berlin Heidelberg 2002

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In Section 2 we recall brieﬂy the spatio-temporal segmentation method of [1,2]. Section 3 introduces our topological reconstruction approach for dealing with occlusions. First we explain how occlusions lead to the fusion of the ﬂat zones corresponding to the diﬀerent objects. Then we outline the decomposition of spatio-temporal objects into connected blocks, which fall into three classes: single-objects, junctions and disjunctions. Single-objects become markers for a modiﬁed 3D watershed algorithm, which takes into account the topological relations between 2D time sections, in order to separate the diﬀerent objects involved in an occlusion. Finally we discuss the oversegmentation and the ways to reduce it. Experimental results are presented in Section 4, where we give the conlusion.

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Segmentation Using Motion Information

Here we summarize our spatio-temporal segmentation method given in [1,2]. Considering a sequence as a single 3-dimensional image, an erosion in the temporal direction modiﬁes the original sequence only at motion boundaries. We can use this information to ﬁnd the moving objects. From these boundaries we reconstruct geodesically [7] the original signal, and this produces ﬂat zones (maximal connected regions where grey level is constant) corresponding to moving objects. With this method we can avoid choosing the structuring element size to create ﬂat zones, which was required in [6]. This reconstruction is performed both for dark and light objects in the scene, using duality in mathematical morphology. From these two reconstructed images, we allocate a region marker for each ﬂat zone, keeping the one which is farthest from the original signal. This processing can produce false detections in some cases. They are removed by our motion ﬁlters. We now describe brieﬂy the construction of ﬂat zones for moving objects. Objects undergoing a typical motion are reconstructed from their motion boundaries. These are computed by taking the diﬀerence between the original sequence S and its erosion in the temporal direction. The shape of the temporal structuring element allows to detect a given motion model. This diﬀerence sequence is ﬁltered by grey-level thresholding and area opening on connected components, to obtain a binary sequence Smb which represents only signiﬁcative motion borders. We notice that these connected components are inside light moving objects. From these components, a new sequence Sm is built, for which we give in each Smb connected component the minimum of its grey-levels from the original sequence. The geodesic reconstruction of the orginal sequence by dilation of the Sm sequence produces ﬂat zones at moving object locations. These large ﬂat zones are used to determine moving objects. The generation of ﬂat zones is applied for both bright and dark moving objects, using the principle of morphological duality. An election stage is performed in order to keep the most signiﬁcant ﬂat zones among bright and dark objects. This new sequence is ﬁltered out by studying the shape of ﬂat zones in space-time. We remove those that do not undergo a given motion model. The remaining ones receive a unique label to diﬀerentiate the objects in the scene (sequence So named object). The ﬂat zones

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produced belong here to space-time, so we have an object tracking across time, and a boundary localization at each time. However our method from [1,2] does not deal with occlusions: occluded objects are fused with occluding ones when they are both lighter (or darker) than the background. We solve this problem in the next section.

3

Topological Reconstruction

In some cases, occlusions cause the creation of the same ﬂat zone for diﬀerent objects in the scene. This drawback comes from our construction method. We name these ﬂat zones multi-objects. The real objects (named single-objects) belonging to the same ﬂat zone can be found by studying the topological structure of the multi-objects in space-time. In a ﬁrst part we describe when the creation of multi-objects occurs, and in a second part we give our method of topological reconstruction. It can determine a single-object in each branch of a multi-object. Branches which don’t contain a single-object are ﬁlled with the labels of the other single-objects. This operation is performed by using a spatio-temporal watershed algorithm with both greylevel and topological changes as hierarchical propagation priorities. 3.1

Generation of Multi-objects

The sequence Sm , from which ﬂat zones are reconstructed, is computed by keeping the minima of the grey-level of the orginal sequence in each connected component of the ﬁltered diﬀerence sequence Sd . So a ﬂat zone recovers only the darkest object at occlusion places. Fig.1 shows the ﬂat zones generated by our method in the case of two crossing bright objects. The sequence Sm , shown as two dark rectangles in Fig.1.c, is computed by keeping the minima of each connected component of the motion border. So only one ﬂat zone is constructed for the two bright objects. This ﬂat zone L2 is a multi-object.

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To diﬀerentiate the objects in this ﬂat zone, we suppose that objects cannot be ubiquitous, so the multi-object shape allows us to determine the number

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of single-objects within it. We apply our topological reconstruction method, see Fig.2. From the sequence object So we decompose each spatio-temporal ﬂat zone into regions where topology is preserved in some sense (deepened below). The study of adjacency between these regions allows to classify them as representing an object or an occlusion place. The latter are ﬁlled with the labels from regions classiﬁed as single-object. So we can allocate a label for each diﬀerent object contained in a unique ﬂat zone from the sequence object. In a ﬁrst part we describe the framework of the detection of objects in ﬂat zones, this step is named connected block decomposition. We give also an algorithm to build them. The propagation of object labels at occlusions places is described in Section 3.4, where we use a modiﬁed version of the Salembier’s region growing algorithm [6]. To overcome the oversegmentation due to the generation of labels we deﬁne a region adjacency graph contraction to reduce the number of regions. T

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Fig. 2. Topological reconstruction step: a) the sequence object So , b) topological invariant region decomposition, c) removal of occlusions zones, d) relabeling using watershed

3.2

Connectivity Block Decomposition

The study of topological changes (i.e., of neigbourhoods of spatial connected components in a multi-objet) can give information about the number of singleobjects in the multi-object. For example, consider a multi-object with a “Y” shape (Fig.2); the temporal dimension is along the vertical axis. We can determine that this object contains at least 2 diﬀerent objects. Indeed the two upper branches point to the presence of two diﬀerent objects. The base of the “Y” is the place of the occlusion between these objects. We now develop a categorization of multi-objects branches. For this we decompose the multi-object into connected blocks. They are connected components which don’t contain a local topological change in the multi-object. A distinction is made within these blocks: those which contain a single-object and the other ones. The blocks of a same object which have more than two adjacent connected blocks in the past or future are considered as blocks that do not contain a single-object. It is the place where the objects move away or approach. We give a more formal deﬁnition of connected blocks and a method to build them. Deﬁnition 1. The temporal section Rt of region R ⊂ E × T at time t is the set Rt = R ∩ (E × {t}).

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Deﬁnition 2. A subset X of region R ⊂ E × T is called a connected block of R iﬀ it is a maximal subset of R such that: a) X = ∅ and X connected, b) for all times, the temporal sections Xt are connected (spatially, because they belong to E) and c) a temporal section Xt−1 or Xt+1 of X which is adjacent to Xt , may not be adjacent to Rt \ Xt .

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Fig. 3. Examples of connected blocks on spatio-temporal objects: a) and b) two counter-examples, c) connected blocks for many conﬁgurations, where bright letters represent single-objects

There is some similarity between our decomposition and Reeb Graphs [3]. Given a smooth function f deﬁned on a smooth surface M , the Reeb Graph of f associates to every connected component of a level set f −1 (t) a point of height t. Smoothly evolving level curves give then rise to arcs in the graph and critical points of the surface produce junction nodes in the graph. In the case of image sequences, the surface M would consist of the spatiotemporal contours of the sequence, and the function f would be time. Then the Reeb Graph would code the evolution of object spatial contours at diﬀerent times. Our representation is simpler, in that we only consider connected components of objects, while we ignore inner contours and holes. 3.3

Connected Block Decomposition Algorithm

To obtain this decomposition, we build a label sequence named Stopo from the begin to the end of the sequence, in a recursive scheme. We label the spatial connected components along time using the labels of connected components already labeled in the past. We use three arrays of labels: – ARRN SO which contains the labels whose connected component descended from a junction (2 spatial connected components in past) or fathering a disjunction (2 in future). This array permits us to know the connected blocks to remove, i.e. the labels not representing a single-object. – ARRP [Lt ] which contains the set of spatial connected component labels at time t−1 which are contiguous to the connected component at time t labeled Lt . – ARRF [Lt−1 ] which contain the set of spatial connected component labels at time t which are contiguous to the connected component at time t − 1 labeled Lt−1 .

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We have: L ∈ ARRP [L ] iﬀ L ∈ ARRF [L]. The labeling is built as follows: At t = 0, a spatial labeling is performed in the time section t = 0. To go to step t + 1: a temporay labeling of spatial connected components of So in time section t + 1 is carried out. The new labels are partially absorbed by the labels of previous time sections already processed. We keep the new label if a birth or topological change occurs. For this we use the array ARRP [] and ARRF []. For each new label L (at time t + 1), we have to deal with several cases: – if ARRP [L ] is empty, a new object appears at time t + 1; it is a birth. This new label is preserved for the next time section. – if ARRP [L ] contains at least two labels: it is a junction. The label L is inserted in ARRN SO because this label indicates a topological change. This label represents the birth of a new connected block. – if ARRP [L ] = {L}, we have to deal with two cases: • if ARRF [L] = {L }, two connected components are neighbours in time. They belong to the same branch of the multi-object (i.e in the same connected block). The new label L take the value of the label L. • if ARRF [L] contains at least two labels, there is a disjunction at time t. The label L is preserved due to the topological change and the creation of a new connected block and the label L is inserted in ARRN SO.

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Fig.4 represents the application of this algorithm in a general scheme and Fig.5 illustrates the local count of neighbouring spatial connected components between two succesive time sections. Now we have a marker only for each single-object in each multi-object. We start from the object sequence So , it is decomposed into connected blocks (sequence Stopo ) as described above. Connected blocks neighbour in past to a junc-

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tion or in future to a disjunction are removed, they are not single-objects (sequence Sso ). Single-object labels are propagated in their respective multi-object to ﬁll the occlusions section. The propagation is based on an extension of watershed using grey-level and topological information, described below. 3.4

Topological Watershed Algorithm

The occlusion regions have been removed during our topological labeling. To reconstruct the segmentation at these places, we use a modiﬁed version of the watershed region growing deﬁned by Salembier [6]. We use grey-level information of the source sequence to allow the tracking of objects across time, and the topological change in multi-objects to synchronize the propagation stage. We must use a double hierarchy (grey-level and topology) to ensure the good propogation of labels in space-time. Indeed, consider Fig.6, we have a multi-object with two labels L1 and L2 of single-objects. The number in each spatial connected component corresponds to the time step in the ﬂooding process. If we use only grey-level information, the L2 label reaches the disjunction before L1 , so at this time section the space ﬂooding is performed whitout taking into account the label L1 . To avoid this drawback, before ﬂooding a time section we synchronize the labels: the labels are propagated normally if and only if they belong to the same connected block. Labels which cross the border of a new connected block are temporarily stopped. When all labels are stopped (i.e they all belong to a connected block border) the ﬂooding process restarts normally. This approach allows the labels to wait for others before ﬂooding in space at a time section.

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To implement this propagation scheme, the use of ordered queues (OQ) is natural to manage this double hierarchy. We use two OQs: OQG and OQT . The ﬁrst one is used to ﬂood the time sections and the other one is used to store labels which are stopped. The hierarchy within these OQs is the similarity between grey-levels of neighbouring pixels. When OQG is empty (i.e., all pixels are stopped) and all occlusion places are not ﬂooded, we swap these two OQs, so that the ﬂooding process can continue with labels synchronized. For instance in Fig.6 the labels are progated using OQG , and when L2 reaches the disjunction time section (step 1), it is stopped and stored in OQT . The label L1 progresses via OQG until it reaches the border between connected blocks (step 4). It is stopped and stored in OQT . The queue OQG is now empty: all labels belong to a border of topological change. We swap the two OQs and the ﬂooding stage restarts (steps 5 and 6). We have described how to ﬁnd objects in a multi-object and how to recover their shape in space-time, but some drawbacks remain. The search for an object, in some cases, generates an over-labeling. We describe in the next section when these cases occur, and how to deal with them. 3.5

Reduction of Oversegmentation

The construction of connected blocks strongly depends on the topology of the multi-object. In two cases, too many connected blocks are constructed. The ﬁrst one is due to topological changes arising from small noisy objects, the second one depends on our connected block decomposition. For instance, when two objects cross each other, their trajectories form together a “X” shape, our block decomposition gives four labels B1 to B4 and an occlusion place C, corresponding to the four branches and the crossing in the “X” shape. We must keep all B labels because we don’t know if they represent the same object before and after the occlusion, indeed some objects can hide other ones which can appear after the occlusion. To avoid the inﬂuence of topological changes due to small deformations, we pre-ﬁlter the object sequence So by removing with an area opening all small spatial connected components in each time section. Block connected decomposition is made on the ﬁltered sequence and the watershed labeling stage is processed on the object sequence using markers obtained with the ﬁltered object sequence. To overcome the over-labeling inherent to some multi-object shapes in spacetime, we have to use more global information; we use the contraction of a region adjacency graph. We deﬁne a model M(R) for each region R. It contains information about it. We use mean grey-level M.g, presence time (birth M.ts and death M.te ), and multi-object membership M.object and size (i.e., the volume in space-time M.N ) to characterise them. An order O(R1 , R2 ) is computed to determine the priority of merging regions. The region merging stage runs as follow: the two regions of the pair with the lowest merging order are merged together, the model M of the union is reestimated, and the merging order is recomputed with the neighbouring regions.

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When regions R1 and R2 merge, the new model M(R1 ∪ R2 ) is: — M.g ← (N1 g1 + N2 g2 )/(N1 + N2 ), M.N ← N1 + N2, — M.ts ← min M(R1 ).ts , M(R1 ).ts and M.te ← max M(R1 ).te , M(R1 ).te , — M.object ← M(R1 ).object (label of So ), where Ni = M(Ri ).N and gi = M(Ri ).g. The merging order is deﬁned as follow: O(R1 , R2 ) = N1 (g1 − g R1 ∪R2 )2 + N2 (g1 − g R1 ∪R2 )2 2 N1 ∆t (R1 ∪ R2 )/∆t (R1 ) − N1 − N2 , 2 × + N2 ∆t (R1 ∪ R2 )/∆t (R2 ) − N1 − N2 where ∆t (Ri ) = M(Ri ).te − M(Ri ).ts + 1 is the region presence time. This term measures the error when two regions merge. It is the product of two factors. The

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ﬁrst one expresses the grey level diﬀerence between regions, and the second one compares the region spatial mean size. In some case, we don’t allow fusion (in putting the merging order to inﬁnity). Regions which belong to diﬀerent multi-objects are not merged. In the same way, regions which share a same presence time cannot be merged because they are not ubiquitous.

4

Results and Conclusion

Fig.7 show the step of our topological reconstruction. From the original sequence Fig.7.a, ﬂat zones are produced by our approach[1,2] (Fig.7.b). It contains three multi-objects. Fig.7.c represents the decomposition into connected blocks, and single-objects blocks are shown in Fig.7.d. The labels are propagated in their respective multi-objects (Fig.7.e) using our double hierarchical watershed algorithm. The over labeling is reduced with our adjacency region graph contraction method (Fig.7.f ). An animated version of Fig.7, as well as other pratical applications (in particular, results on standard motion sequences) can be found at URL http://arthur.u-strasbg.fr/∼agnus/DGCI2002/ . The ﬂat zone generation and topological reconstruction are obtained with low-level morphological operators, that are well suited for fast implementations, thanks to optimal implementations which access each pixel only once. This contrasts with optical ﬂow based methods, which require repeated iterations.

References 1. V. Agnus, C. Ronse, and F. Heitz. Segmentation spatio-temporelle de s´equences d’images. In “12`eme Congr`es Francophone Reconnaissance des Formes et Intelligence Artiﬁcielle”, volume 1, pp. 619–627, Paris, France, Feb. 2000. 2. V. Agnus, C. Ronse, and F. Heitz. Spatio-temporal segmentation using 3d morphological tools. In 15th International Conference on Pattern Recognition, volume 3, pp. 885–888, Barcelona, Spain, Sep. 2000. 3. S. Biasotti, B. Falcidieno, and M. Spagnuolo. Extended Reeb Graphs for Surface Understanding and Description In 9th Discrete Geometry for Computer Imagery Conference, LNCS, Springer Verlag, pp. 185-197, Uppsala, 2000. 4. B. Marcotegui and F. Meyer. Morphological segmentation of images sequences. Mathematical Morphology and its Applications to Image Processing, J. Serra & P. Soille Eds., pp. 101–108, 1994. 5. F. Marqu´es. Temporal stability in sequence segmentation using the watershed algorithm. Mathematical Morphology and its Applications to Image Processing, P. Maragos & R. W. Schafer & M. A. Butt Eds., pp. 321–328, 1996. 6. P. Salembier, P. Brigger, J. R. Casas, and M. Pard´ as. Morphological operators for image and video compression. IEEE Trans. on Image Proc., 5(6): pp. 881–898, June 1996. 7. L. Vincent. Morphological grayscale reconstruction in image analysis: applications and eﬃcient algorithms. IEEE Trans. on Image Proc., vol. 2, pp. 176–201, Apr. 1993.

On the Strong Property of Connected Open-Close and Close-Open Filters Jose Crespo1 , Victor Maojo1 , Jos´e A. Sanandr´es1 , Holger Billhardt1 , and Alberto Mu˜ noz2 1

Laboratorio de Inteligencia Artiﬁcial Facultad de Inform´ atica Universidad Polit´ecnica de Madrid 28660 Boadilla del Monte (Madrid), Spain [email protected] 2 Departamento de Radiodiagn´ ostico Hospital 12 de Octubre Ctra. Andalucia Km. 5’400 28041 Madrid

Abstract. This paper studies connectivity aspects that arise in image operators that process connected components of an input image. The focus is on morphological image analysis (i.e., on increasing image operators), and, in particular, on a robustness property satisﬁed by certain morphological ﬁlters that is denominated the strong-property. The behavior of alternating compositions of openings and closings will be investigated under certain assumptions, especially using a connected component preserving equation. A signiﬁcant result is the ﬁnding that such an equation cannot guarantee the strong property of certain connected alternating ﬁlters. The class of openings and closings by reconstruction should therefore be deﬁned to avoid such situations.

1

Introduction

This paper studies the strong property of morphological ﬁlters that satisfy certain conditions regarding connectivity. The strong property is a robustness property introduced in the morphological ﬁltering framework [1] [2]. Morphological ﬁlters that satisfy this interesting property are more robust to small variations of the input image, such as noise, under certain limits. The strong property is somewhat related to connectivity issues in the sense that some morphological ﬁlter types that do not satisfy the strong-property condition are in fact strong when they are connected. In this work we will study this property for connected ﬁlters that satisfy diﬀerent requirements. A central point will be to study connected ﬁlters that satisfy a well-known additional condition that not only the ﬁlter output is invariant but also each of its connected components. In this paper we will obtain a signiﬁcant result concerning the strong property of alternating ﬁlters when such a condition is used to deﬁne them, in the sense A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 165–174, 2002. c Springer-Verlag Berlin Heidelberg 2002

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that the fact that an alternating ﬁlter is strong does not imply that the dual family under that deﬁnition is strong as well. Of course, it is not the case that the result contradicts the morphological duality principle; instead, the issue is that the condition does not treat symmetrically (and dually) openings and closings. We therefore deﬁne the class of openings and closings by reconstruction in a way to avoid such a problem. Those deﬁnitions have been used previously by the authors of this paper, and this work strongly supports that choice. The outline of the paper is as follows. Section 2 provides some background on mathematical morphology, in which some concepts concerning connectivity are commented. Then, Section 3 studies the strong property and the behavior of connected openings, closings, and alternating ﬁlters that satisfy certain conditions. An important result showing certain problems motivates the deﬁnition of openings and closings by reconstruction to avoid them. Finally, a conclusion section ends the paper.

2 2.1

Some Concepts and Deﬁnitions Basic Notions

Mathematical morphology concerns the application of set theory concepts to image analysis. General references are [3] [1] [2] [4]. A basic set of notions on mathematical morphology can be the following: – Mathematical morphology deals with increasing mappings deﬁned on a complete latice [5] [2]. In a complete lattice there exists an ordering relation, and two basic operations called inﬁmum and supremum (denoted by and , respectively). – A transformation ψ is increasing if and only if it preserves ordering. – A transformation ψ is idempotent if and only if ψψ = ψ. – A transformation ψ is a morphological ﬁlter if and only if it is increasing and idempotent. – An opening (often denoted by γ) is an antiextensive morphological ﬁlter. – A closing (often denoted by ϕ) is an extensive morphological ﬁlter. In all theoretical expressions in this paper, we will be working on the lattice P(E), where E is a given set of points called space and P(E) denotes the set of all subsets of E (i.e., P(E) = {A : A ⊆ E}). In other words, inputs and outputs will be supposed tobe sets or, equivalently, binary functions. In this lattice, the sup and the inf operations are the set union and the set intersection operations, while the order relation is the set inclusion relation ⊆. Even though we will work on the lattice P(E), results are extendable for gray-level functions by means of the so called ﬂat operators [6] [1]. 2.2

The Point Opening γx : The Connected Component Extraction Operator

Let us assume that the space E is provided with a deﬁnition of connectivity. For all pairs of points x, y in E, it is possible to establish whether they are connected

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or not. For example, when the space of points E is R2 or Z2 (associated with the usual connectivity), a pair of points x, y in a set A is said to be connected if there exists a path linking x and y that is also included in A. Connectivity is established more generally in [2] by means of the connected class concept. A connected class C in P(E) is a subset of P(E) such that (a) ∅ ∈ C and for all x ∈ E, {x} ∈ C; and (b) for each family Ci in C, i Ci = ∅ implies i Ci ∈ C. No deﬁnition of neighborhood relationships (i.e., no particular topology) has been assumed for E in the deﬁnition of the connected class C. The subclass Cx that has all members of C that contain x (i.e., Cx = {C ∈ C : x ∈ C}) deﬁnes an opening called point opening [2]. The point opening of a point x, denoted by γx , has as invariant class (i.e., the class formed by those sets that are left unchanged by γx ) Cx ∪ {∅}. For all x ∈ E, A ∈ P(E)

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The operation γx is therefore idempotent (i.e., γx (γx (A)) = γx (A) or, equivalently, γx γx = γx ) and antiextensive (i.e., γx (A) ≤ A or, equivalently, γx ≤ I). When we associate, for example, the operation γx with the usual connectivity in Z2 , the opening γx (A), A ∈ P (Z2 ), can be deﬁned as the union of all paths that contain x and that are included in A. Figure 1 shows an example of the result of γx (A) where the set A comprises the black regions (two connencted components or grains) and x belongs to a connected component of A. When a space E is equipped with the opening γx , connectivity issues in E can be expressed using γx . We can establish, for example, whether or not a set A ∈ P(E) is connected (a set A is connected if and only if A = γx (A), x ∈ A), and whether or not a pair of points x, y belong to the same connected component in A (x, y belong to the same connected component in A if and only if x ∈ γy (A) or, equivalently, if and only if γx (A) = γy (A) = ∅).

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The dual operation of γx is the closing ϕx that is equal to E \ γx (A), for all A ∈ P(E), where \ denotes set diﬀerence and symbolizes the set complementation operator. 2.3

Connected Operators and Connected Component Locality

After establishing the connected class concept in the previous section, let us deﬁne the concept of connected operator. [7] [8] [9] [10]. – A connected operator only extends the input image ﬂat zones (or piecewiseconstant regions). It should be noticed that, from its deﬁnition, a connected operator cannot introduce discontinuities and, therefore, preserves shapes. In a binary framework, connected operators are grain-removing and pore-ﬁlling operations. Connected ﬁlters are connected operators that are idempotent. The fact that connected ﬁlters preserve shapes makes them useful tools for image simpliﬁcation and segmentation purposes [11] [12]. The concept of connected-component (c.c.) local operator, which is deﬁned next, embraces both increasing and non-increasing operators that treat each grain and pore independently of the rest of the input [13]. Deﬁnition 1 Let E be a space equiped with γx . An operator ψ : P(E) −→ P(E) is said to be connected-component local (or c.c. local) if and only if, ∀A ∈ P(E), ∀x ∈ E (a) γx (A) = ∅, γx ψ(A) = ∅ ⇒ ∀B ∈ P(E), γx (A) = γx (B) : γx ψ(B) = ∅. (b) γx (A) = ∅, γx ψ(A) = ∅ ⇒ ∀B ∈ P(E), γx (A) = γx (B) : γx ψ(B) = ∅.

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Connected Filters and the Strong Property

In this section we will study the strong property and its relationship with openings and closings, in particular to their alternating sequential compositions. The strong property [14] [2] is a robustness property satisﬁed by certain morphological ﬁlters. Deﬁnition 2 A ﬁlter Ψ is strong if and only if Ψ = Ψ (I Ψ ) = Ψ (I Ψ ). If a ﬁlter is strong then certain variations (such as noise) of the input do not cause variations in the output. Figure 2 illustrates this be an input concept. Let A set. If Ψ is strong, then for all sets B such that A Ψ (A) ≤ B ≤ A Ψ (A) =⇒ Ψ (A) = Ψ (B). Another way tostate this is saying that Ψ is strong if and only if Ψ is both an -ﬁlter and a -ﬁlter (see [14] [2] for those related concepts). Openings and closings are, respectively, anti-extensive and extensive ﬁlters that satisfy the strong property. In the deﬁnition of this important property does not appear any connectivity reference nor γx in the formula. Nevertheless, in practice the strong property

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Fig. 2. Strong ﬁlter example.

is in some way quite related to connectivity, and, in particular, to connected operators. Some types of ﬁlters are not strong when they are not connected but, on the contrary, satisfy this property if they are connected. This can be the case, for example, of the connected alternating ﬁlters ϕγ and γϕ. In fact, the central point of this section will be to study the strong property under certain conditions for ϕ and γ, and to use it to characterize the class of ﬁlters by reconstruction. In some theoretical studies in mathematical morphology [2] [7] [15], the following equation ψγx ψ = γx ψ

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where ψ is an operator, has served to deﬁne some types of ﬁlters, especially connected ﬁlters. This interesting equation will be studied in detail in the following sections. 3.1

Opening Case

If ψ is an opening γ in equation (2), we have the following property: Property 1 γγx γ = γx γ =⇒ γ is c.c. local. Therefore, such an opening removes grains and treats each one independently from the rest. In fact, if an opening γ is c.c. local and connected, then we can deﬁne γ using the useful concept of trivial opening: γ= γ◦ γx , (3) x

where γ◦ is a trivial opening. Deﬁnition 3 An opening γ◦ is a trivial opening [2] if

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γ◦ (A) =

A, if A satisﬁes an (increasing) criterion ∅, otherwise

Notice that the trivial opening deﬁnition does not imply c.c. locality. In fact, an opening by reconstruction γ can be computed using a non-local trivial opening (and, nevertheless, the resulting opening by reconstruction would of course be local). Thus, for connected openings, it can be said that equation (2) is equivalent to expression (3). 3.2

Closing Case

A diﬀerent situation arises using equation (2) when ψ is a closing ϕ. We introduce the following property: Property 2 A closing ϕ is connected =⇒ ϕγx ϕ = γx ϕ. Property 2 states that any connected closing satisﬁes equation (2), i.e., that grains of the closing output are invariant under the closing. A proof is relatively straightforward. Let E be the space, let ψ be a closing, and let G be a connected component of the output ψ(A), where A is an input set. Then ψ(G) must be equal to G if ψ is idempotent, antiextensive and increasing. Since ψ is connected, the possibilities are (assuming the space E is connected): (a) G, (b) ∅, or (c) E. However, (b) is not possible (ψ is antiextensive), and (c) is not possible either (ψ is increasing and idempotent). Therefore, ψ(G) = G. An important aspect is that equation (2), when ψ is a closing ϕ, does not imply that the closing ϕ is c.c. local. And, therefore, a connected closing that satisﬁes equation (2) (in fact, from property 2, all of them do) cannot always be expressed by the dual of expression (3), which would be ϕ=

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where ϕ◦ is a trivial closing. The deﬁnition of the trivial closing follows (which is the dual of that of the trivial opening). Deﬁnition 4 A closing ϕ◦ is a trivial closing [2] if E, if A satisﬁes an increasing criterion ϕ◦ (A) = A, otherwise 3.3

Alternating Filters ϕγ and γϕ

In this section we will get an important result by investigating the strong property of alternating ﬁlters using equation (2). Later, we will deﬁne the ﬁlter by reconstruction class in a way to avoid this type of situations.

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In [7] [15], it has been shown that if a connected opening γ and a connected closing ϕ satisfy equation (2) ψγx ψ = γx ψ (substituting γ for ψ, and similarly for ϕ), then the alternating ﬁlter ϕγ has the desirable strong property. From property 2, we also can state this as follows. Property 3 Let γ be a connected opening that satisﬁes γγx γ = γx γ, and let ϕ be a connected closing (i.e., γ and ϕ satisfy equation (2)). Then, the alternating ﬁlter ϕγ is strong. However, there is a problem using equation (2) to characterize connected openings and closings because it is not symmetrical, i.e., it does not treat in the same way to openings and closings. We have seen that, for openings, equation (2) means that we can express the opening as in expression (3). However, for closings, equation (2) does not imply that we can express the closing as in expression (4) (which is the dual of expression (3)). Therefore, the following important (but somewhat undesirable) result should not surprise us excessively: Property 4 Let γ and ϕ be, respectively, a connected opening and a connected closing that satisfy equation (2). Then, the alternating ﬁlter γϕ is not necessarily strong. Notice that property 4 does not contradict the morphological duality principle. Let us show and study an example of a case in which γϕ is not strong, under the assumptions of property 4. Let us ﬁrst consider the opening and closing used in the example in Figure 3, which are computed employing reconstruction algorithms [16]. The opening by reconstruction will use as marker the result of the erosion εB , where B is the structuring element that can be seen in Fig. 3(a), along with the input set. The opening γ removes vertical grains of width smaller than that of B, and therefore γ eliminates the small vertical grains at both sides, as shown in Fig. 3(b). The closing ϕ uses as marker the dilation δC where the structuring element C is composed by two particles, as can be seen in Fig. 3(c). The closing ϕ ﬁlls vertical pores of width 1 if there are vertical grains at left and right at a certain distance determined by C (the distance between the central pore and the grains at each side). In Fig. 3(d), the central pore has been ﬁlled. Both γ and ϕ satisfy equation (2), but γ is c.c. local whereas ϕ is not. The example in Figure 4 illustrates property 4. Fig. 4(a) shows the input set (which is the same as that in Fig. 3(a)). The closing ﬁrst ﬁlls the central pore in Fig. 4(b), and thesubsequent opening removes the grains at the sides in Fig. 4(c). The result of A γϕ(A) is shown in Fig. 4(d), where we can notice that the central pore is not ﬁlled, and the grains at both sides are missing. Because the grains at both sides are missing, in Fig. 4(e) the result of the subsequent closing does not ﬁll the central pore. Then, in Fig. 4(f) the last opening leaves the image unchanged, and the ﬁnal result γϕ(A γϕ(A)) is shown. We see clearly that γϕ(A γϕ(A)) = γϕ(A), and that, therefore, γϕ is not strong. The reason to obtain this result, i.e., that ϕγ is strong whereas γϕ is not, when γ and ϕ are connected and satisfy equation (2), is because of the nonsymmetrical nature of equation (2).

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B

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3.4

Openings and Closings by Reconstruction, and the Filter by Reconstruction Class

The “negative” result of property 4 strongly supports that classes of connected openings and closings be deﬁned by dual expressions. An important class should be that deﬁned by expresions (3) and (4). The authors of this paper have used in other works [9] [13] [17] expresions (3) and (4) to deﬁne the important class of openings and closings by reconstruction. Since those expressions are dual of each other, openings and closings are not considered diﬀerently. Deﬁnition 5 An opening γ is an opening by reconstruction if and only if γ = γ γ , where γ◦ is a trivial opening. ◦ x x Deﬁnition 6 A closing ϕ is a closing by reconstruction if and only if ϕ = ϕ ϕ , where ϕ◦ is a trivial closing. ◦ x x The duality of the previous deﬁnitions eliminates problems such as that commented in property 4, and we have that: If γ and ϕ satisfy, respectively, deﬁnitions 5 and 6, then ϕγ is strong and, by duality, γϕ is strong as well. We then normally use the term ﬁlters by reconstruction to denote those ﬁlters composed of openings and closings by reconstruction. Sometimes ﬁlters by reconstruction (in particular, openings and closings by reconstuction) are deﬁned in terms of

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algorithms by reconstruction, which is an alternative and valid way to deﬁne them if the deﬁnition indicates what the markers used in the reconstruction are (in general, algorithms by reconstruction just guarantee the connectedness of the resulting operator). Nevertheless, we prefer deﬁnitions 5 and 6 since they are not linked to any particular implementation.

4

Conclusions

This paper has investigated the strong property of connected morphological ﬁlters, in particular of connected openings, closings and alternating ﬁlters. We have ﬁrst studied a well known condition that guarantees that the connected components of a ﬁlter output are also invariant. This condition does not treat symmetrically openings and closings, and we have obtained a result in which a property satisﬁed by a ﬁlter is not satisﬁed by the dual family under that condition (a result that of course does not stand in contradiction to the morphological duality principle). This result supports that the class of openings and closings by reconstruction should be deﬁned by dual formulae, which prevents results such as that commented in the paper.

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Acknowledgements. This work has been supported in part by “Fondo de Investigaci´on Sanitaria” (Spanish Ministry of Health) and by “Sociedad Espa˜ nola de Radiolog´ıa M´edica”.

References 1. Serra, J.: Mathematical Morphology. Volume I. London: Academic Press (1982) 2. Serra, J., ed.: Mathematical Morphology. Volume II: theoretical advances. London: Academic Press (1988) 3. Matheron, G.: Random Sets and Integral Geometry. New York: Wiley (1975) 4. Heijmans, H.: Morphological Image Operators (Advances in Electronics and Electron Physics; Series Editor: P. Hawkes). Boston: Academic Press (1994) 5. Birkhoﬀ, G.: Lattice Theory. American Mathematical Society, Providence (1984) 6. Maragos, P., Schafer, R.: Morphological ﬁlters — part I: Their set-theoretic analysis and relations to linear-shift-invariant ﬁlters. IEEE Trans. Acoust. Speech Signal Processing 35 (1987) 1153–1169 7. Serra, J., Salembier, P.: Connected operators and pyramids. In: Proceedings of SPIE, Non-Linear Algebra and Morphological Image Processing, San Diego. Volume 2030. (1993) 65–76 8. Crespo, J., Serra, J., Schafer, R.: Image segmentation using connected ﬁlters. In Serra, J., Salembier, P., eds.: Workshop on Mathematical Morphology. (1993) 52–57 9. Crespo, J., Serra, J., Schafer, R.: Theoretical aspects of morphological ﬁlters by reconstruction. Signal Processing 47 (1995) 201–225 10. Heijmans, H.: Connected morphological operators for binary images. Computer Vision and Image Understanding 73 (1999) 99–120 11. Crespo, J., Schafer, R., Serra, J., Gratin, C., Meyer, F.: The ﬂat zone approach: A general low-level region merging segmentation method. Signal Processing 62 (1997) 37–60 12. Crespo, J., Maojo, V.: Shape preservation in morphological ﬁltering and segmentation. In: XII Brazilian Symposium on Computer Graphics and Image Processing, IEEE Computer Society Press, SIBGRAPI 99. (1999) 247–256 13. Crespo, J., Schafer, R.: Locality and adjacency stability constraints for morphological connected operators. Journal of Mathematical Imaging and Vision 7 (1997) 85–102 14. Matheron, G.: Filters and lattices. In Serra, J., ed.: Mathematical Morphology Volume II: theoretical advances. London: Academic Press (1988) 115–140 15. Salembier, P., Serra, J.: Flat zones ﬁltering, connected operators, and ﬁlters by reconstruction. IEEE Transactions on Image Processing 4 (1995) 1153–1160 16. Soille, P.: Morphological Image Analysis: Principles And Applications. SpringerVerlag Berlin, Heidelberg, New York (1999) 17. Crespo, J., Maojo, V.: New results on the theory of morphological ﬁlters by reconstruction. Pattern Recognition 31 (1998) 419–429

Advances in the Analysis of Topographic Features on Discrete Images Pierre Soille EC Joint Research Centre Institute for Environment and Sustainability TP 262, I-21020 Ispra, Italy [email protected]

Abstract. By viewing the grey scale values of 2-dimensional (2-D) images as elevation values above the image deﬁnition domain, geomorphological terms such as crest lines, watersheds, catchment basins, valleys, and plateaus have long been used in digital image processing for referring to image features useful for image analysis tasks. Because mathematical morphology relies on a topographic representation of 2-D images allowing for grey scale images to be viewed as 3-D sets, it naturally oﬀers a wide variety of transformations for extracting topographic features. This paper presents some advances related to the imposition of minima, the lower complete transformation, the hit-or-miss transform, and the extraction of crest lines by a skeletonisation procedure. Keywords. Mathematical morphology, minima imposition, plateau, lower complete transformation, grey scale hit-or-miss, crest lines, grey scale skeletonisation, watersheds

1

Introduction

The topographic representation of grey scale images has long been considered in digital image processing for extracting features such as ridges and valleys [11], watersheds [8], and plateaus. In contrast to approaches based on the analysis of the discrete derivatives of the image, mathematical morphology [20] oﬀers a set theoretic framework which can be considered as a suitable alternative approach in some applications. In this paper, we adopt the morphological framework and revisit several transformations dealing with the processing of topographic features of digital 2-dimensional (2-D) grey scale images. Background deﬁnitions and notations used in this paper are described in [26]. In section 2, we introduce a minima imposition technique based on a carving procedure rather than the usual reconstruction based technique. This approach is a suitable alternative in situations where the creation of ﬂat regions by the reconstruction would impede further processing. We then show in section 3 that the suppression of image plateaus can be achieved by an interpolation technique taking into account the morphology of both the descending and ascending plateau

This work was supported by the EC-JRC EuroLandscape Project.

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borders. This is enabled by a grey weighted geodesic distance called the geodesic time function. The proposed approach leads to an enhanced algorithm for the lower complete transformation which a necessary preprocessing for computing the steepest slope path linking an arbitrary pixel to the regional minimum of its corresponding catchment basin. In section 4, we extend the hit-or-miss transform to the processing of grey scale images by introducing the notion of constrained and unconstrained hit-or-miss. These operations are useful for extracting speciﬁc grey scale neighbourhood conﬁgurations. Before concluding, we show in section 5 that crest lines can be obtained by order independent homotopic thinnings.

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Minima Imposition by Carving

The minima imposition is an image transformation which modiﬁes an input grey scale image in such a way that the only remaining minima are those given by an additional input image called marker image. Note that we refer to an image minimum in the sense of a regional minimum, i.e., a connected component of pixels having the same value and whose external boundary pixels have all a greater value. Denoting by f the input grey scale image, the marker image fm is deﬁned as follows: fm (x) = 0 if x belongs to a marker, tmax otherwise. The minima imposition is usually achieved by performing the following morphoε logical reconstruction by erosion Rε [2, 14]: R(f +1)∧fm (fm ). This procedure is 1 a connected algebraic closing which may create large ﬂat regions because the catchment basins of all unmarked minima are partially ﬁlled in. This is not a problem if one is interested in computing the watersheds of the ﬁltered image. However, there are applications where markers of thin nets can be detected while a watershed could not be used for extracting the nets due to gaps occurring along the net. Indeed, in this situation, the reconstruction based minima imposition removes all parts of the net which do not contain a marker. In addition, if one would like to simulate a ﬂow of water on the topographic representation of the image, the reconstruction based minima imposition has the drawback of creating potentially large ﬂat regions which in turn pose a problem for the determination of ﬂow directions from a local neighbourhood. These problems motivated us to look for an alternative way to impose the minima of an image. Rather than ﬁlling the unmarked regional minima, we propose to carve the image in such a way that these regional minima ﬂow further down. The carving procedure relies also on a ﬂooding simulation but, rather then ﬁlling in unmarked minima, a non-ascending path linking them to their nearest marked minimum is created. The pseudo-metric used for calculating the distance between an unmarked minimum and a marked minimum is based on the ﬂooding paths. The procedure is detailed hereafter. First, all regional minima not marked by the marker image are stored in a binary image. The ﬂooding simulation then starts by inserting into a priority 1

An operator is connected [21, 19] if and only if it coarsens the partition of any given input image, the partitioning being taken in the sense of that induced by the ﬂat zones of the image.

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queue the external boundary pixels of the marker minima, the priority being inversely proportional to the intensity value of the considered pixel. Pixels are then iteratively retrieved from the non-empty queue with the highest priority (i.e., lowest intensity elevation) while inserting their unprocessed neighbours in the priority queue (again considering a priority inversely proportional to their elevation). An additional image is used to store the direction of propagation of the ﬂood at each pixel. Before inserting a pixel in the queue, we check whether it belongs to an irrelevant minimum. If so, the stored directions are used to backtrack the ﬂooding path until we reach a pixel of elevation equal to that of the reached minimum and set all pixels along the path to this elevation. The reached minimum is then discarded from the binary mask of irrelevant minima while inserting all its unprocessed external boundary pixels in the priority queue. The process terminates when the priority queue is empty. By construction, all irrelevant minima are removed by the procedure because each of them can be linked to a marker by a non-ascending path. A comparison between the minima imposition by reconstruction and by carving is performed in Fig. 1 on two samples of a Pan-European digital elevation model (DEM). Given the scale of the input DEM (250 m), a valid assumption is to consider that all minima occurring inside the DEM deﬁnition domain are irrelevant because almost all European catchment basins are connected to the sea. Therefore the ﬁllhole transformation generalised to grey scale images by a reconstruction by erosion can be used to suppress all interior minima as originally proposed in [29]: FILL(f ) = Rfε (fm ), where fm (x) = f (x) if x lies on the border of f , max(f ) otherwise. Figures 1(a)–(b) illustrate that the reconstruction based minima imposition ﬁlls most of the fragments of the stream so that any subsequent ﬂow simulation process will be unable to follow the stream path indicated by lower elevation values in the initial data. Figures. 1(d)–(e) illustrate another problem occurring when a stream goes through a very narrow valley. Indeed, if it happens that the DEM resolution is too coarse to resolve a narrow valley, the ﬁllhole procedure will ﬁll it upstream creating thereby a large ﬂat region suppressing relevant topography in the valley bottom. The images appearing at the bottom of Fig. 1 show that both problems are solved by the carving procedure. Contrary to the minima imposition by reconstruction, the carving procedure is not a connected algebraic closing but merely an anti-extensive and idempotent operation. In contrast to the ﬂooding simulation introduced in [5, 34] for the computation of watershed boundaries and [12] for the computation of drainage networks, the proposed carving algorithm does not require the prior sorting of the image pixels in increasing order of elevation. Indeed, the sorting is achieved by the priority queue which is initialized by the relevant image minima. A similar approach has already been used for the computation of watersheds [3] and contributing drainage areas [30]. As concerns the computation of the binary mask of all regional minima, it is readily obtained thanks to the fast algorithm proposed by Breen and Jones [4], see also [26, p. 169].

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Fig. 1. Minima imposition: Comparison between procedures based on morphological reconstruction by erosion and the proposed carving. The displayed samples were extracted from a Pan-European DEM at the resolution of 250 m. The markers were deﬁned for both procedures as the regional minima of the input DEM connected to its border, according to [29].

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Lower Complete Transformation

In some applications, it may be useful to transform an input image into a lower complete image, i.e., an image where the only pixels having no neighbour with a strictly lower intensity are those belonging to the regional minima of the image [13]: f is lower complete if and only if ∀p ∈RMIN(f ), ∃p ∈ NG (p) | f (p ) < f (p). For example, a lower complete image is necessary for calculating the steepest slope path linking a pixel to a regional minimum. Indeed, steepest slope directions can be calculated for each pixel of a lower complete image. Another potential application is for watershed transformation because, on a lower complete image, the calculation of the distance function on plateaus advocated in [34] can be skipped. An image can be transformed into a lower complete image by adding an auxiliary relief to the plateaus. This relief is deﬁned as the geodesic distance function calculated from the descending border of the plateau (reference set) using the plateaus as geodesic mask. A fast implementation based on priority queues is detailed in [30]. However, this procedure tends to create parallel ﬂow directions. Garbrecht and Martz [9] proposed an alternative procedure ensuring better ﬂow convergence whereby the topography on the plateau is created by adding the inverse geodesic distance from higher terrain to the geodesic distance from lower terrain may itself contain a plateau. However, as noted by these authors, the topography created on the plateau may itself contain a plateau. A solution to this problem is to deﬁne the topography on the plateau as the geodesic time function [23, 25] using the descending border of the plateaus as marker and the inverse of the geodesic distance from higher terrain as grey scale geodesic mask (see also [24]). This procedure is illustrated in Fig. 2 for a 7 × 7 image containing a plateau. Figure 2b shows the inverse geodesic distance function computed on the plateau pixels having no lower neighbour and starting from the neighbour pixels having a higher elevation. However, contrary to the methodology described by Garbrecht and Martz [9], we do not compute the geodesic distance away from lower elevations but the geodesic time function [25] using the inverse of the geodesic distance (Fig. 2b) as geodesic mask and the descending border as marker image. The resulting geodesic time function is displayed in Fig. 2c. By construction, a ﬂow direction is directly deﬁned for all plateau pixels as illustrated in Fig. 2d. The ﬂow direction of a given point is set as the direction of its 8-nearest neighbour point producing the steepest downward slope. As concerns the computation load, fast algorithms for geodesic distance functions are described in [33] for 8-connected and [22] for Euclidean distance calculations. A fast geodesic time function algorithm based on priority queues is described in [26, pp. 202–202].

4

Extraction of Surface Speciﬁc Neighbourhoods

On a binary image, the extraction of speciﬁc neighbourhood conﬁgurations is based on the hit-or-miss transform: The hit-or-miss transformation, HM T , of a

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Fig. 2. Determination of ﬂow directions on a plateau from an artiﬁcial topography created by computing the geodesic time function from the descending border of the plateau (plateau pixels in bold) using the inverse of the geodesic distance away from higher elevations as geodesic mask. In the image of ﬂow directions (d), the ﬂow direction of pixels marked by the sign is set according to the original values of the image.

set X by a composite structuring element B = (BF G , BBG ) is the set of points, x, such that when the origin of B coincides with x, BF G ﬁts X and BBG ﬁts X c: HM TB (X) = {x | (BF G )x ⊆ X, (BBG )x ⊆ X c } = εBF G (X) ∩ εBBG (X c ). (1) This equation could be extended to grey scale images but in this latter case we would combine the erosion of a grey scale image with an erosion of the complement of this image. In addition, due to the increasingness of the HMT, the stacking of the hit-or-miss transformations of the successive cross-sections of a grey scale image does not deﬁne the subgraph of a grey scale image. However, as illustrated in Fig. 3 for a 1-D signal, when we position B at a given pixel x, BF G matches the cross-sections of f from the level 0 up to a given level which we denote by tF G , while BBG matches the complement of the cross-sections of f from the level tmax + 1 down to a given level which we denote by tBG . For example, in Fig. 3a at position x = 8, BF G matches the cross-sections of f from the level 0 up to the level tF G = 5 while BBG matches the complement of the cross-sections of f from the level tmax down to the level tBG = 3. Depending on whether we constrain the SE component containing the origin to match either the foreground (if O ∈ BF G ) or the background (if O ∈ BBG ) of x, we will obtain two diﬀerent deﬁnitions for the grey tone hit-or-miss transform. The ﬁrst will be referred to as the unconstrained hit-or-miss. Accordingly, the previously mentioned constraint is at the basis of the constrained hit-or-miss. Unconstrained hit-or-miss. The output of the unconstrained hit-or-miss, denoted by U HM T , of a grey scale image f by a composite SE B at a position x is deﬁned as the number of cross-sections CS t (f ) such that BF G at x matches CS t (f ) while, simultaneously, BBG at x matches CS t (f ): [U HM TB (f )](x) = card{t | (BF G )x ⊆ CS t (f ), (BBG )x ⊆ CS t (f )}.

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The unconstrained hit-or-miss transform is illustrated on a 2-D image in Fig. 4. Constrained hit-or-miss. The deﬁnition of the constrained hit-or-miss, denoted by CHM T , involves an additional constrain, considered for each pixel position x. Namely, the SE component containing the origin O must match the foreground F G(x) if O ∈ BF G or the background BG(x) if O ∈ BBG (where F G(x) = Tt≥f (x) (f ) and BG(x) = Tt≤f (x) (f )): (BF G )x ⊆ F G(x), if O ∈ BF G . (BBG )x ⊆ BG(x), if O ∈ BBG . As for a graphical representation and looking back to Fig. 3, we only consider the t-connected components2 of pixels marked by a bullet and which have a nonempty intersection with the t-boundary of the subgraph of f . This happens for x = 13 in Fig. 3a and x = 10 in Fig. 3b. In terms of morphological transformations, when O ∈ BF G , (BF G )x ⊆ F G(x) iﬀ f (x) = [εBF G (f )](x). Accordingly, when O ∈ BBG , (BBG )x ⊆ BG(x) iﬀ f (x) = [δBBG (f )](x). Both deﬁnitions of the grey tone hit-or-miss are equivalent in the binary case and come down to Eq. 1. In the grey scale case, they are equivalent only in situations where the component of B containing the origin is restricted to a single pixel. The constrained hit-or-miss is by deﬁnition more restrictive than the unconstrained one as highlighted by the following ordering relationship: CHM TB ≤ U HM TB . The hit-or-miss (whether unconstrained or constrained) by B is complementary to that by Bc : HM TB = HM TBc . Both types of HM T are non-increasing transformations. From a computational point of view, it can be shown that the proposed HMTs can be deﬁned by performing simple conditional tests on the grey scale erosion by BF G and dilation by BBG . In addition, 2

We call t-connectivity the 1-D dimensional connectivity deﬁned along the discrete t-axis for a ﬁxed value of x, i.e., (x1 , t1 ) is t-connected to (x2 , t2 ) iﬀ x1 = x2 and |t1 − t2 | = 1.

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(d) Threshold of U HM TB (f ) for all nonzero values: Tt>0 [U HM TB (f )].

Fig. 4. Grey scale unconstrained hit-or-miss transform extracting the upper corner of each hexagonal cell appearing in the input image. The threshold of the U HM T image for all non-zero values highlights the performance of the detection of the considered surface speciﬁc points.

these latter deﬁnitions are directly suited to the extension to rank operations, similarly to other morphological transformations [27].

5

Crest Lines by Order Independent Thinning

Recent developments about binary order independent homotopic thinnings [16] can be used for extracting crest lines in grey scale images. The principle for anchored grey scale skeletons are detailed in [17] and brieﬂy summarized hereafter. Let us ﬁrst present a generic deﬁnition of an order dependent grey tone thinning algorithm (where N8< (p) denotes the set of neighbours of p whose values are lower than p): for each pixel p of f if p is simple in f add p to the set SimpleSet of simple pixels for each pixel p ∈ SimpleSet (arbitrary sequential scanning order) if p is simple in f f (p) ← max{f (p ) | p ∈ N8< (p)}

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This algorithm will be referred to as Proc1. At the end of Proc1, the input image f contains a thinned image which is homotopic to the original image. Note also that, in accordance with the anti-extensivity property of any thinning, the output image is always less or equal to the input image. Similarly to binary sequential homotopic thinnings, the resulting image depends on the scanning order of the set SimpleSet of simple pixels. This also happens for all algorithms based on grey tone sequential homotopic thinning [10] [26, p. 142] or topological operators [1, 7]. An order independent thinning can be obtained as follows [17]: First, apply Proc1 with all possible sequential scanning orders; second, set the value of each pixel p to the maximum output value at p. In other words, the output image equals to the point-wise maximum of all possible outputs of Proc1, i.e., for all possible sequential scanning orders of the set SimpleSet of simple pixels. In practice, the notion of order independent thinning allows us to avoid testing all possible conﬁgurations by directly testing whether a given simple pixel will be removed by all possible sequential scanning orders. Skeletons are obtained by thinning the image until idempotence. By considering a set of predeﬁned points called anchor points, we obtain anchored skeletons. The algorithm is much faster if each level set of the grey scale image contain at least one anchor point because the Ronse’s 8-deletability test of a connected component of simple pixels [18] can be skipped. The extraction of crest lines on a digital elevation model using the regional maxima as anchor points is illustrated in Fig. 5. Note that the resulting skeleton is not one pixel thick and may even contain very thick regions similarly to the thick watersheds described in [34]. This is due to particular conﬁgurations of the grey scale values such as those illustrated in Fig. 6 and which has been sampled from Fig. 5b in a region where thick skeletal lines occur. The order independent thinning has set a thick region to the elevation 307. However, this thick region, although mostly surrounded by crest lines, cannot be further thinned (i.e., set to either 1 or 3 in this example) because it is itself leading to a crest line. That is, the pixel appearing in a shaded box is not simple. Thick watershed zones (in the sense that one cannot decide whether the thick region drains to a catchment basin or another) always correspond to thick regions of the grey tone skeleton. Finally, it is worth mentioning that recent work on distance ordered homotopic thinning [32, 15, 31] could be made independent of the order used for processing the pixels at the current distance value, simply by using order independent thinning of each distance level.

6

Conclusion and Perspectives

New developments related to the minima imposition, the lower complete transformation, the hit-or-miss transform, and the extraction of crest lines have been proposed. Beyond practical applications to the processing of Pan-European digital elevation models which are currently investigated [28], we believe that these developments will be of interest to other image analysis problems where the searched features correspond to topographic features. Note that the minima

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Fig. 5. Crest lines extracted by order independent homotopic thinning. Left: Shaded view of a 1 km input digital elevation model of the Pyrenees. Right: Support of the skeleton of the DEM using the regional maxima as anchor points.

Fig. 6. Thick skeletal lines extracted from Fig. 5b. The pixels at intensity 1, 2, and 3 belong to the regional minima of the grey tone skeleton, all other pixels belonging to the skeletal lines. The two pixels marked in bold clearly belong to the crest lines, impeding thereby further order independent homotopic thinning of the thick region.

imposition technique based on carving is not incompatible with that based on reconstruction by erosion. Indeed, it is enough to deﬁne two sets of markers, the ﬁrst containing markers leading to a carving while the second to a reconstruction by erosion, the priority queues ensuring an ordered propagation. Finally, it would be of interest to extend the developments of order independent homotopic thinnings to the computation of watersheds and compare this approach with those based on ﬂooding simulations [34], ﬂow simulations [30], and orders [6]. Extensions of the proposed carving, lower complete, and hit-or-miss transformations to 3-D grey scale images are straightforward. However, additional research will be required to extend the notion of order independent homotopic thinnings for producing 3-D grey scale skeletons and watersheds.

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References [1] G. Bertrand, J.-C. Everat, and M. Couprie. Image segmentation through operators based upon topology. Journal of Electronic Imaging, 6(4):395–405, 1997. [2] S. Beucher. Segmentation d’images et morphologie math´ ematique. PhD thesis, Ecole des Mines de Paris, June 1990. [3] S. Beucher and F. Meyer. The morphological approach to segmentation: The watershed transformation. In E. Dougherty, editor, Mathematical morphology in image processing, pages 433–481. Marcel Dekker, New York, 1993. [4] E. Breen and R. Jones. Attribute openings, thinnings, and granulometries. Computer Vision and Image Understanding, 64(3):377–389, 1996. [5] S. Collins. Terrain parameters directly from a digital elevation model. The Canadian Surveyor, 29(5):507–518, December 1975. [6] M. Couprie and G. Bertrand. Tesselations by connection in orders. In Proc. of Discrete Geometry for Computer Imagery’2000, Uppsala, volume 1953 of Lecture Notes in Computer Science, pages 15–26. Springer-Verlag, 2000. [7] M. Couprie, F. Nivando Bezerra, and G. Bertrand. Grayscale image processing using topological operators. In L. Latecki, R. Melter, D. Mount, and A. Wu, editors, Vision Geometry VIII, volume SPIE-3811, pages 261–272, 1999. [8] H. Digabel and C. Lantu´ejoul. Iterative algorithms. In J.-L. Chermant, editor, Quantitative analysis of microstructures in materials sciences, biology and medicine, pages 85–99, Stuttgart, 1978. Dr. Riederer-Verlag GmbH. [9] J. Garbrecht and L. Martz. The assignment of drainage direction over ﬂat surfaces in raster digital elevation models. Journal of Hydrology, 193:204–213, 1997. [10] V. Goetcherian. From binary to grey tone image processing using fuzzy logic concepts. Pattern Recognition, 12:7–15, 1980. [11] R. Haralick. Ridges and valleys on digital images. Computer Vision, Graphics, and Image Processing, 22:28–38, 1983. [12] D. Mark. Automated detection of drainage networks from digital elevation models. Cartographica, 21:168–178, 1984. [13] F. Meyer. Skeletons and perceptual graphs. Signal Processing, 16:335–363, 1989. [14] F. Meyer and S. Beucher. Morphological segmentation. Journal of Visual Communication and Image Representation, 1(1):21–46, September 1990. [15] C. Pudney. Distance-ordered homotopic thinning: A skeletonization algorithm for 3D digital images. Computer Vision and Image Understanding, 72(3):404– 413, December 1998. [16] V. Ranwez and P. Soille. Order Independent Homotopic Thinning. In G. Bertrand, M. Couprie, and L. Perroton, editors, Proc. of Discrete Geometry for Computer Imagery’99, volume 1568 of Lecture Notes in Computer Science, pages 337–346. Springer-Verlag, 1999. [17] V. Ranwez and P. Soille. Order independent homotopic thinning for binary and grey tone anchored skeletons. Pattern Recognition Letters, 2002, Publication pending. [18] C. Ronse. A topological characterization of thinning. Theoretical Computer Science, 43:31–41, 1986. [19] P. Salembier and J. Serra. Flat zones ﬁltering, connected operators, and ﬁlters by reconstruction. IEEE Transactions on Image Processing, 4(8):1153–1160, August 1995. [20] J. Serra. Image analysis and mathematical morphology. Academic Press, London, 1982.

186

P. Soille

[21] J. Serra and P. Salembier, editors. Mathematical morphology and its applications to signal processing, 1993. Universitat Polit`ecnica de Catalunya, Barcelona. [22] P. Soille. Spatial distributions from contour lines: An eﬃcient methodology based on distance transformations. Journal of Visual Communication and Image Representation, 2(2):138–150, June 1991. [23] P. Soille. Morphologie math´ ematique : du relief a ` la dimensionalit´ e —Algorithmes et m´ethodes—. PhD thesis, Universit´e catholique de Louvain ; en collaboration avec l’Ecole des Mines de Paris, February 1992. [24] P. Soille. Generalized geodesic distances applied to interpolation and shape description. In J. Serra and P. Soille, editors, Mathematical Morphology and its Applications to Image Processing, pages 193–200. Kluwer Academic Publishers, 1994 [25] P. Soille. Generalized geodesy via geodesic time. Pattern Recognition Letters, 1235–1240, December 1994 [26] P. Soille. Morphological image analysis: Principles and Applications. SpringerVerlag, Berlin, New York, 1999. Second extended edition to appear in 2002, see also http://ams.egeo.sai.jrc.it/soille [27] P. Soille. On morphological operators based on rank ﬁlters. Pattern Recognition,2002. [28] P. Soille. Carving and adpative drainage enforcement of grid digital elevation models. Water Resources Research, Submitted. [29] P. Soille and M. Ansoult. Automated basin delineation from digital elevation models using mathematical morphology. Signal Processing, 20: 171–182, June 1990. [30] P. Soille and C. Gratin. An eﬃcient algorithm for drainage networks extraction on DEMs. Journal of Visual Communication and Image Representation, 5(2): 181–189, June 1994. [31] S. Svensson, G. Borgefors, and I. Nystr¨ om. On reversible skeletonization using anchor-points from distance transforms. Journal of Visual Communication and Image Representation, 10:379–397, 1999. [32] H. Talbot and L. Vincent. Euclidean skeletons and conditional bisectors. In P. Maragos, editor, Visual Communications and Image Processing, volume SPIE1818, pages 862–876, 1992. [33] B. Verwer, P. Verbeek, and S. Dekker. An eﬃcient cost algorithm applied to distance transforms. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(4):425–429, April 1989. [34] L. Vincent and P. Soille. Watersheds in digital spaces: An eﬃcient algorithm based on immersion simulations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(6):583–598, June 1991.

Morphological Operations in Recursive Neighbourhoods Pieter P. Jonker Pattern Recognition Group, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands [email protected]

Abstract. This paper discusses the use of Recursive Neighbourhoods in Mathematical Morphology. Its two notable applications are the recursive erosion / dilation, as well as the detection of foreground-background changes to be used in skeletonization. The benefit of the latter over an extension of the neighbourhood or the use of sub-cycles is emphasized. Two applications are presented that use the recursive neighbourhood in a 3D surface and a 3D curve anchor-skeleton variant.

1 Introduction In previous papers [3], [4], [5], we have described a general principle for morphological operations on cubic tessellated binary images X N . For the sake of clarity we will briefly repeat this in sections 1 and 2, and in examples use images of dimensions 2 and 3. In section 3 we elaborate on the principle of recursive neighbourhoods. The use of recursive neighbourhoods is not new, however, its principle of work and its possibilities are often not well understood. In section 3 we show that from the four alternatives to circumvent the problem of detecting the topology change of a two pixel/voxel thick N structure in a 3 neighbourhood, as is necessary in skeletonization, the recursive neighbourhood approach is generally the fastest. The recursive neighbourhood detects change, either foreground changed to background or background changed to foreground. This change detection can be fruitfully used in a number of applications, but also give rise to problems, e.g., excessive protrusions in skeletons, when the recursive neighbourhood is not correctly used. In chapter 4 we present two examples of its use. The essence of both examples is the use of a surface, and a curve skeleton, forced through anchor points, whereas its surface and curve protrusions are recursively eroded due to the use of the recursive neighbourhood. From [3], [4], [5], we derived that using a recursive neighbourhood is based on scanning an image with a set of masks that comprises a structuring element S. The masks contain foreground, background and don’t-cares. On each element of the image X an inexact match ( ≅ ) between all masks of the set and the neighbourhood extracted A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 187–196, 2002. © Springer-Verlag Berlin Heidelberg 2002

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from the image can be performed, whereas the result written to the output image Y is the union of all matches. A second input image Z can be used in the same match to perform dyadic operations. Dyadic operations can be used to locally enable or disable the morphological operation, using the points from a second input image. To elaborate on this: Let x, y, and s be the elements of images X, Y, and S. Let S, e.g., be a 3 x 3 structuring element and M k an equivalent 3 x 3 neighbourhood around pixel x k . For binary morphology, the transformation Y ← X ≅ S , is informally: If for any pixel x k in an input image X, its neighbourhood M k matches inexactly with a structuring element S, the pixel y k of output image Y is set to one. If M k doesn’t match S, y k is set to zero. In the inexact neighbourhood match the foreground pixels in S should match with foreground pixels in M k at the same positions AND the background pixels in S should match with background pixels in M k at the same positions, whereas in the don’t care positions of S a match is not required. This inexact match can be extended to: If for any pixel x k in an input image X, its neighbourhood M k matches one mask SSi from a set of masks SS the pixel y k of output image Y is set to one, else to zero. Meaning that the union of all mask matches is taken. A further extension is: If for any pixel x k in an input image X, its neighbourhood M k matches any mask SSi from a given set of masks SS the pixel y k of output image Y is set to one, else set to zero, OR if its neighbourhood matches any mask SiR from a given set of masks SR , the pixel y k is set to zero, else to one. The image transformation Y ← X ≅ S is now implemented with a set of masks S consisting of a subset SS (the SET-masks) and a subset SR (the RESET-masks), one of which may be empty. This means that either a pixel y k is set to zero, if one of the RESET masks fits, or the pixel is set to one, if one of the SET mask fits, where the SET masks are chosen to dominate over the RESET masks. A second input image Z can be used to locally enable/disable the transformation, yielding a dyadic operation. If a "mask-bit" z of a mask of the set S is set to don’t care, the transformation is enabled. If z is do-care, then if the pixel z k of Z matches with z , the operation is disabled, else enabled. Operations that use Z to locally mask-off, or insert seeds, are the propagation operation and the anchor-skeleton. Finally, an operation on an image can be done by performing a transformation with a mask-set once, twice or more, or until the image is idempotent (does not change) under the operation. In many cases practical use can be made of the intermediate results in the output image. This is called (spatial) recursion. For instance, if the transformation is performed by a raster scan over the image, i.e., from top-left to bottom-right, the fact that some neighbours of pixel y k have already obtained a new value can be utilized. For this purpose, the neighbourhood M k of pixel x k is as well as the structuring element S is extended. Both the Local Neighbourhood (LN) and the Recursive Neighbourhood

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(RN) can be used in a neighbourhood matching procedure, referred to as local neighbourhood operations (LNO) and recursive neighbourhood operations (RNO).

LN

x 4k x 3k x 2k

y r3k y r2k y r1k

x 5k x 0k x1k

y r4k y k

x6k x 7k x 8k RN

zk

z

Y

X r k

Z

O Fig. 1. Drawing conventions for a neighbourhood match. Here shown for 2D images and 3 x 3 neighbourhoods. The LN is taken from input image X, the RN from output image Y. For dyadic operations a second input image Z is used.

Figure 1 shows the matching process with a single mask from a set S, whereas the neighbourhood M k is extracted from the three different images. Note that when using a software raster scan over the image, it is beneficial for RNOs to scan in all odd scans from top-left to bottom-right and in all even scans from bottom-right to top-left. In this case, the RN should be transposed to match with the scan direction. RESET

Dilate26cct

RESET

RESET

Detect 26cct

SET

SET

SET

Erode26cct Erode18cct Erode6cct

Fig. 2. Structuring elements for simple operations in X 3 . Opaque elements are don’t cares, light grey are background elements, and dark grey are foreground elements.

Figure 2 shows some structuring elements or mask-sets for simple Local Neighbourhood Operations in X 3 , while Figure 3 shows an example of a dyadic Recursive Neighbourhood Operation; the propagation operation, a recursive conditional dilation: Objects in an image are recursively dilated (the first mask), wherever foreground in image Z and background in X was found (the second mask).

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Skeletonization, Shape Primitives

RESET

RESET

Erosions can be described as a match on foreground area in X 2 and on foreground volume in X 3 . Skeletonization can be seen as conditional erosion [3]. Figure 4 shows that in X 3 a volume is eroded to a curved surface -the surface skeleton-, where after the surface is eroded to a space curve -the curve skeleton-. The condition for the erosion is that surfaces, or curves, should not be eroded. Those conditions are also known as shape primitives in X N . The sets of shape primitives and how they can be found are described in [5]. In 3D one set is called Surf26, which represents the Propagation26cct surface primitives: On all possibilities of a curved 3 Fig. 3. A dyadic RNO in X 3 ; surface to interesect a 3 neighbourhood, one of these masks (or a rotated and/or mirrored version) the propagation operation will hit. The set Curv26 represents likewise the space curve primitives: On all possibilities of a 3 space curve to interesect a 3 neighbourhood one of these masks (or a rotated and/or mirrored version) will hit. Iterating over an image X 3 with the erosion mask Erode26cct of Figure 2 and a set of surface primitives (Surf26) and the set of curve primitives (Curv26) yields a skeleton. Erode26cct erodes surfaces from volumes (it hits only on the core of the volumes, not on their boundaries). The set Surf26 detects surfaces and thus may be used to prevent the erosion of surfaces (the masks only hit on the core of the surface, not on the surface boundaries). The set Curv26 detects curves and can be used to prevent the erosion of curves (the masks only hit on the kernel of the curves). Consequently, as volumes, surfaces and curves cannot be eroded from their kernels, they are eroded from their boundaries. So only closed surfaces and curves will remain.

a)

b)

c)

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Fig. 4. Original1, surface skeleton, curve skeleton, and the topological kernel.

To prevent the erosion of boundaries, from the set Curv26 a set Curv26e can be derived that contains all surface edge situations. Similarly, a set Curve26e can be made from the set Curv26 containing all curve end situations [4],[5]. 1

Courtesy Dr. K. Katada, Fujita Health University Japan

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191

On the Necessity and Use of Recursive Neighbourhoods

A recursive neighbourhood can be used for a number of reasons. First it can be used to erode or dilate objects or background in one or two passes over the image. For instance the masks of Figure 3 can be used to quickly select objects. With objects stored in image X and seed voxels stored in image Z, starting from the seed voxel, a wave front dilation starts over objects that are connected to the seed. Figure 5 shows how such dilation evolves in a 2D situation. The dilation mask (5a) is assumed to make a raster scan over the image from top-left to bottom down. A part of an image X is shown in 5b…5e. In 5b and 5c the mask does not fit and hence the pixels are reset to foreground in the output image Y. When the scan runs over the image at one row lower, shown in Figure 5d and 5e, in situation 5d the pixel swaps value from background to foreground but the swap in 5e depends on which neighbourhood is used. Using the recursive neighbourhood (partially using output image Y), the pixels swaps value, as the mask does not match, but when using the Local Neighbourhood (only using input image X) the mask fits and the pixel remains background. Of course, an unconditioned recursive dilation in a downward scan followed one in an upward scan over the image would fill the whole image. Likewise, a recursive erosion would erode a convex object in two of such scans over the image. Hence its application is found in conditional erosions and dilations. RESET

a)

b)

c)

d)

e)

Fig. 5. Recursive dilation

Secondly, the Recursive Neighbourhood can be used to detect change within a skeletonization procedure2. SET

a)

b)

c)

d)

e)

Fig. 6. Topology check using the local (b, c, d) or recursive (b,c,e) neighbourhood

The problem with a topology check (e.g., a mask) that works on a 3N neighbourhood is that it cannot detect two element (pixel, voxel, …) thick structures. Figure 6 shows a situation in which the mask of Figure 6a is used to detect a straight single pixel thick 2

The first one known to me that used it for this purpose was Hilditch (1969) [2]

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vertical line. This curve primitive (6a) is one of the masks that are used as topology check for a 2D skeleton. A raster scan is assumed. The sequence 6b, c, d shows what happens if the local neighbourhood is used for a topology check. 6b, c, e shows what happens if the recursive neighbourhood is used. In situation 6d the topology is broken, in 6e it is preserved as now the output image is partly used in the mask match. An alternative to the use of the recursive neighbourhood is to use only the Local Neighbourhood, but to erode from one direction at a time -explicitely or implicitely[1]. This technique is called sub-cycling. In the example of Figure 6, the tests of 6c and 6d would then have been done in subsequent cycles through the image, first an erosion from the left, and secondly an erosion from the right. The same effect can be obtained by first processing all odd lines of the image followed by the processing of all even lines in the image. This is often done on parallel hardware such as massively parallel processor arrays [7]. In literature [10], often the terminology sequential algorithm versus parallel algorithm is used for the use of the recursive neighbourhood versus the local neighbourhood with sub-cycles. However, both approaches can be implemented on sequential as well as parallel machines. A third method is the extension of the neighbourhood -for instance in 2D to 3 x 4- to detect the two element thick structures [8]. This lets the number of masks (or topology tests), as well as the number of image elements to be tested grow considerably. When a recursive neighbourhood can be addressed, this is the fastest method, as it diminishes the number of tests within the neighbourhood, the number of topology tests as well as the total number of cycles through the image. When the skeleton is made, the topology checks can be split into checks on the interior of the object and checks on the boundary. In 3D playing with the subsets: erosion mask (Erode26cct), surface detection mask-set (Surf 26), curve detection mask-set (Curv26), the mask-sets for surface ends (Surf26e) and curve ends (Curv26e) skeleton variants can be made. The set {Erode26cct, Surf26, Surf26e, Curv26} is used for the surface skeleton. The set {Erode26cct, Surf26, Curv26, Curv26e} was used to obtain the curve skeleton. The set {Erode26cct, Surf26, Curv26} was used to obtain the last skeleton of Figure 4. The recursive neighbourhood should be used, only for the Surf 26 and Curv26 sets and not for Erode26cc and the Surf26e and Curv26e sets. If the recursive neighbourhood is used for the erosion mask, the erosion will not be performed boundary by boundary like peeling an onion skirt, but the erosion will immediately propagate over the object. The final skeleton will not be on the medial axis, but will lay on the bottoms of the objects, when raster scanning. If the recursive neighbourhood and/or sub-cycling method is also used for the object boundary conditions, e.g., Surf26e and Curv26e, this will lead to the sprouting of spurious protrusions. This can be best explained when using the sub-cycle technique. Due to the first subcycle, locally a noisy boundary may be formed, which may be cancelled out in the subsequent sub-cycle by an erosion from another direction. However, a boundary detection condition mask, may decide that this is the beginning of a protrusion and decides to keep it. So object core conditions should be treated differently from object boundary conditions, to avoid excess sprouting of protrusions. But even then, in the object core conditions, e.g., (Surf26, Curv26), the recursive neighbourhood should

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only be checked to verify if a foreground is changed into background, where a background is expected in the mask. This is shown in Figure 7. SET

a)

b)

c)

d)

Fig. 7. Using the recursive neighbourhood for foreground

Suppose the mask of Figure 7a is used to detect a skeleton curve in 2D. The pixel in 7b is reset to background as the mask does not fit. The pixel in 7c is correctly set to foreground; the mask fits when foreground in the mask (the North pixel) is checked in the input image X and the background in the mask (the West pixel) is verified using the output image Y. 7d Shows what happens if both are verified using the output image Y. The topology is broken. The use of the full recursive neighbourhood in the sense of Figure 7b, can be profitably when a skeleton without end conditions is made. For instance in a skeleton without end-voxel conditions, yielding the topological kernel, as shown in Figure 4d, the space curve conditions are verified in the full recursive neighbourhood. The benefit of this is that the non-closed curves are eroded recursively, i.e., in a few passes through the image as the erosion propagates over the skeleton branches within a single cycle through the image. This makes this skeleton fast.

4

Advanced Examples

In this section two examples are presented of applications that make use of 3D morphological operations, most notable of variants of the 3D skeleton operator and the propagation operation. Both use the recursive neighbourhood in various ways. Consider Figure 8. The original image3 is taken by an acquisition system -based on multiple cameras- that produces a stream of 3D object data [11]. As the system produces imperfect data, each image of the stream has to be processed before rendering texture and colour onto it. The lower images in Figure 8 are cross sections of the legs. The procedure used to clean-up the original is the following sequence: 1) dilate the object with a 26 connected contour 2) dilate the object with a 6 connected contour, twice 3) erode a 26 connected contour from the object 4) erode a 6 connected contour from the object

3

Courtesy Prof. Matsuyama, Kyoto University Japan This is an old stream; currently his 3D streams have a far better quality [11]

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Aerobic girl

original

Aerobic girl processed

5) iteratively erode a 6 connected contour from the object, (Erode6cct) but use also the mask-sets Surf26 and Surf26e, thus performing a few cycles of the surface skeleton. Anchor the original object into the skeleton; with a logic OR operation, after each skeleton cycle the original data points are re-inserted, thus forcing the skeleton through the original data points. Step 1 through 4 performs a 3D closing operation. Due to step 5 the result is a closed single voxel thick surface that goes through the original data points. Like alternating 4 and 8 connected in 2D, the alternation of the 26 and 6 connected erosions/dilations in 3D leads to a better erosion/dilation metric [3]. 6) propagate the edge of the image into the image (Propagation26cct), stop at the object boundary and invert the result. This makes the object solid. 7) extract the contour of the solid object. Steps 6 and 7 remove all inner data points of the object. 8) take a surface skeleton without boundary conditions using {Erode26cct, Surf26}. This finds the closed surface contour only, i.e., it removes sprouting surfaces. The full recursive neighbourhood is used in Surf26 and consequently the boundary surfaces are eroded recursively and two skeleton cycles are sufficient. In Robot Soccer [6],[9] two teams of each 4 autonomous robots play soccer in a field of 5 x 10 meter. One of the issues is to quickly plan collision free paths for the robot. Moreover, if a robot should play together with a teammate, his presence is required on a certain point in space AND time, e.g., to intercept a passed ball. A 3D image X 3 represents the universe of the robot with two space dimensions (x, y) and one time dimension (t). Note that if the orientation of the robot should also be taken into account, the problem should be solved in X 4 (x, y, , t). Figure 9 shows 12 pictures of a soccer field. The soccer field is in this example divided into cubic voxels of 0.3 x 0.3 x 0.6 [m·m·s]. The robot speed is {0, 1, 2} · [0.5 m/s]. The image size [x, y, t] is

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17 x 35 x 35 voxels. The origin of the field is in the centre of the image at t = 0. Assume an attacker robot with ball wants to score, while the goal is defended by two defender robots. The attacker is on point (-6,-6,0) and tries to get to point (6,-2,34) to score, and the defenders are at points (15,7,0) and (10,-6,0) and are perceived by the attacker robot to head for the points (-10,6,34) and (0,0,34). The path planning procedure of the attacker robot consists of the following steps. y t

Fig. 9. Search space

moving objects

reduced search space

collision free path

1) Recursively dilate the start point towards the positive time direction. The resulting cone indicates the points in the field that can be reached by the robot in positive time direction, when using speed 1. See Figure 10a. RESET

SET Fig. 10. a) Fully recursive neighbourhood (reset) mask to propagate a cone in space-time and b) a fully recursive neighbourhood (set) mask to erode objects in space-time

2) Recursively dilate the goal point towards the negative time direction 3) AND the results of 1) and 2). The top row, left column of Figure 9 shows the result, the initial search space. 4) From the two defender robots a speed and heading is perceived by the vision system of the attacker robot. It assumes that for the duration of the path those robots will linearly continue their path. The top row, second column in Figure 9 shows two objects, the linear paths of the defender robots in space-time, dilated one step, to introduce a safety margin. 5) The EXCLUSIVE OR of search space and the moving object images yields the reduced search space of the top row, third column of Figure 9. 6) The curve skeleton without end-curve conditions {Surf26, Curv26}, with start and endpoint of the attacker robot anchored into the skeleton, yield the final collision free path. Note that both Surf26 and Curv26 are used in the full recursive neighbourhood. This enables the fast erosion of sprouting surfaces and space curves. A problem in the planning is that the path to be found may not

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run backwards in time. To prevent this, first all skeleton scans over the image are only applied in the forward time direction. Secondly, the erosion mask is adapted, so that it does not erode backward in time (see Figure 10b). Thirdly, masks in Curv26 that have configurations that allow connectivity’s backward in time are omitted. Thus paths that run backward in time are never found, as their topology is not preserved. 7) As the curve skeleton finds all possibilities to go from start to goal, a simple algorithm is: go left at a branch point if y > 0, right if y < 0, leading the attack over the wings. Alternatively, the distance can be propagated over the branches, to find the shortest of the branches. 8) If no fixed end-time tend is assumed for the attacker robot, tend can be iteratively reduced, until no path is found anymore. The three rows of Figure 9 show the paths at end-times tend = 34, tend = 22 and finally tend = 16, the minimum time for a collision free path. A start value for the end-time is tend=xend-xstart+yend-ystart. 9) For tend = 34, a path is found in 84 msec on a PIII, 600 Mhz .

References 1.

Arcelli C., Cordella L., Levialdi S. (1975) Parallel thinning of binary pictures. Electronic letters 11: pp 140-149 2. Hilditch C.J. (1969) Linear Skeletons from Square Cupboards. In: B. Meltzer and D. Mitchie (eds.), Machine Intelligence, Vol. 4, Edinburgh University Press, 404-420 3. Jonker P.P. (1992) Morphological Image Processing: Architecture and VLSI design. Kluwer Dordrecht, ISBN 90-2012766-7 4. Jonker P.P. (2000), Morphological Operations on 3D and 4D Images: From Shape Primitive Detection to Skeletonization. Proc. 9th Int. Conf., DGCI 2000 (Uppsala, Dec.13-15), Lecture Notes in Computer Science, vol. 1953, Springer, Berlin, 2000, 1371-391. 5. Jonker P.P. (2002), Skeletons in N dimensions using Shape Primitives, Pattern Recognition Letters, April 2002. 6. P.P. Jonker, J. Caarls, and W. Bokhove (2001), Fast and Accurate Robot Vision for Vision based Motion Proc. 4th Int. Workshop on Robocup (Melbourne, Australia, Aug.31-Sep.1, Springer Verlag, Berlin, 149-158. 7. Kyo, S., Okazaki, S., Fujita, Y., Yamashita, N (1997), A Parallelizing Method for Implementing Image Processing Tasks on SIMD Linear Processor Arrays, Proceedings of the IEEE workshop on Computer Architecture for Machine Perception (CAMP 97): 180-184 8. Ma C.M. (1994) On Topology Preservation in 3D Thinning. CVGIP-IU, 59(3), 328-339 9. www. robocup.org , www.robocup.nl 10. Rosenfeld A, Pfalz J.L. (1966) Sequential operations in digital picture processing. Journal of the ACM, 13(4): 471-494 11. Wu X, Wada T, Matsuyama T (2001) Real-time Active 3D Object Shape Reconstruction for 3D Video. Proc. of the 4th Int. Workshop on Cooperative Distributed Vision, March 22-24, Kyoto, Japan: 455-474. [email protected]

Computing the Diameter of a Point Set Gr´egoire Malandain and Jean-Daniel Boissonnat INRIA, 2004 route des lucioles, BP 93, 06 902 Sophia-Antipolis Cedex, France {gregoire.malandain,jean-daniel.boissonnat}@sophia.inria.fr

Abstract. Given a ﬁnite set of points P in Rd , the diameter of P is deﬁned as the maximum distance between two points of P. We propose a very simple algorithm to compute the diameter of a ﬁnite set of points. Although the algorithm is not worst-case optimal, it appears to be extremely fast for a large variety of point distributions.

1

Introduction

Given a set P of n points in Rd , the diameter of P is the maximum Euclidean distance between any two points of P. Computing the diameter of a point set has a long history. By reduction to set disjointness, it can be shown that computing the diameter of n points in Rd requires Ω(n log n) operations in the algebraic computation-tree model [PS90]. A trivial O(n2 ) upper-bound is provided by the brute-force algorithm that compares the distances between all pairs of points. In dimensions 2 and 3, better solutions are known. In the plane, it is easy to solve the problem optimally in O(n log n) time. The problem becomes much harder in R3 . Clarkson and Shor gave a randomized O(n log n) algorithm [CS89]. This algorithm involves the computation of the intersection of n balls (of the same radius) in R3 and the fast location of points with respect to this intersection. This makes the algorithm less eﬃcient in practice than the brute-force algorithm for almost any data set. Moreover this algorithm is not eﬃcient in higher dimensions since the intersecd tion of n balls of the same radius has size Θ(n 2 ). Recent attempts to solve the 3-dimensional diameter problem led to O(n log3 n) [AGR94,Ram97b] and O(n log2 n) deterministic algorithms [Ram97a,Bes98]. Finally Ramos found an optimal O(n log n) deterministic algorithm [Ram00]. All these algorithms use complex data structures and algorithmic techniques such as 3-dimensional convex hulls, intersection of balls, furthest-point Voronoi diagrams, point location search structures or parametric search. We are not aware of any implementation of these algorithms. We suspect that they are very slow in practice compared to the brute-force algorithm, even for large data sets. Some of these algorithms could be extended in higher dimensions. However, this is not worth trying since the data structures they use have sizes that depend exponentially on the dimension: e.g. the size of the convex hull of n points of Rd d can be as large as Ω(n 2 ). Our algorithm works in any dimension. Moreover, it does not construct any complicated data structure; in particular, it does not require that the points are A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 197–208, 2002. c Springer-Verlag Berlin Heidelberg 2002

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in convex position and therefore does not require to compute the convex hull of the points. The only numerical computations are dot product computations as in the brute-force algorithm. The algorithm is not worst-case optimal but appears to be extremely fast under most circumstances, the most noticeable exception occuring when the points are distributed on a domain of constant width, e.g. a sphere. We also propose an approximate algorithm. Independently, Har-Peled has designed an algorithm which is similar in spirit to our algorithm [Har01]. We compare both methods and also show that they can be combined so as to take advantage of the two.

2

Deﬁnitions, Notations, and Geometric Preliminaries

We denote by n the number of points of P, by h the number of vertices of the convex hull of P, and by D the diameter of P. δ(·, ·) denotes the Euclidean distance, and δ 2 (·, ·) the squared Euclidean distance. A pair of points of P is called a segment. The length of a segment pq is the euclidean distance δ(p, q) between p and q. A segment of length D is called maximal. For p ∈ P, F P (p) denotes the subset of the points of P that are at maximal distance from p. The segment joining two points p and q is called a double normal if p ∈ F P (q) and q ∈ F P (p). If pq is a maximal segment, pq is a double normal. The converse is not necessarily true. Observe that the endpoints of a maximal segment or of a double normal belong to the convex hull of P. Observe also that, if the points are in general position, i.e. there are no two pairs of points at the same distance, the number of double normals is at most h/2. B(p, r) denotes the ball of radius r centered at p, Σ(p, r) its bounding sphere. The ball with diameter pq is denoted by B[pq] and its boundary by Σ[pq]. Since the distance between any two points in B[pq] is at most δ(p, q), we have: Lemma 1 If p, q ∈ P and if pq is not a maximal segment, any maximal segment must have at least one endpoint outside B[pq]. As a corollary, we have: Lemma 2 If p, q ∈ P and if P \ B[pq] = ∅, pq is a maximal segment of P and δ(p, q) is the diameter of P.

3

Computation of a Double Normal

Algorithm 1 below repeatidly computes a furthest neighbour of a point of P until a double normal DN is found. To ﬁnd a furthest neighbour of p ∈ P, we simply compare the distances between p and all the other points in P (F P scan). Point p is then removed from P and won’t be considered in further computations.

Computing the Diameter of a Point Set 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

199

procedure DoubleNormal( p, P ) // p is a point of P ∆20 ← 0 i ← 0 repeat // F P scan increment i ∆2i ← ∆2i−1 P ← P \ {p} // remove p from P from any further computation ﬁnd q ∈ F P (p), i.e. one of the furthest neighbours of p if δ 2 (p, q) > ∆2i then ∆2i ← δ 2 (p, q) and DN ← pq p←q until (∆2i = ∆2i−1 ) return DN Algorithm 1: Computes a double normal.

Lemma 3 Algorithm 1 terminates and returns a double normal. Proof. ∆i can only take a ﬁnite number of diﬀerent values and strictly increases: this ensures that the algorithm terminates. After termination (after I iterations) we have q ∈ F P (p) and all the points of P belong to B(p, δ(p, q)). Since ∆I−1 = δ(p, q), all the points of P belong also to B(q, δ(p, q)) and therefore p ∈ F P (q). ✷ After termination of Algorithm 1, the original set P has been replaced by a strictly subset P since some points have been removed from P (line 6 of algorithm 1). By construction, the returned segment pq is a double normal of the reduced set P (lemma 3), and it is also a double normal of the original set P. Lemma 4 The only numerical operations involved in Algorithm 1 are comparisons of squared distances. Lemma 5 Algorithm 1 performs at most h F P scans and takes Θ(nh) time. Proof. The upper bound is trivial since all the points q that are considered by Algorithm 1 belong to the convex hull of P and all points q are distinct. As for the lower bound, we give an example in the plane, which is suﬃcient to prove the bound. Consider a set of 2n + 1 points p0 , . . . , p2n placed at the vertices of a regular polygon P (in counterclockwise order). For i > 0, we slightly move the pi outside P along the ray Opi by a distance εi for some small ε < 1. Let pi be the perturbed points It is easy to see that the farthest point from pi is always pi+n mod (2n+1) except for pn+1 . Therefore, the algorithm will perform F P scans starting successively at pσ0 , . . . , pσ2n+1 where σi = i × n (modulo 2n + 1). ✷ Although tight in the worst-case, the bound in lemma 5 is very pessimistic for many point distributions. This will be corroborated by experimental results.

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Iterative Computation of Double Normals

Assume that Algorithm 1 has been run and let Q = P \ B[pq]. If Q = ∅, pq is a maximal segment and δ(p, q) is the diameter of P (lemma 2). Otherwise, we have to determine whether pq is a maximal segment or not. Towards this goal, we try to ﬁnd a better (i.e. longer) double normal by running Algorithm 1 again, starting at a point in Q rather than in P, which is suﬃcient by lemma 1. Although any point in Q will be ﬁne, experimental evidence has shown that choosing the furthest point from Σ[pq]1 is usually better. Algorithm 2 below repeats this process further until either Q becomes empty or the current maximal distance ∆ does not increase. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13:

∆2 ← 0 stop ← 0 pick a point m ∈ P repeat // DN scan DoubleNormal(m, P) // yields a double normal pq of length δ(p, q) if δ 2 (p, q) > ∆2 then ∆2 ← δ 2 (p, q) and DN ← pq Q ← P \ B[pq] if Q = ∅ then ﬁnd m ∈ Q a furthest point from Σ[pq] else stop ← 1 // terminates with Q = ∅. until Q = ∅ or stop = 1 return DN ← pq, ∆2 ← δ 2 (p, q) Algorithm 2: Iterated search for double normals.

Lemma 6 Algorithm 2 can be implemented so that the only numerical computations are comparisons of dot products of diﬀerences of points. Lemma 7 Algorithm 2 performs O(h) DN scans. Its overall time-complexity is O(nh). Proof. The ﬁrst part of the lemma comes from the fact that the algorithm enumerates (possibly all) double normals by strictly increasing lengths. Let us prove now the second part of the lemma. Each time Algorithm 1 performs a F P scan starting at a point p (loop 3-11), p is removed from further consideration (line 6). Moreover, except for the ﬁrst point p to be considered, all these points belong to the convex hull of P. It follows that the total number of F P scans is at most h + 1. Since each F P scan takes O(n) time, we have proved the lemma. ✷ 1

This point is the furthest point from

p+q 2

outside B[pq].

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5

201

Diameter Computation

Assume that Algorithm 2 terminates after I iterations. Since, at each iteration, a new double normal is computed, the algorithm has computed I double normals, noted pi qi , i = 1, . . . , I, and we have δ(p1 , q1 ) < . . . < δ(pI−1 , qI−1 ).. Each time Algorithm 1 is called, some points are removed from the original data set. We rename the original data set P (0) and denote by P (j) the set of points that remain after the j-th iteration, i.e. the one that computes pj qj . Hence set P (i) is strictly included in P (i−1) . Moreover, each segment pi qi is a double normal for all the sets P (j) , j = i − 1, . . . , I 2 . It is easily seen that, at each iteration j, the length of the computed double normal pj qj is strictly greater than the distances δ(x, F P (x)) computed so far, or equivalently, than the lengths of all the segments in P \ P (j) × P since Algorithm 1 removed the corresponding x from P . When Algorithm 2 terminates, we are in one of the two following cases : Case 1 : δ(pI , qI ) > δ(pI−1 , qI−1 ) and Q = P (I) \ B[pI qI ] = ∅. In this case, pI qI is a maximal segment of P: by lemma 2, it is a maximal segment of P (I) , and, as mentionned above, no segment with an endpoint in P \ P (I) can be longer. Case 2 : δ(pI , qI ) ≤ δ(pI−1 , qI−1 ). In this case, P (I−1) \ B[pI−1 qI−1 ] was not empty before the computation of [pI , qI ]. We have to determine whether pI−1 qI−1 is a maximal segment or not. Thanks to lemma 1, if a longer double normal exists, one of its endpoints lies in P (I) \ B[pI−1 qI−1 ]. If this last set is empty, which is checked by Algorithm 3, pI−1 qI−1 is a maximal segment of P. Required: P (I) and pI−1 qI−1 (provided by Algorithm 2) 1: Q ← P (I) \ B [pI−1 qI−1 ] 2: if Q = ∅ then 3: pI−1 qI−1 is a maximal segment of P Algorithm 3: Checks whether Q = P (I) \ B[pI−1 qI−1 ] = ∅.

If Q = P (I) \ B[pI−1 qI−1 ]

= ∅, we have to check whether there exists a maximal segment with an endpoint in this set. To search for such maximal segments, we propose two methods. For clarity purpose, we will write P instead P (I) in the following. 5.1

Exhaustive Search Over Q × P

The ﬁrst method (Algorithm 4) simply considers all segments in Q × P. 2

Strictly speaking, as pi and qi do not belong to P (j) , we should say that pi qi is a double normal for all the sets P (j) ∪ {pi , qi }, j = i − 1, . . . , I.

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Required: ∆2 (provided by Algorithm 2) and Q (provided by Algorithm 3) 1: if Q = ∅ then // Exhaustive search with an endpoint in Q 2: for all points pi ∈ Q do 3: for all points pj ∈ P do 4: if δ 2 (pi , pj ) > ∆2 then 5: ∆2 ← δ 2 (pi , pj ) 6: return ∆2 Algorithm 4: Exhaustive search over Q × P.

5.2

Reduction of Q

As it might be expected and is conﬁrmed by our experiments, the observed total complexity is dominated by the exhaustive search of the previous section. It is therefore important to reduce the size of Q. For that purpose, we propose to reuse all the computed segments pi qi , i = 1, . . . , I − 2, and pI qI . Principle. Assume that we have at our disposal an approximation ∆ of the diameter of set P and a subset Q ⊂ P that contains at least one endpoint of each maximal segment longer than ∆ (plus possibly other points). To identify such endpoints in Q (i.e. to ﬁnd the maximal segments longer than ∆), we may, as in Algorithm 4, exhaustively search for a maximal segment over Q × P. The purpose of this section is to show how this search can be reduced. Under the assumption that the diameter of P is larger than ∆, we know, from lemma 1, that any maximal segment will have at least one endpoint outside any ball of radius ∆/2. Consider such a ball B of radius ∆/2. The exhaustive search over Q × P can then be reduced to two exhautive searches associated to a partition of Q into Q ∩ B and Q \ B . More precisely, if p ∈ Q, searching for a point q such that δ(p, q) > ∆ reduces to searching q in P \ Q \ B if p belongs to B , and searching q in P otherwise. This way, instead of searching over Q × P, we search over (Q ∩ B ) × (P \ Q \ B ) and (Q \ B ) × P, therefore avoiding searching a maximal segment in (Q ∩ B ) × (P ∩ B ). B should be chosen so as to maximize the number of points in P ∩ B , which reduces the cost of searching over (Q ∩ B ) × (P \ Q \ B ). The idea is to reuse the already found segments pi qi (which are double normals of P) and to iteratively i center the balls of radius ∆/2 at the points pi +q 2 . Algorithm. Assume that Algorithm 2 terminates under case 2, yielding the segment pmax qmax (i.e. pI−1 qI−1 ) of length ∆ = δ(pmax , qmax ) which is considered as an estimation of the diameter. Moreover, we assume that the set Q computed by Algorithm 3 is not empty. All the double normals pi qi that have been found by Algorithm 2, except pI−1 qI−1 , are collected into a set S.

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Required: ∆2 = δ 2 (pmax , qmax ) and S provided by Algorithm 2 Required: Q(0) = Q provided by Algorithm 3 1: for all segments pi qi ∈ S, i = 1 . . . |S| do i 2: B ← B pi +q , ∆/2 2 3: d2 ← max δ 2 (p, q) (q, p) ∈ Q(i−1) ∩ B × P \ Q(i−1) \ B 4: 5: 6: 7: 8: 9:

if d2 > ∆2 then // A better diameter estimation was found ∆ 2 ← d2 Add segment pq to set S Q(i) ← Q(i−1) \ B // new set Q if Q(i) = ∅ then return ∆2 // diameter has been found

Algorithm 5: Iterative reduction of Q by successive examination of all segments pi qi .

If Algorithm 5 terminates with Q(|S|)

= ∅, one still must run Algorithm 4 with Q = Q(|S|) , i.e. the exhaustive search over Q(|S|) × P.

6

Diameter Approximation def

Our algorithm provides a lower bound√ ∆ = ∆min on the diameter. It also provides an upper bound ∆max = ∆min 3. Indeed, let pq be the double normal whose length is ∆min . All the points of P belong to the intersection of the two balls of radius ∆min centered at p and q. With only slight modiﬁcations, our algorithm can also be used to compute a better approximation of the diameter. More precisely, for any given ε, we provide an interval [∆min , ∆max ] of length ≤ ε that contains the true diameter. Since the algorithm provides a lower bound ∆, we simply need to ensure that ∆ + ε is an upper bound of the true diameter. We will just indicate where the necessary modiﬁcations must take place. First, during the iterative search of double normals (line 9 in Algorithm 2) the ball centered at p+q 2 and passing through the furthest point m contains all the points of P. The diameter ∆max of that ball is given by →− → + ∆2 ∆2max = 4 − mp. mq 2

where ∆ = δ(p, q). Therefore, when ∆2max ≤ (∆ + ε) , we have found an εapproximation of the diameter and we stop. Second, the intermediate step (Algorithm 3) checks if P (I) contains the endpoint of some potential maximal segment. Here the set Q has to be replaced by I−1 ∆+ε P (I) \ B pI−1 +q , . 2 2 A better estimate than ∆ + ε of the upper bound ∆max is then obviously pI−1 + qI−1 2 × max δ ,q q∈R 2

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pI−1 + qI−1 ∆ + ε pI−1 + qI−1 ∆ with R = P (I) ∩ B , \B , . 2 2 2 2 If Q is empty, we stop. The exhaustive search over Q × P described in Algorithm 4 will possibly update both ∆ and ∆max . Required: ∆2 provided by algorithm 2, Required: Q, ∆2max and provided by modiﬁed algorithm 3 1: if Q = ∅ then // Exhaustive search with an endpoint in Q 2: for all points pi ∈ Q do 3: for all points pj ∈ P do 4: if δ 2 (pi , pj ) > ∆2 then 5: ∆2 ← δ 2 (pi , pj ) 6: if δ 2 (pi , pj ) > ∆2max then 7: ∆2max ← δ 2 (pi , pj ) 2 8: return ∆ and ∆2max Algorithm 6: Modiﬁed exhaustive search over Q × P.

Finally, in Algorithm 5 (line 2), we will use ∆max instead of ∆ and update both ∆2 and ∆2max when necessary (lines 4-6).

7

Experiments

We conduct experiments with diﬀerent point distributions in Rd : Volume based distributions: in a cube, in a ball, and in sets of constant width (only in 2D); Surface based distributions: on a sphere, and on ellipsoids; and with real inputs3 The interested reader will ﬁnd detailed results and discussion in [MB01] for our own method.

8

Comparison with Har-Peled’s Method

The most comparable approach to ours is the one developed very recently by S. Har-Peled [Har01]. Although it is similar in spirit, Har-Peled’s algorithm is quite diﬀerent from ours. We ﬁrst summarize his method and then compare experimentally the two methods. Since the two methods have diﬀerent advantages and drawbacks, it is worth combining them, leading to good hybrid algorithms with more stable performances. 3

Large Geometric Models Archive, http://www.cs.gatech.edu/projects/large models/, Georgia Institute of Technology.

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Table 1. CPU times for 3D volume based synthetic distributions.

Inputs Cube Points 10,000 our method 0.01 HPM - original 0.01 HPM - our implementation 0.02 hybrid method #1 0.01 hybrid method #2 0.02

Running time in seconds 3D Volume Based distributions Cube Cube Ball Ball Ball 100,000 1,000,000 10,000 100,000 200,000 0.19 0.53 0.04 0.79 1.20 0.18 1.96 0.31 18.16 53.88 0.18 1.92 0.20 5.12 20.57 0.18 2.00 0.13 2.25 5.26 0.35 1.50 0.07 1.05 3.29

In his approach, Har-Peled recursively computes pairs of boxes (each enclosing a subset of the points). He throws away pairs that cannot contain a maximal segment. To avoid maintaining too many pairs of boxes, Har-Peled does not decompose a pair of boxes if both contain less than nmin points (initially set to 40 in HarPeled’s implementation). Instead, he computes the diameter between the two corresponding subsets using the brute-force method. Moreover, if the number of pairs of boxes becomes too large during the computation (which may be due to a large number of points or to the high dimension of the embedding space), nmin can be doubled: however, doubling nmin increases the computing time. Diﬀerently from our method, Har-Peled’s algorithm depends on the coordinate axes (see table 2). Table 2. CPU times for 3D surface based synthetic distributions. The points sets in the second and the third columns are identical up to a 3D rotation.

Inputs

Ellipsoid (regular) Points 1,000,000 our method 1.34 HPM - original 1.78 HPM - our implementation 1.81 hybrid method #1 1.82 hybrid method #2 2.30

Running time in seconds 3D Surface Based distributions Ellipsoid Ellipsoid Sphere Sphere Sphere (rotated) 1,000,000 1,000,000 10,000 100,000 200,000 2.02 1.61 1.08 358.21 not computed 3.84 37.70 2.13 95.49 328.90 3.51 23.88 0.63 39.97 166.26 3.38 6.38 0.33 6.99 16.75 3.10 1.79 0.44 8.58 19.75

We provide an experimental comparison of both approaches, using the original Har-Peled’s implementation4 which only works for 3D inputs. In order to 4

Available at http://www.uiuc.edu/˜sariel/papers/00/diameters/diam prog.html.

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be able to deal with inputs in higher dimensions, we have re-implemented his algorithm, following the same choices that were made in the original implementation. 8.1

Hybrid Methods

It should be ﬁrst notice that both methods can easily be modiﬁed to compute the diameter between two sets, i.e. the segment of maximal length with one endpoint in the ﬁrst set and the other in the second set. Both methods have quadratic parts. Ours with the ﬁnal computation over Q × P, and Har-Peled’s one when computing the diameter for a pair of small boxes. We have implemented two hybrid methods that combines Har-Peled’s method and ours. We ﬁrst modiﬁed Har-Peled’s algorithm by replacing each call to the brute-force algorithm by a call to our algorithm. We also tested another hybrid method where we modiﬁed our algorithm by replacing the ﬁnal call to the bruteforce algorithm by a call to the ﬁrst hybrid method. The two hybrid methods can be tuned by setting several parameters. The experimental results presented here have been obtained with the same values of the parameters. The results show that the hybrid methods are never much worse than the best method. Moreover, their performances are more stable and less sensitive to the point distribution. Table 3. CPU times on real inputs. Running time in seconds Inputs Bunny Hand Dragon Buddha Blade Points 35,947 327,323 437,645 543,652 882,954 our method 5.73 0.29 8.51 172.91 0.49 Har-Peled’s method (HPM) - original 0.08 0.45 0.90 0.72 1.00 HPM - our implementation 0.07 0.43 0.89 0.69 0.94 hybrid method #1 0.07 0.41 0.86 0.67 0.90 hybrid method #2 0.10 0.32 1.37 1.09 0.50

9

Discussion

Our method is based on the computation of double normals. Computing a double normal appears to be extremely fast under any practical circumstances and in any dimension (despite the quadratic lower bound of lemma 5). Moreover, the reported double normal is very often the true maximal segment. This is not too much surprising since, on a generic surface, the number of double √ normals is ﬁnite and small. In any case, having a double normal provides a 3-approximation of the diameter in any dimensions.

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However, even if the reported double normal is a maximal segment s, it may be costly to verify that this is indeed the case. A favourable situation is when the point set is contained in the ball B of diameter s. The bad situation occurs when there are many points in set P \ B since we verify that none of these points is the endpoint of a maximal segment. This case occurs with sets of constant width but also with some real models: e.g. bunny, dragon and buddha (see tables [MB01]. For these three cases, the ﬁrst double normal found by the algorithm was the maximal segment. The second found double normal was shorter. After Algorithm 3, Q contains respectively 1086, 2117, and 2659 points for the bunny, dragon, and buddha models. For both the bunny and the buddha, the second double normal was very close to the ﬁrst one, then very few points were removed from Q (respectively 7 and 36), and most of the points of Q will undergo the ﬁnal quadratic search. This explains why there is a so little diﬀerence between our method with and without the reduction of Q for these two models [MB01]. For the dragon model, the second double normal is quite diﬀerent from the ﬁrst one, hence the noticeable improvement of our method with the reduction of Q. Table 4. CPU times for synthetic distributions in higher dimensions.

Inputs Points our method HPM - our implementation hybrid method #1 hybrid method #2 our method HPM - our implementation hybrid method #1 hybrid method #2 our method HPM - our implementation hybrid method #1 hybrid method #2 our method HPM - our implementation hybrid method #1 hybrid method #2

Running time in seconds volume distributions surface distributions Cube Ball Regular Ellipsoid Ellipsoid Sphere 100,000 100,000 100,000 100,000 100,000 Dimension = 6 0.31 36.95 0.11 0.33 not computed 0.85 466.44 0.97 0.87 465.08 0.67 77.20 0.79 0.73 118.06 0.66 63.31 0.19 0.65 142.38 Dimension = 9 0.89 128.02 0.51 0.52 not computed 139.23 568.99 264.96 590.14 569.08 17.42 135.90 44.54 67.27 232.39 1.21 121.91 1.25 16.03 302.86 Dimension = 12 3.87 445.03 1.08 7.88 not computed 629.37 651.56 648.88 650.98 647.74 44.45 354.14 58.53 56.11 511.41 19.72 380.41 13.00 24.62 745.60 Dimension = 15 10.99 798.66 7.26 20.31 not computed 734.69 735.26 731.76 733.70 737.51 64.49 610.70 69.11 90.35 701.18 44.37 782.20 21.30 70.41 1120.57

Har-Peled’s method does not suﬀer from this drawback. However, it depends on the coordinate axes (since the boxes are aligned with the axes) and on the dimension d of the embedding space. The ﬁrst hybrid method compensates for the quadratic search between small boxes (boxes containing less than nmin points), i.e. one major drawback of original Har-Peled’s method.

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The second hybrid method compensates for the major drawback of our method, by building pairs of boxes from Q × P.

References [AGR94]

N. M. Amato, M. T. Goodrich, and E. A. Ramos. Parallel algorithms for higher-dimensional convex hulls. In Proc. 35th Annu. IEEE Sympos. Found. Comput. Sci., pages 683–694, 1994. [Bes98] S. Bespamyatnikh. An eﬃcient algorithm for the three-dimensional diameter problem. In Proc. 9th Annu. ACM-SIAM Symp. Discrete Algorithms, pages 137–146, 1998. [CS89] K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry. Discrete Comput. Geom., 4:387–421, 1989. [Har01] S. Har-Peled. A practical approach for computing the diameter of a pointset. In Symposium on Computational Geometry (SOCG’2001), pages 177– 186, 2001. [MB01] Gr´egoire Malandain and Jean-Daniel Boissonnat. Computing the diameter of a point set. Research report RR-4233, INRIA, Sophia-Antipolis, July 2001. http://www.inria.fr/rrrt/rr-4233.html. [PS90] F.P. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Springer Verlag, October 1990. 3rd edition. [Ram97a] E. Ramos. Construction of 1-d lower envelopes and applications. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pages 57–66, 1997. [Ram97b] E. Ramos. Intersection of unit-balls and diameter of a point set in R3 . Comput. Geom. Theory Application, 8:57–65, 1997. [Ram00] Edgar A. Ramos. Deterministic algorithms for 3-D diameter and some 2-D lower envelopes. In Proc. 16th Annu. ACM Sympos. Comput. Geom., pages 290–299, 2000.

Shape Representation Using Trihedral Mesh Projections Llu´ıs Ros1 , Kokichi Sugihara2 , and Federico Thomas1 1 Institut de Rob` otica i Inform` atica Industrial, CSIC-UPC. Llorens Artigas 4-6, 08028 Barcelona, Spain, [email protected],[email protected], 2 Dept. of Mathematical Engineering and Information Physics, University of Tokyo. 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan. [email protected].

Abstract. This paper explores the possibility of approximating a surface by a trihedral polygonal mesh plus some triangles at strategic places. The presented approximation has several attractive properties. It turns out that the Z-coordinates of the vertices are completely governed by the Z-coordinates assigned to four selected ones. This allows describing the spatial polygonal mesh with just its 2D projection plus the heights of four vertices. As a consequence, these projections essentially capture the “spatial meaning” of the given surface, in the sense that, whatever spatial interpretations are drawn from them, they all exhibit the same shape, up to some trivial ambiguities.

1

Introduction

A polygonal mesh is a piecewise linear 2-manifold made up with planar polygonal patches, glued along the edges, and possibly containing holes. A polygonization method is an algorithm able to construct a polygonal mesh approximating a given surface. The literature on polygonization methods, mainly on triangulations, is vast (see [3] for a recent survey on triangulations and algorithms to simplify them). In general, the main goal is to obtain meshes that are close to the surface within a known error, as a way to understand and represent the surface shape [7]. Other goals have been to increase the speed of polygonization and the ability of the polygonizer to satisfy some constraints in the solution (e.g., one might request the most accurate approximation using a given number of line segments or triangles). In general, a polygonal mesh cannot be reconstructed from its projection onto a plane because inﬁnitely many meshes generate exactly the same projection. For example, for the triangular mesh projection in ﬁgure 1, there are many diﬀerent reconstructions, as illustrated. The ﬁrst two seem to have no meaning; but, actually, there is a rather “hidden” meaningful reconstruction: Nefertiti’s face! Can we obtain a spatial mesh approximating Nefertiti’s face in such a way that its projection still keeps its spatial meaning? A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 209–219, 2002. c Springer-Verlag Berlin Heidelberg 2002

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Fig. 1. Arbitrary reconstructions of this triangulated projection have no spatial meaning. But actually, a very specific one of them really does: it shows Nefertiti's face.

There is a class of meshes whose projections fully determine the spatial shape once the heights of four vertices are given. We call these projections unequzvocal because their reconstructions represent essentially the same object, up to some trivial ambiguities. For example, the projection in figure 2a unequivocally represents a truncated tetrahedron, as seen in figures 2d, e, and f. Observe that it suffices to set the heights of P , Q , T and R to determine those of S and U , using the fact that all cofacial vertices must be coplanar and, hence, S must lie on the face-plane R P Q S , and U on S Q T U . One of our goals is then to approximate any given surface with a polygonal mesh yielding unequivocal projections that uniquely identify the spatial shape up to the trivial ambiguities produced by changing the heights of only four vertices. Section 2 presents the trihedral polygonal mesh, the model we use to this end, and shows how its projections are unequivocal in the sense given above. Nevertheless, we need to go beyond this goal if this representation is to be useful. Consider what happens if the (x,y) vertex positions in figure 2a are slightly altered (figure 2b). The new projection no longer represents a correct truncated tetrahedron for, to be so, the edges joining the two triangular faces, when extended, should be concurrent at the apex of the (imaginary) original tetrahedron. Equivalently, note that once P, Q , R and T are given, the height of U is overconstrained, for it can be calculated from both the coplanarity of SQTU or that of R P T U . For generic vertex positions, the two values of this height do not necessarily coincide, and the only spatial reconstruction that keeps cofacial vertices coplanar is a trzvzal one, with all vertices lying on a single plane [5,6]. This makes the four provided heights inconsistent between each other. In sum, the consistency of the four heights only holds at very specific positions of the vertices and inevitable discretization errors will make this representation useless. This problem is common in Computer Vision [8] and Computer Graphics [12, 101, and mathematical characterizations of generically consistent projections are given in [11,9]. The way we use to make this representation robust against these

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Fig. 2. A truncated tetrahedron (a) and three possible reconstructions (d, e, f). The slightest perturbation destroys the correctness of the projection (b), but this can be avoided adding new triangular faces (c).

errors follows from this observation: if the height of a vertex in a projection is overconstrained because the vertex lies on several planes that ﬁx it, we just introduce new triangular faces around it for preventing this to occur (ﬁgure 2c). Section 3 gives a fast algorithm to this end, derived from this observation, using the so-called T/TT-transformations. Section 4 describes a complementary optimization step that properly places these transformations to minimize the reconstruction errors by reducing the problem to a cyclic AND/OR graph search. We ﬁnally conclude in section 5.

2

Trihedral Polygonal Meshes

Trihedral meshes, i. e., those where all vertices have exactly three incident faces, produce unequivocal projections. Indeed, ﬁgure 3 shows that in them, after ﬁxing the planes of two adjacent faces, we have enough data to derive the heights of the remaining vertices. Clearly, the heights of the bold vertices ﬁx the shadowed face-planes and the heights of other vertices on them. At this point, any other surrounding face has three vertices whose height is known and, so, its plane can be ﬁxed too. The same argument can be iteratively applied and the result is a height propagation reaching all vertices in the projection. In the schematic representation of this height propagation (ﬁgure 3) every face f receives three incoming arrows from the three vertices that ﬁx it. The derivation of heights for the rest of vertices on f is indicated with outgoing arrows from f . The result is a tree-shaped structure spanning all vertices and faces. In this tree, a path from any of the initial four vertices to any other vertex will be hereafter referred to as a propagation wave. Note that, height propagations

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where a face is ﬁxed from three (almost) collinear vertices must be avoided. Section 4 gives a way to compute propagations eluding these collinearities. A trihedral mesh approximating a convex or concave surface can be readily obtained by distributing a set of random points all over the surface and computing its tangent planes at these points. This leads to a plane arrangement whose upper envelope –if the surface is convex– or lower envelope –if it is concave– provides a good mesh approximation of the surface. Since the tangent plane orientations are random, any three of such planes meet in a single point, and hence the mesh is trihedral. Alternatively, a trihedral mesh approximation of a piece of concave or convex surface can be obtained by starting with a rough mesh approximation and iteratively applying a bevel-cutting [2] and/or a corner-cutting [1] operation to attain the desired approximation. Obviously, the situation becomes much more complex when concavities and convexities are simultaneously present. The ﬁrst step in these cases would be to decompose the surface into patches having congruent signs for the maximum and minimum curvatures at all their points. If this is done for a general C ∞ surface, we would get patches labeled (+, +), (+, −), (−, +) or (−, −) separated by curves which could be labeled with (−, 0), (+, 0), (0, −), or (0, +), and isolated points (actually, maxima or mimima) which would be labeled with (0, 0). Saddle points would be also labeled with (0, 0) but they would apFig. 3. A height propagation start- pear as intersections of separating curves. 2 ing at four pre-speciﬁed (bold) vertices. If we extend this treatment to C surfaces, Several vertices can have an overcon- we could get entire patches with one of strained height. the above nine possible labels. For example, all plane patches would have the label (0, 0). Patches labeled with (+, +) or (−, −) represent fully convex or concave patches and thus they can be polygonized as described above. Patches labeled as (∗, 0) or (0, ∗) can be polygonized by locating random points along the direction of maximum curvature. Patches (0, 0) would only require a single point on them. Unfortunately, the treatment of (+, −) or (−, +) patches remains as an open problem for us. The connection between polygonized patches can be obtained by computing tangent planes on points along their common boundaries. In sum, the polygonization we propose can be done, ﬁrst for each patch by generating the tangent planes in a suﬃciently high density, and next by connecting them using the tangent planes generated along their common boundaries.

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Fig. 4. (a and b) T and TT-transformations. (c) Overhanged and self-intersecting reconstructions induced by T-transformations at locally non-convex faces.

3

T and TT-Transformations

In a trihedral mesh a projection is overconstrained because any of its vertices lies on three faces and, potentially, up to three propagation waves can determine a height at the same time. However, as done in ﬁgure 2c, this can be avoided by adding triangular faces. To this end, we ﬁrst compute an arbitrary height propagation spanning all vertices, and check which of them receives more than one wave. We then take one overconstrained vertex v at a time and prevent all but one waves from reaching v as follows. To stop the wave getting v from face f , we apply either of these two transformations (ﬁgure 4a and b): – A T-transformation, which places a new edge joining the two neighboring vertices of v in f , say vl and vr . – A TT-transformation, which places a new vertex v on f near v and the three new edges (v , v), (v , vl ) and (v , vr ). After either transformation, f cannot constrain the height of v anymore. Also, the added triangles are innocuous because all heights can still be determined from the four initial ones. Which transformation is preferred depends on the geometry of face f around vertex v. If all points inside the triangle vl vr v belong to f , we say that f is locally convex at v. So, for situations where f is locally convex at v, simplicity prevails and T-transformations are enough (ﬁgure 4a). When local nonconvexities are present (ﬁgure 4b), T-transformations yield occluded or partially occluded crossing edges whose spatial reconstructions have overhanged parts, or self-intersecting faces (ﬁgure 4c). Here, TT-transformations are preferred for they can avoid this. An observation complements the strategy. In an overconstrained vertex v, either two or three incoming propagation waves arrive. If no more than one of

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Fig. 5. A projected dodecahedron (a) together with a height propagation (b) and the T-transformations it yields (c). A protruded tetrahedron (d) and two possible corrections: (e), involving TT-transformations, and (f), involving only T-transformations.

them comes through a locally non-convex face, then we can always drop the incidence constraint in this vertex just with T-transformations: we just leave the eventual “bad” wave to determine the height of v and stop the others with T-transformations. This completes the description of a one-sweep algorithm removing overdetermination. As an example, ﬁgures 5a-c show a projected dodecahedron before and after applying T-transformations. In general, when the approximated surface is uniformly convex, or uniformly concave, all faces of the resulting trihedral polygonal mesh will be locally convex, and hence T-transformations will suﬃce. However, even when local nonconvexities exist at the faces, there still might be some height propagations where only T-transformations suﬃce. In ﬁgure 5e, for example, an algorithm computing an arbitrary propagation can be forced to use TT-transformations, whereas with a proper search, a robust projection is obtained only with T-transformations (ﬁgure 5f). But one certainly ﬁnds correct projections where no propagation strictly using T-transformations can be found [5, Section 8.4].

4

Optimal Propagations and Cyclic AND/OR Graphs

The algorithm in the preceeding section corrects the incidence structure by ﬁnding an arbitrary height propagation and inserting a T or a TT-transformation whenever a vertex height is determined by two or more faces. However, arbitrary propagations might travel along “degenerate paths” where the planes for some of the faces are determined by three aligned (or almost aligned) vertices. Clearly,

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these degenerate propagations must be avoided if we want to minimize the errors during the reconstruction of the spatial shape from the initial set of four heights. This section provides an algorithm to ﬁnd height propagations that avoid these degeneracies by formulating the problem as that of ﬁnding the least cost solution of a cyclic AND/OR graph [4]. We now recall some preliminary concepts about this kind of graphs. An AND/OR directed graph G, can be regarded as a hierarchic representation of possible solution strategies for a major problem, represented as a root node, r, in G. Any other node v represents a subproblem of lower complexity whose solution contributes to solve the problem at hand. There are three types of nodes: AND nodes, OR nodes and TERMINAL nodes. Every node v has a set S(v) of successor nodes, possibly empty, to which it is connected in either of two ways: – An AND node v is linked to all nodes si ∈ S(v) through directed AND arcs (v, si ), meaning that the subproblem for v can be trivially solved once all subproblems for the nodes in S(v) have been solved. – An OR node v is linked to all nodes si ∈ S(v) through directed OR arcs (v, si ), meaning that the subproblem for v can be trivially solved once any one of the subproblems for the nodes in S(v) has been solved”. – A TERMINAL node represents a yet-solved or trivial subproblem and has no successors. With this setting, a feasible solution to the problem becomes represented as a directed subgraph T of G verifying: – r belongs to T . – If v is an OR node and belongs to T , then exactly one of its successors in S(v) belongs to T . – If v is an AND node and belongs to T , then every successor in S(v) belongs to T . – Every leaf node in T is a TERMINAL node. – T contains no cycle, it is a tree. One can also assign a cost c(u, v) > 0 to every arc (u, v) in G and ask for the solution T with minimum overall cost C(T ) = (u,v)∈E(T ) c(u, v), where E(T ) is the set of arcs of T . Note that, as deﬁned, G can contain cycles. This turns out to be the main diﬃculty for this optimization problem, which, in the past, was usually tackled by a rather ineﬃcient trick: “unfolding” the cycles and applying standard AND/OR search methods for acyclic graphs. However, explicit treatment of cycles has recently been considered, and an eﬃcient algorithm is achieved in [4]. The search for an optimal height propagation is next reduced to this model. This amounts to (1) constructing an AND/OR graph Ghp whose feasible solutions deﬁne a height propagation, and (2) deﬁne a cost function that promotes non-degenerate propagations over degenerate ones.

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Fig. 6. AND/OR subgraphs for the propagation rules. AND nodes are indicated by joining all their emanating arcs. (a) Constructed subgraph translating rule R2 for a quadrilateral face. Dummy-face nodes are shadowed in grey. Note that, actually, there is only one vertex node for each vertex in the trihedral mesh, but for clarity they are here duplicated. (b) Propagation waves reaching a vertex. (c) Subgraph for rule R3, with an arc for each of the possibilities in (b).

4.1

Feasible Height Propagations

A height propagation can be deﬁned by the following rules, with the given straightforward translation into AND/OR subgraphs. R1: Four selected vertices of the projection trigger the propagation. For this, we put a TERMINAL node for each of the triggering vertices. R2: Every face in the polygonization can be determined once the heights of any three of its vertices are determined. If deg(f ) denotes the number of vertices ) of face f , then there are cf = deg(f possible combinations of three vertices 3 determining f . If we put a node in Ghp for every vertex, except for the four triggering ones, then this rule is translated by adding an OR node for every face, linked to cf new “dummy-face” AND nodes, each representing one of the above combinations. Each dummy-face node is in turn linked with arcs to the three involved vertices in the combination. Figure 6 gives a schematic representation. The newly introduced vertex nodes have not been assigned a type yet. This type is induced by the following rule. R3: Except for the initial four vertices, the height of every other vertex is determined once one of its incident faces has a determined plane. This implements the fact that the propagation wave ﬁxing the height of a vertex can come from any of its three incident faces (ﬁgure 6b). This rule can be represented by setting each vertex node as OR type, and linking it to the face nodes of its incident faces ﬁgure 6c. R4: The height propagation must reach all vertices. For this, we add a root AND node r to Ghp and link it to all vertex nodes. Note that a feasible solution tree of Ghp provides instructions to derive a height propagation that reaches all vertices, starting at the four pre-speciﬁed heights.

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217

Cost Function

In order to penalize propagations using sets of almost-aligned vertices, we proceed as follows. Consider a height propagation that ﬁxes a face-plane f from the point coordinates of three previously ﬁxed vertices vi , vj and vk . We can simply penalize the corresponding arcs in Ghp emanating from f by giving them a cost that is inversely proportional to the area of the triangle deﬁned by vi , vj and vk in the projection. The rest of arc costs are actually irrelevant, but need to be positively deﬁned [4]. In sum, for every directed arc (u, v) we deﬁne its cost as follows: 1. c(u, v) = 1/det(v1 , v2 , v3 ), if u is a dummy-face AND node and v is any one of its descendants. Here, vi , vj and vk are the homogeneous coordinates of the vertices associated with the three descendants of u. 2. c(u, v) = 1, if u is an OR node. 3. c(u, v) = 1, if u is the root AND node. Once the least cost solution T is found, the projection can be made robust to slight vertex perturbations as follows. At a vertex v receiving more than one propagation wave, we put a T/TT-transformation on all faces ﬁxing v, except on the one in the propagation wave represented in T . 4.3

Complexity Analysis

The worst-case complexity of computing the optimal solution of a cyclic AND/OR graph with n nodes is O(n3 ) [4]. We now prove that the number of nodes in Ghp grows linearly with the number of vertices of the trihedral polygonal mesh. Let e, v and f be the number of edges, vertices and faces of the given mesh. Then, 2e = 3v because the mesh is trihedral. Moreover, if the mesh has h holes, with “the outside” of the mesh counting as a hole too, then Euler’s relation says that v − e + f = 2 − h. From these two equalities the number of faces of the mesh can be written in terms of the number of vertices and holes, f = v+4 2 − h. Let us now count the number of nodes added by each of the rules R1,...,R4: – Rule R1 adds four vertex nodes. – Rule R2 adds one OR node for each face, amounting to f = v+4 2 − h = O(v) total nodes, assuming a constant number of holes. Also, for every face f this ) rule adds cf = deg(f dummy-face AND nodes. Although this number is 3 clearly in the worst case O(deg(f )3 ), if we divide the sum of face degrees by the number of faces, the average face degree is six, at an increasing number of randomly placed vertices in the mesh: 3v 6v allf aces deg(fi ) = v+4 = , f v + 4 − 2h − h 2 which will keep the number of dummy-face AND nodes linearly growing: 6 v+4 f = 20 − h = O(v). 3 2

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– Rule R3 adds a linear number of OR vertex nodes. – Rule R4 only adds one AND node, the root. Up to now we have assumed that the four vertices triggering the propagation are a priori selected. But other height propagations starting at other four vertices could yield better height propagations. To test all possibilities, we do not need to repeat the AND/OR search for every diﬀerent combination of four vertices. Indeed, note that these vertices just ﬁx the planes of the faces they belong to. So, any other set of four vertices on these faces will yield the same optimal propagations, provided that two of them lie on the common edge. We can equivalently think of pairs of faces triggering the propagation and use their face nodes as TERMINAL in Ghp . The choice of TERMINAL vertices (instead of TERMINAL faces) was done to be coherent with previous explanations. In sum, if one wants to search over all possible starting places of propagation, then for each pair of adjacent faces the AND/OR search needs to be repeated. This amounts to solve e = 32 v optimization problems in the worst case, meaning that the overall complexity will be O(v 4 ), under the assumption that the face degree is six.

5

Conclusion

We have shown how trihedral mesh projections can capture the spatial shape of a given object’s surface, up to some trivial ambiguities. We have also presented a local strategy that takes a trihedral projection as input and places some triangular faces at strategic places until it is made robust to perturbations in its vertex coordinates. Finally, we have found how to put these triangles so that the spatial reconstruction is performed in the most accurate way possible, avoiding height propagations along degenerate paths. Although we can deal with an important range of surfaces, no algorithm has been devised yet to obtain trihedral meshes approximating surfaces with saddlecrests or saddle-valleys. This constitutes a main issue for further research. Acknowledgements. The authors wish to thank Carme Torras and Pablo Jimenez for fruitful discussions around the AND/OR search strategy. This work has been partially supported by the Spanish Ministry of Education under grant FPI94-46634232, by the Spanish CICYT under contracts TIC960721-C02-01 and TIC2000-0696, and by the Grant-in-Aid for Scientiﬁc Research of the Japanese Ministry of Education, Science, Sports and Culture.

References 1. C. Boor. Cutting corners always works. Computer Aided Geometric Design, 4:125– 131, 1987. 2. D. Fox and K. I. Joy. On polyhedral approximations to a sphere. In IEEE Int. Conf. on Visualization, pages 426–432, 1998.

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3. P. Heckbert and M. Garland. Survey of polygonal surface simpliﬁcation algorithms. In Multiresolution Surface Modeling Course SIGGRAPH’97, May 1997. Available at http://www.cs.cmu.edu/˜ph. 4. P. Jim´enez and C. Torras. An eﬃcient algorithm for searching implicit AND/OR graphs with cycles. Artiﬁcial Intelligence, 124:1–30, 2000. 5. L. Ros. A Kinematic-Geometric Approach to Spatial Interpretation of Line Drawings. PhD thesis, Polytechnic University of Catalonia, May 2000. Available at http://www-iri.upc.es/people/ros. 6. L. Ros and F. Thomas. Overcoming superstrictness in line drawing interpretation. IEEE Trans. on Pattern Analysis and Machine Intelligence. Accepted for publication. Preliminary scheduled for Vol 24, No. 4, April 2002. 7. W. Seibold and G. Wyvill. Towards an understanding of surfaces through polygonization. In IEEE Int. Conf. on Visualization, pages 416–425, 1998. 8. K. Sugihara. An algebraic approach to shape-from-image problems. Artiﬁcial Intelligence, 23:59–95, 1984. 9. K. Sugihara. Machine Interpretation of Line Drawings. The MIT Press, 1986. 10. K. Sugihara. Resolvable representation of polyhedra. Discrete and Computational Geometry, 21(2):243–255, 1999. 11. W. Whiteley. Some matroids on hypergraphs with applications to scene analysis and geometry. Discrete and Computational Geometry, 4:75–95, 1988. 12. W. Whiteley. How to design or describe a polyhedron. Journal of Intelligent and Robotic Systems, 11:135–160, 1994.

Topological Map Based Algorithms for 3D Image Segmentation Guillaume Damiand and Patrick Resch LIRMM, 161 rue Ada, 34392 Montpellier Cedex 5, France {damiand,resch}@lirmm.fr

Abstract. One of the most commonly used approach to segment a 2D image is the split and merge approach. In this paper, we are deﬁning these two operations in 3D within the topological maps framework. This mathematic model of regions segmented image representation allows us to deﬁne these algorithms in a local and generic way. Moreover, we are deﬁning a new operation, the coreﬁning, which allows to treat big images. They are cut into small units, treated separately, then the result of each of them are combined to reconstruct the ﬁnal representation. These three operations let us view eﬃcient 3D segmentation algorithms, which is a diﬃcult problem due to the size of data to treat.

1

Introduction

The region segmentation is a diﬃcult problem that was studied in many diﬀerent works in 2 dimensions. It consists in making a partition of an image into connected sets of pixels verifying an homogeneity criterion, and that we are calling regions. The main approaches for the region segmentation are the split and merge methods. The top-down approach [16,13] consists in taking big regions and cutting into smaller and smaller regions. The bottom-up approach [6,11] is the opposite approach, which begins with many small regions that are merged into bigger and bigger regions. At last the mixed approach [12,17] consists in mixing the two previous ones, possibly making again the process until for example, result stabilization. These approaches require a “good” model of images representation. Many works in 2 dimensions [10,7,8,3] have shown that topological maps are a model that allows to introduce these segmentation algorithms in an eﬃcient way. Moreover, thanks to that model we can also deﬁne many processing algorithms allowing to access or to modify the result of this segmentation. Recently, the topological maps have been extended in dimension 3 [5,2]. Indeed, more and more domains need to work in dimension 3, as medical imagery, geology, or industry. In order to carry out segmentation algorithms as in dimension 2, we need to deﬁne basic operations on topological maps. This problem turns out to be more diﬃcult than in dimension 2, because ﬁrst now we have an additional dimension, that sets down new problems, but also because it is much A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 220–231, 2002. c Springer-Verlag Berlin Heidelberg 2002

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more diﬃcult to represent and to make visual objects in 3 dimensions. These visualization problems act as a brake to the comprehension and the development of new algorithms. Moreover, there are complexity constraints, in memory space as well as in execution time, that are much more important than in dimension 2. Indeed, the treated data quantity is much more important in dimension 3. It requires us to deﬁne very eﬃcient algorithms to use them on big images. In this paper, we present the two algorithms of merge and split on the 3D topological maps. These two algorithms are the basic operations for the regions segmentation. The use of the topological maps let us deﬁne these algorithms in a generic way, because they work on any conﬁguration, and in a local way because they treat the map element by element, only looking the direct neighborhood of the current element. These two properties make these algorithms more simple to understand but more eﬃcient in complexity too. We also present the algorithm of coreﬁning that allows us to treat big 3D images in parallel. We are going to cut this image into several small units, we are going to segment each unit in parallel, then we will reconstruct all the image with this units, using the coreﬁning operation. Because of a lack of space and to not lose ourselves into technical details we are just presenting the principle of the algorithms and the main ideas of these three operations. For more details about these operations, we can report to [18,19]. We are ﬁrst presenting Section 2 the combinatorial maps then the topological maps that are combinatorial maps verifying speciﬁc properties. Then we are presenting our three algorithms : merge Section 3, split Section 4 and coreﬁning Section 5. We are describing their principle and the diﬀerent cases we have met. At last, we are concluding and presenting some perspectives Section 6.

2

Topological Maps Recall

Topological maps allow to represent the nD regions segmented images. They encode at the same time the topology and the geometry of images. Topological maps are combinatorial maps with particular properties. So we are beginning by recalling the notion of combinatorial map. This is just a short reminder; a more detailed description can be found in [5,1]. 2.1

Combinatorial Maps

Combinatorial maps are a mathematical model of representation of space subdivisions in any dimension. They were introduced in the sixties by [9], at ﬁrst as a planar graph representation model, and extended by [14] in dimension n to represent orientable or not-orientable quasi-manifold. Combinatorial maps encode space subdivisions and all the incidency relations. They are made of abstract elements, called darts, on which are deﬁned application, called βi . We are giving here the combinatorial map deﬁnition in n dimensions, that we can ﬁnd for example in [15].

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Deﬁnition 1 (combinatorial maps). Let n ≥ 0. A n combinatorial map, (or n-map) is an (n + 1)-uplet M = (B, β1 , . . . , βn ) where : 1. 2. 3. 4.

B is a ﬁnite set of darts; β1 is a permutation on B; ∀i, 2 ≤ i ≤ n, βi is an involution on B; ∀i, 1 ≤ i ≤ n − 2, ∀j, i + 2 ≤ j ≤ n, βi ◦ βj is an involution.

In this deﬁnition, there is an application βi for each space dimension which puts in relation two i-dimensional cells. When two darts are linked with βi , we say that they are βi -sewed. Each space cell is implicitly represented by a set of darts. We can see ﬁgure 1.a an example of an image and ﬁgure 1.b the corresponding combinatorial map. Each dart is represented by a segment, the β1 relation by light grey arrows and the β2 relation by dark grey arrows. β1 put in relation a dart and the next dart of the same face. For example, the light grey face of the image is represented by four β1 -sewed darts in the map. The adjacency between this face and the dark grey face is represented by two darts β2 -sewed together. We are using the simpliﬁed representation (ﬁgure 1.c) that does not represent explicitly the applications, because it is more understandable.

a. A 2D object.

b. The corresponding map.

c. Simpliﬁed representation.

Fig. 1. An object and the corresponding combinatorial map represented by two diﬀerent way.

2.2

The Topological Maps

In the combinatorial maps framework, several representations of a single object exist. We want a unique characterization of objects, for example to make easier isomorphisms. That is the main goal of the topological maps. They are mathematical model of 3D segmented images representation, which encode all incidency and adjacency relations. They represent interpixel elements composing the edge of boundary faces of an 3D image. Moreover, they are minimal, stable for rigid transformations, and they characterize the image’s objects with their topology. We remind that a boundary face is a surface between two neighbouring regions. We can see on ﬁgure 2.a an example of a 3D image, and on ﬁgure 2.b its boundary faces. The construction of a topological map is progressive: at the

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beginning all image’s voxels are encoded with a combinatorial map, then we simplify this map with successive mergings. The corresponding topological map of our example is shown on ﬁgure 2.c. We can see that each face of the map encodes a boundary face of the image.

a. A 3D image

b. Its boundary faces.

c. The topological map.

Fig. 2. A 3D image and the corresponding topological map.

Inclusions of volumes are represented by a inclusion tree where each node corresponds to a region and its son nodes correspond to included regions. The root is R0 , the inﬁnite region which rounds up the image. When there is a hole on a face, the connexion between exterior and interior borders of this face is represented with a virtual edge. This is a one degree edge (that is adjacent twice to the same face). Such an edge should have been removed during the construction of the topological map, but have been retained to conserve the map connected, and each face homeomorphic to a topological disk. These edges are also useful to represent closed faces (without border), as we can see for the torus example shown ﬁgure 3.

a. A torus.

b. Intermediary step.

c. The topological map.

Fig. 3. The topological map of a torus.

This torus is represented with a single closed boundary face. An intermediate step of the topological map’s construction is shown on ﬁgure 3.b, where we can see two virtual edges (in grey) which keep the upper and lower faces connected.

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The topological map shown ﬁgure 3.c is only composed of virtual edges. We obtain the classic minimal representation of the torus composed of one face, one vertex and two edges. Combinatorial maps encode only the topological part of our model. To encode the geometry of the corresponding image, we are using a geometrical model, which links a geometrical face to each topological face of the map . We are calling embedding this geometrical model. This distinction between topology and embedding allows the diﬀerentiation of treatments and sometimes enables a hierarchization of these treatments. Indeed, some operations only work on the topological model, others only on the geometrical model, and some on both. We can see ﬁgure 4.a the topological map of two adjacent objects and ﬁgure 4.a the embedding of these objects. Each dart of the topological is linked with the border

3

a c

b β2

d

β2

β3

1

2

Fig. 4. Embedding example of a topological map.

of an embedding surface. For our example, the dart named a of the topological map is linked with the surface named 1, the two darts named b and c are linked with the same surface 2 because they belong to the same topological face, and the dart named d is linked with the surface 3.

3

Merge

The merging operation consists, starting from two adjacent regions R1 and R2 , to gather them into a single region R’, union of the two ﬁrst regions. Algorithm 1 make this operation on a topological map. It is local because it treats independently each dart of R1 . There are two diﬀerent cases depending if the current dart belongs to a virtual edge or not. In the ﬁrst case, we are testing if the deletion of this edge disconnects the map into several connected components. If it happends, we are modifying the inclusion tree this way. In the second case, we are just sewing the two faces adjacent to the currently treated edge together, and we are deleting this edge. This operation is made for all the darts of the boundary faces, so the two regions are merged and boundary faces are ﬁnally destroyed. At last, the topological map is simpliﬁed, because operations could have made the map incoherent or not minimal. For

Topological Map Based Algorithms for 3D Image Segmentation

225

Algorithm 1 Merge 3D Data: Two adjacent regions R1 and R2 . Result: The two regions are merged into R2 . foreach dart b of the region R1 do if b belongs to the boundary between R1 and R2 then if b belongs to a virtual edge then foreach dart t of the orbit < β1 , β2 > of β1 (b) and β13 (b) do if all regions of β3 (t) are included into R1 ∪ R2 then Daughter(R2 )=Daughter(R2 )+regions of the connected component of β3 (t) Destroy the virtual edge incident to b; else β2 -sew(β2 (b), β32 (b)); Destroy β3 (b) and b; Simpliﬁcation of R2;

that, ﬁrst we are removing degree 2 vertex and edges. Then, as virtual edges have just a topological existence, they can be moved to reduce the edge number. These simpliﬁcation step guarantee the minimality of the topological map. Moreover, during this step, we are using the Euler characteristic to keep invariable all topological characteristics of the map. We can see ﬁgure 5 an example of the merging operation. Figure 5.a shows a topological map that represent three adjacent objects, before the merging of R1 and R2 . Figure 5.b represents the map obtained after the merging. We have destroyed the face between R1 and R2 and β2 -sewn each other darts incident to this face. We can see on this ﬁgure that this map is not a topological map because

R1

R2

a. A topological map.

R2 b. After the merging.

c. After the simpliﬁcation.

Fig. 5. An example of Merge 3D

it is not the minimal representation. Indeed, we have two edges incident to only two faces. These edges can be removed without lost of topological information. This is done by the simpliﬁcation step of our algorithm. After the removal of these edges, we obtain some vertices incident to only two diﬀerent edges. They

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can be also removed without lost of information. Finally, we obtain the map shown ﬁgure 5.c that is the topological map representing two adjacent objects. We can note that this simpliﬁcation step can be performed during the merging, but this leads to a less understandable algorithm, but also with a more eﬃcient complexity. We chose to present here the more simple algorithm, even though its complexity is a little higher, to make easier its comprehension.

4

Split

This operation is the opposite of the merging one. It consists in spliting a region by a separation face. This face is composed of a list of vertices which represent the intersections of a plane and the edges of the map. Constraints have been ﬁxed on this separation face to simplify the algorithm and to limit the number of diﬀerent cases to treat. A simple solution to do complex splits is to combine several simple splits and merge operation to obtain the wanted result. The principle of the algorithm is ﬁrst to insert a new boundary face into the embedding then the corresponding boundary face in the topological map which represents the region. Algorithm 2 Split 3D Data: A list List of vertex representing the separation face A region R. Result: R is splited in two regions along the separation face. foreach dart b of R do foreach dart t of the embedding of b do foreach vertex S of List do InsertVertexPlongement(S,t); if t is on an edge then InsertVertexMap(S,b) if the separation face cuts an edge then separate the concerned boundary face; Create 2 new boundary faces and β2 -sewed them to the previous separated boundary face; else if the separation face cuts a closed face then Create 4 faces composed each by a single dart a, b, c and d; Insert a in the closed face; β2 -sew(a, b); β2 -sew(c, d); β3 -sew(b, c); else // The separation face cuts a not closed face F without cutting its edge; Create 4 faces Fa , Fb , Fc andFd copyings of F ; β3 -sew Fc and Fd and link Fa et Fb with a virtual edge; Insert the 4 faces between F and the face which was β2 -sewed to F ; Simplify the map;

Topological Map Based Algorithms for 3D Image Segmentation

227

There are three diﬀerent cases : 1. If the separation face of R1 cuts the edge of an already existing boundary face, like the example shown ﬁgure 6.a. New vertices will be inserted into this boundary face then they have to be separated with inserted edges between these vertices. At last the new boundary face can be inserted in the middle of the region. It is sewed on this new edge. We can see on ﬁgure 6.c the resulting topological map of the operation of spliting. β2 β2 R1

R1

β3

R1

R1 a. Two adjacents cubes.

b. The corresponding map.

R2 c. After the split.

Fig. 6. Case 1 of Split 3D

2. If the separation face does not cut any edge of already existing boundary faces (ﬁgure 7.a). A new face is inserted which is composed by four half-faces. These half-faces are copying of cutted region’s faces. Two of this half-faces are linked with an virtual edge because they represent the same surface as it is shown on ﬁgure 7.b.

R1 R1

a. Two adjacents cubes.

β3

R 1’

β3

β2

β2

β2

b. After the split.

β2

Fig. 7. Case 2 of Split 3D

3. If the separation face cuts a closed face. A closed face is only represented with virtual edges in the topological map. There is an example on ﬁgure 8.b with its embedding in a. To obtain the minimal map, the new inserted boundary faces are only composed with one dart and one of this faces has to be directly

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β1 -sewed to the closed face. The result of the split for a torus is shown on ﬁgure 8.c. We can verify with the Euler formula that the characterization of the torus is preserved. There is now 1 vertex, 3 edges and 2 faces, which gives a genius of 1.

β2

β2 β β

2

β

2

β2

a. A torus and the separation face.

b. The corresponding map.

3

β2

c. After the split.

Fig. 8. Case 3 of Split 3D

5

Coreﬁning

One method to segment a big 3D image consists in cutting it into several parts and segmenting each part in parallel. During the reconstruction of the image, we have to glue back the diﬀerent segmented parts. The operation which performs this glue back is the coreﬁning. To design the two faces of each volume which are going to be put in contact, we are passing two embedding darts to the algorithm. The faces which contain these darts gives the coordinates of the corresponding planes (as we works with interpixel). In a ﬁrst time, we build the new boundary faces. For that, we extract the embeddings of the two faces belonging to the two planes, because we consider that the initial image was cut in regular parts. We are building the map of these new border faces A and B. This construction can be diﬀerent if the new boundary face cuts a closed face or the border of an existing boundary face, in a similar way that the three diﬀerent cases explained for the split. We can see ﬁgure 9.a an example of this ﬁrst step. In a second step, we are inserting the intersection vertices of the two faces A and B in the embedding of the two maps of these two faces. These two maps have now exactly the same vertices, topological as well as geometrical. We are β3 -unsewing and destroying faces belonging to R0 which are previously β3 -sewn to A and B. Then, for each dart of A, we are looking for a similar dart in B (a dart which connects the same vertices). If such a dart does not exist, we are adding in B a copy of this dart. And we are exactly doing the same operation for

Topological Map Based Algorithms for 3D Image Segmentation

a. Step 1.

b. Step 2.

229

c. Final result.

Fig. 9. Coreﬁning principle.

each dart of B, with eventually adding some copies in A. Then we are obtaining two faces A and B totally similar, as we can see on ﬁgure 9.b. We just have to β3 -sew these two faces. For that, we are using the same coverage of the darts of A and B, and β3 -sew each couple of darts. We can see on ﬁgure 9.c the result of this ﬁnal operation. Algorithm 3 Coreﬁning 3D Data: Two maps M1 and M2 and two darts d1 and d2 of their embeddings. Result: The maps M1 and M2 are β3 -sewn by the faces containig d1 and d2 . Step 1: foreach region R of M1 (resp. M2 ) do Building the new boundary face A (resp. B) of R; Extracting this embeding and set it to A (resp. B); Step 2: Inserting intersection vertices of A and B into the faces A and B; foreach dart d of A do if it does not exist in B a dart similar to d then Add a copy of d in B; foreach dart d of B do if it does not exist in A a dart similar to d then Add a copy of d in A; foreach dart a and b of A and B joining the same vertices do β3 -sew(a, b);

6

Conclusion

In this paper, we have presented the basic operations for the 3D segmentation : merge and split plus an interesting operation in a parallelization goal : coreﬁning. These operations are deﬁned on the 3D topological map, a mathematical model of 3D images representation. The two ﬁrst operations were already presented

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in [4], but our approach diﬀers from this solution by our aims to deﬁne local algorithms. The last operation is very interesting, because it allows us to segment big images by working in parallel onto diﬀerent small parts of the image. Such a segmentation is very diﬃcult to perform if we want to do in a direct way. We have totally used all the topological map properties to obtain algorithms the most generic as possible. We are thus obtaining operations working on every possible case. Moreover, these algorithms operate in a local way, which simpliﬁes a lot the diﬀerent treatments. Indeed, we are treating each element without particular order, and we just have to look at the direct neighborhood of the current element. These diﬀerent advantages allow us to obtain algorithms relatively simple and understandable, but also to keep a good complexity. Now, we have to implement these three operations in our computer software, to obtain ﬁnally some 3D segmentation algorithms. We are currently working to use these algorithms to perform segmentation reﬁnement. Our approach consists in starting from a ﬁrst segmentation, performed in a classical way on the voxel matrix with some algorithms used in the signal treatment research. Then, we are computing the topological map corresponding to this ﬁrst segmentation. We can thus perform some treatments onto this map to reﬁne this segmentation : automatic treatments, for example to remove small regions or particular conﬁgurations, or interactifve treatments made by an expert. These results are under development to obtain a software for cerebral tumor diagnostic. Moreover, we also work on some other operations, that could be combination of these three basic operations, but also some speciﬁc ones as the chamfering or boolean operations.

References [1] Y. Bertrand, G. Damiand, and C. Fiorio. Topological encoding of 3d segmented images. In Discrete Geometry for Computer Imagery, number 1953 in Lecture Notes in Computer Science, pages 311–324, Uppsala, Sweden, december 2000. [2] Y. Bertrand, G. Damiand, and C. Fiorio. Topological map: Minimal encoding of 3d segmented images. In Workshop on Graph based representations, pages 64–73, Ischia, Italy, may 2001. IAPR-TC15. [3] J.-P. Braquelaire and L. Brun. Image segmentation with topological maps and inter-pixel representation. Journal of Visual Communication and Image Representation, 9(1):62–79, march 1998. [4] J.-P. Braquelaire, P. Desbarats, and J.-P. Domenger. 3d split and merge with 3-maps. In Workshop on Graph based representations, pages 32–43, Ischia, Italy, may 2001. IAPR-TC15. [5] J.-P. Braquelaire, P. Desbarats, J.-P. Domenger, and C.A. W¨ uthrich. A topological structuring for aggregates of 3d discrete objects. In Workshop on Graph based representations, pages 193–202, Austria, may 1999. IAPR-TC15. [6] R. Brice and C.L. Fennema. Scene analysis using regions. Artiﬁcial intelligence, 1:205–226, 1970. [7] L. Brun. Segmentation d’images couleur a ` base Topologique. Th`ese de doctorat, Universit´e Bordeaux I, d´ecembre 1996.

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[8] L. Brun and J.-P. Domenger. A new split and merge algorithm with topological maps and inter-pixel boundaries. In The ﬁfth International Conference in Central Europe on Computer Graphics and Visualization, february 1997. [9] R. Cori. Un code pour les graphes planaires et ses applications. In Ast´erisque, volume 27. Soc. Math. de France, Paris, France, 1975. [10] J.P. Domenger. Conception et impl´ementation du noyeau graphique d’un environnement 2D1/2 d’´ edition d’images discr` etes. Th`ese de doctorat, Universit´e Bordeaux I, avril 1992. [11] C. Fiorio and J. Gustedt. Two linear time union-ﬁnd strategies for image processing. Theoretical Computer Science, 154:165–181, 1996. [12] S.L. Horowitz and T. Pavlidis. Picture segmentation by a directed split-andmerge procedure. In Proc. of the Second International Joint Conf. on Pattern Recognition, pages 424–433, 1974. [13] C.H. Lee. Recursive region splitting at hierarchical scope views. Computer Vision, Graphics, and Image Processing, 33:237–258, 1986. [14] P. Lienhardt. Subdivision of n-dimensional spaces and n-dimensional generalized maps. In 5th Annual ACM Symposium on Computational Geometry, pages 228– 236, Saarbr¨ ucken, Germany, 1989. [15] P. Lienhardt. Topological models for boundary representation: a comparison with n-dimensional generalized maps. Computer Aided Design, 23(1):59–82, 1991. [16] R. Ohlander, K. Price, and D.R. Reddy. Picture segmentation using a recursive region splitting method. Computer Graphics and Image Processing, 8:313–333, 1978. [17] M. Pietikainen, A. Rosenfeld, and I. Walter. Split and link algorithms for image segmentation. Pattern Recognition, 15(4):287–298, 1982. [18] P. Resch. Algorithmes pour la manipulation des cartes topologiques en 2 et 3 dimensions. M´emoire de dea, Universit´e Montpellier II, june 2001. [19] P. Resch. Algorithmes pour la manipulation des cartes topologiques en 2 et 3 dimensions. Annexe technique, Universit´e Montpellier II, june 2001.

On Characterization of Discrete Triangles by Discrete Moments ˇ c Joviˇsa Zuni´ Computer Science, Cardiﬀ University, Queen’s Buildings, Newport Road, PO Box 916, Cardiﬀ CF24 3XF, Wales, U.K. [email protected]

Abstract. For a given real triangle T its discretization on a discrete point set S consists of points from S which fall into T . If the number of such points is ﬁnite, the obtained discretization of T will be called discrete triangle. In this paper we show that the discrete moments having the order up to 3 characterize uniquely the corresponding discrete triangle if the discretizationing set S is ﬁxed. Of a particular interest is the case when S is the integer grid, i.e., S = Z2 . Then the discretization of a triangle T is called digital triangle. It turns out that the proposed characterization preserves a coding of digital triangles from an integer grid of a given size, say m × m within an O(log m) amount of memory space per coded digital triangle. That is the theoretical minimum. Keywords. Digital triangle, digital shape, coding, moments.

1

Introduction

The basic motivation for this paper was recovering a simple and eﬃcient characterization of digital triangles. By digital triangles we mean digital (binary) pictures of real triangles, or more formally, a digital triangle D(T ) is the set consisting of integer points which fall into a real triangle T : D(T ) = {(i, j) | (i, j) ∈ T,

i, j are integers} = {(i, j) | (i, j) ∈ T ∩ Z2 } .

That is the most usual digitization scheme for planar regions. But, sometimes the digitization (i.e., discretization) is made by using another “discretizationing” set than it is Z2 . Some other examples of discrete presentation of real objects are: Discrete images on the hexagonal grid, radar images, images made on statistically distributed set of points, e.t.c. Because the method presented here can be applied to discretizations on diﬀerent sets we start with

The author is also with the Mathematical institute of Serbian Academy of Sciences, Belgrade.

A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 232–243, 2002. c Springer-Verlag Berlin Heidelberg 2002

On Characterization of Discrete Triangles by Discrete Moments

. . . . .. . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . .. .. . . . . . . .. . . . . . .. . . .. .

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Fig. 1. A discretization of the triangle ABC consisting of 20 points is shown above.

a general deﬁnition of a discrete triangle D(T ) from a ﬁxed discrete point set S (see Fig. 1). Thus, we deﬁne D(T ) = {(x, y) | (x, y) ∈ T ∩ S} . Through the paper, it will be assumed but not mentioned, all discrete triangles consist of a ﬁnite number of points. For an illustration, the discretizations on the set consisting of all points with the coordinates which are rational numbers (i.e., S = Q2 ) are not considered. It is not convenient to use a real triangle to represent its discretization. In that case any discrete triangle can be represented by inﬁnitely many real triangles, since there is a continuum of real triangles which have the same discretization on a given set of points. Depending on S, it can be diﬃcult to answer which diﬀerent real triangles have diﬀerent discretizations. By the way, if we have a binary picture of a triangular objects, the “original triangle” is usually unknown. Consequently, the characterization of discrete triangles by a real triangle with a given discretization requires a procedure for reconstruction of an original triangle from its discretization. In the next section we give a characterization of discrete triangles which is simple and fast for any choice of the discretizationing set S. For such a characterization we will use, so called, discrete moments. Precisely, the discrete moment µp,q (X) of a ﬁnite set X is deﬁned as: µp,q (X) = xp · y q . (x,y)∈X

The moment µp,q (X) has the order p + q.

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We prove that ten discrete moments having the order up to 3 are enough for a unique characterization of discrete triangles discretized on a ﬁxed discrete set. In Section 3 we give some performance analysis of the proposed characterization if the discretization is made on a squared integer grid of a given size, i.e., on Z2 ∩ [0, m − 1]2 , for some integer m. Z2 ∩ [0, m − 1]2 , will be called the m × m integer grid. It turns out again that the use of moments is a powerful tool in image analysis ([3], [8]). Section 4 contains concluding remarks. Through the paper a ﬁnite set means that the set consists of a ﬁnite number of points. Also, a unique characterization and coding will have the same meaning. We shall say that a function f (x, y) separates sets S1 and S2 if f (x, y) has the diﬀerent sign in the points of S1 and S2 . For example, (x, y) ∈ S1 implies f (x, y) > 0, while (x, y) ∈ S2 implies f (x, y) < 0.

2

Characterization of Discrete Triangles

In this section it will be shown that the discrete moments having order up to 3 match uniquely the discrete triangles presented on a ﬁxed set S. We start with the following theorem. Theorem 1. Let S1 and S2 be two ﬁnite planar sets. If there exists a function of the form f (x, y) = αp,q · xp · y q , (1) p+q≤k

where p, q ∈ {0, 1, . . . , k} and αp,q are arbitrary real numbers, such that f (x, y) separates S1 \ S2 and S2 \ S1 then µp,q (S1 ) = µp,q (S2 ) is equivalent to

for all non negative integers

p, q,

with

p + q ≤ k,

S1 = S2 .

Proof. If S1 = S2 then the corresponding discrete moments are equal obviously. What we have to prove is: The equalities of the corresponded moments of the order up to k preserve S1 = S2 . We prove that by a contradiction. Let

xp · y q = µp,q (S1 ) = µp,q (S2 ) =

(x,y)∈S1

xp · y q

(x,y)∈S2

holds for all non negative integers p and q satisfying p + q ≤ k, and for some diﬀerent ﬁnite sets S1 and S2 . Since S1 = S2 we can assume S1 \ S2 is non empty, else we can start with the non empty S2 \ S1 . Further, because there exists a function f (x, y) of the form (1) f (x, y) =

p+q≤k

αp,q · xp · y q ,

On Characterization of Discrete Triangles by Discrete Moments

235

which separates S1 \ S2 and S2 \ S1 . Let be f (x, y) > 0 for (x, y) ∈ S1 \ S2 , while (x, y) ∈ S2 \ S1 implies f (x, y) < 0. Then    0< f (x, y) = αp,q · xp · y q  (x,y)∈S1 \S2

=



p+q≤k

−



=

 αp,q ·

p+q≤k

αp,q · xp · y q 

(x,y)∈S1 ∩S2

αp,q · xp · y q  

 αp,q ·

(x,y)∈S2 \S1

The contradiction

xp · y q  −

 



p+q≤k

0<0

 xp · y q 

(x,y)∈S1 ∩S2

αp,q · xp · y q  =

xp · y q 

αp,q ·





(x,y)∈S1 ∩S2

p+q≤k

(x,y)∈S2

αp,q ·

p+q≤k





xp · y q  −

(x,y)∈S1

p+q≤k

=





(x,y)∈S1 ∩S2

p+q≤k







αp,q · xp · y q  +

(x,y)∈S1 \S2

p+q≤k

=





p+q≤k

(x,y)∈S1 \S2

f (x, y) ≤ 0.

(x,y)∈S2 \S1

ﬁnishes the proof.

[]

Now, we can prove the main result of the paper. Theorem 2. Let a discrete point set S be given. Then, any discrete triangle D(T ) from S is uniquely determined by ten discrete moments µp,q (D(T )),

where

p + q ≤ 3,

and

p, q

are non negative integers.

Proof. We will use the previous theorem specifying that k = 3 and S1 and S2 are discrete triangles D(T ) and D(T1 ). If we consider T and T1 as planar compact regions then the set intersection T ∩T1 is either the empty set or the convex region whose boundary is the polygon having 6 vertices at most. In the case of T ∩ T1 = ∅ the proof is trivial because there always exists a straight line which separates T and T1 and consequently the same line separates D(T ) \ D(T1 ) and D(T1 ) \ D(T ). If the mentioned line is deﬁned by α · x + β · y + γ = 0 then (due to Theorem 1) the proof follows by setting f (x, y) = α · x + β · y + γ. If T ∩ T1 = ∅ then it is easily to conclude that all possible situations are characterized by Figures 2-9. Namely, if we consider the number of edges (i.e.,

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the number of vertices) of T ∩ T1 and mutual order between edges which belong to the boundary of T and those which belong to the boundary of T1 we distinguish the following cases: i) if T ∩ T1 has 6 vertices, the situation corresponds to Figures 2 and 3; ii) if T ∩T1 has 5 vertices we recognize two possibilities corresponded to Figures 4 and 5; iii) if T ∩ T1 has 4 vertices, we recognize three essentially diﬀerent situations corresponded to Figures 6, 7, and 8; iv) if T ∩ T1 has 3 vertices we have one nontrivial situation presented on Fig. 9. At all Figures 2-9, the triangle T with vertices A, B, and C is ﬁxed, while the triangle T1 varies. Triangle T1 is drown by dashed lines. The statement follows due to Theorem 1, since the existence of a function of the form (1) which separates the set diﬀerences of any two diﬀerent discrete triangles D(T ) and D(T1 ) is shown in all possible situations. For a straight line l, let αl · x + βl · y − γl = 0 be an equation which deﬁnes l. In the capture of any ﬁgure, a function f (x, y) which separates D(T ) \ D(T1 ) and D(T1 ) \ D(T ) is described by suitable chosen lines and the product of their equations.

. . w . . .. . . . . . .. . .. . C . . .. . . . .. . . . . .u . . . . . . . . . .A . . . . . . . . . . . . . . . . . . . . . . . . . v . . .. . . . . . . . . . . . . . . . .. . . . .. . . . . B . .. . . .. . a

b

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Fig. 2. f (x, y) = (αu · x + βu · y − γu ) · (αv · x + βv · y − γv ) · (αw · x + βw · y − γw ) is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b).

On Characterization of Discrete Triangles by Discrete Moments

237

. . . . .. . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . .. . . . . . .. . . .. . C

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Fig. 3. f (x, y) = (αu · x + βu · y − γu ) · (αv · x + βv · y − γv ) is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b).

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Fig. 4. f (x, y) = (αu · x + βu · y − γu ) · (αv · x + βv · y − γv ) is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b).

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. . . . .. . . .u . . .. . . . . C . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . A . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . .. . . . B . . .. . . .. . a

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Fig. 5. f (x, y) = αu · x + βu · y − γu is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b) .

. . . . .. . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . u . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . .. . v . .. . C

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Fig. 6. f (x, y) = (αu · x + βu · y − γu ) · (αv · x + βv · y − γv ) is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b) .

On Characterization of Discrete Triangles by Discrete Moments

239

. . u . . .. . . . . . .. . .. . . . .. . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . . . . . .. . . . . . .. . . .. . C

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Fig. 7. f (x, y) = αu · x + βu · y − γu is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b) .

. . . . .. . . . . . .. . .. . . .. . . . . . . . . . . . . . . . . . .. . . u. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . .. . . . . . .. . . . .. C

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Fig. 8. f (x, y) = αu · x + βu · y − γu is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b) .

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. . . . . . . . . C . . . . . . . . . . .. . . . .. . . . . . . . . . . . u . . . .A . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . .. . . . . B . .. . . .. . a

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Fig. 9. f (x, y) = αu · x + βu · y − γu is a separating function for D(T ) \ D(T1 ) (points labeled by a), and D(T1 ) \ D(T ) (points labeled by b). []

3

On Digital Triangles

As it is mentioned, we use a digital triangle instead of a discrete triangle if the discretization is made on the integer grid, i.e., S = Z2 . In this section we analyze the storage complexity if the proposed characterization (coding scheme) is applied. We have the following theorem. Theorem 3. Any digital triangle presented on the m × m integer grid can be coded by O(log m) bits. That is the asymptotic minimum. Proof. The proof follows from the simple fact µp,q (D(T )) ≤ (m − 1)5 , which implies that less than 10 · log(m − 1)5 = O(log m) bits are enough for the storage of the discrete moments µp,q (D(T )) under the assumptions: p + q ≤ 3 and D(T ) is a digital triangle from the (m × m)-integer grid. On the other side, since any of m2 pixels can be a discretization of a real triangle with a suﬃciently small area, it follows that O(log m) bits per coded digital triangle is a lower bound for the storage complexity. [] Remark. An equivalent formulation of the previous theorem is: If a real triangle is presented on a digital picture of a given resolution r (i.e., there are r pixels per unit) then O(log r) bits are suﬃcient for the storage.

On Characterization of Discrete Triangles by Discrete Moments

. . . . . .

1.5 ln 3

. . . . . .

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Fig. 10. Digitalization of the triangle T having the vertices ( π3 , ln 3), (4, 3), and √ ( 30, 32 ) by applying the resolutions r=1 and r = 2. The “new” points appearing in D(T ) after the increase of resolution are denoted by +.

Of course, this is much better than storage of all ≈ area of (T ) · r2 = O(r2 ) pixels belonging to D(T ), but also, it is better than the storage by using the Freeman 8-chain code. Namely, it is straightforward that O(r) bits are necessary for the storage of D(T ) by the Freeman code of the “digital” boundary of D(T ). A few examples given in Table 1, conﬁrm that. From the table, it can be seen that for a relatively small resolution r = 1, i.e., one pixel is the measure unit (see Figure 10), the proposed code requires 76 bits (since 19 characters from the set {0, 1, 2, . . . 9} should be stored) while 18 bits (for the storage of 6 characters from {0, 1, 2, . . . , 7}) are suﬃcient for Freeman coding (i.e., 8-chain coding). The situation is changing if the resolution increase. For example, if r = 100, i.e., there are 100 pixels per measure unit, the proposed code (99 characters from {0, 1, 2, . . . 9}) requires 396 bits while the Freeman code consisting of 884 characters from {0, 1, 2, . . . 7} requires 2652 bits. If r = 500 the amounts of bits are are 504 and 13284, for r = 1000, they are 556 and 35432, respectively. Of course, the dominance of the new code is more obvious for higher values of r and in the limit case it is in accordance with the previous asymptotic estimates.

4

Concluding Remarks

In this paper an eﬃcient characterization of digital triangles is given. The described characterization is simple and asymptotically optimal with respect to the

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√ π 3 Table 1. The triangle T with vertices ( , ln 3), (4, 3), and ( 30, ) is 3 2 presented on digital pictures having diﬀerent resolutions. The code of D(T ) presented here and the size of the Freeman code of the boundary of D(T ) are given. applied the proposed code resolu- µ0,0 , µ1,0 , µ0,1 , µ2,0 , µ1,1 , µ0,2 , tion r µ3,0 , µ2,1 , µ1,2 , µ0,3

Freeman code length

1

3, 11, 7, 41, 26, 17 155, 98, 64, 43

6

2

15, 107, 57, 803, 410, 229 6281, 3082, 1664, 975

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218

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1000 3620, 401408, 213717, 47566142, 24222891, 13226957 5924570114, 2902540215, 1526584563, 856530219

100

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36185, 12693880, 6752390, 4759918368, 2420790884, 1320567596 1876452768418, 917747474042, 482006394242, 270164847422 884

used memory space. The method can be applied to discretization of triangles on diﬀerent “discretizationing” sets, as well. The corresponding discrete moments having order at most 3 are suﬃcient for a unique determination of the given discrete triangle. Let us mention that the problem of eﬃcient representation is already studied for digital straight line segments ([1], [7]), digitized circular arcs ([11]), digital ellipses ([13]), and digital polynomial segments ([12]). So, by the coding scheme proposed here a fast comparison of digital triangles and an eﬃcient storage of them are preserved. The recognition and reconstruction problems are not studied here. For digital n-gons the recognition problem is considered in the literature but only for digital squares ([2], [9]). The problem of optimal coding of digital polygons for an arbitrary n seems to be diﬃcult one even in the case of digital convex polygons. Namely, is it known ([4]) that any optimal coding is within

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243

O(m2/3 ) bits per coded digital convex polygon from the m × m−integer grid but such a coding scheme is still unknown. The problem of a fast computation of the discrete moments for digital triangles is omitted from the paper because a number of papers were already devoted to developing fast algorithms of moment computation for 2D objects ([5], [6]). Also, a general approach for moment calculation for polygons and line segments is given in [10].

References 1. Dorst, L., Smeulders, A. W. M.: Discrete representation of straight lines. IEEE Trans. Pattern Anal. Machine Intell. 6 (1984) 450-463. 2. Forchhammer, S., Kim, C. E.: Digital squares. IEEE Trans. Pattern Anal. Machine Intell. 22 (1988) 672-674. 3. Hu, M., Visual pattern recognition by moment invariants, IRE Trans. Inf. Theory 8 (1962) 179–187. ˇ c, J.: On the number of digital convex polygons in4. Ivi´c, A., Koplowitz, J., Zuni´ scribed into an (m, m)-grid. IEEE Trans. on Information Theory. 40 (1994) 16811686. 5. Jiang, X. Y., Bunke, H.: Simple and fast computation of moments. Pattern Recognition. 24 (1991) 801-806. 6. Leu, J.-G.: Computing a shape’s moments from its boundary. Pattern Recognition. 24 (1991) 949-957. 7. Lindenbaum, M., Koplowitz, J.: A new parametrization of digital straight lines. IEEE Trans. Pattern Anal. Machine Intell. 13 (1991) 847-852. 8. Mamistvalov, A. G.: n-Dimensional moment invariants and conceptual mathematical theory of recognition n-dimensional solids. IEEE Trans. Pattern Anal. Machine Intell. 20 (1998) 819-831. 9. Nakamura, A., Aizawa, K.: Digital squares. Computer Vision, Graphics Image Processing 49 (1990) 357-368. 10. Singer, M. H.: A general approach to moment calculation for polygons and line segments. Pattern Recognition. 26 (1993). 1019-1028. 11. Worring, M., Smeulders, A. W. M.: Digitized circular arcs: Characterization and parameter estimation. IEEE Trans. Pattern Anal. Machine Intell. 17 (1995) 587598. ˇ c, J., Acketa, M. D.: A general coding scheme for families of digital curve 12. Zuni´ segments. Graphical Models and Image Processing 60 (1998) 437-460. ˇ c, J., Sladoje, N.: Eﬃciency of Characterizing Ellipses and Ellipsoids by Dis13. Zuni´ crete Moments. IEEE Trans. Pattern Anal. Machine Intell. 22 (2000) 407-414.

Weighted Distance Transforms for Images Using Elongated Voxel Grids Ida-Maria Sintorn and Gunilla Borgefors Centre for Image Analysis, Swedish University of Agricultural Sciences L¨ agerhyddv¨ agen 17, SE-752 37 Uppsala, Sweden {ida,gunilla}@cb.uu.se

Abstract. In this paper we investigate weighted distance transforms in 3D images using elongated voxel grids. We use a local neighbourhood of size 3 × 3 × 3 and assume a voxel grid with equal resolution along two axes and lower along the third. The weights (local distances) in the local neighbourhood are optimized by minimizing the maximum error over linear trajectories, which is a completely digital approach. General solutions are presented, as well as numerical solutions for the cases when the voxels are 1.5 and 2.58 times longer in one direction. Integer solutions for both real and integer scale factors are presented. As an application example, the medial axis of an object is computed in an image with elongated voxels and compared to the medial axis computed on the same image interpolated to equal resolution along all axes.

1

Introduction

Three-dimensional imaging systems often generate images with unequal resolution along the diﬀerent axes. The case investigated here is when the resulting images consist of elongated voxels, where two sides are equal and the third is longer. This is generally true for tomographic, and many microscopic, images. To be able to apply standard image operations, the images are usually interpolated in the longer direction to give cubic voxels. This interpolation step results in a much larger data set without adding any new information compared to the original image. It also vastly increases the computational load for any operation applied. Distance transforms (DTs), are used in many various image operations. Therefore, a DT that could be applied to the original image with elongated voxels would be a useful tool. A distance transformation is applied to a binary image, where it computes the distance from each voxel in the foreground (background) to the nearest voxel in the background (foreground). The image is converted to a grey-level image, as each voxel in the foreground (background) is given the value of the distance. As in 2D, the DT is computed by propagating local distances over the image using two raster scans [7]. When computing a weighted DT (WDT) the local distances used (in our case those in a 3 × 3 × 3 mask) are given diﬀerent weights, so that the resulting DT becomes as rotation independent as possible. A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 244–254, 2002. c Springer-Verlag Berlin Heidelberg 2002

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The Euclidean distance is rotation independent up to digitization eﬀects. Therefore, the weights in the WDT are chosen to minimize the diﬀerence between the WDT and the Euclidean distance. An Euclidean distance transform for images with elongated voxels (or pixels) is possible to construct, but we have not seen it in the literature. In any case, an Euclidean distance transform is inconvenient for many applications and it is also more computationally demanding than the WDT. Integer approximations of the weights in the WDT are often calculated, since integers are usually preferred in image processing. WDTs in square and cubic grids have been thoroughly investigated [2,3]. For rectangular grids, WDTs have been optimized using two diﬀerent optimization criteria. Coquin and Bolon [4], use the diﬀerence between the Euclidean distance and the weighted distance over circular trajectories as their optimization criterion. Bolon, Vila, and Auzepy [1], and Sintorn and Borgefors [8], optimize over linear trajectories. In three dimensions, WDTs in parallelepipedic grids have so far only been optimized using circular trajectories, [5], but without any discussion of constraints on the local distances, or of the diﬀerent cases that occur. (Not all combinations of local distances give the expected results.) In this paper, we optimize over linear trajectories, which is a completely digital approach. In Section 2, we describe our optimization method and in Section 3, we give resulting WDTs and show an application example.

2

Method

The working space here is a binary 3D image digitized in an elongated voxel grid. Each voxel has height and width equal to 1, and length Λ ≥ 1. This is the most common case in tomographic and microscopic images. Fig. 1 shows the local distances in the ﬁrst octant of a 3 × 3 × 3 neighbourhood. The optimal local distances a, b, c, d, and e need to be determined for each Λ. Our optimization criterion is to minimize the maximum error between the 3 × 3 × 3 WDT and the digital Euclidean distance in a cubic image of size ΛM × ΛM × M , see Fig. 2. d a a

e b

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Fig. 1. Local distances for 3 × 3 × 3 WDTs in rectangular grids. (Two views of the same neighbourhood.)

All sets of local distances do not give useful WDTs. The shortest path between two voxels is the set of local distances used to connect the voxels that gives the minimal total cost (sum of weights). Some constraints need to

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be applied to a, b, c, d, and e to prohibit the shortest path between two voxels to zig-zag over the image, which would result in extremely complex distance expressions and serious violations of the triangular inequality for the resulting distance measure. Deﬁnition 1: Assume that two voxels can be connected by only one type and one direction of local steps. If these steps constitute a minimal path deﬁning the distance, the WDT is semi-regular. If it is the only minimal path the WDT is regular. If Λ is assumed to be ≥ 1, the following constraints need to be fulﬁlled for the WDT to be semi-regular: a ≤ b,

a ≤ c,

b ≤ d,

c ≤ d,

d ≤ e,

2e ≤ b + 2d,

b ≤ 2a,

d ≤ a + c. (1)

These inequalities are not suﬃcient to determine unique expressions for the distance values. The path from the origin to the points (1,2,1) and (1,1,2) can be formed in two ways fulﬁlling the regularity criteria stated above. The point (1,2,1) can be reached either by steps a + b or d + b, and the point (1,1,2) can be reached by steps e + c or d + d. These ambiguities give rise to four cases, which are listed below. The only case further investigated here is denoted Case 1. Case Case Case Case

1:e+a≤d+b 2:e+a≤d+b 3:d+b≤e+a 4:d+b≤e+a

and e + c ≤ 2d, and 2d ≤ e + c, and e + c ≤ 2d, and 2d ≤ e + c.

(2)

Optimization is performed in one half of the ﬁrst octant, with one object voxel in (0,0,0). Due to symmetry, the rest of the volume need not be considered. The seven other octants are symmetric to the ﬁrst one, and the two halves of the ﬁrst octant are symmetric since the case when x = ΛM is symmetric to the case when y = ΛM . The expression for the computed distance is diﬀerent in diﬀerent parts of the volume. The three distance expressions become: DT (x, y, z) = ax + (b − a)y + (e − b)z for DT (x, y, z) = ax + (d − a)z + (e − d)y for DT (x, y, z) = cz + (d − c)x + (e − d)y for

0 ≤ z ≤ y ≤ x, 0 ≤ y ≤ z ≤ x, 0 ≤ y ≤ x ≤ z.

(3)

We can assume that the maximal diﬀerence to the EDT occurs on the border of the image, i.e., for x = ΛM or z = M . This results in ﬁve diﬀerence equations. Fig. 2 shows the areas that need to explored and also were the ﬁve diﬀerence equations occur.

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Fig. 2. Geometry of the WDT in the 3 × 3 × 3 neighbourhood case. Two orthogonal cuts through an ΛM × ΛM × M image are shown.

Diﬀ1 (ΛM, y, z) = aΛ + (b − a)y + (e − b)z − Λ2 M 2 + y 2 + Λ2 z 2 for 0 ≤ z ≤ M, z ≤ y ≤ ΛM = Area 1, Diﬀ2 (ΛM, y, z) = aΛM + (d − a)z + (e − d)y − Λ2 M 2 + y 2 + Λ2 z 2 for 0 ≤ y ≤ M, y ≤ z ≤ M = Area 2, Diﬀ3 (x, y, M ) = ax + (b − a)y + (e − b)M − x2 + y 2 + Λ2 M 2 for M ≤ x ≤ ΛM, M ≤ y ≤ x = Area 3, Diﬀ4 (x, y, M ) = ax + (d − a)M + (e − d)y − x2 + y 2 + Λ2 M 2 for M ≤ x ≤ ΛM, 0 ≤ y ≤ M = Area 4, Diﬀ5 (x, y, M ) = cM + (d − c)x + (e − d)y − x2 + y 2 + Λ2 M 2 for 0 ≤ x ≤ M, 0 ≤ y ≤ x = Area 5.

(4)

To optimize the WDT, we must ﬁnd the a, b, c, d, and e that minimize the maximum of these expressions. For each of them the maximum can occur at the end points of the intervals, or where the derivative is equal to zero. This results in 39 error equations, of which 23 are diﬀerent. These are listed below. The top index shows in what area the error equation appears, and the lower index is a consecutive numbering of all error equations. All Eji have been divided by M . Area 1 : E11 = Λ(a − 1) for (ΛM, 0, 0), E21 = Λ(a − √ 1 − (b − a)2 for (ΛM, ymax , 0), 1 E3 = Λ(b − 2) for (ΛM, ΛM, 0), E41 = Λ(a − 1 − (e − a)2 /(1 + Λ2 ) for (ΛM, zmax , zmax ), 2 − Λ2 (b − a)2 − (e − b)2 E51 = Λa − √ Λ for (ΛM, ymax , zmax ), 1 E6 = Λb − 2 Λ2 √ − (e − b)2 for (ΛM, ΛM, zmax ), 2+1 E71 = Λa + e − a − √ 2Λ for (ΛM, M, M ), 1 E8 = Λa + e√− b − Λ 2 1 − (b − a)2 for (ΛM, ymax , M ), E91 = Λ(b − 3) + e − b for (ΛM, ΛM, M ).

(5)

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Area 2 : 2 E10 = E11 , 2 E11 = Λa − √Λ2 − (d − a)2 for (ΛM, 0, zmax ), 2 E12 = Λ(a − 2 + d − a for (ΛM, 0, M ), 2 E13 = E41 , 2 E14 = Λa − Λ2 − Λ2√ (e − d)2 − (d − a)2 for (ΛM, ymax , zmax ), 2 E15 = Λa + d − a − Λ 2 1 − (e − d)2 for (ΛM, ymax , M ), 2 E16 = E71 .

(6)

Area 3 : √ 3 E17 = e − Λ2 + 2 3 E18 = ∅, 3 2 E19 = E71 = E16 , 3 E20 = ∅, 3 E21 = ∅, 3 E22 = E81 , 3 E23 = E91 .

(7)

for

(M, M, M ),

Area 4 : √ 4 E24 = d − Λ2 + 1 for (M, 0, M ), 4 E25 = ∅, 4 2 E26 = E12 , 4 E27 = d − (1 + Λ2 )(1 − (e − d)2 ), for 4 E28 = ∅, 4 2 E29 = E15 , 4 3 E30 = E17 , 4 E31 = ∅, 4 2 3 E32 = E71 = E16 = E19 .

(M, ymax , M ),

Area 5 : 5 E33 = c − Λ for (0, 0, M ), 5 E34 = c − Λ 1 − (d − c)2 for (xmax , 0, M ), 5 4 E35 = E24 , √ 5 E36 = c −Λ/ 2 2 − (e − c)2 for (ymax , ymax , M ), 5 E37 = cΛ 1 − (d − c)2 − (e − d)2 for (xmax , ymax , M ), 5 4 E38 = E27 , 5 3 4 E39 = E17 = E30 .

(8)

(9)

Numerical experimentation shows that the minimum of these expressions occurs when 3 4 5 − E11 = −E31 = E51 = −E17 = −E24 = −E33 .

(10)

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249

Solving these equations give the following expressions for the optimal a, b, c, d, and e, and the maximal diﬀerence from the Euclidean distance, denoted maxdiﬀ. √ √ √ √ 3Λ2 + Λ( 2 − 2 − Λ2 + 2) + Λ2 + 2 + 1 − 2 a(Λ) = + 5Λ2 − 2Λ + 1 √ √ √ √ √ Λ2 (4 Λ2 + 2(2 2 − 1 + Λ) + 5Λ2 (2 2 − 3) + 2Λ(3 − 4 2) + 6 2 − 19) , 5Λ2 − 2Λ + 1 √ b(Λ) = a(Λ) − 1 + 2, c(Λ) = Λa(Λ), √ d(Λ) = Λa(Λ) − Λ + √ 1 + Λ2 , e(Λ) = Λa(Λ) − Λ + Λ2 + 2, maxdiﬀ(Λ) =| Λ − Λa(Λ) | .

(11) These solutions are valid for all Λ ≥ 1. With Λ = 1 this becomes exactly the solution for the cubic grid found in [3]. The solution when a ≡ 1 is needed when integer approximations with integer scale factors are calculated. The system to be solved then becomes ∗3 ∗4 ∗5 − E3∗1 = E5∗1 = −E17 = −E24 = −E33 ,

(12)

which gives: a∗ (Λ) = 1√ √ √ √ 2 2Λ2 − ( Λ2 + 2 + 2)Λ + Λ2 + 2 b∗ (Λ) = + 3Λ2 − 2Λ + 1 √ √ √ √ √ √ √ Λ2 (6 2 − 9)Λ2 + 2 2 Λ2 + 2 − 4 2 + 2)Λ − 7 + 2 2 Λ2 + 2 + 2 2) , 3Λ2 − 2Λ + 1 √ ∗ ∗ c (Λ) = Λ(b +√1 − 2), √ d∗ (Λ) = Λ(b − √ 2) + √ Λ2 + 1, e∗ (Λ) = Λ(b − 2) + Λ2 + 2, √ maxdiﬀ(Λ) =| Λ( 2 − b) | .

(13) Note that we do not get c∗ (1) = a∗ (1), i.e., we do not get the same solution as for the cubic grid for Λ = 1. This is correct. We have only constrained a∗ , not c∗ . With a∗ = 1, the other local distances, including c∗ , decrease to compensate for the forced increase in a∗ . Integer approximations are found by multiplying the optimal local distances by a scale factor and rounding to the nearest integer. If only integers are considered as scale factors the solution with a ≡ 1, b∗ , c∗ , d∗ , e∗ , is used, since a then will serve as the scale factor. Besides integer scale factors for the a ≡ 1 solutions, real scale factors ranging from 1 to 20, with three decimals, are investigated here, as integer scale factors give poor results for Λ > 1. In this case the solution a(Λ), b(Λ), c(Λ), d(Λ), and e(Λ) should be used.

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In practice, for each integer approximation of a an integer neighbourhood of b is checked, and for each integer approximation of b an integer neighbourhood of c, etc, is searched. For each set of integer local distances the maximal diﬀerence is computed from the error equations (5) − (9). Each solution is, of course, also examined to make sure that it fulﬁlls the regularity criteria (1) and (2), Case 1.

3

Results

In this section Tables of solutions and an illustrating example are presented. In Tables 1 and 2 the optimal, real, and integer local distances for Λ = 1.5, and Λ = 2.58 are shown, respectively. The sizes of Λ were chosen to represent a small and a fairly large Λ. Λ = 2.58 is also the Λ occurring in the image used in the application example. In the Tables, the solutions for aopt , and a ≡ 1 are presented, as well as the best integer solutions for integer and real scale factors up to 20. Only solutions better than those for lower scale factors are listed. As can be seen by comparing the Tables, the error (maxdiﬀ) grows with increasing Λ, as can be expected. It is interesting to compare our optimal values to those in [5], which optimizes the maximal error over circular trajectories. Their optimal local distances for Λ = 1.5 become: a = 0.9237, b = 1.3063, c = 1.3855, d = 1.6652, e = 1.9042, and for Λ = 2.58: a = 0.8953, b = 1.2662, c = 2.3099, d = 2.4774, e = 2.6342. As can be seen in the Tables, the two methods generate very similar local distances. A comparison of the errors for the two methods is irrelevant, since they use diﬀerent optimization criteria. As mentioned above, the error grows rapidly with increasing Λ. This means that the calculated weighted distances diﬀer quite a lot compared to the Euclidean distances. Hence, this WDT is not a good choice when true distances are desired. The WDT also becomes more rotation dependent than desired. In many applications, however, only relative distances are needed and rotation is not an issue. The use of parallelepipedic WDTs can then save much time and memory. As an example, the medial axis (MA), also known as the set of centres of maximal balls, of an object is calculated on an image with diﬀerent resolution along the z-axis as well as on the same image interpolated to cubic voxels. The MA in turn, can, e.g. be used for image description, analysis, and compression. The binary test image comes from a magnetic resonance angiography image, where the arteries have been segmented. The size of the original image is 280 × 430 × 96 voxels (11.6 Mbyte) and each voxel is of size 1 × 1 × 2.58 mm3 . Interpolated to equal resolution along all three axes the image size becomes 280 × 430 × 250 voxels (30Mbyte). The number of voxels in the original image is 416,423 compared to 1,084,929 for the interpolated image. Fig. 3 shows projections, middle slices, and middle slices of the MA of the interpolated and original images. All images are from the right (z − y) side of the volume to show the diﬀerence in resolution in the z-direction. Hence, the big bend in the middle of the images is the aorta bending towards the person’s back. Fig. 3a) and 3d) show simple 2D-projections of the volume, where the grey levels represent the depths of the object voxels.

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Table 1. Integer approximations for Λ = 1.5. scale factor a b c d e 1.999 2 3 3 3 4 2.160 2 3 3 4 4 3.103 3 4 5 5 6 4.507 4 6 7 8 9 5.266 5 7 8 9 10 6.666 6 9 9 11 13 8.847 8 12 12 15 17 16.558 15 22 23 28 32 18.727 17 25 26 32 36 . . . . . . ∞ aopt bopt copt dopt eopt opt 0.9085 1.3227 1.3628 1.6656 1.9243 1 1 1 2 2 2 2 2 3 3 3 4 3 3 4 4 5 6 6 6 8 8 10 11 7 7 9 10 12 13 8 8 11 11 13 15 9 9 12 12 15 17 15 15 20 20 25 28 16 16 21 21 26 30 . . . . . . ∞ 1 b∗ c∗ d∗ e∗ opt* 1 1.2887 1.3118 1.6145 1.8733

maxdiﬀ 0.302 0.210 0.191 0.169 0.163 0.153 0.144 0.141 0.139 . 0.1372 0.1372 0.621 0.303 0.253 0.228 0.204 0.202 0.200 0.195 0.190 . 0.1882 0.1882

For the interpolated image with equal resolution along all sides the common <3,4,5> WDT, in our notation 3-4-3-4-5, maxdif f = 10.01%, was used, and for the original image with Λ = 2.58 the 6-9-16-17-18 WDT was chosen from Table 2. It was chosen because its error is rather close to the optimal error. To get a smaller error, the local distances have to be increased rather signiﬁcantly, making the voxel values uncomfortably large. The MA is computed from the WDT by local tests using a Look-up table, which is speciﬁc for each set of weights used. The Look-up tables store, for each voxel value of the WDT, the smallest values of the neighbours that prohibit the voxel to be part of the MA. The Look-up tables were calculated by the method presented by Remy and Thiel in [6]. To produce the Look-up tables, they ﬁrst calculate the WDT from each point in the ﬁrst 1/48th of the volume to the origin. The rest of the volume can be omitted from the calculations due to symmetry. Then they scan through this WDT and for each voxel and all directions of local steps they search for the smallest value that prohibit the voxel in focus to be part of the MA. This is done by comparing the value in the WDT at a position one mask step away from the voxel with the value of the voxel plus the weight of the mask step. Their method simultaneously tests whether the directions in

252

I.-M. Sintorn, G. Borgefors Table 2. Integer approximations for Λ = 2.58. scale factor a b c d e maxdiﬀ 1.999 2 3 4 5 5 0.584 2.301 2 3 6 6 6 0.338 4.601 4 6 11 12 12 0.337 6.882 6 9 16 17 18 0.331 17.105 15 22 39 42 45 0.330 . . . . . . . ∞ aopt bopt copt dopt eopt 0.3225 opt 0.8750 1.2892 2.2575 2.4445 2.6197 0.3225 1 1 1 2 2 2 1.069 2 2 3 4 5 5 0.582 3 3 4 7 7 8 0.546 4 4 5 9 10 11 0.481 6 6 8 13 14 15 0.446 8 8 10 18 19 20 0.442 15 15 19 33 35 38 440 19 19 24 41 45 48 0.435 . . . . . . . ∞ 1 b∗ c∗ d∗ e∗ 0.4263 opt* 1 1.2490 2.1537 2.3408 2.5159 0.4263

the WDT mask are enough to produce the correct Look-up table, or if extra directions must be added to compute the table. The only changes that were needed to make the method applicable to a parallelepipedic grid was to modify the mask and to search 1/16th of the volume instead of 1/48th. For the original image the MA consists of 110,067 voxels while the MA for the interpolated image consists of 207,013 voxels. The calculation of the MA of the original image required 1.12 CPUs compared to 2.83 CPUs for the interpolated image. Hence, it is almost exactly a factor Λ faster to calculate the MA in the original image instead of in the interpolated one. A further bonus is of course that no interpolation is needed.

4

Conclusions

Optimal weighted distance transforms for a 3×3×3 neighbourhood in 3D images with elongated voxel grids have been investigated. The results presented are valid for all elongated voxel ratios as long as two sides are of equal length, which is often the case for tomographic and microscopic images. Numerical results for grids with voxels of size 1×1×Λ, when Λ equals 1.5 and 2.58 have been presented. The new WDTs are useful in applications where relative distances are needed. As an example the MA was computed on a medical image with elongated voxels and compared to the MA computed on the same image interpolated to cubic voxels. To get a better approximation to the Euclidean distance transform a larger

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Fig. 3. 2D-projection, middle slice, and middle slice of computed medial axis of segmented arteries from a magnetic resonance angiography image. All images are produced from the right (z − y) side of the volume to show the diﬀerence in resolution in the z-direction. The lower row shows images from the original volume with voxel dimension 1 × 1 × 2.58. The upper row shows images from the same volume but interpolated to voxel size 1 × 1 × 1. The grey levels in c) and f) represent the voxel values in the MA. The resulting MA are quite similar in both cases (when interpolation is not considered).

neighbourhood could of course be considered. The asymmetrical neighbourhood 5 × 5 × 3, used in [5], should then be a good way to compensate for the largest errors, without increasing the computation times as much as with a complete

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5 × 5 × 5 neighbourhood. However, to calculate such local distances many more constraints would have to be considered to ensure that the WDTs are wellbehaved (semi-regular) and that the equations used for optimization really are the ones valid in the Case investigated. In summary, the errors become quite large when distance transforms are computed in images with elongated voxels, unless the elongation factor is small. In fact, a Λ of 1.57 is enough to double the optimal maxdiﬀ compared to a cubic grid. The WDTs presented here are good choices for small Λ in all cases and for larger Λ in cases where rotation dependence and true distances are unimportant. In other cases, interpolation is the only feasible option.

References 1. P. Bolon, J. L. Vila, and T. Auzepy. Operateur local de distance en mailliage rectangulaire. In Proc. 2eme Colloque de Geometrie Discrete en Imagerie: Fondements et Applications, Grenoble, France, pages 45–56, Sept. 1992. 2. G. Borgefors. Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34:344–371, 1986. 3. G. Borgefors. On digital distance transforms in three dimensions. Computer Vision and Image Understanding, 64(3):368–376, Nov. 1996. 4. D. Coquin and P. Bolon. Discrete distance operator on rectangular grids. Pattern Recognition Letters, 16:911–923, 1995. 5. D. Coquin, Y. Chehadeh, and P. Bolon. 3D local distance operator on parallelepipedic grids. In Proc. 4th Discrete Geometry for Computer Imagery, Grenoble, France, pages 147–156, Sept. 1994. 6. E. Remy and E. Thiel. Computing 3d medial axis for chamfer distances. In Borgefors, Nystr¨ om, and S. di Baja, editors, Discrete Geometry for Computer Imagery, volume 1953 of Lecture Notes in Computer Science, pages 418–430. Springer-Verlag, Dec. 2000. 7. A. Rosenfeld and J. L. Pfaltz. Sequential operations in digital picture processing. Journal of the Association for Computing Machinery, 13(4):471–494, Oct. 1966. 8. I.-M. Sintorn and G. Borgefors. Weighted distance transforms in rectangular grids. In Ardizzone and Ges` u, editors, Proc. 11th International Conference on Image Analysis and Processing (ICIAP 2001), Palermo, Italy, pages 322–326. IEEE Computer Society, sep 2001.

Robust Normalization of Shapes 1

2

Javier Cortadellas , Josep Amat , and Manel Frigola

3

1

Departament d’Electrònica, Enginyeria La Salle, Universitat Ramon Llull Pso. Bonanova 8, 08022- Barcelona, Spain [email protected] 2 IRI - Institut de Robòtica e Informàtica, Universitat Politècnica de Catalunya Llorens Artigas 4-6, 08028 - Barcelona, Spain [email protected] 3 Departament d’Enginyeria de Sistemes, Automàtica i Informàtica Industrial Universitat Politècnica de Catalunya C. Pau Gargallo 5, 08028 Barcelona, Spain [email protected]

Abstract. The normalization of a binary shape is a necessary step in many image processing tasks based on image domain operations. When one must deal with deformable shapes (due to the projection of non-rigid objects onto the image plane or small changes in the position of the view point), the traditional approaches doesn’t perform well. This paper presents a new method for shape normalization based on robust statistics techniques, which allows to keep the location and orientation of shapes constant independent of the possible deformations they can suffer. A numerical comparison of the sensitivity of both methods is used as a measure to validate the proposed technique, together with a ratio of areas between the non-overlapping regions and the overlapping regions of the normalized shapes. The results presented, involving synthetic and real shapes, show that the new normalization approach is much more reliable and robust that the traditional one.

1 Introduction The normalization of a binary shape is a necessary step in many image processing tasks based on image domain operations. It is even a crucial step when the main goal of a computer vision system is the discrimination of objects depending on their visual appearance. In such systems, the way an object is represented arises as an important question to be solved. Although other appearance properties such as color or texture can be exploited, shape can be used as a powerful tool for describing and recognizing objects, if a reliable image segmentation method is available. The shape of an object refers to its profile and physical structure. In this paper, shapes are considered binary images that come from the projection of 2D or 3D objects onto a 2D image plane (they are also usually called silhouettes [14]). There are basically four ways of representing this important characteristic: by means of boundaries, regions, moments or structural representations. Among them, the description based on the region that an arbitrary shape occupies is the simplest one. The benefits of this simple description are that it has a straight meaning for human visual perception systems, it avoids the computation of shape descriptors [3,4,5], which can be ambiguous, and finally, as the description lies on the image domain, it is A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 255–266, 2002. © Springer-Verlag Berlin Heidelberg 2002

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possible to use specialized hardware architectures like [13] that operate over the image domain at very high speeds. However, the main drawback for this kind of representation is that, when used in shape recognition tasks, it needs to be normalized with respect to similarity transformations (translation, rotation and scaling) in order to grant a unique representation of the projection of an object onto an image plane. Traditionally, the centroid (also called center of gravity) and the angle of the axis of the least moment of inertia have been used to normalize the localization and orientation of the shapes [2,3,4,5]. A standard method consists of rotating the shape around the centroid so that the axis previously mentioned, called principal axis, has some predefined orientation. This normalization does not present difficulties when dealing with shapes of rigid objects that have always been projected onto the image plane from the same point of view. However, problems arise when one must deal with deformable shapes, produced by small changes of perspective or alterations of the physical structure of non-rigid objects, which produces significant changes of the periphery of their shape. In many applications such as shape recognition or registration, it would be desirable that two different shapes of a same object class share to a certain extent (depending on the degree of deformation) the same location and orientation, because this fact would ease further processing tasks, but the classical normalization approaches doesn’t perform well when dealing with such deformed shapes. The main contribution of this paper is the development of a reliable technique for accurately estimating the robust centroid and the robust principal axis of deformable shapes, making use of robust statistics [1,7,8]. Size normalization is not treated in this paper because it is assumed that objects with different scale must be discriminated. The structure of the paper is as follows: after this introduction, the traditional approach for normalizing shapes, based on the centroid and the principal axis, is briefly described. After pointing out some of the existing problems of this approach, a new robust normalization method is developed. Then, by means of a similarity measure based on the discrepancy of overlapping areas, the proposed method is compared with the traditional normalization technique. Some results involving synthetic and real shapes are also presented and finally, the conclusions are summarized.

2 Normalization of Shapes. Classical Method There are a number of different methods that have been used to determine the orientation of a shape [3], like the maximum Feret’s diameter – the line between the two points on the periphery that are furthest apart - or the major axis of an ellipse fitted to the contour of the shape silhouette. Among these methods, those that take into account all the pixels of the shape have often worked better than those based on contour or boundary representations, because they are less influenced by the presence or absence of a single pixel around the periphery. One of the most widely used methods among those that consider the whole shape is related to the computation of the principal axis. It is important to mention that the main goal of this work is not to find a meaningful axis in terms of visual perception but to find an axis insensitive to alterations of the shape due to admissible deformations. This axis would allow different deformed shapes of a same object to be all aligned in a fixed direction.

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2.1 Principal Axis Based Orientation Method In this method, the centroid is used to normalize the location and the orientation is defined as the angle of the least moment of inertia axis [5]. This axis corresponds to the line about which it takes the least amount of energy to spin an object of like shape. It is called principal axis, and can be regarded as the line that “best fits” the shape. It is obtained by minimizing, with respect to the angle θ, the sum of the squared distances of each point of the shape to an axis that, passing through the centroid of the shape, has a slope with value tan θ.

θ Pi

di

Fig. 1. Arbitrary shape with its principal axis, which is the line that minimizes the sum of the distances (di) of all the points of the shape (Pi). Assuming discrete images, this points are the coordinates of the pixels of the shape.

The result of this minimization is [3,5]:

θ=

 2 µ11  1  tan −1  2  µ 20 − µ 02 

(1)

where µ are the central moments of the shape. Being I a binary-valued picture, and S = {(x,y) | I(x,y) =1} the set of pixels representing a two-dimensional shape, for each pair of nonnegative integers (j,k), the above central moments of S are given by [5] jk

µ jk =

∑ (x − x ) ( y − y ) ( ) j

k

(2)

x , y ∈S

where ( x, y ) are the coordinates of the centroid of the shape, also used for location normalization. Moments can be given a physical interpretation by regarding S as an area composed of a set of point masses located at the points (x,y), and thus providing useful information about the spatial arrangement of the points of S. The standard method for normalizing S with respect to rotation is to rotate it in such a way that its principal axis has some standard orientation, say vertical. The main problem of this principal axis approach is that, although it is quite insensitive to small variations on the boundary of the shape (i.e. due to discretization noise), it becomes very sensitive when the shape varies significantly (i.e. due to structural changes of the object it represents). This fact can be seen in Fig. 2.

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(a)

(b)

(c)

Fig. 2. Principal Axis estimation of a deformable object: (a) θ = 30.47º, centroid =(64,61), (b) θ = 13.92º, centroid=(61,65). (c) Overlapping of the two previous images. Notice the difference between the two shapes and its centroids.

This exemple shows two instances (a and b) of the same class of object “elephant”. Despite of the fact that both shapes have a very similar structure, - and thus it would be desirable that they share, to a certain extent, the same orientation and centroid -, there is a difference of more than 16 degrees between both principal axis and of 7 pixels between the centroids (using the chessboard metric). These normalization results would be unacceptable in a template matching framework for instance, because the missalignment would cause an important decrease of the similarity measure. The reason for these differences is that a region on the periphery of the shape has changed (in this case, the position of the elephant’s trunk). This change has biased the centroid and orientation estimation.

3 Robust Normalization of Shapes Considering the problem stated above, the objective now is to reduce the sensitivity of the centroid and principal axis estimation to shape deformations. To achieve this goal, it is assumed that not all the regions of a shape are equally important when computing the principal axis. This means that pixels that are far away from the tendency of the bulk of the pixels or that are in regions easily subjected to changes should be downweighted, instead of letting each one vote equally in the estimation. This weighting can be dealt by means of robust statistics. Robust estimation has been successfully used to solve many computer vision, signal processing and statistical problems [1,6,9,10]. There are many robust procedures to tackle these problems, such as Least Median Squares, Ransac, Mestimator, etc (see [7,10] for a review). In this paper, an M-estimator is chosen since the solution can be found with a continuous derivative process, which is very appealing. In order to apply robust statistics to the principal axis estimation, the problem has been reformulated. It can be shown [3] that the orientation computed in (1) coincides with the direction of the eigenvector corresponding to the larger eigenvalue of this matrix:

Robust Normalization of Shapes

 µ 20   µ11

µ11   µ 02 

259

(3)

Following a similar formulation as in [15], this eigenvector estimation problem can be related to the minimization of the following energy function, based on a quadratic norm: N r r r r E { v , ci } = ∑ d i − m − v ·ci i =1

2

(4) 2

r r T T where d i = (x i , y i ) are the coordinates of the N pixels of the shape, m = ( x , y ) is v the centroid, ci are the linear coefficients that minimize the projection error and v is the eigenvector associated to the largest eigenvalue of Eq. 3. It is well known that in a least squares framework, this energy function minimization would not be robust to outliers, which can be interpretated as the pixels that are not close to the principal axis. Changes in the position of these pixels would cause significant variations in the orientation estimation. For this reason, the quadratic function in the energy function for the least square estimation is replaced by a pfunction, which assigns smaller weights for constraints with larger residues. Two p functions commonly used in computer vision are the Lorentzion function and the Geman & McClure’s function given as follows [6,7,9]:  x2  ρ LO ( x,σ ) = log 1 + 2  ,  2σ 

ρ GM (x, σ ) =

x2 σ + x2

(5)

2

where x is the residue of the data constraint and σ is the scale parameter. When using the p-function for robust estimation in an error minimization framework (for example, line fitting), the influence of each data constraint to the solution is characterized by an influence function, which is the derivative of the pfunction. If the derivatives of the above two p-functions are taken, it is straightforward to see that the influence functions decrease as the magnitude of the residue increases [1]. For the least-square estimation, the influence function is linearly increasing as the magnitude of the residue increases. Therefore the least-square estimation is more sensitive to outliers than the robust estimation. Incorporating the p-function into Eq. 4 yields the following new energy function

(

N r rT r E{ v , ci } = ∑ ρ ei ·ei ,σ i =1

r

)

(6)

where eri = d i − mr − vr·ci is the error vector and σ is the scale parameter. In this paper, the Geman-McClure p-function has been used. It must also be remarked that unlike [6], intra-sample outliers have not been taken into account, because it has no meaning that only the x or the y coordinate of a distant pixel were an outlier.

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In order to minimize (6), the relation between robust statistics and IRLS (Iterative Reweighted Least-Squares) has been exploited [6], reformulating the previous robust estimation problem in terms of Weighted Least Squares: N r r r r E{v , ci } = ∑ wi d i − m − v ·ci

(

i =1

r

) (d T

i

r r − m − v ·ci

with weights wi related to the derivative of the p-function by rT r dρ ( ei ·ei , σ ) rT r d ( ei ·ei ) wi = rT r ei ·ei

)

(7)

(8)

An efficient way of computing the minimization of Eq. (8) consists of applying Alternated Least Squares. The iterative scheme, which can be verified taking the v partial derivatives with respect to v and ci, would be: r

N

r m(k + 1) =

∑ w (k )·d (k ) i

i =1

i

(9)

n

∑1 i =1

r

∑ w (k )·c (k )·(d (k ) − m(k ) ) N

r v ( k + 1) =

i =1

i

i

n

∑ w (k )·c i =1

ci ( k + 1) =

r

i

i

2 i

(k )

r r r v ( k ) T ⋅ d i ( k ) − m( k ) r r v ( k ) T ·v ( k )

(

(10)

)

(11)

It is important to notice that the robust centroid of the shape (9) is also iteratively computed taking into account the weights wi associated to each location of pixels of the shape. It is also worth mentioning here that at every iteration k the weights are updated following Eq. (8). A solution can be usually reached in less than 10 iterations, and the computational cost of each iteration is just O(n). The scale parameter σ can be automatically estimated using the Median Absolute Deviation (MAD) [7], which can be viewed as a robust statistical estimate of the standard deviation. 3.1 Shape Dependent Weighting Function As it has been shown in the previous section, it is necessary to weight the contribution of the pixels to the principal axis computation. These weights should penalize those pixels that are far away from the tendency of the bulk of the pixels or that are more likely to vary due to shape deformations. However, Equation (8) does not consider the later dependence on the shape, as it can be seen in Fig. 3.

Robust Normalization of Shapes d2 P2

261

d2 P2

P d1 1

P d1 1

(a)

(b)

Fig. 3. Two instances of a deformable shape, with the desired principal axis and the distances of two pixels to this axis.

This example shows two instances of a deformable shape. It would be desirable that both have the same principal axis, regardless of small changes of the position of the tail. In Fig. 3(a) the pixel P2 is closer to the axis than P1, so according to Eq. (8), its influence on the final solution is bigger. However, in Fig. 3(b) pixel P2 and pixel P1 are at the same distance, so both have now the same influence on the final solution. A small change in the position of the tail can causes significant differences in its associated weights, affecting the centroid and orientation estimation. It is clear that the tail of this shape is more likely to vary, and that the influence of P2 in comparison with the influence of P1 should be downweighted in both figures. In order to cope with these situations and to add more meaning to the weights, the original weights wi are modified by multiplying them by a shape dependent weighting function M(xi,yi)β, whose value depends on the location of each pixel i of the shape. These new weights Ωi are:

Ω i = wi ·(M ( xi , y i ) )

β

(11)

where β is a constant factor that controls the influence of the periphery on the robust principal axis estimation. The shape dependent weighting function function M(xi,yi)β must penalize those regions of the shape which are more likely to vary due to deformations or changes of the shape. As it is assumed that the major alterations occur on the periphery of the shapes, a measure of the distance between the pixels of the shape and its boundary is a good candidate for being M(xi,yi). In this paper, the chessboard distance metric [3,11] has been used to compute M(xi,yi). It is defined as follows:

{

d ch ( pi , p j ) = max x i − x j , y i − y j

}

(12)

for pixels pi= (xi, yi) ∈ S and pj= (xj, yj) ∈ SB where SB denotes the set of shape pixels that are in the boundary of the shape. According to this, M(xi,yi) is defined for each pixel of the shape as the distance dch from that pixel to the nearest pixel on the boundary. There are efficient algorithms for computing distance maps based on the

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chessboard distance metric that can be used to generate M(xi,yi) [2,3]. Figure 4 shows two examples of shape dependent weighting functions M(xi,yi) for a deformable shape. The β factor depends on the dimension of the image, i.e. the size of the shape. The bigger the β factor is, the more down weighted are the outer pixels of the shape in relation to the inner pixels. Consequently, this factor also downplays the effect of spurious edges and noise due to binarization on the boundary of the shape. In the experiments, a value between 0.5 and 2 worked well, being 1.4 the standard value used for images or 128x128 pixels.

Fig. 4. Above, two shapes of the same deformable object class “teddy bear”. Below, their corresponding shape dependent weighting functions M(xi,yi) showing that the parts more likely to vary (arms and legs) have smaller associated weights.

4 Evaluation Method In order to measure the effectiveness of the proposed robust orientation and localization method in comparison with the traditional principal axis estimation, three parameters are carried out. First, the difference between the centroid estimation of a a shape (COGDIFF) that is subjected to a deformation, and has been computed using the chessboard metric as COGDIFF = | x1-x2 | + | y1 - y2 |. The second is the change in the orientation estimation (θDIFF) when a shape has been deformed and has been computed for both the standard method (T) and the robust method (R) as θDIFF = θ1 – θ2 . Finally, a similarity measure (δ) is carried out for pairs of shapes once they have been aligned using the previous estimations. This third parameter has been introduced to validate the use of this normalization procedure in a deformable object recognition framework based on shapes, and it is closely related to the work of P. Rosin [12]. Given the areas shown in Figure 5: R the difference between the original shape and the modified shape, D the difference between the modified and the original shape, and T the overlapped region common to both shapes, then the similarity measure δ is

Robust Normalization of Shapes

δ = 1−

R+D T

263

(13)

This parameter measures the normalized discrepancies between the areas of the aligned shapes and has a maximum equal to 1 in case that both the original and the deformed shape are the same. In contrast to the simple sum of the overlapping area, which would be equivalent to a template matching approach, this measure is valid even when due to the deformation, one shape encloses the other. The main difference between this measure and those from [12] is that here the objective is not to measure shape properties but to recognize them. Hence, the errors have been normalized by the intersection of areas instead of the whole area of the reference shape to provide better selectivity when it comes to recognize shapes.

(a) R

(b) D

(c) T

Fig. 5. Areas (in black) used in the similarity measure based on the discrepancy method, applied to the deformable shape shown in Fig. 4.

5 Results The robust and the traditional normalization method have been applied to a set of deformable shapes related to recognizable objects. Some of these shapes are shown in Fig. 6. The procedure has been as follows: for each shape, both the standard method based on moments and the robust normalization method have been carried out. This results in two estimates, the centroid and the orientation for both methods. Then each shape has been rotated around its centroid in order to align the principal axis vertically and it is shifted to place the centroid of the shape centered in the image. Table 1 condenses some results about the reliability of the robust method. The left column indicates the pair of shapes whouse parameters have been compared. This table clearly suggests that robust normalization (R) is much less sensitive to changes of the shape than the traditional approach (T). The maximum difference between the orientation estimation of a shape that has been deformed is of less than 3º, in comparison with the 29º of the classical method. There is also a significant improvement in the stability of the centroid estimation, as can be seen in the comparison between shapes h1 and h2. The similarity measure has also verified that normalizing shapes with respect to robust estimates results in a better resemblance measurement between shapes, although the selectivity was not as high as expected. This is due to the fact that great part of the pixels of two silhouettes

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of the same class of object (those sorrounding the centroid) overlap even when the difference of orientation is significant. These results also show that when two shapes that belong to different object classes are compared, the similarity measure is quite low, thus it is nonsense to compare in this cases the robust and traditional estimation methods. Table 1. Comparison between the standard (T) and robust (R) normalization of shapes of deformable objects.

θDIFF a1/a2 b1/b2 c1/c2 d1/d2 e1/e2 f1/f2 g1/g2 h1/h2 i1/i2 a1/f2 e1/b2 f1/i2 i1/d2

T 16.64º 12.54º 14.1º 6.25º 29.05º 2.21º 7.56º 9.98º 6.59º -

R 2.61º 1.26º 2.97º 0.42º 0.01º 0.19º 2.88º 0.27º 1.47º -

δ

COGDIFF T 7 2 6 7 4 2 9 10 6 -

R 1 1 1 2 0 1 3 1 1 -

T 0.61 0.58 0.59 0.72 0.50 0.76 0.53 0.61 0.42 0.47 0.5 0.39 0.33

R 0.88 0.84 0.77 0.90 0.91 0.80 0.63 0.72 0.74 0.42 0.52 0.48 0.28

6 Conclusions and Further Work The classical location and orientation normalization method, based on the centroid and the least moment of inertia axis, is not appropriate when dealing with deformable shapes, because it is based on a least squares minimization framework that is not robust to outliers. They can be interpretated as pixels that are far away from the tendency of the bulk of the data, and have a strong influence on the estimation, so minor changes of their position have a great impact on the final solution. By means of a reformulation in terms of an energy function, it has been possible to incorporate robust statistics and make the estimations much less sensitive to variations of the shape. This goal has been attained by using a shape dependent weighting function, which penalizes those pixels of the shape that are more likely to be subjected to changes due to deformations, i.e. the periphery of the shape. Finally, to prove the validity of the proposed method, a similarity measure has been introduced. It is based on the discrepancy between the areas of two shapes, once they have been aligned with both the traditional and the robust normalization methods.

Robust Normalization of Shapes

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Fig. 6. The set of shapes used for evaluation purposes

The experimental results have demonstrated the superior performance of the robust approach with respect to the traditional one, even when shapes suffer important structural changes. Although the similarity measure is able to differentiate among the deformable shapes and validates the proposed normalization method, it turned out to be not as selective as it would be desirable. It is important to remark that the method tolerates deformations of the shapes to a certain extent, over which it can be considered that the shape is scarcely resembling to the original one and thus, the normalization procedure results in erroneous estimations. This paper does not deal with size normalization. It has been assumed that different instances of the same object are available, so changes of size are only due to deformations, which have been treated. Future research will focus on the scale normalization.

References [1]

M. J. Black and P. Anandan. “The robust estimation of multiple motions: parametric and piecewise-smooth flow-fields”. Computer Vision Image Understanding, Vol. 63, No. 1, pp. 75-104, 1996. nd [2] J. C. Russ. The Image Processing handbook, 2 . edition. CRC Press, 1995. [3] A. Rosenfeld, A. C. Kak. Digital Picture Processing, Vols. 1 & 2. Academic Press, 1982. [4] A. K. Jain, Fundamentals of Digital Image Processing. Prentice Hall, 1989. [5] R. M. Haralick, L. G. Shapiro. Computer and Robot Vision, Vols. I&II. AddisonWesley, 1993. [6] F. de la Torre, M. Black. “A Framework for Robust Subspace Learning”. Accepted for Int. Journal of Computer Vision, 2002. [7] F. Hampel, E. Ronchetti, P. Rousseeuw, W. Stahel. Robust Statistics: The Approach based on Influence Functions. Wiley, 1986. [8] P. J. Huber. Robust Statistics. Wiley, 1981 [9] S. Geman, D. McClure. “Statistical methods for tomo-graphic image reconstruction”. Bulletin of the Inter-national Statistical Institute. Vol. LII, pp. 4-5, 1987. [10] D. Mintz, P. Meer. “Robust Estimators in Computer Vision: an Introduction to Least Median of Squares Regression”. Artificial Intelligence and Computer vision, Y.A. Feldman, A. Bruckstein (Eds.). Elsevier, 1991. [11] G. Borgefors. “Distance transformations in arbitrary dimensions”. CVGIP, Vol.27, pp.321-345, 1984

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[12] P.L. Rosin. “Measuring Shape: Ellipticity, Rectangu-larity, and Triangularity”. 15th Int. Conf. Pattern Recognition, Barcelona, Spain, vol. 1, pp. 952-955, 2000. [13] J. Cortadellas, J.Amat, “Image Associative Memory”. 15th Int. Conf. Pattern Recognition, Barcelona, Spain, vol. 3, pp. 638-641, 2000. [14] V. Bruce, P.R. Green, M.A. Georgeson. Visual Perception. Physiology, Psychology and rd Ecology. 3 . Ed., Psychology Press, 1996. [15] L. Xu, A. Yuille, ”Robust Principal Component Analysis by Self-Organizing Rules Based on Statistical Physics Approach”, IEEE Trans. on Neural Networks, Vol. 6, Nº 1, January 1995

Surface Area Estimation of Digitized 3D Objects Using Local Computations Joakim Lindblad and Ingela Nystr¨ om Centre for Image Analysis, Uppsala University L¨ agerhyddv¨ agen 17, SE-75237 Uppsala, SWEDEN joakim,[email protected]

Abstract. We describe surface area measurements based on local estimates of isosurfaces originating from a marching cubes representation. We show how improved precision and accuracy are obtained by optimizing the area contribution for one of the cases in this representation. The computations are performed on large sets (approximately 200,000 3D objects) of computer generated spheres, cubes, and cylinders. The synthetic objects are generated over a continuous range of sizes with randomized alignment in the digitization grid. Sphericity, a scale invariant measure of compactness, allows us, in combination with the improved surface area estimate, to distinguish among the test sets. Keywords: shape analysis, marching cubes, isoperimetric inequality, accuracy, rotation invariance

1

Introduction

In quantitative shape analysis, it is important to know the precision and accuracy of the measurement, and if there are any restrictions on the input domain where the reliability of the measure does not hold. One aim of this paper was to study how 3D shape can be measured in terms of compactness. In 2D image analysis, P 2/A is a commonly used measure of compactness which may be derived from the classical isoperimetric inequality [2]. An “inverse” to compactness is the circularity of objects. This concept is extendable to higher dimensions. Hence, in 3D image analysis, one feature of interest when distinguishing among diﬀerent classes of objects is sphericity. We deﬁne this as a dimensionless ratio between enclosed volume and surface area. The problem that arises is that, in image analysis, we are given only a digitized version of the original continuous object. Under these conditions how reliable is our measure? Note that there is a diﬀerence in measuring digital objects and measuring digitized objects. The former class of objects exists solely in the digital world, and exact measures can be calculated. The latter represents the digitization of continuous original objects. The aim of the measure is not to ﬁnd properties of the digital version, but rather to estimate properties of the continuous original. in the methods applied. Also, the measurement on the digitized A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 267–278, 2002. c Springer-Verlag Berlin Heidelberg 2002

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object can never become more than an estimate, as information is irreversibly lost in the digitization process. This paper focuses on how the surface area can be measured by an approximating isosurface generated by the Marching Cubes approach [13]. Our aim is to ﬁnd a way to obtain accurate estimates with high precision based on local computations, avoiding strong assumptions about the object of study. We also verify the robustness of the sphericity measurement for small to large convex objects with randomized alignment in the digitization process.

2

Surface Area of 3D Objects

In 2D image analysis, perimeter has been, and sometimes still is, measured in terms of cumulative distance from pixel centre to pixel centre. This is straightforward to accomplish using the Freeman chain code [5], but results in incorrect √ measurements. Note that the weights 1 for horizontal and vertical steps and 2 for diagonal steps are not optimal when measuring digitized line segments [6]. By assigning optimized weights [17,1,3] to the steps, more accurate perimeter measurements are obtained. If the boundary is represented by the pixel edges between object and background, steps can be taken from pixel edge to pixel edge, taking advantage of optimized weights, with promising results [12]. This corresponds to Marching Squares, a 2D equivalent of Marching Cubes. In 3D image analysis, when measuring surface area, an analogous ﬁrst approach would be to connect voxel centre to voxel centre and add the resulting areas. This is not simple and would, as in the 2D case, produce incorrect measurements. In a second approach, where the surface is represented by the faces of the voxels at the boundary between object and background [15], the number of faces gives an eﬃcient and simple estimation of the surface area, but this is an overestimate [14]. By approximating the boundary between object and background with a triangular representation, e.g., the one obtained from the marching cubes algorithm, more correct surface area estimates are obtained. There exist recent publications [7,9] that have studied the problem of surface area estimation. Our approach is diﬀerent in that we exploit the simplicity of working in small neighbourhoods and base our estimates on local computations, still sacriﬁcing neither precision nor accuracy.

3

Triangular Isosurface Representation

The voxel representation of object and background has a close relation to polygonal representations; a digital surface can be transformed directly into a triangulated isosurface [10,11]. An m-cube (short for Marching Cube), is the cube bounded by the eight voxels in a 2 × 2 × 2 neighbourhood. Hence, each corner of the m-cube corresponds to a voxel. If object and background are considered, the possible number of conﬁgurations of the eight voxels is 256. Each conﬁguration consists of zero to four

Surface Area Estimation of Digitized 3D Objects Using Local Computations

(a) Case 0

(b) Case 0

(c) Case 1

(d) Case 1

(e) Case 2

(f) Case 2

(g) Case 8

(h) Case 9

269

Fig. 1. m-cubes of 2 × 2 × 2 voxels, where voxels denoted by a • are inside the object and the other voxels are outside. The isosurfaces for the conﬁgurations consist of zero to four triangles.

triangles constituting the isosurface through the speciﬁc m-cube. The conﬁgurations are often grouped into symmetry and complementary cases, resulting in 14 (or 15) cases [13,16]. See Figures 1 and 2 for examples. Two of these conﬁgurations do not represent a boundary situation and have no triangles. These correspond to the case when the m-cube is placed completely outside or completely inside the object (Figures 1(a) and 1(b), respectively). We number the cases according to previous literature [13], except for cases 11 and 14 which we group into one case 11. Some of the cases can be triangulated in diﬀerent ways, e.g., case 11, which is illustrated in Figures 2(a) and 2(b). We use the more symmetric triangulation in Figure 2(b). The approximated isosurface in the m-cube is computed from some interpolation of the voxel values. The interpolation results in intersection points on the edges of the m-cube for the triangle vertices. In the following, we will use the simplest case, where the intersection points are positioned midway along the m-cube edges. The correct connection among intersection points is ambiguous for some conﬁgurations of voxels. See Figures 2(c) and 2(d) for an example. This classical problem in the original marching cubes algorithm was pointed out by D¨ urst in 1988 [4]. The problem requires a careful consideration when a closed surface is desired. It has been examined and solved (e.g., [16,8]). For this study it should be suﬃcient to assign a contributing area for each conﬁguration.

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(a) Case 11

(b) Case 11

(c) Case 3

(d) Case 3

Fig. 2. (a), (b): Some m-cubes can be triangulated in diﬀerent ways (vertices moved from the midpoint for visibility reasons). We use the more symmetric version (b). (c), (d): Shown are the triangles we use for these ambiguous m-cube conﬁgurations.

4

Sphericity

In 2D image analysis, P 2/A is a commonly used measure for compactness, or circularity, of objects. The concept behind this dimensionless ratio is extendable to higher dimensions. In 3D image analysis, a feature of interest when distinguishing among diﬀerent classes of objects would, hence, be the dimensionless ratio of surface area to volume of an object. For a consistent and natural deﬁnition of this class of ratios, we choose the following for the 2D case and correspondingly for the 3D case: Circularity, C: perimeter of a circle, enclosing the same area A as the object, divided by the perimeter P of the object. √ 4πA C= (1) P Sphericity, S: surface area of a sphere, enclosing the same volume V as the object, divided by the surface area A of the object. √ 3 36πV 2 S= (2) A In the continuous case, these measures are in the range [0, 1] given by the isoperimetric inequality. The upper bound is reached only by the ball of the corresponding dimension. For the nD case, the dimension of the ratio is lengthn−1 /lengthn−1 . An alternative would be to deﬁne the ratio as lengthn /lengthn , but we choose the one formulated above as it more closely reﬂects the dimension of the predominant part of the measurement.1 The reason for inverting the P 2/A expression is that circularity should reach its maximum for a true circle. 1

The measure P 2/A suﬀers from the fact that the perimeter has a more predominant eﬀect on the measure than the area.

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5

271

Experiments

We have generated sets of synthetic convex objects. Each set consists of a certain shape digitized in diﬀerent sizes, rotations, and positions, with respect to the digitization grid. The objects are generated in the continuous space and then discretized using Gauss centre point digitization, where the digitized object is deﬁned to be the set of all grid points (voxel centres) contained in the continuous set. All units are given with respect to the sampling grid, i.e., edge length, face area, and volume of voxels are all equal to 1. The surface area contribution for each of the 256 marching cube conﬁgurations, AM C(i) , i = 0..255, is precomputed by summing the areas of the included triangles and stored in a lookup table [14]. The histogram describing the number of each of the 256 conﬁgurations, ni , is computed for each digitized object. The surface area of the speciﬁc object is then calculated as A=

255

ni AM C(i)

(3)

i=0

Of interest is to plot the ratio of estimated area to true area versus the size of the object. The variance within a given size, for diﬀerent rotations and placement of that object, should capture digitization eﬀects. We are also interested in the sphericity, computed as described in Section 4. In a recent publication, the enclosed volume is computed for the marching cubes representation in an incremental fashion from the surface [14]. How this would aﬀect the sphericity estimate needs further studies. Here, the volume V is computed as a simple voxel count of the object, which is a good estimate for large objects. 5.1

Bias

Our main assumption is that, on a local scale, objects are fairly ﬂat. The surface of a large sphere is a good sampling of planes in all directions. Therefore, our ﬁrst test object is a Euclidean sphere. This also happens to be the object with maximal sphericity. The digitization of a sphere centered at (x0 , y0 , z0 ) ∈ R3 of radius r ∈ R is generated by the following equation 1 if (x − x0 )2 + (y − y0 )2 + (z − z0 )2 r2 f (x, y, z) = , (x, y, z) ∈ Z3 (4) 0 otherwise See Figure 3 for sphere examples, where r is 4, 10, and 25, respectively, presented as renditions of their marching cubes triangulations. Figure 4 shows three surface area estimates for digitized spheres of increasing radius. The function describes the mean value for a given radius and the error bars indicate the corresponding smallest and largest estimates, where the surface area is computed according to Eq. (3). If the surface area is computed with the triangle conﬁguration of Figures 5(a) and 5(b) for case 5 (area contribution of 5ﬂat = 1.150), this gives the largest overestimate (8.8%). From now on, we instead use the triangle

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(a) radius 4

(b) radius 10

(c) radius 25

Fig. 3. Three spheres presented as renditions of their marching cubes triangulations. Area of digitized spheres 1.14

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1.12 1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 Case 5 flat Case 5 slanted Unbiased

0.92 0.9 0

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Fig. 4. Three surface area estimates divided by the true surface area, for 27,000 digitized spheres of increasing radius.

(a) Case 5

(b) Case 5

(c) Case 5

(d) Case 5

Fig. 5. The triangular representations in (a) and (b) may seem as the simplest for this conﬁguration and is possibly the most commonly used. The representations in (c) and (d) approximate a smoother surface. (Third • not visible in (a) and (c).)

Surface Area Estimation of Digitized 3D Objects Using Local Computations

(a) side 4

(b) side 10

273

(c) side 40

Fig. 6. Three cubes of diﬀerent sizes presented as renditions of their marching cubes triangulations. (a) is aligned with the digitization grid. (b) and (c) have the same orientation, rotated (15◦ , −20◦ , 25◦ ), but represent diﬀerent sizes.

conﬁguration of Figures 5(c) and 5(d) for case 5 (area contribution of 5slanted = 1.122), for which the overestimate is slightly reduced (8.0%). To obtain an unbiased estimate for randomly aligned planar surfaces, we choose to divide the area by the average overestimate for large digitized spheres. Using 5slanted the bias term ξ becomes 1.080 (supposed convergence value from Figure 4). This also gives good convergence to the true surface area. 5.2

Alignment Invariance

If we wish to study the behaviour of speciﬁcally aligned planar surfaces, the sphere is not useful as it represents every alignment. We seek alignment invariance for large planar surfaces, which here is represented by cubes, rotated and positioned in diﬀerent ways. When applying the same surface area estimator for cubes, the results divided by the bias term ξ = 1.080 are (on average) an underestimate of the true surface area. This is due to the cutting of corners and edges. The cube in Figure 6(a), aligned with the axes of the digitization grid, is small enough to illustrate this eﬀect. For large objects the eﬀect can be neglected, though. The area measurements for the cubes also contain a large variance. This is due to the diﬀerent alignments of the faces of the cube. The surface area becomes too large when the cube is rotated in some direction that is not well captured by the triangles of the marching cubes triangulation, e.g., as in Figures 6(b) and 6(c). The increased variance can be seen in Figure 7. The approach of a simple scaling by the bias term is not suﬃcient for good convergence. We wish to reduce the variance of the estimate. Our approach is to identify the cases for which the area estimate is extreme and to correct for these errors. Therefore, we study the relative frequency of the m-cube conﬁgurations

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Estimated value / True value

1.12

Case 5 slanted Unbiased

1.1 1.08 1.06 1.04 1.02 1 0.98 0.96 0.94 0.92 0.9 0

20

40

60 80 Side length

100

120

140

Fig. 7. Two surface area estimates divided by the true surface area, for 80,000 digitized cubes of increasing size.

for cubes of side 100 with diﬀerent alignment, and examine the situations giving rise to the largest under- and overestimates, respectively. From Figure 8(a), we interpret that some of the m-cube conﬁgurations, e.g., case 5, have not been assigned a value agreeing well with the truth. We have already noted two possible triangulations for case 5 (Figure 5), none of which approximates a ﬂat isosurface through the m-cube. By plotting the relative maximum error against a scale factor used on mcube case 5 (Figure 8(b)), we see that there is an optimum for the scale factor 0.892. The sum of the three triangle areas of m-cube conﬁguration case 5 equals 1.122. If we, instead of assigning this triangle area contribution, assign the value 1.122 ∗ 0.892 = 1.001 to case 5, we reduce the relative maximum error from 7.1% to 4.3% and the coeﬃcient of variation (CV, standard deviation divided by mean) from 1.6% to 1.2%. Note, however, that the scale factor 0.892 is optimal only if case 5 solely is allowed to vary, but the scale factor will change if other cases are also scaled to further improve the surface area estimate. 5.3

Results

Our surface area estimator can be summarized in the following: Starting with the surface area from the marching cubes triangulation, we assign to case 5 area contribution 1.001 and divide by the overall bias term ξ = 1.046. The change of area contribution assigned to case 5 requires a change of the bias term, to acquire an unbiased estimate for randomly aligned planar surfaces as described in Section 5.1. To verify our surface area estimates, we will study objects of other shape than spheres and cubes. Here, we choose cylinders with the same height as diameter, a shape resembling the other two, but which still should be possible to distinguish

Surface Area Estimation of Digitized 3D Objects Using Local Computations

0.8

0.1 Minimum area case All cases Maximum area case

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(a)

0 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 Scale factor for case 5

1

(b)

Fig. 8. (a) Relative frequency of the diﬀerent m-cube cases for 30,000 digitized cubes, of size 100 × 100 × 100, of random position and orientation. (b) Maximum error and coeﬃcient of variation for the estimated area of digitized cubes and cylinders, for diﬀerent scale factors for the area contribution of m-cube case 5.

as diﬀerent. As done previously, the cylinders are generated in a continuous range of sizes with randomized alignment in the digitization grid. For the surface area estimate of cylinders (radius 50), the relative maximum error is reduced from 3.8% to 2.7%, and the CV from 1.1% to 0.8% (Figure 8(b)), using the described method. Surface area, volume, and sphericity for the three test sets are presented in Figure 9. The plots compare well with expectations. There is still a fair amount of variance in the surface area measurements, but the estimation performance has signiﬁcantly improved compared to the uncorrected marching cubes area estimate. The volume estimate is robust and show no surprises. The sphericity agrees well, at least for objects down to radius 5, with the sphericity in the continuous case, where a sphere, a cube, and the described cylinder have values 1, 0.806, and 0.874, respectively. The sphericity measure manages to totally separate the three test sets.

6

Discussion and Conclusions

This paper describes surface area measurements based on local estimates of isosurfaces originating from a marching cubes representation. We show how improved precision and accuracy can be obtained by optimizing the area contribution for one of the cases in this representation. Our results indicate improved

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Fig. 9. Measurements for digitized spheres, cubes, and cylinders of increasing size. Top: Surface area estimates divided by the true surface area. Middle: Volume estimates divided by the true volume. Bottom: Sphericity estimates.

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robustness when measuring compactness, or sphericity, to distinguish classes of objects. Non-convex and more natural shapes should be studied in further work. Another future challenge is to analyze all of the m-cube conﬁgurations and assign proper weights to them in order to obtain accurate measurements with high precision. This would be an optimization process in 13 dimensions; there are 13 cases of area contributions. Our surface area computations are performed for a binary marching cubes representation, where the isosurface intersects the marching cube midway between the corners. Further studies on a grey-level marching cubes representation, where the isosurface is approximated according to some interpolation of the grey-level values of the boundary voxels, seems worth pursuing after the promising results for the binary case. Acknowledgements. We thank Prof. Gunilla Borgefors, Centre for Image Analysis, Uppsala, Sweden, and Prof. Jayaram K. Udupa, Medical Image Processing Group, University of Pennsylvania, Philadelphia, for helpful comments on our work.

References 1. G. Borgefors. Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34:344–371, 1986. 2. S.-s. Chern. Studies in Global Geometry and Analysis, volume 4 of Studies in Mathematics, pages 25–29. The Mathematical Association of America, Washington, DC, 1967. 3. L. Dorst and A. W. M. Smeulders. Length estimators for digitized contours. Computer Vision, Graphics and Image Processing, 40:311–333, 1987. 4. M. J. D¨ urst. Letters: Additional reference to ”Marching Cubes”. In Proceedings of ACM SIGGRAPH on Computer Graphics, volume 22(2), pages 72–73, Apr. 1988. 5. H. Freeman. Boundary encoding and processing. In B. S. Lipkin and A. Rosenfeld, editors, Picture Processing and Psychopictorics, pages 241–266. Academic Press, 1970. 6. F. C. A. Groen and P. W. Verbeek. Freeman code probabilities of object boundary quantized contours. Computer Graphics and Image Processing, 7:391–402, 1978. 7. Y. Kenmochi and R. Klette. Surface area estimation for digitized regular solids. In L. J. Latecki, R. A. Melter, D. M. Mount, and A. Y. Wu, editors, Vision Geometry IX, pages 100–111. Proc. SPIE 4117, 2000. 8. Y. Kenmochi, K. Kotani, and A. Imiya. Marching cubes method with connectivity. In Proc. of IEEE Int. Conference on Image Processing (ICIP’99), pages 361–365, 1999. 9. R. Klette and H. J. Sun. Digital planar segment based polyhedrization for surface area estimation. In C. Arcelli, L. P. Cordella, and G. Sanniti di Baja, editors, Visual Form 2001, volume 2059 of Lecture Notes in Computer Science, pages 356– 366. Springer-Verlag, 2001. 10. J.-O. Lachaud and A. Montanvert. Digital surfaces as a basis for building isosurfaces. In Proc. of 5th IEEE Int. Conference on Image Processing (ICIP’98), volume 2, pages 977–981, Chicago, IL, 1998.

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11. J.-O. Lachaud and A. Montanvert. Continuous analogs of digital boundaries: A topological approach to iso-surfaces. Graphical Models, 62:129–164, 2000. 12. J. Lindblad. Perimeter and area estimates for digitized objects. In Proceedings of SSAB (Swedish Society for Automated Image Analysis) Symposium on Image Analysis, pages 113–117, Norrk¨ oping, Sweden, Mar. 2001. Available from the author. 13. W. E. Lorensen and H. E. Cline. Marching Cubes: A high resolution 3D surface construction algorithm. In Proceedings of the 14th ACM SIGGRAPH on Computer Graphics, volume 21(4), pages 163–169, July 1987. 14. I. Nystr¨ om, J. K. Udupa, G. J. Grevera, and B. E. Hirsch. Area of and volume enclosed by digital and triangulated surfaces. In S. K. Mun, editor, Medical Imaging 2002: Visualization, Image-Guided Procedures, and Display. Proc. SPIE 4681. Accepted for publication. 15. J. K. Udupa. Multidimensional digital boundaries. Graphical Models and Image Processing, 56(4):311–323, July 1994. 16. A. Van Gelder and J. Wilhelms. Topological considerations in isosurface generation. ACM Transactions on Graphics, 13(4):337–375, 1994. 17. A. M. Vossepoel and A. W. M. Smeulders. Vector code probability and metrication error in the representation of straight lines of ﬁnite length. Computer Graphics and Image Processing, 20:347–364, 1982.

An Abstract Theoretical Foundation of the Geometry of Digital Spaces Gabor T. Herman Department of Computer Science, The Graduate Center, City University of New York 365 Fifth Avenue, New York, NY 10016, USA [email protected]

Abstract. Two approaches to providing an abstract theoretical foundation to the geometry of digital spaces are presented and illustrated. They are critically compared and the possibility of combining them into a single theory is discussed. Such a theory allows us to state and prove results regarding geometrical concepts as they occur in a digital environment independently of the speciﬁcs of that environment. In particular, versions of the Jordan Curve Theorem are discussed in this general digital setting.

1

Introduction

One kind of digitization of (three-dimensional) space is obtained by tessellating it into cubes. Then there are natural notions of adjacencies between cubes determined, for example, by the sharing of a single face. However, the adjacencies may also be deﬁned using edges and/or vertices. Tessellations of arbitrarydimensional spaces into arbitrary polyhedra similarly give rise to various notions of adjacencies of these polyhedra. Alternatively, we may study a grid of points in an N -dimensional space and consider certain points adjacent by whatever criterion seems desirable to us. If we view such models as appropriate for capturing the notion of a digital (as opposed to continuous) space, then we see that the suitable underlying mathematical concept is a (possibly inﬁnite) graph, which is a collection of vertices - corresponding to the above-mentioned spatial elements (spels, for short) - some pairs of which are considered adjacent. Our aim is to introduce geometrical concepts at this very general level, so that we get away from the speciﬁc consequences of the choices of tessellations and adjacencies. If we can prove nontrivial theorems in such a general setting, then these theorems will have nontrivial corollaries in all the speciﬁc manifestations of the general theory. Certain continuous geometrical concepts have an immediate natural equivalent in a digital space. If ρ denotes the adjacency, then a ρ-path is a ﬁnite sequence of spels each but the last of which is ρ-adjacent to the one following it. A ρ-connected set S can then be deﬁned as one in which for any pair of spels in S there is a ρ-path entirely in S from one to the other. A simple closed ρ-curve C is a nonempty ﬁnite ρ-connected set such that for each element in C there are A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 279–288, 2002. c Springer-Verlag Berlin Heidelberg 2002

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exactly two other elements in C ρ-adjacent to it. Other notions are more diﬃcult to capture in a natural way. For example, the digital equivalent of the continuous notion of a simply connected set (one in which every simple closed curve can be continuously deformed into a point) is not so obviously deﬁnable: we need somehow to capture the digital correspondent of “continuously deformed.” In this article we discuss two (somewhat related) attempts at providing such abstract theoretical foundation to the geometry of digital spaces. We give a concise, but rigorous, description of both, including samples of theorems that have been proved. We follow this up with a discussion of the diﬀerences between the two approaches and speculate on a possible synthesis into a single theory. Finally we mention where we see the interesting open problems.

2

Digital Spaces

The material in this section is based on the approach presented in [1]. There a digital space is deﬁned as a pair (V, π), with V an arbitrary nonempty set (of spels) and π a symmetric binary relation (called the proto-adjacency) on V such that V is π-connected. A trivial example is when V consists of all the cubes into which space is tessellated and two cubes are in the relation π if, and only if, they share a single face. In mathematical notation, this is the digital space (Z 3 , ω3 ) deﬁned, more generally for any positive integer N (the number of dimensions), by Z N = {(c1 , · · · , cN ) | cn ∈ Z, for 1 ≤ n ≤ N },

(1)

with Z being the set of integers, and ωN the binary relation on Z N satisfying (c, d) ∈ ωN ⇔

N

|cn − dn | = 1.

(2)

n=1

This trivial example illuminates the thinking underlying [1]: π has the alternative interpretation to just being an adjacency, a (c, d) in π can also be thought to represent the surface element (surfel, for short) facing from the spel c to the spel d. Thus any nonempty subset of π is referred to as a surface in (V, π). The boundary between subsets O and Q of V, deﬁned as ∂(O, Q) = {(c, d) | (c, d) ∈ π, c ∈ O and d ∈ Q},

(3)

is a surface provided that it is not empty. The fact that π is a set of ordered pairs allows us to deﬁne, for any surface S, its immediate interior II(S) and its immediate exterior IE(S): II(S) = {c | (c, d) ∈ S for some d in V },

(4)

IE(S) = {d | (c, d) ∈ S for some c in V }. (5) (0) (K) We say that a π-path c , · · · , c crosses S if there is a k, 1 ≤ k ≤ K, such that either (c(k−1) , c(k) ) ∈ S or (c(k) , c(k−1) ) ∈ S. The surface S is said

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to be near-Jordan if every π-path from an element of II(S) to an element of IE(S) crosses S and it is said to be Jordan if it is near-Jordan and no nonempty proper subset of it is near-Jordan. This terminology is justiﬁed by the following theorem, which makes use of the notions of interior I(S) and exterior E(S) of S deﬁned by I(S) = {c ∈ V | there is a π − path connecting c to an element of II(S) which does not cross S},

(6)

E(S) = {c ∈ V | there is a π − path connecting c to an element of IE(S) which does not cross S}.

(7)

Theorem 1. A Jordan surface S in a digital space (V, π) has the following properties. 1. 2. 3. 4.

S = ∂(I(S), E(S)). I(S) ∪ E(S) = V and I(S) ∩ E(S) = ∅. Both I(S) and E(S) are π-connected. Every π-path from an element of I(S) to an element of E(S) crosses S.

This theorem, which is Corollary 3.3.6 of [1], says that a Jordan surface S has properties reminiscent of those indicated by the Jordan Curve Theorem for simple closed curves in the plane: S is the boundary between its interior and its exterior, which do not intersect but contain all the spels between them, they are both π-connected, but one cannot get from the interior to the exterior by a π-path without crossing S. It is worthy of note that near-Jordanness is quite powerful by itself: with the possible exception of the third one, a near-Jordan surface has all the properties listed in Theorem 1 (as can be seen from Lemmas 3.2.1 and 3.2.2 of [1]). While it is quite impressive that such powerful-looking results can be stated (and proved) after only just a very few deﬁnitions, their practical usefulness is limited by the fact that it maybe very diﬃcult (if not impossible) to check for an arbitrary surface whether or not it is near-Jordan. For this reason, [1] introduces a more desirable “local” property which under some circumstances implies near-Jordanness. A surfel (c, d) is said to cross S if exactly one of (c, d) ∈ S or (d, c) ∈ S. The surface S is said (where N is a positive integer) if, for any to be N -locally-Jordan π-path P = c(0) , · · · , c(K) such that (c(0) , c(K) ) ∈ S and 2 ≤ K ≤ N + 1, the number of surfels among (c(0) , c(1) ), · · · , (c(K−1) , c(K) ) that cross S is odd. N locally-Jordanness does not by itself imply near-Jordanness; we need to introduce two further conditions: one on the digital spaces (they have to be in some sense simply connected) and one on the surfaces (they have to be certain kinds of boundaries). We now discuss both of these conditions. If P = c(1) , · · · , c(m) , d(0) , · · · , d(n) , e(1) , · · · , e(l) (8) and

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P = c(1) , · · · , c(m) , f (0) , · · · , f (k) , e(1) , · · · , e(l)

(9)

are π-paths such that f (0) = d(0) , f (k) = d(n) , and 1 ≤ k + n ≤ N + 2,

(10)

then P and P are said to be elementarily N-equivalent. The digital space is said to be N -simply connected if, for any π-path c(0) , · · · , c(K) such that c(K) = c(0) , there is a sequence of π-paths P0 , · · · , PL (L ≥ 0) such that P0 = c(0) , · · · , c(K) , (0) PL = c and, for 1 ≤ l ≤ L, Pl−1 and Pl are elementarily N -equivalent. This is a useful concept, as indicated by Theorem 6.3.5 of [1]: Theorem 2. For any positive integer N , (Z N , ωN ) is 2-simply connected. The kind of boundaries in which we are particularly interested appear in binary pictures over the digital space (V, π). These are deﬁned as triples (V, π, f ), with f a function mapping V into {0, 1}. Those spels which map into 0 are called 0-spels and those which map into 1 are called 1-spels. In order to give the intuitively desired interpretation to objects in binary pictures, we are forced to consider simultaneously more than one adjacency. For example, in the two binary pictures shown below (in which the spels are from Z 2 ), the 1-spels form a letter O and a letter C respectively. 0 0 0 0 0 0 0 0 0

0 0 0 1 1 1 0 0 0

0 0 1 0 0 0 1 0 0

0 1 0 0 0 0 0 1 0

0 1 0 0 0 0 0 1 0

0 0 1 0 0 0 1 0 0

0 0 0 1 1 1 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 1 1 1 0 0 0

0 0 1 0 0 0 1 0 0

0 1 0 0 0 0 0 1 0

0 1 0 0 0 0 0 1 0

0 0 1 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

Note that neither the O nor the C forms an ω2 -connected set. On the other hand, if we deﬁne the binary relation δN on Z N by N (c, d) ∈ δN ⇔ 0< |cn − dn | ≤ 2 and, for 1 ≤ n ≤ N, |cn − dn | ≤ 1 , n=1

(11) then both the O and the C are δ2 -connected, but the inside of the O is also δ2 connected to its outside. For such reasons, it is customary and useful to consider diﬀerent adjacencies for the 0-spels and for the 1-spels. In [1] a symmetric binary relation ρ on V is called a spel-adjacency if π ⊆ ρ. If κ and λ are spel-adjacencies, then a surface S is called a κλ-boundary in the binary picture (V, π, f ) if there is κ-component O of 1-spels and a λ-component

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Q of 0-spels, such that S = ∂(O, Q). In the previously-presented binary pictures over the digital space (Z 2 , ω2 ), the letter O forms a single δ2 -component and its inside and outside each forms a single ω2 -component. Hence there are two δ2 ω2 -boundaries in the binary picture on the left. On the other hand, there is only one δ2 ω2 -boundary in the binary picture on the right. For reasons explained quite in detail in early chapters of [1] (they have to do with being able to prove that certain computer procedures for multidimensional image processing perform as desired), it is useful to choose spel-adjacencies so that every non-empty κλ-boundary is κλ-Jordan, in the sense that it is nearJordan, its interior is κ-connected and its exterior is λ-connected. (Note that in view of Theorem 1, a Jordan surface is always a ππ-Jordan.) For this we need the concept of a tight spel-adjacency ρ, which is the property deﬁned as having that, for all (c, d) in ρ, there exists a π-path c(0) , · · · , c(K) from c to d such that, for 0 ≤ k ≤ K, either (c(0) , c(k) ) ∈ ρ or (c(k) , c(K) ) ∈ ρ (or possibly both). Clearly, both ωN and δN are tight spel-adjacencies in (Z N , ωN ). The following is Theorem 6.2.7 of [1], it holds for all positive integers N . Theorem 3. Let κ and λ be tight spel-adjacencies in an N -simply connected digital space (V, π). A κλ-boundary is in a binary picture over (V, π) is κλJordan if, and only if, it is N -locally Jordan. The advantage of this theorem as compared to Theorem 1 is that the desirable property of being κλ-Jordan follows from the property of being N -locally Jordan, which appears to be a condition that is easier to check than the condition of being near-Jordan. We now show that under some circumstances this appearance very much corresponds to reality. We call a π-path c(0) , c(1) , c(2) , c(3) , c(0) a unit square if both c(0) = c(2) (1) (3) and c = c . An unordered pair {κ, λ} of spel-adjacencies in a digital space is said to be a normal pair if, for any unit square c(0) , c(1) , c(2) , c(3) , c(0) , we have (c(0) , c(2) ) ∈ κ or (c(1) , c(3) ) ∈ λ or both. It is easy to prove that, for any positive integer N , {δN , ωN } is a normal pair in (Z N , ωN ) (Theorem 6.3.8 of [1]). Theorem 4. If {κ, λ} is a normal pair of spel-adjacencies in a digital space and S is a κλ-boundary in a binary picture over the digital space, then S is 2-locally Jordan. This result (which is Lemma 6.3.3 of [1]) together with Theorems 2 and 3 implies that, for any positive integer N , every δN ωN -boundary in any binary picture over (Z N , ωN ) is δN ωN -Jordan. To emphasize what this means in the special case of tessellating space into cubes, we spell out in full its consequences in that space (Corollary 6.3.9 of [1]). Theorem 5. Let A be a nonempty proper subset of Z 3 . Let O be a δ3 -component of A and Q be an ω3 -component of Z 3 \A, such that ∂(O, Q) is not empty. Then there exists two uniquely deﬁned subsets I and E of Z 3 with the following properties.

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1. 2. 3. 4. 5.

G.T. Herman

O ⊆ I and Q ⊆ E. ∂(O, Q) = ∂(I, E). I ∪ E = Z 3 and I ∩ E = ∅. I is a δ3 -connected subset of Z 3 and E is an ω3 -connected subset of Z 3 . Every ω3 -path connecting an element of I to an element of E crosses ∂(O, Q).

From the point of view of our discussion here, the important aspect of this theorem is not what it says but rather the fact that the general results discussed earlier immediately yield similar theorems for other tessellations of three and other dimensional spaces and even for digital spaces which are obtained by means other than tessellating a Euclidean space. Many speciﬁc examples of this are presented in [1].

3

Generic Axiomatized Digital Surface-Structures

The material in this section is based on the approach presented in [2]. That paper aims at providing an axiomatic foundation of digital topology; it succeeds in this only partially inasmuch that it deals only with discrete structures which model subsets of the Euclidean plane and of other surfaces. Its main advantage over the approach of the previous section is its treatment of the digital version of “continuously deformed”: instead of having the awkward hierarchy of elementary N -equivalences, the allowable deformations are embedded into the very deﬁnitions of the structures, in the form of some “loops.” A basic diﬀerence between the conventions of [1] and [2] is that in the latter adjacencies are represented by unordered pairs. In order to accommodate the terminology of [2], we deﬁne a proto-edge in a digital space to be any two-element set {c, d} such that (c, d) is a surfel. A 2D digital complex is deﬁned as a triple (V, π,L), where (V, π) is a digital space and L is a set of simple closed π-curves (its elements are called loops) such that the following conditions hold: 1. V contains more than one spel and if (c, d) ∈ π, then c = d. 2. For any two distinct loops L1 and L2 , L1 ∩ L2 is either empty, or consists of a single spel, or is a proto-edge. 3. No proto-edge is included in more than two loops. 4. Each spel belongs to only a ﬁnite number of proto-edges. If L2×2 is the set of all c(0) , c(1) , c(2) , c(3) such that c(0) , c(1) , c(2) , c(3) , c(0) is a unit square in (Z 2 , ω2 ), then (Z 2 , ω2 ,L2×2 ) is a 2D digital complex. Now we see why we use the term 2D digital complex: if we tried to do the same for the space (Z 3 , ω3 ) we would violate Condition 3 of the deﬁnition, since each proto-edge would be included in four loops. For an arbitrary spel-adjacency ρ in (V, π), let the P in (8) and the P in (9) be two ρ-paths. They are said to be elementarily loop-equivalent in (V, π,L) if 1. either there is a proto-edge {c, d} such that one of d(0) , · · · , d(n) and (0) f , · · · , f (k) is c and the other is c, d, c,

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2. or f (0) = d(0) , f (k) = d(n) , and there is a loop which contains d(0) , · · · , d(n) and f (0) , · · · , f (k) . A ρ-path c(0) , · · · , c(K) such that c(K) = c(0) , is said to be ρreducible in (V, π,L) if there is a sequence of ρ-paths P0 , · · · , PL such that P0 = c(0) , · · · , c(K) , PL = c(0) and, for 1 ≤ l ≤ L, Pl−1 and Pl are elementarily loop-equivalent in (V, π,L). The 2D digital complex (V, π,L) is said to be simply connected if every π-path c(0) , · · · , c(K) such that c(K) = c(0) is π-reducible in it. A spel c of (V, π,L) is called interior if every proto-edge that contains c is included in two loops. A 2D digital complex is called a pseudomanifold if all its spels are interior. (Trivially, every proto-edge of a pseudomanifold is included in exactly two loops.) It is easy to see that (Z 2 , ω2 ,L2×2 ) is a pseudomanifold. Two loops L and L of a 2D digital complex are said to be adjacent if L ∩ L is a proto-edge. A subset L of L is said to be strongly connected if, for any two loops L and L of L , there exists a sequence L0 , · · · , LK of loops in L such that L0 = L, LK = L and, for 1 ≤ k ≤ K, Lk−1 and Lk are adjacent. The 2D digital complex (V, π,L) is said to be strongly connected if L is strongly connected. A spel c is said to be a singularity of a 2D digital complex if the set of all loops that contain c is not strongly connected. The following is Proposition 3.7 of [2]. Theorem 6. A 2D digital complex that is both simply connected and strongly connected has no singularities. For an arbitrary spel-adjacency ρ in (V, π), let C be a simple closed ρ-curve. A ρ-path c(0) , · · · , c(|C|) , where |C| denotes the number of elements in C, which is such that c(|C|) = c(0) and C = c(1) , · · · , c(|C|) is called a ρ-parameterization of C. (Note that this exists.) We say that in this parameterization c(k−1) precedes c(k) and c(k) follows c(k−1) , for 1 ≤ k ≤ |C|. Let L and L be two adjacent loops of a 2D digital complex and let {c, d} ∈ L∩ L . We say that a π-parameterization of L is coherent with a π-parameterization of L if c precedes d in one of the π-parameterizations and c follows d in the other. A 2D digital complex (V, π,L) is said to be orientable if there is a function Ω with domain L such that: 1. For each loop L in L, Ω(L) is a π-parameterization of L. 2. For all pairs of adjacent loops L and L , Ω(L) is coherent with Ω(L ). It is easily seen that (Z 2 , ω2 ,L2×2 ) is orientable. A generic axiomatized digital surface-structure (or GADS, for short) is a pair G = ((V, π,L), (κ, λ)), where (V, π,L) is a 2D digital complex (called the complex of G, whose spels, proto-edges and loops are also referred to as the spels, protoedges and loops of G) and κ and λ are spel-adjacencies in (V, π) that satisfy: Axiom 1. If (c, d) ∈ κ ∪ λ, then c = d. Axiom 2. If (c, d) ∈ (κ ∪ λ) \ π, then some loop contains both c and d. Axiom 3. If {c, d} is a subset of a loop L, but it is not a proto-edge, then (a) (c, d) ∈ λ if, and only if, L \ {c, d} is not κ-connected and (b) (c, d) ∈ κ if, and only if, L \ {c, d} is not λ-connected.

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(Note if in the underlying complex it is the case that, for every unit square (0) that c , c(1) , c(2) , c(3) , c(0) , c(0) , c(1) , c(2) , c(3) is a loop, then Axiom 3 implies that {κ, λ} is a normal pair of spel-adjacencies.) The following is Theorem 6.1 of [2]. Theorem 7. Let C be a simple closed (κ ∩ λ)-curve contained in a loop L of a GADS ((V, π,L), (κ, λ)). Then C has one of the following properties: 1. For all distinct c and d in C, (c, d) ∈ κ. 2. For all distinct c and d in C, (c, d) ∈ λ. A GADS ((V, π,L), (κ, λ)) is a subGADS of a GADS ((V , π ,L ), (κ , λ )) if: 1. V ⊆ V , π ⊆ π and L⊆L . 2. For all L ∈L, κ ∩ L2 = κ ∩ L2 and λ ∩ L2 = λ ∩ L2 , where L2 denotes the set of ordered pairs of elements of L. (It is a consequence of this deﬁnition and Axiom 2 that if ((V, π,L), (κ, λ)) is a subGADS of ((V , π ,L ), (κ , λ )), then κ ⊆ κ and λ ⊆ λ .) A GADS assumes the properties of its complex; thus a GADS is said to be simply connected, strongly connected or orientable if its complex is simply connected, strongly connected or orientable, respectively. Thus we can state the following, which is Proposition 5.1 of [2]. Theorem 8. A subGADS of a simply connected GADS is orientable. We are now ready to state Theorem 8.1 of [2], which is a very general version of the Jordan Curve Theorem for GADS. In it we make use of the concept of a pGADS, which is a GADS whose complex is a pseudomanifold. Theorem 9. Let ((V, π,L), (κ, λ)) be a GADS that is a subGADS of an orientable pGADS ((V , π ,L ), (κ , λ )) whose complex has no singularities. Let P be a κ-parameterization of a simple closed κ-curve C in V such that: 1. C is not included in any loop in L. 2. Every spel in C is an interior spel of (V, π,L). 3. P is κ -reducible in (V , π ,L ). Then V \ C has exactly two λ-components and, for each spel c in C, Nλ (c) (the set of all spels λ-adjacent to c) intersects both of these λ-components. To illustrate the applicability of this theorem, we discuss its implication for ((Z 2 , ω2 ,L2×2 ), (δ2 , ω2 )), which is easily checked to be a GADS. Based on previously made remarks, we see that it is in fact an orientable pGADS and it is easy to see that its complex has no singularities. In this application of Theorem 9 we can use this same GADS for the two GADS mentioned in that theorem. Furthermore, it is not diﬃcult to prove (similarly how our Theorem 2 is proved in [1]) that every δ2 -path is δ2 -reducible in (Z 2 , ω2 ,L2×2 ). Recognizing that a simple closed δ2 -curve in Z 2 which contains at least four spels cannot be contained in an element of L2×2 and that every spel of a pGADS is interior, all this proves the following (which is purposely stated to resemble Theorem 5):

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Theorem 10. Let C be a simple closed δ2 -curve in Z 2 which contains at least four spels. Then there exists two uniquely deﬁned nonempty subsets I and E of Z 2 with the following properties. 1. 2. 3. 4. 5.

I ⊆ Z 2 \ C and E ⊆ Z 2 \ C. For every c in C, both I ∩ Nω2 (c) and E ∩ Nω2 (c) are nonempty. I ∪ E ∪ C = Z 2 and I ∩ E = ∅. Both I and E are ω2 -connected subsets of Z 2 . Every ω2 -path connecting an element of I to an element of E contains an element of C.

Again, the important aspect of such a theorem is that it is just one example of many similar theorems that can be derived from Theorem 9 for a variety of GADS. Some examples of interesting GADS are given in [2].

4

Discussion

Although the approaches of the two previous sections are clearly related, there are many diﬀerences between them. An inessential one is expressed by Condition 1 of the deﬁnition of a 2D digital complex; such restrictions are not made in the deﬁnition of a digital space. However, nothing interesting can be said without this restriction which cannot be said in its presence and so there is no harm in restricting our study of digital spaces to those which satisfy Condition 1. A more interesting diﬀerence is due to the use of proto-edges rather than surfels. The concept of surfel is not even mentioned in [2], everything there is developed in terms of proto-edges. That material has been rewritten for Section 3 so as to make it notationally consistent with the previous section. However, there is more than notation at stake here. The use of surfels allowed us to deﬁne in a natural way the concept of a surface. The corresponding concept in GADS is a simple closed curve, which is a very diﬀerent sort of animal: surfaces in digital spaces are sets of surfels, while curves in GADS are sets of spels. The consequences of this diﬀerence in approach become evident when comparing Theorems 5 and 10. Since both approaches have been utilized in the literature, it seems desirable to develop a theory capable to deal with them simultaneously. To investigate this, let us start with a situation in which all the conditions of Theorem 9 are satisﬁed (as a speciﬁc illustration consider Theorem 10 and the letter O on the left of the previously shown ﬁgure as the simple closed δ2 -curve C). Creating a binary picture in which the 1-spels are elements of C (which is κ-connected), we see that there are exactly two κλ-boundaries, namely ∂(C, Q1 ) and ∂(C, Q2 ) with Q1 and Q2 being the two λ-components of V \ C. Under what additional conditions on the GADS in Theorem 9 are these two boundaries guaranteed to be κλ-Jordan? (That they can indeed be such is illustrated by the corresponding boundaries of the O in the ﬁgure; see also the more general statement after Theorem 4.) If they were, then we would have a way of going from C to surfaces which approximate it and have desirable properties.

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On the other hand suppose that we have a κλ-Jordan surface S in the digital space of the underlying complex of a GADS. Under what conditions is II(S) a simple closed κ-curve? If it is and if the other conditions of Theorem 9 are also satisﬁed, then one of the λ-components of V \ II(S) implied by Theorem 9 has to be E(S) and the other has to be I(S) \ II(S). (This implies that I(S) itself is λ-connected, in addition to being κ-connected, giving us one necessary condition.) Symmetrical conditions would be obtained for insuring that IE(S) is a simple closed λ-curve. One might even investigate the circumstances under which IE(S) is a simple closed κ-curve, but this seem less likely to lead to useful results for a κλ-Jordan surface S. A major disadvantage of the GADS-based approach is its restriction to 2D digital complexes and consequently (in that approach) to curves to play the role of “surfaces which separate space into two components.” An attempt has been made in [1] to introduce more general structures to fulﬁll such a role. In any digital space (V, π), a nonempty subset of P of V is called a spel-manifold if it satisﬁes the following three conditions: 1. P is π-connected. 2. For each c ∈ P , Nπ (c) \ P has two π-components, 3. For each c ∈ P and for each d ∈ Nπ (c)∩P , Nπ (d) has a nonempty intersection with both π-components of Nπ (c) \ P . This deﬁnition is half satisfactory in the sense that it is the case that [1, Theorem 7.3.1] if P is a spel-manifold in a digital space (V, π), then V \ P has at most two π-components, but it is not guaranteed to have more than one π-component, even if we restrict ourselves to rather special digital spaces [1, Theorem 7.3.2]. It appears desirable to ﬁnd some nice conditions on a set of spels which would guarantee that it is a κλ-manifold in the sense that it is κ-connected and its complement has exactly two λ-components. Another intriguing approach is to generalize the notion of a 2D digital complex to N dimensions. Even though it is not too diﬃcult to envision how to do this satisfactorily (an inductive deﬁnition is a possibility), it is less clear how such a generalization can be combined with spel-adjacencies; in particular, Axiom 3 seems to be very anchored to the two-dimensional environment. In summary, the previous two sections have illustrated that powerful results can be proven in an abstract framework; these results have immediate consequences in the many speciﬁc theories that have been put forward to study geometry in a digital framework. However, as discussed in this ﬁnal section, not all aspect of such a theoretical foundation have yet been satisfactorily resolved.

References 1. Herman, G.T.: Geometry of Digital Spaces. Birkh¨ auser, Boston Basel Berlin (1998) 2. Fourey, S., Kong, T.Y., Herman, G.T.: Generic Axiomatized Digital SurfaceStructures, submitted for publication in Disc. Appl. Math. (preliminary version appeared in Electronic Notes in Theoretical Computer Science 46 (2001) see http://www.elsevier.nl/gej-ng/31/29/23/86/27/show/Products/notes/index.htt)

Concurrency of Line Segments in Uncertain Geometry Peter Veelaert Hogent, Schoonmeersstraat 52, 9000 Ghent, Belgium, [email protected]

Abstract. We examine the derivation of consistent concurrency relations in uncertain geometry. This work extends previous work on parallelism and collinearity. We introduce the concept of a metadomain, which is deﬁned as the set of parameter vectors of lines passing through two domains, where a domain is deﬁned as the uncertainty region of the parameter vector of a line segment. The intersection graph of the metadomains is introduced as the primary tool to derive concurrency relations.

1

Introduction

We consider the concurrency of digital line segments within the framework of uncertain geometry, a geometric theory introduced to model the uncertainty of positions as well as geometric properties of objects in a digital image. In uncertain geometry points are equipped with uncertainty regions. The notion of an uncertainty region coincides with the use of a structuring element in the discretization by dilation scheme developed by Heijmans and Toet [1]. The definition of concurrency was introduced previously together with deﬁnitions for digital collinearity and parallelism [2]. One of the main characteristics of uncertain geometry is that properties such as parallelism, collinearity and concurrency are not necessarily consistent in the same way they are consistent in Euclidean geometry, and that somehow we must restore this consistency wherever it is needed in an application. In uncertain geometry when the line A is parallel with B, and B with C, then this does not necessarily imply that A is also parallel with C [2,3]. Thus, for the sake of consistency, after establishing a set of geometric relations between objects in uncertain geometry, we must repartition the objects, discard some relations as well as add new ones to obtain a consistent set of relations. This repartitioning can be accomplished by an optimal grouping process as was done for collinearity and parallelism in previous work [3,4,5]. As for parallelism, for example, the grouping process led to consistency by extracting cliques from an interval graph, in which each interval represented the uncertainty about the slope of a line segment. Since concurrency concerns triples of lines, a direct approach to optimal grouping would involve the extraction of cliques from a hypergraph, where A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 289–300, 2002. c Springer-Verlag Berlin Heidelberg 2002

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the hyperedges represent concurrent triples [6]. Here, to simplify the grouping process, we reduce the conditions imposed on concurrency, similar to the way collinearity was replaced by the concept of weak collinearity to simplify the extraction of collinear groups of line segments [5]. What we propose in this paper is one approach to geometric reasoning when the position and parameter vectors are unprecise or uncertain. Also in robotics, mechanical design and computer vision, there is a need to deal with uncertainty. The models proposed there include the use of ﬁnite precision arithmetic [7], the use of probability density functions [8,9], the use of tolerance zones for mechanical parts [10], and signiﬁcance measures for geometric relations [11]. However, these methods have not yet been integrated into a larger mathematical framework, what this work is aimed at. In Section 2 we brieﬂy sketch how we derive concurrency relations. In Section 3 we give more details on the computation of a metadomain. Section 4 examines the extraction of consistent relations. We conclude the paper in Section 4.

2

Concurrency and Line Transversals

During discretization the precise knowledge about the position of geometric objects is lost. We model this uncertainty by an uncertainty region that we associate with each point. The discretization process that coincides naturally with this notion of uncertainty is the discretization by dilation scheme developed by Heijmans and Toet [1]. The structuring element Ap used in this scheme coincides with our notion of an uncertainty region associated with a point p. Furthermore, to keep the complexity of the computations acceptable, as model for the uncertainty of the position of a digital point p = (x, y), we often use a very simple uncertainty region, i.e, the vertical line segment Cp (τ ), which comprises all points (x, b) ∈ IR2 that satisfy y − τ /2 ≤ b < y + τ /2. Here τ is a positive real number, which controls the uncertainty. Also to simplify the exposition, we restrict ourselves to the concurrency of straight lines of the form y = αx + β, where the slope α satisﬁes −1 < α < 1. We assume that each set S contains at least two points with distinct x-coordinates. We shall also not discuss how to deal with lines of slope |α| ≥ 1, which can be done along the same lines as discussed in [2,4] for parallelism. Deﬁnition 1. A ﬁnite digital set S ⊂ Z 2 is digitally straight if there is a Euclidean straight line that cuts all uncertainty segments Cp (τ ), p ∈ S. We call such a set also a digital line segment. Deﬁnition 2. Let S1 , . . . , Sn be a ﬁnite collection of digital line segments. We deﬁne the following properties: – The sets {Si : i = 1, . . . , n} are called digitally collinear if there exists a common Euclidean straight line A that cuts the uncertainty segments of all the sets Si ;

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– The sets in {Si : i = 1, . . . , n} are called digitally parallel if there exist n Euclidean straight lines A1 , . . . , An that are parallel and such that for i = 1, . . . , n the line Ai cuts the uncertainty segments of the set Si ; – The sets in {Si : i = 1, . . . , n} are called digitally concurrent if there exist n Euclidean straight lines A1 , . . . , An that meet in a common point and such that for i = 1, . . . , n the line Ai cuts the uncertainty segments of the set Si ; In the remainder the notions of collinearity, parallelism and concurrency are used in the sense of Deﬁnition 2, unless we specify explicitly that we are using the Euclidean deﬁnition. When the sets S1 , . . . , Sn are digitally concurrent we denote this as conc(S1 , . . . , Sn ). The parameter vectors of the Euclidean lines passing through the uncertainty regions of a digital line segment, deﬁne a new kind of uncertainty region, also called the domain of the line segment. Deﬁnition 3. Let S be a ﬁnite digital set that contains at least two points with distinct x-coordinates, and let τ be the uncertainty parameter. Then the domain of S, denoted as domx S, is the set of all parameter vectors (α, β) ∈ IR2 that satisfy the following system of inequalities: − τ /2 ≤ αxi + β − yi ≤ τ /2, (xi , yi ) ∈ S.

(1)

Note that a domain is deﬁned here as a convex, closed polygon. The properties of Deﬁnition 2 can be reformulated in terms of domains [2]. Two digital line segments are collinear when their domains intersect. Two line segments are parallel when the intervals that result from the projection of the domains upon the α-axis have a none-empty intersection. Finally, three line segments are concurrent if there is a straight Euclidean line passing through their domains. Attaching uncertainty regions to points and parameter vectors leads to additional levels of uncertainty. In fact, the introduction of domains can proceed indeﬁnitely. Thus, we may consider the uncertainty of the parameter vector of a Euclidean line passing through two or more domains, and in this way construct a metadomain. One of the strong points of the use of uncertainty regions is that on each level the uncertainty depends in a direct way on the uncertainties introduced at the ﬁrst level, that is, on positions of image points. Given a collection of digital line segments and their domains, we shall perform the following steps to ﬁnd out whether the line segments are concurrent or not: 1. 2. 3. 4.

compute the metadomains for pairs of domains; construct an intersection graph of the metadomains; ﬁnd a consistent grouping on the vertices of the intersection graph; compute the metadomains for the groups.

We illustrate these steps further in Fig. 1. In Fig. 1(a) we are given a collection of digital line segments which have been extracted from a digital image by use of the Ransac method [12]. The domains of these segments are shown in Fig. 1(b), where each domain corresponds to one line segment in Fig. 1(a). Large line segments correspond to small domains, because there is less uncertainty regarding their slope and height. Note, for example, that the domain of line segment G is

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Fig. 1. (a) Scanned image of handmade line drawing, (b) domains, and (c) intersection graph.

much smaller than the domain of F , because G actually consists of two parts as shown in Fig. 1(a). Because the line segments were drawn by hand, to be able to extract at least some geometric relations, we have chosen a relatively large value for the uncertainty parameter used to compute the domains, i.e, τ = 5. Once they have been computed, the domains can serve as new uncertainty regions, for which we want to ﬁnd line transversals, as line transversals determine concurrency relations. Therefore, for each pair of non-parallel digital line segments we derive a metadomain formed by the parameter vectors of those Euclidean lines that cut their domains. Deﬁnition 4. Let A and B be two digital line segments with domains domx A and domx B. In addition assume that A and B are not-parallel, that is, domx A and domx B can be separated by a Euclidean line parallel to the β-axis. The metadomain consists of all parameter vectors (p, q) such that the Euclidean line β − αp − q = 0

(2)

cuts domx A as well as domx B in the αβ-plane. The points of a metadomain can be identiﬁed with the points in the original image space. In fact, since the Euclidean line (2) cuts both domains, it contains

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Fig. 2. Metadomains for pairs of lines.

two points (α1 , β1 ) and (α2 , β2 ) such that (x, y) = (p, q) lies on a line β1 − α1 x − y = 0 which cuts all uncertainty regions of points in A, as well as on the line β2 − α2 x − y = 0 which cuts all uncertainty regions of B. In this sense the metadomain is the uncertainty region of the intersection point (p, q) of a Euclidean line passing through the uncertainty regions of A and a Euclidean line passing through the uncertainty regions of B. Therefore, the metadomains of the domains in Fig. 1(b) can be superimposed on the original image, as done in Fig. 2. Next, we construct the intersection graph of the metadomains, as shown in Fig. 1(c). Each metadomain is represented by a vertex. When two metadomains intersect, they are joined by an edge. Thus the existence of an edge refers to the fact that there is a Euclidean line cutting either three or four domains. The precise number of domains cut depends on whether the two domain pairs have one domain in common or not. For example, since the vertices AG and BG are adjacent, there must be a straight line cutting the domains A, B and G. Deriving concurrency relations directly from the intersection relations of the metadomains would almost always lead to inconsistency. Instead, as will be explained later, we perform a consistent grouping of the vertices of the intersection graph of the metadomains. We shall prove that we can obtain consistency when we extract cliques from the graph such that two cliques do not have a common neighbor. Next, for each clique we compute the common intersection of the metadomains that correspond to the vertices of the clique. These intersections are shown in Fig. 3. They form new uncertainty regions, representing the uncertainty regarding the position of the intersection point of three or more concurrent lines. For example, the region ABF corresponds to the uncertainty of the intersection point of the three Euclidean lines passing through the uncertainty regions of A, B and F . Each clique gives rise to a concurrency relation, pro-

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Fig. 3. Uncertainty regions for intersection points.

vided the intersection of the metadomains that are involved is non-empty. The cliques of the graph in Fig. 1(c) yield the following candidates for the concurrency relations: conc(A, B, F ), conc(A, B, G), conc(A, B, H), conc(A, B, D, E), conc(C, E, H) and conc(C, E, G). Finally, to improve consistency we may discard illegal cliques that contain two or more parallel lines. Since C and E are parallel, but not collinear, the only relations that remain are conc(A, B, F ), conc(A, B, G), conc(A, B, H). They can coexist since A and B are in fact collinear.

3

Derivation of Convex Metadomains from Domain Pairs

Since the construction of the intersection graph of the metadomains is greatly simpliﬁed when the metadomains are convex, an important question is the following: Given two disjoint convex domains, when is their metadomain also convex? To examine this, given two domains A and B we introduce the lines and halﬂines shown in Fig. 4(a). Here V denotes a Euclidean line of minimal slope cutting A as well as B. Similarly, we let W denote a transversal of maximal slope. In addition, we choose a point au ∈ (V ∩ A) common to the line V and the set A, as well as a point bl ∈ (V ∩ B). Let Au denote the halﬂine starting at au parallel to the y-axis and extending in the direction of increasing y-values. Similarly we introduce the points bu , al for the transversal of maximum slope W , and the halﬂines Al , Bu and Bl , as shown in Fig. 4. We then have the following result.

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Fig. 4. A non-vertical line that does not cut Au , Al , Bu , Bl , must cut A and B.

Lemma 1. Let A and B be two domains that are convex polygons, and let Al , Au , Bl and Bu be deﬁned as in Figure 4. If there is a transversal cutting either Bl or Bu then one of the supporting lines of B cuts A. Proof. Without loss of generality we suppose there is a transversal L of A and B that cuts the halﬂine Bu and that does not pass through any of the vertices of B, as shown in Fig. 4. We choose an arbitrary point p ∈ A ∩ L, and we deﬁne c as the point c ∈ L ∩ B such that the Euclidean line segment cp has only c in common with B. Let M denote the supporting line of B that passes through c. Since B is convex, the slope of M must be smaller than the slope of W , but larger than the slope of L. Hence M cuts Bu as well as the domain A. Proposition 1. Let A and B be two domains that are convex polygons. If the supporting lines of one domain do not cut the other domain then their metadomain is a convex polygon. Proof. Let al = (pal , qal ), . . . , bu = (pbu , qbu ) be the coordinates of the endpoints. By Lemma 1 if no supporting line of either A or B cuts the other set, then there is no transversal line cutting one or more of the halﬂines Al , Au , Bl or Bu . On the other hand, any line that cuts none of these four halﬂines is a transversal of A and B. Therefore the metadomain in the uv parameter space is determined by the four inequalities  v − qal − upal > 0    v − qbl − upbl > 0 (3) v − qau − upau < 0    v − qbu − upbu < 0. Hence the domain is convex. Metadomains are not always convex. It is easy to construct a counterexample as follows. By the identiﬁcation of the metadomain with the uncertainty region of an intersection point it follows that the metadomain can also be considered as the intersection of two collinearity regions as deﬁned in [2]. Furthermore,

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Fig. 5. Non-convex uncertainty region for the intersection point of two lines.

since collinearity regions are not necessarily convex, it is not diﬃcult to ﬁnd two collinearity regions whose common intersection is not convex, as shown in Fig. 5. In [2] a collinearity region of a digital set S is deﬁned as the set consisting of those points p in IR2 for which there is at least one Euclidean line passing through p as well as through the uncertainty regions associated with the points of S. In Fig. 5 the intersection of the two collinearity regions is shown as a gray, non-convex region. Since metadomains are not necessarily convex, to construct the intersection graph of Fig. 1(c), we replace the metadomain by its convex hull, which is done as follows. First, we compute the parameter vector of each line that passes through any vertex of the ﬁrst domain and any vertex of the second domain, next we take the convex hull of the parameter vectors. This approach overestimates uncertainty. When the conditions of Proposition 1 are met, however, which for real lines is often the case, then the metadomain coincides with its convex hull without any overestimation of uncertainty.

4

Consistency and Grouping

From the metadomains we construct an intersection graph, in which two vertices are joined by an edge when the two metadomains have a non-empty intersection. Next, the connectivity of the intersection graph is used to ﬁnd larger groups of metadomains that have a common non-empty intersection. The basic idea is that when two vertices are joined by an edge then this points to the existence of a line crossing three or four domains, and therefore also points to a concurrency relation for digital line segments, i.e., the line segments whose domains have given rise to the metadomains that correspond with the two vertices. The extension of this idea is to extract cliques from the intersection graph, since the vertices of a large clique will often correspond to a large collection of metadomains that have a common non-empty intersection, and thus indirectly it will correspond to a concurrency relation that involves a large number of line segments. There is no guarantee, however, that the metadomains of clique always have a non-empty intersection. Fig. 6(a) shows an exception, where we have six line segments, and the metadomains AB, CD and EF which are

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Fig. 6. Clique whose metadomains have no common intersection.

Fig. 7. Minimum clique covering leads to inconsistencies.

shown here as the uncertainty regions of the six line segments. Fig. 6(b) shows a subgraph of the intersection graph induced by the vertices AB, CD, EF , which form a clique. Nonetheless, the intersection of the metadomains is empty. Such cases are rare, however. In fact, they can only occur when the union of the three metadomains forms a topological torus. In this work, whenever such a situation occurred we solved this problem by taking the convex hull of the hole of the torus as the uncertainty region of the intersection point. A second problem is that most groupings, and therefore cliques also, lead to inconsistent geometric relations. This is illustrated in Fig. 7. A minimum clique covering algorithm may propose, for example, the cliques AC − AB − BC and AD − BD and thus the concurrent relations conc(A, B, C), conc(A, B, D). Since these two triples share two line segments (A and B), we should also have conc(A, B, C, D), which is not the case here, because some edges are missing, and therefore some of the metadomans that are involved do not have a common non-empty intersection. Informally we can deﬁne a set of relations between digital line segments as consistent if the set contains no relations that cannot occur in Euclidean geometry. A further requirement would be that for suﬃciently small values of the uncertainty parameter τ the relations found in uncertain geometry should coincide with Euclidean geometry. Note, however, that consistency is violated already at the lowest level, since parallelism and collinearity are not transitive in

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uncertain geometry. It follows that we can only restore consistency upto a certain degree. For example, even when we restore the usual properties of parallelism, collinearity and concurrency, more involved properties such as Pappus’ Theorem may still not hold. As for concurrency, in this work we restrict ourselves to the removal of the most obvious inconsistencies: (A B) ∧ (A = B) ∧ conc(A, B, C),

(4)

(A

B ) ∧ (A

C ) ∧ (B

C ) ∧ conc(A, B, C) ∧ conc(B, C, D) ∧ ¬conc(A, B, D). (5) To remove inconsistencies of the form (4) or (5) we shall use the following result, which gives a suﬃcient, but not necessary, condition. Lemma 2. Let G be the intersection graph of the metadomains of a collection of digital line segments A, B, . . .. Let S be a set of disjoint cliques of G such that the pathlength between two vertices belonging to distinct cliques in S is always larger than two. If in the set of concurrency relations implied by the cliques in S we have two relations of the form conc(A, B, C, . . .) and conc(A, B, P, . . .), then we have A B. Proof. Suppose we have conc(A, B, C, . . .) and conc(A, B, P, . . .). Assume that A B . Then AB appears as a vertex in the intersection graph. Since there is a clique whose vertices imply the relation conc(A, B, C, . . .), either AB must belong to this clique, or there must be vertices AX and BY in the clique, where X and Y can even be identical. Since AX and BY are adjacent, there is a common transversal cutting the four domains A, X, B and Y . Hence, both AX and BY must also be adjacent to AB. Similarly, in the clique generating the relation conc(A, B, P, . . .), there must be vertices AU and BV that are also adjacent to AB. In this case both cliques would have a common neighbor. Since we exclude such cliques, it follows that the assumption A

B cannot be true. In fact, two relations of the form conc(A, B, C, . . .) and conc(A, B, P, . . .) can only arise when AB is not a vertex of the graph, or in other words when A B. In particular, A may even be collinear with B. To ﬁnd consistent concurrency relations, we use a greedy algorithm which is based on Lemma 2. First, we extract a maximum clique and remove all its neighbors as well as the neighbors of these neighbors from the intersection graph of the metadomains. By proceeding with the next maximum clique in the graph, we construct thus a list of cliques such that no two cliques share a common neighbor. Next, from each clique we derive the implied concurrency relation. For example, a clique with vertices AB, CD, AE implies conc(A, B, C, D, E). The result is shown in Fig. 3, where we draw the uncertainty regions for the intersection points of a number of concurrency relations, i.e., the region ABH for the relation conc(A, B, H). Finally, we remove those concurrency relations which involve one or more pairs of digital line segments that are parallel but not collinear. Since C and E are parallel, only conc(A, B, F ), conc(A, B, G), conc(A, B, H) are left as valid concurrency relations.

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With the above algorithm we can avoid inconsistencies of the form (4) as well as (5). Suppose, for example, that we have found the concurrency relations conc(A, B, C) and conc(A, B, D). Because the extracted cliques satisfy the conditions of Lemma 2, it follows that A B, which excludes an inconsistency of the form (5). In addition, if A is not collinear with B, then both concurrency relations will be discarded in the ﬁnal step of the algorithm where we examine parallel pairs. This excludes an inconsistency of the form (4). If A and B are collinear, however, then the concurrency relations will not be discarded because they can coexist without any contradiction.

5

Concluding Remarks

We presented a method for the derivation of concurrency relations in uncertain geometry. The removal of inconsistencies is a major part in this approach. Without doubt, there is still no unique decisive procedure that guarantees the removal of all possible geometric inconsistencies. In this work we have proposed one approach, which gives satisfactory results on real images, but still has some shortcomings. In particular, the following important questions have been barely examined. How can we obtain consistency when we consider distinct types of relations? For example, if A is collinear with D while A, B and C are concurrent, then consistency requires that B, C and D must also be concurrent. Upto now the consistency of properties such as parallelism, collinearity and concurrency have only been examined separately. Furthermore, in real applications we must combine the extraction of linear structure with other properties such as proximity and symmetry. Also this question has not been suﬃciently examined yet. Finally, the uncertainty is controlled by the uncertainty parameter τ . When analyzing relations in a digital image, we often derive geometric relations for different values of τ . It is a natural requirement that these relations should behave in a consistent way when we increase or decrease τ .

References 1. H. Heijmans and A. Toet, “Morphological sampling,” CVGIP: Image Understanding, vol. 54, pp. 384–400, 1991. 2. P. Veelaert, “Geometric constructions in the digital plane,” J. Math. Imaging and Vision, vol. 11, pp. 99–118, 1999. 3. P. Veelaert, Parallel line grouping based on interval graphs, Proc. of DGCI 2000, vol. 1953 of Lecture Notes in Computer Science, pp. 530–541. Uppsala, Sweden: Springer, 2000. 4. P. Veelaert, “Graph-theoretical properties of parallelism in the digital plane,” submitted. 5. P. Veelaert, “Collinearity and weak collinearity in the digital plane,” Digital and Image Geometry, vol. 2243 of Lecture Notes in Computer Science, pp. 434–447, Springer, 2001. 6. P. Veelaert, “Line grouping based on uncertainty modeling of parallelism and collinearity,” in Proceedings of SPIE’s Conference on Vision Geometry IX, (San Diego), pp. 36–45, SPIE, 2000.

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7. V. J. Milenkovic, “Veriﬁable implementations of geometric algorithms using ﬁnite precision arithmetic,” in Geometric Reasoning (Kapur and Mundy, eds.), pp. 377– 401, Cambridge: MIT Press, 1989. 8. H. F. Durrant-Whyte, “Uncertain geometry in robotics,” IEEE Trans. Robotics Automat. , pp. 23–31, 1988. 9. H. F. Durrant-Whyte, “Uncertain geometry,” in Geometric Reasoning (Kapur and Mundy, eds.), pp. 447–481, Cambridge: MIT Press, 1989. 10. A. Fleming, “Geometric relationships between toleranced features,” in Geometric Reasoning (Kapur and Mundy, eds.), pp. 403–412, Cambridge: MIT Press, 1989. 11. D. Lowe, “3-d object recognition from single 2-d images,” Artiﬁcial Intelligence, vol. 31, pp. 355–395, 1987. 12. M. A. Fischler and R. C. Bolles, “Random sample consensus: A paradigm for model ﬁtting with applications to image analysis and automated cartography,” CACM, vol. 24, pp. 381–395, 1981.

Discretization in 2D and 3D Orders Michel Couprie, Gilles Bertrand, and Yukiko Kenmochi Laboratoire A2 SI, ESIEE Cit´e Descartes, B.P. 99 93162 Noisy-Le-Grand Cedex France, {coupriem,bertrand,kenmochy}@esiee.fr

Abstract. Among the diﬀerent discretization schemes that have been proposed and studied in the literature, the supercover is a very natural one, and furthermore presents some interesting properties. On the other hand, an important structural property does not hold for the supercover in the classical framework: the supercover of a straight line (resp. a plane) is not a discrete curve (resp. surface) in general. We follow another approach based on a diﬀerent, heterogenous discrete space which is an order, or a discrete topological space in the sense of Paul S. Alexandroﬀ. Generalizing the supercover discretization scheme to such a space, we prove that the discretization of a plane in R3 is a discrete surface, and we prove that the discretization of the boundary of a “regular” set X (in a sense that will be precisely deﬁned) is equal to the boundary of the discretization of X. This property has an immediate corollary for half-spaces and planes, and for convex sets. Keywords: discretization, topology, orders, supercover, discrete surfaces

1

Introduction

An abundant literature is devoted to the study of discretization schemes. Let E be an “Euclidean” space, and let D be a “discrete” space related to E. Typically, one can take E = Rn and D = Zn (n = 2, 3), but we do not limit ourselves to this case. A discretization scheme associates, to each subset X of E, a subset D(X) of D which is called the discretization of X. Diﬀerent discretization schemes have been proposed and compared with respect to some fundamental geometrical, topological and structural properties. We may, for example, ask the following questions: if X ⊆ E is the image of X by a symmetry, is D(X ) the image of D(X) by the same symmetry ? If X is connected, is D(X) connected (in some sense) ? And if X is a curve, is D(X) a curve (in some sense) ? In this paper, we consider the discretization scheme called supercover, and we focus on some structural properties. Consider E = R2 , for simplicity, and let D be the set of all closed squares in E with side 1 and the vertices of which have integer coordinates (the elements of D are often called pixels). Let X be a subset of E, the supercover of X is the set of all the pixels that have a non-empty intersection with X. The supercover has many interesting properties, which have been studied by several authors [9,10,2,1,8,21,22]. In particular, Andr`es [1] proposed an analytical characterization of the supercover of straight lines, and more generally A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 301–312, 2002. c Springer-Verlag Berlin Heidelberg 2002

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for hyperplanes and for simplices in higher dimensions. Also, Ronse and Tajine showed that the supercover is a particular case of Hausdorﬀ discretization [21, 22]. But the supercover has also a drawback for thin objects such as straight lines. If a straight line δ in R2 goes through a point with integer coordinates, then the supercover of δ contains the four pixels that cover this point - this conﬁguration is called a “bubble” (Fig. 1(a)). An extreme case is when δ is horizontal or vertical, and hits elements of Z2 (Fig. 1(b)): the supercover of such a line is 2-pixel thick. Thus, the supercover of a straight line cannot be seen as a discrete curve.

(a) (b) Fig. 1. (a): A straight line segment and its supercover (shaded), which contains a “bubble” (set of four pixels sharing a common vertex). (b): A horizontal line segment that has a 2-pixel thick supercover.

Another popular discretization scheme for lines, called grid intersection digitization [17,23], does guarantee that the discretization of a straight line δ is a digital curve, in the sense of the digital topology [16]. A proof of this property can be found in [15]. The drawback of this discretization scheme is its lack of symmetry: for any intersection of δ with a pixel boundary, the pixel vertex which is closest to this intersection is chosen as an element of the discretization of δ, and if the intersection is at equal distance between two vertices, then an arbitrary choice is made (for example, the rightmost or upmost vertex). This drawback is shared by other discrete models for straight lines and planes, the Bresenham’s model [7], the naive model [20] and the standard model [1]. On the other hand, the supercover does not suﬀer from this lack of symmetry. An attempt to solve the problem of “bubbles”, which seems to be the price payed for symmetry, has been made in [8] with the notion of minimal cover. Let X be a subset of R2 . Any set S of pixels, such that X is included in the union of the elements of S, is called a cover of X. Let S be a cover of X, we say that S is a minimal cover of X if there is no other cover of X which is a proper subset of S. We see in Fig. 2 that the minimal cover of certain straight lines is “thinner” than the supercover, but we see also that the minimal cover is not unique in general. We follow another approach based on a diﬀerent, heterogenous discrete space which is an order, or a discrete topological space in the sense of Paul S. Alexandroﬀ [3]. Such spaces have been the subject of intensive research in the recent past, not only from the topology point of view [13,18,11,5], but also in relation with discretization and geometrical models [14,25]. The discrete space D that

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(a) (b) Fig. 2. (a): A straight line segment and its minimal cover (shaded). (b): A horizontal line segment and one of its possible minimal covers.

we will consider is a partition of the Euclidean space E, composed (in the case of E = R2 ) of open unit squares, unit line segments and singletons. The fact that D is a partition of E leads to a fundamental property: for any subset X of E, the supercover of X (relative to D) is the unique minimal cover of X. We will focus on this discretization scheme, and discuss only the 3D case in the sequel (corresponding results in 2D are particular cases). The two main contributions of this paper are the following results. (i) We prove that the discretization of a plane in R3 is a discrete surface. (ii) We prove that the discretization of the boundary of a “regular” set X (in a sense that will be precisely deﬁned) is equal to the boundary of the discretization of X. This property has an immediate corollary for half-spaces and planes, and for convex sets.

2

Basic Notions on Orders

In this section, we recall some basic notions relative to orders (see also [13,5,6]). If X is a set, P(X) denotes the set composed of all subsets of X, if S is a subset of X, S denotes the complement of S in X. If S is a subset of T , we write S ⊆ T , the notation S ⊂ T means that S is a proper subset of T , i.e. S ⊆ T and S = T . If γ is a map from P(X) to P(X), the dual of γ is the map ∗γ from P(X) to P(X) such that, for each S ⊆ X, ∗γ(S) = γ(S). Let δ be a binary relation on X, i.e., a subset of X × X. We also denote by δ the map from X to P(X) such that, for each x of X, δ(x) = {y ∈ X, (x, y) ∈ δ}. We deﬁne δ as the binary relation δ = δ \ {(x, x); x ∈ X}. An order is a pair (X, α) where X is a set and α is a reﬂexive, antisymmetric, and transitive binary relation on X. An element of X is also called a point. The set α(x) is called the α-adherence of x, if y ∈ α(x) we say that y is α-adherent to x. We illustrate these general notions on orders with the example of Fig. 3, which is composed of the following elements : two triangles t1 , t2 ; ﬁve edges e1 , e2 , e3 , e4 , e5 ; and four vertices v1 , v2 , v3 , v4 . Here, we deﬁne the order relation α by: α(t1 ) = {t1 , e1 , e2 , e3 , v1 , v2 , v3 }; α(t2 ) = {t2 , e2 , e4 , e5 , v2 , v3 , v4 }; α(e1 ) = {e1 , v1 , v2 }; α(e2 ) = {e2 , v2 , v3 }; α(e3 ) = {e3 , v1 , v3 }; α(e4 ) = {e4 , v3 , v4 }; α(e5 ) = {e5 , v2 , v4 }; and for i = 1 . . . 4, α(vi ) = {vi }. Let (X, α) be an order. We denote by α the map from P(X) to P(X) such that, for each subset S of X, α(S) = ∪{α(x); x ∈ S}, α(S) is called the α-

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v1

e1

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Fig. 3. An example for the basic notions on orders.

closure of S, ∗α(S) is called the α-interior of S. A subset S of X is α-closed if S = α(S), S is α-open if S = ∗α(S). In our example of Fig. 3, let S be the set {t1 , e1 , e5 , v2 }. We see that α(S) = {t1 , e1 , e2 , e3 , e5 , v1 , v2 , v3 , v4 } = S, thus S is not α-closed. We can also see that ∗α(S) = {t1 , e1 } = S, thus S is not α-open. On the opposite, α(t1 ), {e2 , e5 , v2 , v3 , v4 }, {v1 } for example are α-closed, and {t1 }, {t1 , t2 , e2 } for example are α-open. Let (X, α) be an order. We denote by β the relation β = {(x, y); (y, x) ∈ α}, β is the inverse of the relation α. We denote by θ the relation θ = α ∪ β. The dual of the order (X, α) is the order (X, β). Notice that ∗α(S) = {x ∈ S; β(x) ⊆ S}, and ∗β(S) = {x ∈ S; α(x) ⊆ S}. In our example of Fig. 3, β(v2 ) = θ(v2 ) = {v2 , e1 , e2 , e5 , t1 , t2 }; β(e2 ) = {e2 , t1 , t2 }; θ(e2 ) = {v2 , v3 , e2 , t1 , t2 }; β(t1 ) = θ(t1 ) = {t1 }. The set composed of all α-open subsets of X satisﬁes the conditions for the family of open subsets of a topology, the same result holds for the set composed of all β-open subsets of X. These topologies are P.S. Alexandroﬀ topologies, i.e., topologies such that every intersection of open sets is open [3]. An order (X, α) is countable if X is countable, it is locally ﬁnite if, for each x ∈ X, θ(x) is a ﬁnite set. A CF-order is a countable locally ﬁnite order. Let (X, α) be a CF-order. Let x0 and xk be two points of X. A path from x0 to xk is a sequence x0 , x1 , ..., xk of elements of X such that xi ∈ θ(xi−1 ), with i = 1, ..., k. A CF-order (X, α) is connected if for all x, y in X, there is a path from x to y. If (X, α) is an order and S is a subset of X, the order relative to S is the order |S| = (S, α ∩ (S × S)). We will use a general deﬁnition for curves and surfaces which has been used in several works (see e.g. [11,6]). This notion is close to the notion of manifold used by Kovalevsky [18]; nevertheless it does not involve the necessity to attach a notion of dimension to each element of X, which allows to have a simpler deﬁnition (in particular, no use of isomorphism is made). Let |X| = (X, α) be a non-empty CF-order. - The order |X| is a 0-surface if X is composed exactly of two points x and y such that y ∈α(x) and x ∈α(y). - The order |X| is an n-surface, n > 0, if |X| is connected and if, for each x in X, the order |θ (x)| is an (n − 1)-surface. - A curve is a 1-surface, a surface is a 2-surface.

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In our example of Fig. 3, the orders relative to the following sets: {v1 , e1 , v2 , e5 , v4 , e4 , v3 , e3 } and {v1 , e1 , v2 , t2 , v3 , e3 }, are both curves. Conversely, the order depicted in Fig. 3 is not a surface, since for example, θ (v1 ) = {e1 , t1 , e3 } is not a curve.

3

An Order Associated to Rn

Let R be the set of real numbers. We consider the families of subsets of R named G01 , G11 and G 1 such that: G01 = {{p + 12 }, p ∈ Z}, G11 = {]p − 12 , p + 12 [, p ∈ Z}, G 1 = G01 ∪ G11 . A subset R of Rn which is the cartesian product of exactly m elements of G11 and n − m elements of G01 is called an m-gel of Rn . n For a given integer m, we denote by Gm the set of all m-gels of Rn , and we n n denote by G the union of all the sets Gm , for all m = 0 . . . n. An element of G n is called a gel. For example, with n = 2, a 0-gel is a singleton (a set containing a single point), a 1-gel is a line segment which does not contain its extremities (either of the form {p + 12 }×]q − 12 , q + 12 [ or ]p − 12 , p + 12 [×{q + 12 }), and a 2-gel is an open square. We remark that, according to the “standard” topology of Rn , only the n-gels are open subsets of Rn (they are open hypercubes), and that only the 0-gels are closed subsets of Rn (they are singletons). For 0 < m < n, an m-gel is neither open nor closed. On the opposite, all pixels (see section 1) are closed subsets of R2 . Notice also that G n is a partition of Rn , this is not the case with the covering of R2 with pixels. Let x be a gel, we denote by cl(x) the closure of x (according to the “standard” topology of Rn ). We consider the order (G n , α) deﬁned by: ∀x, y ∈ G n , y ∈ α(x) if y ⊆ cl(x). For example, with n = 2, let x be an open square (a 2-gel). Then, α(x) is composed of x itself, of the four line segments that border x (without the vertices), and of the four singletons containing each a vertex of cl(x). Notice that these orders are equivalent to those obtained in the framework of connected ordered topological spaces introduced by E.D. Khalimsky [12]. As far as we know, the ﬁrst mention of (G n , α) as a discrete topological space can be found in the classical topology textbook by P.S. Alexandroﬀ and H. Hopf [4], as one of the ﬁrst examples used to illustrate the notion of a topological space.

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Generalized Covers and Supercovers

Let F be a family of subsets of Rn (n ≥ 1). We say that the family F covers Rn if Rn is equal to the union of all the elements of F. In the following, we will consider the families G n and Gnn . Notice that G n does cover Rn , but Gnn does not. Let R be any subset of Rn , we say that a subset S of F is an F−cover of R, if R is included in the union of all the elements of S (this deﬁnition generalizes the notion of cover in [8], but is diﬀerent from the notion of cover discretization in [22]). Let F be a family of subsets of Rn , and let R be any subset of Rn . We consider the hit and miss transforms as deﬁned in [24]. The hit of R in F, denoted by F(R), is the set of all the elements of F which intersect R: F(R) = {x ∈ F, x ∩ R = ∅}. In a dual way, we may consider the set ∗F(R) composed of all elements of F which are included in R. If F is a family that covers Rn , then F(R) is called the F−supercover of R. The F−supercover is obviously a particular case of F−cover, and is uniquely deﬁned for any given R. If we choose n = 2 and F equals the set of all pixels, we retrieve the notion of supercover presented in the introduction. In this paper, we focus on supercovers based on the family G n . The fact that n G is a partition of Rn leads to several interesting properties. Furthermore: Property 1 Let F be a family of sets covering Rn . Then, the two following propositions are equivalent: (i) for any subset R of Rn , the F-supercover of R is the unique minimal F-cover of R. (ii) F is a partition of Rn . This property is a direct consequence of Prop. 2, which is stated in the more general framework of binary relations. Let A, B be two sets. A relation Γ from A to B is a subset of the cartesian product A × B. If (a, b) ∈ Γ , we also write that (b, a) ∈ Γ −1 , that b ∈ Γ (a) and that a ∈ Γ −1 (b), and we say that b is a successor of a and that a is a predecessor of b. We say that the relation Γ is surjective if each b in B has at least one predecessor, and Γ is a map from A to B if each a in A has a unique successor. Let R be a subset of A, we write Γ (R) = ∪a∈R Γ (a). Let A, B be two sets, let Γ be a relation from A to B. We say that Γ deﬁnes a covering of A by B if both Γ and Γ −1 are surjective, i.e. if each element of A has at least one successor and each element of B has at least one predecessor. Let A, B be two sets, let Γ be a relation deﬁning a covering of A by B. Let R be a subset of A. We say that S ⊆ B is a Γ -cover of R if R ⊆ Γ −1 (S). Furthermore, we say that the Γ -cover S is minimal if there is no other Γ -cover of R strictly included in S. The set Γ (R) is called the Γ -supercover of R. It is obviously a Γ -cover of R, which is uniquely deﬁned for any given R, but in general it is not a minimal Γ -cover of R. For example, if we take A = Rn and choose for B a family of sets covering Rn , and deﬁne Γ (x) as the set of elements of B which hit x, then we retrieve

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the notions and the results of the beginning of this section. In particular, the following property generalizes Prop. 1. Property 2 Let A, B be two sets, let Γ be a relation deﬁning a covering of A by B. Then, the two following propositions are equivalent: (i) for any subset R of A, the Γ -supercover of R is the unique minimal Γ -cover of R. (ii) Γ is a map from A to B. Proof: (ii) ⇒ (i). Let S = Γ (R). Is S minimal ? Suppose that there exists another Γ -cover S strictly included in S, and let s be an element of S \ S . Since S = Γ (R), there exists an x in R such that s ∈ Γ (x), and since Γ is a map, we have Γ (x) = {s}. Thus there is an element x of R which has no successor in S , a contradiction. Is S the unique minimal Γ -cover ? Suppose that there exists another minimal Γ -cover S = S. Since both S and S are minimal, there must exist at least an element s in S \ S and an element s in S \ S. This leads to the same contradiction. (i) ⇒ (ii). Suppose that (i) and that Γ is not a map. Then, there exists an element x of A that has either 0 or more than two successors. As Γ −1 is a surjection, x has at least one successor. Let y, z be two distinct successors of x, and let us consider the set R = {x}. The set {y} is a strict subset of Γ (R) which is also a Γ -cover of R, a contradiction. The following properties can also be easily proved. They are mentioned in [9] for the particular case of R2 and a covering with pixels. Notice that the condition that the relation Γ is a map is not required. Property 3 Let A, B be two sets, let Γ be a relation deﬁning a covering of A by B. Then, ∀R, S ⊆ A we have: (i) Γ (R ∪ S) = Γ (R) ∪ Γ (S) (ii) Γ (R ∩ S) ⊆ Γ (R) ∩ Γ (S) (iii) R ⊆ S ⇒ Γ (R) ⊆ Γ (S) Furthermore, if Γ is a map deﬁning a covering of A by B, then the cardinality of Γ (R) is less or equal to the cardinality of R, for any R subset of A.

5

Properties

This section contains the main results of the paper. We ﬁrst show that an analytic characterization of G 3 -supercovers of planes can be given, by adapting the result of Andr`es for the “classical” supercover. Such an analytical characterization is essential to design fast algorithms that generate discrete planar objects. We consider the plane π deﬁned by: π = {(x, y, z) ∈ R3 /ax + by + cz + d = 0}, where a, b, c, d belong to R. The following result, adapted from Andr`es ([1], Th. 18) gives us an analytical characterization of the elements of G33 (π). Remind that G33 (π) is the set of the elements (open cubes) of G33 that have a non-empty intersection with π.

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(a)

(b)

(c)

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(m) (n) (o) Fig. 4. The ﬁfteen ways for a plane to hit a 3-gel and its θ-neighborhood. The 1-gels and 0-gels that are hit by the plane are highlighted. The 2-gels (squares) that are hit by the plane have not been highlighted, in order to preserve the readability of the ﬁgure.

Property 4 Let a, b, c, d ∈ R, ab = 0 or bc = 0 or ac = 0, let π = {(x, y, z) ∈ R3 /ax + by + cz + d = 0}. Let (p, q, r) ∈ Z3 , we denote by Gpqr the 3-gel ]p − 12 , p + 12 [ × ]q − 12 , q + 12 [ × ]r − 12 , r + 12 [. Then, G33 (π) = {Gpqr ∈ G33 , − |a|+|b|+|c| < ap + bq + cr + d < |a|+|b|+|c| }. 2 2 Let x0 ∈ R, let π = {(x, y, z) ∈ R3 /x = x0 }. If x0 − 12 ∈ Z, then G33 (π) = ∅, else G33 (π) = {Gpqr ∈ G33 , |p − x0 | ≤ 12 }. For planes deﬁned by y = y0 or z = z0 , a similar statement holds. In order to have a complete characterization of the elements of G 3 (π), we must characterize also the elements of G23 (π), of G13 (π) and those of G03 (π). Let s = {(p, q, r)} be an element of G03 . We denote by π(s) the index which characterizes the position of s relative to π:   −1 if ap + bq + cr + d < 0, 0 if ap + bq + cr + d = 0, π(s) =  +1 if ap + bq + cr + d > 0 Property 5 Let a, b, c, d ∈ R, let π = {(x, y, z) ∈ R3 /ax + by + cz + d = 0}. a) If b = c = 0 and ad − 12 ∈ Z, then:

1 1 G03 (π) = {{( −d a , q + 2 , r + 2 )}, q, r ∈ Z}, and

1 1 1 −d 1 G13 (π) = {{ −d a } × ]q − 2 , q + 2 [ × {r + 2 }, q, r ∈ Z} ∪ {{ a } × {q + 2 } × ]r − 12 , r + 12 [, q, r ∈ Z} and

Discretization in 2D and 3D Orders

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(n) (o) Fig. 5. Neighborhoods of a 3-gel (open cube)

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(b) (c) (d) Fig. 6. Neighborhoods of a 2-gel (square)

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1 1 1 1 G23 (π) = {{ −d a } × ]q − 2 , q + 2 [ × ]r − 2 , r + 2 [, q, r ∈ Z}. Similar characterizations are obtained for the cases a = b = 0 and a = c = 0. b) Other cases: let v ∈ G03 , let i ∈ G13 , with α(i) = {v1 , v2 }, and let s ∈ G23 . Then: v ∈ G03 (π) iﬀ π(v) = 0, i ∈ G13 (π) iﬀ (π(v1 ) = π(v2 ) = 0) or (π(v1 ).π(v2 ) < 0), s ∈ G23 (π) iﬀ at least two vertices of s have diﬀerent non-null indices.

Now we are ready to state the ﬁrst main result of this paper. It says that the discretization of a plane is a surface, in the sense deﬁned in section 2. This result can be easily transposed in 2D, where it states that the discretization of a straight line is a curve. Property 6 Let a, b, c, d ∈ R, let π = {(x, y, z) ∈ R3 /ax + by + cz + d = 0}. The order |G 3 (π)| is a surface. The proof of Prop. 6 involves the examination of the diﬀerent conﬁgurations of a plane π hitting an open cube, a square, a line segment and a single point, and

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(a) (b) (c) (d) (e) (f) Fig. 7. (a,b,c): neighborhoods of a 1-gel (segment), (d,e,f): neighborhoods of a 0-gel (singleton)

their respective θ-neighborhoods. For the open cube, these conﬁgurations (up to rotations and symmetries) are only 15, they are depicted in Fig. 4. We can easily check (see Fig. 5) that for each of these conﬁgurations, the θ -neighborhood of any 3-cube in G 3 (π) is a curve. For the cases of a single point, a line segment and a square, the numbers of possible conﬁgurations (up to rotations and symmetries) are respectively 3, 3 and 5. Figs. 6 and 7 shows the θ -neighborhood of such an element in each possible conﬁguration, again we can verify that it forms a curve. Our second main result states that, for any “regular” object (in a sense that will be deﬁned below), the boundary operator commutes with the discretization operator. Let (O, α) be an order, and let P be a subset of O. We deﬁne the θ−boundary of P in O (or simply the boundary of P ) as the set B(P ) of elements p of P such that θ(p) ∩ P = ∅. Let X be a subset of R3 . Let C0 be the unit closed cube centered at the origin: C0 = [− 12 , 12 ]3 . Let Cu be the translation of C0 by the vector u of R3 . The set X is morphologically open by the structuring element C0 if X is equal to the union of all the translations Cu of C0 which are included in X (see [24]). Notice that this notion is close to the notion of a par(r, +)-regular set deﬁned by Latecki in the continuous plane[19]. The (topological) closure of X is denoted by cl(X), and the boundary of X is deﬁned by b(X) = cl(X) ∩ cl(X). Property 7 If X is a closed subset of R3 , then B(G 3 (X)) ⊆ G 3 (b(X)). Furthermore, if cl(X) is morphologically open by C0 , then the boundary of the discretization of X is equal to the discretization of the boundary of X, in other words, B(G 3 (X)) = G 3 (b(X)). A corollary can be immediately derived from this property, concerning planes and half-spaces. We consider the closed half-space γ = {(x, y, z) ∈ R3 /ax + by + cz + d ≥ 0}, where a, b, c, d belong to R. The boundary of γ is the plane π deﬁned by: π = {(x, y, z) ∈ R3 /ax + by + cz + d = 0}. Then we have: Corollary 8 G 3 (b(γ)) = G 3 (π) = B(G 3 (γ)) From this, we can easily deduce a more general corollary which holds for any convex and closed set: Corollary 9 If X is a convex closed subset of R3 , then B(G 3 (X)) = G 3 (b(X)). Fig. 8 illustrates these properties in 2D.

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Fig. 8. Illustration of Prop. 7 and Cor. 9 in 2D. We see the discretizations of three objects : a disc, a convex polygon and a third convex set. The boundaries of these objects appear as continuous solid lines. The discretizations of the boundaries, which coincide with the boundaries of the discretizations, are represented by light gray squares, black segments and black dots.

Acknowledgements. The authors wish to thank both reviewers for their interesting and useful comments.

References 1. E. Andr`es, Mod´elisation analytique discr`ete d’objets g´eom´etriques, Th`ese de HDR, Universit´e de Poitiers (France), 2000. 2. E. Andr`es, C. Sibata, R. Acharya, “Supercover 3D polygon”, Conf. on Discrete Geom. for Comp. Imag., Vol. 1176, Lect. Notes in Comp. Science, Springer Verlag, pp. 237-242, 1996. 3. P.S. Alexandroﬀ, “Diskrete R¨ aume”, Mat. Sbornik, 2, pp. 501-518, 1937. 4. P.S. Alexandroﬀ, H. Hopf, Topologie, Springer Verlag, 1937. 5. G. Bertrand, “New notions for discrete topology”, 8th Conf. on Discrete Geom. for Comp. Imag., Vol. 1568, Lect. Notes in Comp. Science, Springer Verlag, pp. 216226, 1999. 6. G. Bertrand, M. Couprie, “A model for digital topology”, 8th Conf. on Discrete Geom. for Comp. Imag., Vol. 1568, Lect. Notes in Comp. Science, Springer Verlag, pp. 229-241, 1999. 7. J. Bresenham, “Algorithm for computer control of digital plotter”, IBM System Journal, Vol. 4, pp. 25-30, 1965.

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8. V.E. Brimkov, E. Andr`es, R.P. Barneva, “Object discretization in higher dimensions”, 9th Conf. on Discrete Geom. for Comp. Imag., Vol. 1953, Lect. Notes in Comp. Science, Springer Verlag, pp. 210-221, 2000. 9. J.M. Chassery, A. Montanvert, G´eom´ etrie discr`ete en imagerie, Herm`es, Paris, France, 1991. 10. D. Cohen-Or, A. Kaufman, “Fundamentals of surface voxelization”, Graphical models and image processing, 57(6), pp. 453-461, 1995. 11. A.V. Evako, R. Kopperman, Y.V. Mukhin, “Dimensional Properties of Graphs and Digital Spaces”, Jour. of Math. Imaging and Vision, 6, pp. 109-119, 1996. 12. E.D. Khalimsky, “On topologies of generalized segments”, Soviet Math. Doklady, 10, pp. 1508-1511, 1969. 13. E.D. Khalimsky, R. Kopperman, P. R. Meyer, “Computer Graphics and Connected Topologies on Finite Ordered Sets”, Topology and its Applications, 36, pp. 1-17, 1990. 14. R. Klette, “m-dimensional cellular spaces”, internal report, University of Maryland, CAR-TR-6, MCS-82-18408, CS-TR-1281, 1983. 15. R. Klette, “The m-dimensional grid point space”, Computer vision, graphics, and image processing, 30, pp. 1-12, 1985. 16. T.Y. Kong and A. Rosenfeld, “Digital topology: introduction and survey”, Comp. Vision, Graphics and Image Proc., 48, pp. 357-393, 1989. 17. J. Koplowitz, “On the performance of chain codes for quantization of line drawings”, IEEE Trans. on PAMI, 3, pp. 180-185, 1981. 18. V.A. Kovalevsky, “Topological foundations of shape analysis”, in Shape in Pictures, NATO ASI Series, Series F, Vol. 126, pp. 21-36, 1994. 19. L.J. Latecki, Discrete representation of spatial objects in computer vision, Kluwer Academic Publishers, 1998. 20. J-P. Reveill`es, G´eom´ etrie discr`ete, calcul en nombres entiers et algorithmique, Th`ese d’´etat, Universit´e Louis Pasteur, Strasbourg (France), 1991. 21. C. Ronse, M. Tajine, “Hausdorﬀ discretization of algebraic sets and diophantine sets”, 9th Conf. on Discrete Geom. for Comp. Imag., Vol. 1953, Lect. Notes in Comp. Science, Springer Verlag, pp. 216-226, 2000. 22. C. Ronse, M. Tajine, “Hausdorﬀ discretization for cellular distances, and its relation to cover and supercover discretizations”, Journal of Visual Communication and Image Representation, Vol. 12, no. 2, pp. 169-200, 2001. 23. A. Rosenfeld, A.C. Kak: Digital picture processing, Academic Press, 1982. 24. J. Serra, Image Analysis and Mathematical Morphology, Academic Press, 1982. 25. J. Webster, “Cell complexes and digital convexity”, Digital and image geometry, Vol. 2243, Lect. Notes in Comp. Science, Springer Verlag, pp. 268-278, 2002.

Deﬁning Discrete Objects for Polygonalization: The Standard Model Eric Andres Universit´e de Poitiers Laboratoire IRCOM-SIC, Bˆ at. SP2MI, BP 30179, 86962, Futuroscope Cedex, France

Abstract. A new description model, called the standard model, for discrete linear objects in dimension n is proposed. Standard objects are tunnel-free and (n-1)-connected. The discrete objects are deﬁned analytically as union of intersections of half-spaces. The standard 3D polygons are well suited for polygonalization. This is the main reason why this model has been developed.

1

Introduction

Polygonalization of discrete objects is one of the major research problems of the discrete geometry community for many years now. The main approach used in practice is the “marching cubes” [16] type approach. In this approach, local neighbourhoods of voxels are replaced by Euclidean polygons. This means that the resulting number of polygons is proportional to the number of boundary voxels of the discrete object. As the number of boundary voxels is usually extremely important, people tend to apply simpliﬁcation schemes to reduce that number of polygons and thus loose approximation quality. For a couple of years now, another approach, we call a discrete analytical polygonalization, is investigated by a number of research groups. In this approach, the aim is to decompose the boundary of discrete objects into discrete analytical polygons and then these discrete polygons into Euclidean polygons. By discrete analytical polygons we understand discrete polygons that are not deﬁned as set of discrete points but by an analytical description that is independent of the number of discrete points composing it. The aim here is to decompose the boundary of a discrete object into a number of discrete polygons that isn’t, in general, directly proportional to the number of boundary voxels. Potential applications can be found in compression, visualization, transformations, medical imaging, ... Numerous papers have brought new insight and new ideas on how to tackle this diﬃcult problem. However, only a small part of the problem has been solved. In this paper we propose a new element to the problem that was missing so far, the deﬁnition of a discrete analytical polygon. Several authors have proposed algorithms that determine if a given set of discrete points belong to the same Reveill`es analytical discrete plane [17,2]. At the same time they provide the analytical description (two inequalities) of the discrete plane [9,13,12]. Some others A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 313–325, 2002. c Springer-Verlag Berlin Heidelberg 2002

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propose a description of the equivalence class of all the possible discrete analytical planes the set of boundary points belongs to [11,19]. This only leads to decompositions of the boundary of discrete object into planar sections. All these diﬀerent approaches couldn’t go beyond a planar decomposition since nobody knew how to deﬁne discrete polygon and therefore nobody could propose a complete discrete polygonalization algorithm. Existing discrete polygons such as the ones proposed by A. Kaufman [14] are well suited for visualization purposes but can’t be used for polygonalization: they are not planar (points do not all belong to a same discrete plane), not topologically consistent (sometimes with holes) and not analytically deﬁned. As already stated we propose in this paper a deﬁnition of a 3D discrete analytical polygon but also, more generally, a discrete analytical model for all linear objects in dimension n (discrete points, m-ﬂats and geometrical simplices). To the best authors knowledge, it is the ﬁrst time that a discrete model is proposed that deﬁnes discrete objects in arbitrary dimensions. The new discrete analytical model proposed is called the standard model and is derived from the supercover model [1,3,4,18,5]. In fact, a standard object is obtained by a rather simple rewriting process of the inequalities deﬁning analytically a supercover object [5]. The name “standard” model derives from the “standard plane” introduced by J. Fran¸con [10]. The standard model is called a discrete analytical model because the discrete objects (points, m-ﬂats, simplices) are deﬁned analytically by a ﬁnite number of inequalities that is independent of the set of discrete points of the object. For instance, a 3D standard triangle is deﬁned by 17 or less inequalities independently of its size. The model we propose has many interesting properties. It has been shown that the standard model is in fact a 0-discretisation of Brimkov, Andres and Barneva [6,7] and therefore is (n − 1)-connected and tunnel-free. In our notation, in 3D, our 2-connectivity corresponds to the classical 6-connectivity. This means that our model ﬁts particularly well polygonalization approaches such as the one proposed by L. Papier and J. Fran¸con [12] in Khalimski-Kovalesky spaces [15, 10]. The model is by deﬁnition geometrically consistent: for instance, the vertices of a 3D standard polygon are 3D standard points, the edges of a 3D standard polygons are 3D standard line segments and the 3D standard polygon is a piece of a 3D standard plane. In section 2, we start by introducing the main notations of the paper, before brieﬂy recalling the main properties of the supercover model. In section 3, we introduce the orientation convention that forms the basis of the deﬁnition of the standard model before we formally deﬁne the standard model. In section 3.3, the main properties of the standard objects are presented especially the tunnelfreeness and the (n − 1)-connectivity. In Section 3.5, we examine the diﬀerent classes of standard linear objects to see how the deﬁnition is translated in practice and how the diﬀerent inequalities deﬁning the objects are established. We conclude in section 4.

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315

Preliminaries Basic Notations

Most of the following notations correspond to those given by Cohen and Kaufman in [8] and those given by Andres in [5]. We provide only a short recall of these notions. Let Zn be the subset of the nD Euclidean space Rn that consists of all the integer coordinate points. A discrete (resp. Euclidean) point is an element of Zn (resp. Rn ). A discrete (resp. Euclidean) object is a set of discrete (resp. Euclidean) points. A discrete inequality is an inequality with coeﬃcients in R from which we retain only the integer coordinate solutions. A discrete analytical object is a discrete object deﬁned by a ﬁnite set of discrete inequalities. An m-ﬂat is a Euclidean aﬃne subspace of dimension m. Let us consider a set P of m + 1 linearly independent Euclidean points P 0 , . . . , P m . We denote Am (P ) the m-ﬂat induced by P (i.e. the m-ﬂat containing P ). We denote S m (P ) the geometrical simplex of dimension m in Rn induced by P (i.e. the convex hull of P ). For S = S m (P ) a geometrical simplex, n we denote S = Am(P ) the corresponding m-ﬂat. For a n-simplex S = S (P ), i n−1 i we denote E S, P is the half-space of boundary A P \ P that contains P i. We denote pi the i-th coordinate of a point or vector p. Two discrete points p and q are |pi − qi | ≤ 1 for 1 ≤ i ≤ n, and nk-neighbours, with 0 ≤ k ≤ n, if n k ≤ n − i=1 |p − q |. The voxel V (p) ⊂ R of a discrete nD point p is deﬁned i 1 i 1 1 1 by V (p) = p − , p + × · · · × p − , p + . For a discrete object F , 1 1 n n 2 2 2 2 V (F ) = V (p). p∈F

A k-path in a discrete object A is a sequence of discrete points all in A such that consecutive pairs of points are k-neighbours. A discrete object A is k-connected if there is a k-path between two arbitrary points of A. A k-component is a maximal k-connected set. Let D be a subset of a discrete object E. If E \ D is not k-connected then D is said to be k-separating in E. Let E be a k-separating discrete object in Zn such that Zn \ E has exactly two k-components. A k-simple point in E is a discrete point p such that E \ p is k-separating. A k-separating discrete object in Zn is called k-minimal if it does not contain any k-simple points. Let us consider two objects F of dimension n and G of dimension m. The Cartesian product of F and G is deﬁned by F × G = {(f, g) |f ∈ F, g ∈ G }. The Minkowski sum of F and G is deﬁned by F ⊕ G = {f + g |f ∈ F, g ∈ G }. We denote σ n the set of all the permutations of {1, . . . , n}. Let us denote Jnm the set of all the strictly growing sequences of m integers all between 1 and n: Jnm = {j ∈ Zm |1 ≤ j1 < j2 < . . . < jm ≤ n }. This deﬁnes a set of multi-indices. Let us consider an object F in the n-dimensional Euclidean space Rn , with n > 1.

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The orthogonal projection is deﬁned by: πi (F ) = {(q1 , . . . , qi−1 , qi+1 , . . . , qn ) |q ∈ Rn } , for 1 ≤ i ≤ n; πj (F ) = (πj1 ◦ πj2 ◦ · · · ◦ πjm ) (F ) , for j ∈ Jnm . The orthogonal extrusion is deﬁned by: εj (F ) = πj−1 (πj (F )) , for j ∈ Jnm . We deﬁne an axis arrangement application rj , for j ∈ Jnm , by: rj : Rn → Rn x →xσj (1) , xσj (2) , . . . , xσj (n) where the permutation σj ∈ σ n is deﬁned by: for 1 ≤ i ≤ m, σj (ji ) = i. σj = else σj (kr ) = i so that kr < kr+1 and kr = js for all 1 ≤ r ≤ n − m and for all 1 ≤ s ≤ m. The axis arrangement application has been speciﬁcally so that it designed veriﬁes the two following properties: πj (F ) = π(1,2,... ,m) rj−1 (F ) and εj (F ) = rj ε(1,2,... ,m) rj−1 (F ) for all F in Rn and j ∈ Jnm . 2.2

Recalls on Supercover

A discrete object G is a cover of a Euclidean object F if F ⊂ V (G) and ∀p ∈ G, V (p) ∩ F = ∅. The supercover S (F ) of a Euclidean object F is deﬁned by S (F ) = {p ∈ Zn |V (p) ∩ F = ∅ }. S (F ) is by deﬁnition a cover of F . It is easy to see that if G is a cover of F then G ⊂ of F can be deﬁned S(F ). Thesupercover in diﬀerent ways: S (F ) = F ⊕ B ∞ 12 ∩ Zn = p ∈ Z n d∞ (p, F ) ≤ 12 where B ∞ (r) if the ball centered on the origin, of radius r for the distance d∞ . This links the supercover to mathematical morphology [18]. The supercover has many properties. Let us consider two Euclidean objects F and G, and a multi-index j ∈ Jnm , then: S (F ) = α∈F S (α), S (F × G) = S (F ) × S (G), rj (S (F )) = S (rj (F )), πj (S (F )) = S (πj (F )) and εj (S (F )) = S (εj (F )) = rj (Zm × S (πj (F ))) [5]. Deﬁnition 1. (Bubble) A k-bubble, with 1 ≤ k ≤ n, is the supercover of a Euclidean point that has exactly k half-integer coordinates. A half-integer is a real l+ 12 , with l an integer. A k-bubble is formed of 2k discrete points. Deﬁnition 2. (Bubble-free) The cover of an m-ﬂat is said to be bubble-free if it has no k-bubbles for k > m. The cover of a simplex S is said to be bubble-free if S is bubble-free.

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There are two types of bubbles in the supercover of an m-ﬂat F . The k-bubbles, for k ≤ m, are discrete points that are part of all the covers of F . If we remove any of these points, the discrete object isn’t a cover anymore. In the k-bubbles, for k > m, there are discrete points that are “simple” points. The aim of this paper is to propose discrete analytical objects that are bubble-free by removing one of the points as illustrated by the ﬁgure. Lemma 1. A discrete point p belongs to a k-bubble, k > m, of the supercover of an m-ﬂat F if and only if there exists a point α ∈ F with k half-integer coordinates such that p ∈ S (α). The proof of this lemma is obvious.

3

Standard Model

The aim of this paper is to propose a new cover class, called the standard cover. This cover is so far only deﬁned for linear objects in all dimensions. This discrete analytical model has been designed to conserve most of the properties of the supercover, to be bubble-free and (n − 1)-connected. 3.1

Orientation Convention

The standard model, contrary to the supercover, is not unique. It depends on the choice of an orientation convention. We need one orientation convention per dimension Rm , m > 0. This choice must then remain unchanged for all the primitives handled. The choice of an orientation convention per dimension has to be coherent with the operator π. This means that we want the following property to be veriﬁed: St (πj (F )) = πj (St (F )) . If this is not the case, we won’t have correct modeling properties. In general, with arbitrary orientation conventions there is no reason for this property to be veriﬁed. It can sometimes be tricky to ﬁnd a “good” set of orientation conventions. We propose a set of orientation conventions, denoted On and called the basic orientation conventions, that verify the above mentioned property. Deﬁnition 3. (Standard orientation) n Let us consider a discrete analytical half-space E : i=1 Ci Xi ≤ B and the basic orientation convention On . We say that E has a standard orientation if : – C1 > 0; – or if C1 = 0 and C2 > 0; . – ..

– or if C1 = · · · = Cn−1 = 0 and Cn > 0. If E has not a standard orientation then we say that E has a supercover orientation.

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We consider from now on, without loss of generality, only the basic orientation conventions for all n > 0. All the standard primitives are deﬁned with these basic orientation conventions. The basic orientation conventions are coherent with respect to the operators π. After πj , for j ∈ Jnm , the orientation convention On in Rn becomes On−m in Rn−m . 3.2

Standard Model Deﬁnition

We now have gathered all the elements we need to deﬁne the standard discretisation model of linear objects in Rn . Deﬁnition 4. (Standard Model) Let F be a linear Euclidean object in Rn whose supercover is described ann alytically by a ﬁnite set of inequalities Fk : i=1 Ci,k Xi ≤ Bk . The standard model St (F ) of F , for the basic orientation convention On , is the discrete ob ject described analytically by a ﬁnite set of discrete inequalities Fk obtained by substituting each inequality Fk by Fk deﬁned as follows:

– If Fk has a standard orientation then Fk : n – else Fk : i=1 Ci,k Xi ≤ Bk .

n

i=1

Ci,k Xi < Bk ;

This deﬁnition is algorithmically easy to set up. Once a discrete analytical description of an object is available, the transition from the supercover model to the standard model and vice-versa is trivial. 3.3

Properties

We are now going to present the most important properties of standard objects. Let us consider a Euclidean linear object F of topological dimension m in Rn . We have by deﬁnition St (F ) ⊂ S (F ) even more precisely, if p ∈ S (F ) \ St (F ), then d∞ (p, F ) = 12 . A standard object is a supercover object from which we have removed some discrete points. These points are all at a distance 12 from the Euclidean primitive. We have St (F ) = S (F ) if no point with at least m + 1 half-integer coordinates belongs to the boundary of F . The diﬀerences between the supercover of F and the standard model of F are located in the k-bubbles of F , for k > m. One of the immediate consequences of this is that the standard model remains a cover: F ⊂ V (St (F )). It is because of this property that the standard model is also sometimes called standard cover. The standard model retains most of the set properties of the supercover. It is easy to deduce from deﬁnition 4, that if we have two Euclidean linear objects F and G in Rn , then: St (F ∪ G) = St (F ) ∪ St (G) ; St (F ∩ G) ⊂ St (F ) ∩ St (G) F ⊂ G ⇒ St (F ) ⊂ St (G) ; St (F × G) = St (F ) × St (G) ; St (πj (F )) = πj (St (F )) ; St (εj (F )) = εj (St (F )) .

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The ﬁrst property ensures that we’ll be able to construct complex discrete objects out of basic elements such as simplices. These last properties are characteristic of correct orientation conventions. The properties are only veriﬁed if the orientation conventions are deﬁned for all dimensions lower or equal to n and if they are coherent with respect to the operator π. This is the case for the basic orientation conventions Ok , for k ≤ n. It is important to notice that, in general, St (F ) = α∈F St (α). This property of the supercover is not conserved. We have St (F ∪ G) = St (F ) ∪ St (G) for a union of a ﬁnite number of objects. This comes simply from the fact that the standard model is not deﬁned for an analytical description that has an inﬁnite number of discrete inequalities. One simple example for that is given by the 2D line D: x1 − x2 = 0. The standard model of

the line is St (D) : −1 ≤ x1 − x2 < 1 while α∈F St (α) = x ∈ Z2 |x1 − x2 = 0 . One of the main properties of the standard model concerns the connectivity and the tunnel-freeness: Theorem 1. (connectivity and tunnel-freeness) Let F be a Euclidean linear object of topological dimension m in Rn . Its standard model St (F ) is (n − 1)-connected and tunnel-free. The standard model is a particular case of k-discretisations as introduced by Brimkov, Andres and Barneva in [6,7]. It is shown that the standard model is in fact a 0-discretisation (Theorem 3 in [7]) and that 0-discretisations are (n − 1)connected and tunnel-free (proposition 3 in [6] and theorem 4 in [7]). Another property proved in [18,7] is that the standard model minimizes the Hausdorﬀ distance with the Euclidean object. See [18,6,7] for details. 3.4

Description of Standard Primitives

We’ll examine now the discrete analytical description of the diﬀerent classes of standard linear primitives (half-space, point, m-ﬂat and m-simplex) and how they can be computed. Our purpose here is to propose a discretisation scheme that can be used is practical applications. By deﬁnition 4, every analytical description of a standard linear primitive is based on the analytical description of a standard half-space. That is the one we present ﬁrst. We deduce from it the formulas for the standard point, m-ﬂat and m-simplex in the sections that follow. Standard Half-space. The standard half-space is given by : Proposition 1. (Standard half-space) n Let us consider a Euclidean half-space E : i=1 Ci Xi ≤ B. The standard model St (E) of E, according to an orientation convention, is analytically described by:

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– If E has a standard orientation then n

n n i=1 |Ci | St (E) = p ∈ Z Ci pi < B + ; 2 i=1 – else

St (E) =

n

n i=1 |Ci | p∈Z Ci pi ≤ B + . 2 i=1 n

The proposition is an immediate extension to dimension n of results on the supercover [1,3,4,5] and of deﬁnition 4. Standard point. The analytical description of a standard point can easily be deduced from the one of the standard half-space. It is however interesting to notice that the standard discretisation of a Euclidean point is always composed of one and only one discrete point contrary to what happens with a supercover discretization of a Euclidean point that can be formed of 2k points, 0 ≤ k ≤ n. Proposition 2. (Standard point) Let us consider a Euclidean point α ∈ Rn and the basic orientation convention On . The standard model St (α) of α is the discrete point: 1 1 St (α) = α1 − , . . . , αn − 2 2 The proof is obvious. In ﬁgure 1, the cross represents the Euclidean point. The black dot represents the corresponding discrete standard point. The square with the doted lines represent the zone covered by the 4 inequalities corresponding to the analytical description of a standard point.

Fig. 1. Diﬀerent conﬁgurations of 2D standard points.

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Standard m-ﬂat. One of the consequences of the properties St (πj (F )) = πj (St (F )) and St (εj (F )) = εj (St (F )) is that the formulas that lead to the discrete analytical description of a standard m-ﬂat or standard m-simplex are simple transpositions of the formulas that have been established for the supercover [5]. Proposition 3. (Standard m-ﬂat) Let us consider an m-ﬂat F in Rn and the basic orientation conventions Ok , for all k > 0. a) If F is a 0-ﬂat in Rn , we apply proposition 2, b) If F is a (n − 1)-ﬂat, we apply twice proposition 1, c) else the analytical description of the standard model of F is given by: St (F ) = St (εj (F )) = rj (Zm × St (πj (F ))) j∈Jn n−1−m

j∈Jn n−1−m

We reapply then, recursively, proposition 3 on St (πj (F )) for all j ∈ Jnn−1−m . This proposition is composed of several steps corresponding to the algorithm that yields the analytical description of the standard model of an m-ﬂat. Let us discuss step c). The formula St (F ) = St (εj (F )) alone is not suﬃcient j∈Jn n−1−m

to describe the standard m-ﬂat, with 0 < m < n−1, since εj (F ) is not necessarily a hyperplane in Rn . We might even have F = εj (F ) for some j ∈ Jnn−1−m . The way around this problem is to examine πj (F ) in Rm+1 . The new orientation convention for Rm+1 after πj is Om+1 . We have diﬀerent cases that occur: – If πj (F ) is a hyperplane in Rm+1 then εj (F ) is a hyperplane in Rn . We do not actually need to consider πj (F ). We could directly use case b) on St (εj (F )). – If πj (F ) is a point in Rm+1 then we consider case a) in Rm+1 , with the basic orientation convention, and formula rj (Zm × St (πj (F ))). – If πj (F ) is a k-ﬂat, 0 < k < m, in Rm+1 then we consider again case c), with the basic orientation convention. We have, by deﬁnition, St (πj (F )) = St (εj (πj (F ))). We repeat the operation described in case c) for j ∈Jm+1 m−k

πj (πj (F )) in Rk+1 . We know that this process ends since each time we repeat case c) we consider a new object in a space of strictly lower dimension. Standard Simplex. Let us ﬁnish with the formulas describing a standard simplex. These formulas are a direct transposition of the formulas obtained for the supercover [5].

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Fig. 2. Supercover and Standard 3D line.

Proposition 4. (Standard Simplex)

Let us consider a simplex S = P 0 , . . . , P m of dimension m and the basic orientation conventions Ok , for all k > 0. The standard model of S is deﬁned by: n n i a) If m = n then St (S) = St E S, P ∩ St (ε (S)) ; j i=0 j=1 n b) If m = n − 1 then St (S) = St S ∩ j=1 St (εj (S)) ; c) If m ≤ n − 2 then St (S) = St (εj (S)) . j∈Jn n−m−1

Let us just recall some notations. If S is a simplex of dimension m then we denote S the m-ﬂat containing all thepoints deﬁning S. If S is a simplex of dimension n in Rn then we denote E S, P i the half-space that contains P i and of boundary the (n − 1)-ﬂat {P 0 , . . . , P i−1 , P i+1 , . . . , P m }. Figure 3 shows three views of a standard 3D triangle: Figure 3(a) presents a classical, voxel view, of the standard triangle; ﬁgure 3(b) presents the same triangle in a K2 -space representation; ﬁnally ﬁgure 3(c) represents what we have called the analytical view and represents the 17 inequalities describing the standard 3D triangle.

4

Conclusion

We have deﬁned in this paper the standard model for half-spaces, m-ﬂats and m-simplices in dimension n. This is, to the authors best knowledge, one of the ﬁrst times that discrete primitives are described analytically in dimension n. The standard model is geometrically consistent, deﬁned analytically and standard objects are tunnel-free and (n − 1)-connected. It seems to us that the path

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towards discrete polygonalization will be much easier with standard polygons than with other notions and this for several reasons: the standard objects are all topologically consistent. This is not the case for the discrete na¨ive model for instance. Most planar recognition algorithms have been designed to recognize discrete na¨ive plane pieces [9,19]. However, it is not very diﬃcult to show that if the model is geometrical consistent then 3D discrete na¨ive edges of polygons aren’t connected in general [5] and 3D na¨ive vertices might be composed of zero discrete points. This will make it very diﬃcult to perform any polygon reconstruction process. Designing a polygon reconstruction algorithm is the next step towards polygonalization. Indeed, the following steps need to be accomplished in order to perform a discrete polygonalization: ﬁrst we need to decompose the boundary of the discrete object into discrete plane pieces. This part, as we have recalled in the introduction, has been realised by several diﬀerent approaches in the past. Secondly, one needs to describe the plane pieces as discrete polygons. This supposes that we know what a discrete polygon is (our paper) and that we perform some edge and vertex recognition algorithm. This is still an open and somehow diﬃcult question. In order to facilitate the implementation and the test of such polygonalization algorithms, we are developing a discrete modeling tool, called SpaMod (for Spatial Modeler), at the University of Poitiers (France). The standard model, as well as the supercover model, are part of the discrete objects models handled by Spamod. Spamod is still in the preliminary stages of its development, however the images of Figure 3 have been produced with this software.

Fig. 3. Standard triangle with (a) Voxel view (b) K2 -space view (c) Analytical view

Acknowledgments. The images of Figure 3 have been produced in Spamod with help of the algorithms developed by Martine Dexet.

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References 1. E. Andres, C. Sibata and R. Acharya, Supercover 3D Polygon, in: Proc. 6 th Int. Workshop on Discrete Geometry for Computer Imagery, Lyon (France), Lecture Notes in Computer Science, vol. 1176 (Springer, Berlin-Heidelberg, 1996) 237-242. 2. E. Andres, R. Acharya and C. Sibata, Discrete Analytical Hyperplanes, Graphical Models and Image Processing 59-5 (1997) 302-309. 3. E. Andres, Ph. Nehlig and J.Fran¸con, Tunnel-free supercover 3D polygons and polyhedra, in: Proc. Eurographics ’97, Budapest (Hungary), Computer Graphics Forum 16-3 (1997) C3-C13. 4. E. Andres, Ph. Nehlig and J. Fran¸con, Supercover of Straight Lines, Planes and Triangles, in: Proc. 7 th Int. Workshop on Discrete Geometry for Computer Imagery, Montpellier (France), Lecture Notes in Computer Science, vol. 1347 (Springer, Berlin-Heidelberg, 1997) pp. 243-253. 5. E. Andres, Mod´elisation analytique discr`ete d’objets g´eom´etriques, Habilitation (in french), Laboratoire IRCOM-SIC, University of Poitiers (France), 8 Dec. 2000. 6. V.E. Brimkov, E. Andres, R.P. Barneva, Object discretizations in high dimensions and membership recognition, Proc. 9 th Int. Workshop on Discrete Geometry for Computer Imagery, Uppsala (Sweden), Lecture Notes in Computer Science, vol. 1953 (Springer, Berlin-Heidelberg, 2000) pp. 210-221.. 7. V.E. Brimkov, E. Andres, R.P. Barneva, Object discretizations in high dimensions, accepted for publication in Pattern Recognition Letters. 8. D. Cohen and A. Kaufman, Fundamentals of Surface Voxelization, Graphical Models and Image Processing 57-6 (1995). 9. I. Debled-Renesson, J-P. Reveill`es, A linear algorithm for digital plane recognition, 4th int. workshop in discrete geometry for computer imagery, Grenoble (France), sept. 1994. 10. J. Fran¸con, Discrete Combinatorial Surfaces, Graphical Models and Image Processing, vol. 57, Janv. 1995. 11. J. Fran¸con, J-M. Schramm, M. Tajine, Recognizing arithmetic straight lines and plane, 6th int. workshop in discrete geometry for computer imagery, Lyon (France), LNCS n◦ 1176, nov. 1996, pp. 141-150. 12. J. Fran¸con, L. Papier, Polyhedrization of the boundary of a voxel object, in: Proc. 8 th Int. Workshop on Discrete Geometry for Computer Imagery, Marne-laVall´ee (France), Lecture Notes in Computer Science, vol. 1568 (Springer, BerlinHeidelberg, 1999) 425-434. 13. Y. G´erard, Local conﬁgurations of digital hyperplanes, 8th int. Workshop on discrete geometry for computer imagery, Marne-la-Vall´ee (France), LNCS n◦ 1568, mars 1999, pp. 365-374. 14. A. Kaufman, An algorithm for 3D scan conversion of polygons, in: Proc. of Eurographics ’87, Amsterdam, (1987). 15. V. Kovalesky, Digital geometry based on the topology of abstract cell complexes, 3rd Discrete Geometry Conference in Imagery, Strasbourg (France), 1993, pp. 259284. 16. W. Lorensen, H. Cline, Marching Cubes: a high resolution 3d surface construction algorithm. SIGGRAPH’87, Anaheim (USA). Computer Graphics J., vol. 21, n◦ 4, Jul. 1987, pp. 163-169. 17. J-P. R´eveilles, G´eom´etrie Discr`ete, calculs en nombres entiers et algorithmique, State Thesis (in french), D´epartement d’Informatique, Universit´e Louis Pasteur, Strasbourg (France), 1991.

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18. M. Tajine, D. Wagner, C. Ronse, Hausdorﬀ discretizations and its comparison to other discretization schemes, in: Proc. 8 th Int. Workshop on Discrete Geometry for Computer Imagery, Marne-la-Vall´ee (France), Lecture Notes in Computer Science, vol. 1568 (Springer, Berlin-Heidelberg, 1999) 399-410. 19. J. Vittone, J-M. Chassery, (n − m)-cubes and Farey nets for na¨ive planes understanding, 8th int. workshop in discrete geometry for computer imagery, Marne-laVall´ee (France), LNCS n◦ 1568, mars 1999, pp. 76-87.

Visibility in Discrete Geometry: An Application to Discrete Geodesic Paths David Coeurjolly Laboratoire ERIC Universit´e Lumi`ere Lyon 2 5, av. Pierre Mend`es-France 69676 BRON CEDEX [email protected]

Abstract. In this article, we present a discrete deﬁnition of the classical visibility in computational geometry. We present algorithms to compute the set of pixels in a non-convex domain that are visible from a source pixel. Based on these deﬁnitions, we deﬁne discrete geodesic paths in discrete domain with obstacles. This allows us to introduce a new geodesic metric in discrete geometry.

Introduction In discrete geometry, many Euclidean geometric tools are redeﬁned to take into account speciﬁcities of the discrete grid. In this article, we propose a deﬁnition of the classical Euclidean visibility based on discrete objects. The interest is double: on one hand we extend the discrete geometry with a new tool and on the other hand, since this visibility allows us to deﬁne discrete geodesic paths and discrete shortest paths, we have a practical tool needed by many applications in medical imaging or image analysis to estimate geodesic distance in non-convex domains. The visibility deﬁnition we propose is based on classical Discrete Straight Lines (DSL for short). Many algorithms exist for the DSL recognition problem. Some of these approaches are based on chain code analysis [24], on links between the chain code and arithmetical properties of DSL [6,7], on links between the chain code and the feasible region in the dual -or parameter- space [9,15,23] and others on linear programming tools such that Fourier-Motzkin’s algorithm [10]. All these algorithms present a solution either to decide if a given set of pixels is a discrete straight segment (DSS for short) or to segment a discrete curve into DSS, or both. In our case, the problem is quite diﬀerent, we want to decide if there exits a DSS between two pixels in a non-convex domain. We present deﬁnitions and algorithms to compute the set of pixels which are visible from a source. Then, we deﬁne a notion of discrete geodesic path and a metric associated to such path based on this visibility deﬁnition. We also proposed an eﬃcient implementation of the geodesic distance labelling from a source pixel. A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 326–337, 2002. c Springer-Verlag Berlin Heidelberg 2002

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Visibility Notions and Deﬁnitions

Let us denote D a discrete domain, that is a n−connected set of pixels. We ¯ the complement of D, we call this set indiﬀerently the background or denote D the set of obstacles. In the following, we consider D a 8-connected domain. In this domain, we deﬁne the discrete visibility by analogy to the continuous deﬁnition. Deﬁnition 1 (Discrete Visibility) Let s and t be two pixels in D, we deﬁne the discrete visibility as a binary relationship v : D → D such that we have v(s, t) if and only if there exists a 8-connected discrete straight segment from s to t whose pixels belong to D Before introducing the visibility problem in non-convex domain, we recall classical parameter space characterizations of DSL [15,16,23]. If we consider an Euclidean straight line y = αx + β, the digitization of this line using the Grid Intersect Quantization (see [11] for a survey on digitization scheme) is the set of discrete points such that: ∆(α, β) = {(x, y) ∈ Z2 | −

1 1 ≤ αx + β − y < } 2 2

Note that all classical digitization schemes (GIQ, Object Boundary Quantization or Background Boundary Quantization) can be used and such a choice will not interfere in our algorithms. We choose the GIQ scheme because of its symmetry properties. In the parameter space of the previous deﬁnition, we can describe the set of Euclidean straight lines whose the digitization contains a pixel p(x, y): Sp = {(α, β) ∈ R2 | −

1 1 + y ≤ αx + β < + y} 2 2

A pixel in D deﬁnes a strip in the (α, β)-space delimited by two lines L1 : αx + β − y ≥ − 12 and L2 : αx + β − y < 12 . If we want to know if a set of pixels belongs to a DSL, a classical way is to compute the intersection in the (α, β)-space of strips associated to each pixel. If the feasible domain is not empty, it describes all DSL containing the pixels (cf ﬁgure 1 for an example). In the following, we deﬁne the domain S(s, t) associated to pixels s and t the, intersection Ss ∩ St . In order to compute the visibility in non-convex domains, the main idea is to check in the dual space if domains associated to obstacle pixels do not hide the current pixel t from the source s. 1.2

Visibility Domain

Let o denote an obstacle pixel. If we want to describe the set of Euclidean straight lines whose digitizations do not contain o, we also introduce a strip in

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y (3,2)

(0,0)

x

Fig. 1. An example of S(s, t) domain with pixels (0,0) and (3,2), the S(s, t) domain in the parameter space deﬁned by inequations : {β < 1/2, β ≥ −1/2, β < −3α + 5/2, β ≥ −3α + 3/2}.

the parameter space such that the inequations are reversed. Hence, an obstacle o ¯ 1 (o) : αx+β−y < −1/2 and L ¯ 2 (o) : αx+β−y ≥ 1/2. is associated to constraints L If we want to know if this obstacle blocks the visibility from s to t, we just have ¯ 1 (o) ∩ L ¯ 2 (o). If to compute in the (α, β)-space L1 (s) ∩ L2 (s) ∩ L1 (t) ∩ L2 (t) ∩ L this intersection is empty then t is not visible from s. More generally, if we consider a non-convex domain D and a set of obstacle ¯ such that all point abscissas are pixels O = {oi }i=1..n that is a restriction of D between the abscissa of s and the abscissa of t (all other points can be omitted for the visibility problem). We have the lemma: Lemma 1 Let s be the source and t a pixel in D, t is visible from s in D if and only if: ¯ 1 (oi ) ∩ L ¯ 2 (oi ) S(s, t) ∩ L =∅ i=1..n

The proof of this lemma can be deduced by the visibility deﬁnition and by construction of S. Obviously, we do not have to consider all obstacle pixels. We ﬁrst deﬁne: Deﬁnition 2 A pixel o in O is called “blocking pixel” for the visibility problem v(s, t) if: ¯ 1 (o) ∩ L ¯ 2 (o) S(s, t) ∩ L = S(s, t) and the abscissa of o is between the abscissa of s and t. These blocking pixels are those which interfere in the visibility problem. Nonblocking pixels in O can be removed from the v(s, t) test. We can characterize the shape of the domain when a blocking pixel modiﬁes it: Lemma 2 If o is a blocking pixel for the v(s, t) problem, either the domain ¯ 1 (o) ∩ L ¯ 2 (o) is empty or it has only one connected component. S(s, t) ∩ L Proof: we consider the domain S(s, t) and a blocking pixel o such that o, s and t ¯ 1 (o) are not collinear (in that case, the domain is empty). We show that either L ¯ 2 (o) crosses the domain. We have diﬀerent cases (cf ﬁgure 2-a) that induce or L

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two components but the left and the middle cases are excluded because they imply that the abscissa of o denoted xo is not between xs and xt and thus, o is not a blocking pixel according to deﬁnition 2. As the matter of fact, if xo is ¯ 1 (o) is between the slope of L1 (s) and the between xs and xt , then the slope of L slope of L1 (t). By construction of the strips, the vertical distance between L1 ¯ 1 with the and L2 is equal to 1. Hence, in ﬁgure 2-b, the intersection in a of L vertical line going through b implies that b must be outside the interval [a, b] on ¯ 2 is greater than the slope of the edge cb, L ¯2 the vertical line. Since the slope of L ¯ cannot cross the domain. Same idea can be applied when L2 crosses the domain. ¯ 1 (o) ∩ L ¯ 2 (o) Hence, all cases of the ﬁgure 2-a are impossible and thus, S(s, t)) ∩ L has only one connected component. ✷

b’ c b

L 2

a’ a

(a)

L 1

(b)

Fig. 2. a) Diﬀerent cases that induce two connected components, the left case and the middle case are impossible by deﬁnition of blocking points. The third case must be taken into account b) illustratcﬁon of the proof of lemma 2.

¯ 1 (resp. L ¯ 2 ) of an obstacle crosses According to this lemma, if a straight line L ¯ ¯ the domain, the other constraint L2 (resp. L1 ) can be removed for the visibility ¯ 1 crosses the domain is above the problem. Geometrically, an obstacle such that L ¯ 2 crosses the S(s, t) domain Euclidean segment [s, t] and an obstacle such that L is beneath the segment [s, t] (cf ﬁgure 3 for an example). We denote U(s, t) the set of blocking pixels above [s, t] and L(s, t) the set of blocking pixels beneath the segment [s, t]. 1.3

Visibility Algorithm

In this section, we present algorithms that compute the equivalence class associated to the visibility binary relationship of a source s. We propose two algorithms, the ﬁrst one computes the equivalence class with the visibility deﬁnition given above, and the second one introduces a new visibility deﬁnition that is a restriction of the previous one but the associated algorithm complexity justiﬁes this new version of the visibility. The ﬁrst algorithm we propose is a really straightforward computation of the visibility. Indeed, we can use classical linear programming tools to solve the

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L 1 (c)

L 1 (d)

d t

L 2 (a)

c b s

a

L 2 (b) Fig. 3. Visiblity domain associated to a set of blocking pixels. The black feasible region in the parameter space is the visibility domain associated to grey pixels constrained with the black blocking pixels.

linear inequation system given by obstacle constraints. Such tools are for example the Fourier-Motzkin [10] system simpliﬁcation algorithm, the Simplex algorithm or the Megiddo’s algorithm [17]. Note that the complexity of the Megiddo’s algorithm is linear in the number of inequations but the problem comes with the dimension of the system. In our case, the constraint system is in dimension 2 and thus the implementation of the Megiddo’s algorithm is tractable with a complexity bounded by 4n where n is the number of inequations. We consider a source s, a domain D. We label all pixels in D using a breadthﬁrst tracking of the domain using for example the 8-adjacency. During the propagation process, if we meet an obstacle we store its coordinates in a list O. At each pixel visited in the breadth-ﬁrst tracking, we extract from O the set of blocking pixels and we solve the visibility problem using the Megiddo’s algorithm. Straightforward visibility algorithm Input: a domain D and a source s Output: the set of pixels which satisfy v(s, t) Let Q be a FIFO queue Let O be the obstacle list Append last(s,Q) While Q is not empty t:=remove ﬁrst(Q) For each 8-neighbor n of t not labelled closed or visible If n is an obstacle then Append(n,O) else Let B be the set of blocking points of O according to the pixel n ¯ 1 or L ¯ 2 the constraints of each point of B Compute the linear inequation system S with L If M egiddo(S) = ∅ then Label n as visible Append last(n,Q) else Label n as closed //n is not visible and the point is closed endFor endWhile

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If we denote n the number of pixels in D and m the number of obstacles in O, each step in the while loop has got a complexity bounded by O(m). Hence, the global cost of this algorithm is O(nm). Due to the diﬃculties to provide an eﬃcient data structure to propagate blocking points from a point to its neighbors, this algorithm has a quite important complexity and is not eﬃcient for the geodesic computation. Thus, we propose a new deﬁnition of the discrete visibility which is a weak version of the deﬁnition presented above but that leads to an eﬃcient algorithm for the visibility computation and the discrete geodesic problem. Deﬁnition 3 (Weak Discrete Visibility) Let s and t two pixels in D, we deﬁne the weak discrete visibility as a binary relationship v ∗ : D → D such that we have v ∗ (s, t) if and only if there exists an Euclidean straight line going ¯ between s and t. through s and whose digitization contains t and no pixels in D Instead of considering the inequation associated to s, we constraint the set of Euclidean lines to go through s. This new deﬁnition restricts the previous one and make the visibility not be a symmetric binary relationship. However, this deﬁnition allows an eﬃcient data structure for the visibility test. We suppose that all obstacle pixels are sorted by polar angles using s as the origin. Using this data structure and the above deﬁnition, we have the following property. Proposition 1 Given a set of obstacles sorted by polar angles of center s and a point t, we denote u the minimum of U(s, t) and l the maximum of L(s, t). We have: ¯ 1 (u) ∩ L ¯ 2 (l) v ∗ (s, t) ⇔ S ∗ (s, t) ∩ L =∅ where S ∗ denotes the new domain associated to the weak visibility which is now a segment in the parameter space. Hence, instead of considering all blocking pixels, we just have to test two characteristic pixels given by a polar sort. The proof of this property is a straightforward application of the visibility deﬁnition. Note that the polar sort can be done with integer arithmetic. We can present the algorithm associated to this deﬁnition: Weak visibility algorithm Input: a domain D and a source s Output: the set of pixels which satisfy v(s, t) Let Q be a FIFO queue Let O be the obstacle list sorted in a polar trigonometric order of center s Append last(s,Q) While Q is not empty t:=remove ﬁrst(Q) For each 8-neighbor n of t not labelled closed or visible If n is an obstacle then Append sort(n,O) else Let (u, l) be the localization of n in the sorted set O ¯ 1 (u) ∩ L ¯ 2 (l) If S ∗ (s, t) ∩ L = ∅ then Label n as visible Append last(n,Q)

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endFor endWhile

Label n as closed //n is not visible and the point is closed

The visibility test has got a constant time cost and according to the data structure, both localization and obstacle insertion have a cost in O(log(m)). Thus, the global cost of this algorithm is O(nlog(m)). Moreover, the cone (s, u, l) associated to a point t can be propagated for both localization and insertion to reduce the expected complexity of the algorithm that makes this labelling very eﬃcient.

2

Discrete Shortest Path and Discrete Geodesic Metric

Based on these deﬁnitions of the visibility, we can deﬁne discrete shortest paths and discrete geodesic paths. 2.1

Deﬁnition and Previous Works

We ﬁrst remind some classical facts on discrete metrics that approximate the Euclidean one. All discrete metrics are based on: – either a mask approach where elementary steps in the neighborhood graph are weighted in order to approximate the Euclidean distance of these steps. For example, elementary steps of the Manhattan distance (or d4 ) are horizontal or vertical moves weighted to 1, the chess-board distance (or d8 ) also considers diagonal moves weighted to 1. More generally, chamfer metrics ﬁrst list elementary moves and then associate weights to each move (see [1,22] for initial works) ; – or a vector approach that leads to exact Euclidean metric where displacement vector (dx, dy) is stored and then the distance can be exactly computed d = dx2 + dy 2 but the main goal is to design distance map algorithm that only deal with the integer displacements [5,20,4]. For the discrete geodesic problem, the mask based approach leads to eﬃcient algorithms because a weighted graph can be computed from the metric and the adjacency graph of the domain D and thus, classical shortest path algorithms can be applied such as the Dijkstra’s graph search algorithm [19]. In the following, we use the data structure and the implementation of the geodesic mask given by Verwer et al. [21]. The authors describe an bucket sorting implementation of the Dijkstra’s graph search algorithm which leads to a uniform cost algorithm. In [4], Cuisenaire proposes a region growing Euclidean distance transform using the same structures but the bucket are indexed by the square distance dx2 + dy 2 . For all the visible pixels from the source, this algorithm provides a

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good estimation of the Euclidean distance metric. This algorithm is not error-free but we will discuss this point later. In [18,2], Moreau presents an algorithm for the geodesic metric problem based on a discrete arc chain code propagation scheme but some operations to maintain the data structure are expensive. In our case, we use a uniform cost data structure from which we can extract arc chain code but the visibility property is propagated instead of iso-metric points. 2.2

Algorithm

The main idea of our discrete geodesic algorithm is the following: for all pixels which are visible from the source, we do not have any problem to compute their distance because it exists a discrete straight line between the source and these points and thus, we can compute the displacement vector and return dx2 + dy 2 . If a pixel p is not visible, we start a new visibility computation such that p is a new source and each pixel t such that v(p, t) will be labelled by the distance from p to the source plus the distance between p and t. More formally, we have the following purely discrete deﬁnition of a geodesic path in D: Deﬁnition 4 (Discrete Geodesic Path) A discrete geodesic path between a point t and a source s is a sequence of pixels in D denoted {pi }i=0..n+1 with p0 = s and pn+1 = t such that: v(pi , pi+k )

iﬀ

k = {−1, 0, 1}

with i = 1..n

And such that the geodesic distance dgeodes (s, t) is minimal. The geodesic distance is deﬁned by: dgeodes (s, t) =

n

deuc (pi , pi+1 )

i=0

where deuc (a, b) denotes the Euclidean distance between pixels a and b. The discrete geodesic path is thus a 8-connected curve that is segmented into DSS by construction. The metric we associate to this curve have been intensively studied and both the stability and multigrid convergence have been proved [14, 13,3]. In order to design an eﬃcient algorithm based on the Verwer’s bucket structure [21], we consider rounded geodesic distance to index the buckets: a pixel p belongs to the bucket d if and only if: dgeodes (s, p) = d This estimated metric is still consistent for the Verwer’s algorithm (A∗ algorithm) because it satisﬁes the triangular inequality [18,2]: for a, b, c ∈ R

a + b ≥ c ⇒ a + b ≥ c

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For a computational eﬃciency of the algorithm, we implement the v ∗ visibility. Hence, at each pixel p in the buckets d, we associate a data structure that contains: its coordinates, the current source pixel pi such that v(pi , p) and the distance dgeodes (s, pi ). We also have an obstacle data structure associated to each new source. Each obstacle list contains the set of obstacles sorted by polar angles met during the visibility propagation associated to each source. We can know present the discrete geodesic algorithm. Note that some steps of this pseudo-code are not detailed for sake of clarity. Discrete Geodesic Algorithm Input: a domain D, a source s and a goal g Output: the geodesic distance for each pixel of D Let Bucket[i] be an array of FIFO queues Let O[i] be an array of double-linked list of obstacles Let d denotes the current bucket (d:=0) Append last(s,Bucket[d]) While there is no more pixel in buckets If the bucket d is empty then increment(d) t:=remove ﬁrst(Bucket[d]) For each 8-neighbor n of t not labelled closed or visible If n is an obstacle then Add n to the obstacle list associated to the source of p else Let (u, l) be the localization of n in the sorted set O[i] associated to the current source If n is visible then Label n as visible Compute the geodesic distance d of n Append last(n,Bucket[d’]) if d > d else Label n as closed Initialization of new source n whose obstacle list is empty Compute the geodesic distance d of n Append last(n,Bucket[d’]) if d > d endFor endWhile

3

Experiments and Discussions

In our experiments, we compute the geodesic distance labelling of a binary image according to the coordinates of a source. In ﬁgure 4, we present the distance labelling with three metrics: d4 , d8 and dgeodes in various domains. Geodesic distances are represented using a circular gray scale map in order to check the wave front propagations. In ﬁgures 5, instead of labelling the pixels according to their distance, pixels with the same color belong to the same equivalence class for the visibility problem. An illustration of these ﬁgures can be the minimum number of guards needed to control a room and the visibility associated to each guard (the ﬁrst guard is given here). In ﬁgure 6, we present discrete geodesic metric on a blood vessel network. The domain is computed using a segmented angiography image. Using this geodesic distance algorithm, we naturally would like to apply this algorithm to compute the discrete Voronoi diagram or the Euclidean distance

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Fig. 4. From the left column to the right column: the discrete domains and the source point (isolated white pixels), the geodesic labelling using d8 , the geodesic labelling using d4 , the geodesic labelling using dgeodes .

Fig. 5. Global visibility graph: each pixel with the same color are in the same visibility equivalence class, source points of domains are the same of ﬁgure 4.

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Source

Fig. 6. Application of the geodesic labelling in medical imaging: left An angiography image, middle binary image when blood vessels are segmented and right the geodesic labelling.

transform just considering multiple sources. Since this algorithm use a local propagation scheme (as the Cuisenaire’s algorithm [4]), the classical Danielsson’s algorithm errors are not solved in this approach. Hence, this algorithm presents a solution to this problem but errors may occur.

4

Conclusion

In this article, we have presented a discrete deﬁnition of the visibility in classical computational geometry. This deﬁnition is based on well known discrete objects (DSS) and is computed only with integers. Based on this deﬁnition, we have presented several algorithms to solve several problems: if we want to decide if there exist DSS between two pixels, we have a cost linear in the number of obstacle pixels O(m); if we want to label all pixels in a domain visible from a source, we have an algorithm in O(nm). Using the weak visibility deﬁnition, we reduce the complexity of both algorithms respectively to O(log(m)) and O(nlog(m)). We also have presented a deﬁnition of discrete geodesic paths and an algorithm that compute the geodesic distance of each point in the domain according to a source. This article also introduces open problems: is it possible to ﬁnd an eﬃcient data structure for the straightforward visibility algorithm ? How to generalize this approach for 3D domains and for discrete surfaces ? For this last problem, solutions exist in mask based approaches [12,8] but for the proposed method, discrete straight lines in 3D are well studied [3] and thus similar visibility algorithm is expected.

References 1. G. Borgefors. Distance transformations in digital images. Computer Vision, Graphics, and Image Processing, 34(3):344–371, June 1986.

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2. J.P. Braquelaire and P. Moreau. Error free construction of generalized euclidean distance maps and generalized discrete vorono¨i diagrams. Technical report, Universit´e Bordeaux, Laboratoire LaBRI, 1994. 3. D. Coeurjolly, I. Debled-Rennesson, and O. Teytaud. Segmentation and length estimation of 3d discrete curves. In Digital and Image Geometry. to appear, Springer Lecture Notes in Computer Science, 2001. 4. O. Cuisenaire. Distrance Transformations : Fast Algorithms and Applications to Medical Image Processing. PhD thesis, Universit´e Catholique de Louvain, oct 1999. 5. P.E. Danielsson. Euclidean distance mapping. CGIP, 14:227–248, 1980. 6. I. Debled-Rennesson. Etude et reconnaissance des droites et plans discrets. PhD thesis, Th`ese. Universit´e Louis Pasteur, Strasbourg, 1995. 7. I. Debled-Rennesson and J.P. Reveill`es. A linear algorithm for segmentation of digital curves. In International Journal of Pattern Recognition and Artiﬁcial Intelligence, volume 9, pages 635–662, 1995. 8. G. Sanniti di Baja and S. Svensson. Detecting centres of maximal discs. Discrete Geometry for Computer Imagery, pages 443–452, 2000. 9. L. Dorst and A.W.M. Smeulders. Decomposition of discrete curves into piecewise straight segments in linear time. In Contemporary Mathematics, volume 119, 1991. 10. J. Fran¸con, J.M. Schramm, and M. Tajine. Recognizing arithmetic straight lines and planes. Discrete Geometry for Computer Imagery, 1996. 11. A. Jonas and N. Kiryati. Digital representation schemes for 3d curves. Pattern Recognition, 30(11):1803–1816, 1997. 12. N. Kiryati and G. Sz´ekely. Estimating shortest paths and minimal distances on digitized three-dimension surfaces. Pattern Recognition, 26(11):1623–1637, 1993. 13. R. Klette and J. Zunic. Convergence of calculated features in image analysis. Technical Report CITR-TR-52, University of Auckland, 1999. 14. V. Kovalevsky and S. Fuchs. Theoritical and experimental analysis of the accuracy of perimeter estimates. In Robust Computer Vision, pages 218–242, 1992. 15. M. Lindenbaum and A. Bruckstein. On recursive, o(n) partitioning of a digitized curve into digital straigth segments. IEEE Transactions on PatternAnalysis and Machine Intelligence, PAMI-15(9):949–953, september 1993. 16. M. D. McIlroy. A note on discrete representation of lines. Atandt Tech. J., 64(2, Pt. 2):481–490, February 1985. 17. N. Megiddo. Linear programming in linear time when the dimension is ﬁxed. Journal of the ACM, 31(1):114–127, January 1984. 18. P. Moreau. Mod´elisation et g´en´eration de d´egrad´es dans le plan discret. PhD thesis, Universit´e Bordeaux I, 1995. 19. J. Piper and E. Granum. Computing distance transformations in convex and nonconvex domains. Pattern Recognition, 20:599–615, 1987. 20. I. Ragnemalm. Contour processing distance transforms, pages 204–211. World Scientiﬁc, 1990. 21. B. J. H. Verwer, P. W. Verbeek, and S.T Dekker. An eﬃcient uniform cost algorithm applied to distance transforms. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-11(4):425–429, April 1989. 22. B.J.H Verwer. Local distances for distance transformations in two and three dimensions. Pattern Recognition Letters, 12:671–682, november 1991. 23. J. Vittone and J.M. Chassery. Recognition of digital naive planes and polyhedization. In Discrete Geometry for Computer Imagery, number 1953 in Lecture Notes in Computer Science, pages 296–307. Springer, 2000. 24. L.D. Wu. On the chain code of a line. IEEE Trans. Pattern Analysis and Machine Intelligence, 4:347–353, 1982.

Multi-scale Discrete Surfaces Jasmine Burguet and R´emy Malgouyres LLAIC1 - IUT D´epartement Informatique BP 86 63172 AUBIERE CEDEX {Burguet,Remy.Malgouyres}@llaic.u-clermont1.fr

Abstract. In this article, we ﬁrst propose a method to discretize a surface represented by a polyhedron. Then, we deﬁne a data structure used to work on such a discrete surface and that allows us to consider multi-scale discrete surfaces. Then, we explain how to perform easily and quickly boolean set operations on this data structure. Finally, we expose a method to reconstruct the whole surface and we display some results obtained from the boolean set operations. Keywords: discretization, multi-scale, discrete surface, boolean set operations.

1

Introduction

In this article, we work on surfaces of the objects, instead of working directely on the volumes. Indeed, the data sizes are then generally smaller. We explain the contexts in which our work make sense in Geometric Modeling. ([4], [10]). In the ﬁeld of Constructive Solid Geometry (CSG), the principle is to construct complicated objects from elementary ones (like spheres or cubes), using boolean set operations such as union, intersection or set diﬀerence. These complicated objets are not explicitely encoded, but are represented by trees which nodes are boolean set operations and leaves are elementary objects. There is no method to display the objects of CSG that could allow us to perform real-time navigation programs. But such programs (OpenGL for example) are available for surfaces represented by polyhedra. So, it could be interesting to compute explicitely the surfaces of the objects of CSG. In the ﬁeld of Computer Assisted Design (CAD) (see [5]), the aim is to construct automatically real objects with machines, and it also could be useful to use a representation of the surfaces of these objects. The algorithms used to compute the set operations on continuous surfaces encounter problems in some cases, for example when the considered surfaces are tangent. Our idea is to have a discrete step, since the discrete surfaces are naturally adapted to perform boolean set operations. In [1], a complete method to discretize polyhedrons, to compute the boolean set operations on the obtained discrete surfaces and to polyhedrize these ones is discribed. But a multi-scale deﬁnition of the discrete surfaces is necessary. For example, if we want to compute the union of a little nail and a wall, it is interesting to be able to represent A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 338–349, 2002. c Springer-Verlag Berlin Heidelberg 2002

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these two objects with diﬀerent scales: a small one for the nail and a bigger one for the wall. The notion of multi-scale discrete surface is one major originality of our work. The main purpose of this article is ﬁrst to propose a quick method to obtain a discrete multi-scale approximation of a surface made of polyhedrons, then to deﬁne a data structure to represent such discrete surfaces, whatever its size and the discretization level, and ﬁnally to compute the boolean set operations on such data structures. The discretization process and the so-obtained data structure are presented in Section 3. The discretization method is derived from the z-buﬀer algorithm (see [3]). To represent a discrete surface, we use the notion of quad-trees, that is quickly discribed in the ﬁrst Section (see [2], [9]), at diﬀerent levels. Using the structure we deﬁne, we know the surfels orthogonal to a given direction, one of the axes of the coordinate space. In Section 4, we explain the method to compute set operations on the discrete surfaces, and some results are proposed in Section 4.4. Finally, from our representation, we have to construct the missing surfels, i.e. the lateral surfels. The method used is discribed in Section 5.

2 2.1

Basic Notions Classical Notion of Discrete Surface

A voxel v is a point of Z3 . We can represent such a voxel by a cube centered on its coordinates. A discrete 3D object is a set of voxels. We can deﬁne adjacency relation on the voxels: two voxels are said to be 6-adjacent if they share a common face. Let us deﬁne our discrete surfaces. First, a surfel is a pair of 6-adjacent voxels. We can see a surfel as a square shared by two 6-adjacent voxels. Then, the surface of a given object O is the set of surfels {(v1 , v2 )/v1 ∈ O, v2 ∈ O}. We want the discrete surfaces to have the Jordan property. Indeed, this property allows us to deﬁne an object from its surface, and so the suface separates an interior (the object) and an exterior (its complementary). In [6] and [7], the properties that the elements composing the object and the ones of its complementary have to satisfy to have the Jordan property are exposed. In [8], G.T. Herman deals with the needed properties of the elements of the surface of the object. In the following, we work on surfaces without considering the object itself. Furthermore, the surfels composing a given surface will not always have the same width. So, we will use an adapted tool to our multi-scale surfaces: quad-trees. 2.2

Quad-Tree

The quad-trees ([2]) were originally used to represent binary images and this method is based on successive sub-divisions. If an image I is not entirely made of 0’s or entirely made of 1’s, we subdivide I into four quadrants (squares), and so on until each quadrant is only composed of 0’s or 1’s. We represent the successive divisions by a quaternary tree. The root is the whole image, each node

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1

0 0 0 10 0 1 0 1

Fig. 1. Example of the quad-tree of a binary image.

x z O y

Fig. 2. The bounding box and the coordinate space.

represents a quadrant of the 2D image, and the leaves are 1 or 0, depending on the values of the pixels composing the quadrant (see Figure 1). In the following, we will use the same ordering of the children of a node. In our case, we will use quad-trees to represent the 3D space according to one of the coordinate axes. This choice is motivated by the multi-scale character of our discrete surfaces. Moreover, the structures used to represent two surfaces with diﬀerent sizes will be the same.

3

Discretization and Deﬁnition of the Surfaces Data Structures

After the discretization of a polyhedron, the result will be a discrete surface that will be composed of surfels of the same width, as deﬁned in Section 2.1. In the following, we denote by Oxy the plane containing the origin O of the coordinate space and the axes x and y. First, we compute a bounding box B which contains the whole initial polyhedron. In the following, we assume that the width of the box is a power of 2 denoted by CB . Furthermore, the origin of the coordinate space is a vertex of the bounding box and the axes are according to the edges of the box (see Figure 2). We also assume that the width of the surfels composing our obtained discrete surface is a power of 2, with range from 1 to CB . We consider a surface represented by a closed polyhedron P . We want to compute a discrete approximation of P, in other words a set of surfels which is close to P. Let us denote by wP the wanted width of each surfel after the discretization of P . The idea is the following: ﬁrst we build a 2-dimensional array A of lists of ﬂoating points. The dimensions of A are (CB /wP ) × (CB /wP ). Each cell of the array represents a square, of width equal to w, in Oxy . For each i, j such that 0 ≤ i ≤ CB and 0 ≤ j ≤ CB , a list Lij contains the depths z of the intersections of the line deﬁned by x = i, y = j, and the faces of the surface. To compute these lists, we use a method derived from the z-buﬀer algorithm ([3]): we project each face F of the polyhedron on the plane z = 0. Then we use a polygon ﬁlling algorithm: for each projected face Fp , we cover the integer points k = (ki , kj ) contained in Fp and we compute the intersections of the corresponding lines x = ki and y = kj with F , that we put in Lij . Once we have computed the lists of ﬂoating points, we can construct the surfels of the surface, considering integer approximations d of the intersections

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Fig. 3. The surfels of the tree that represents the discrete version of a sphere.

represented by elements of Lij . The value of d depends on whether we enter into the surface or we get out (we use a parity rule), and on the width w of the surfels composing the discretized surface. If we enter, then d = z/w × w, and d = z/w × w if we get out. On Figure 3, the pictures show the result of the surfels obtained by the discretization algorithm. Note that only surfels which are orthogonal to the z axis are computed; this enables us to reconstruct a voxelized volume, still using a parity rule. 3.1

A Quad-Tree of Virtual Surfels

For each discretized surface, we have an array that contains some surfels composing this surface. But our motivation is to work on surfaces that have diﬀerent scales, and to obtain multi-scale surfaces. An array have a static nature and is not adapted. Then we decide to use a better tool for our purpose: quad-trees. First, let us consider the bounding box B of a ﬁxed surface (see above). We build a quad-tree of virtual surfels as follows: the root of the tree represents a face F of B (the one that is contained in Oxy ), we subdivide F to build a quadtree, and we stop the construction of a branch when we reach the size of the surfels of the surface. The leaves of this tree are lists of special structures, the quad-tree of surfels (see Figure 4 and Section 3.2). Note that there are two kinds of quad-trees: quad-trees of virtual surfels and quad-trees of surfels. Considering that the depth of the root is 0, the surfels of the list of a leaf at a depth d in the tree have a size equal to CB /2d , and the position of these surfels according to the plane Oxy depends on the position of the list in the virtual tree. In order to build only useful parts of the tree, we stop the construction of the branches which end with empty lists in A (see Figure 5). Note that the virtual tree of depth n has almost 22×n leaves. In the following, we call virtual tree the tree of virtual surfels. 3.2

Construction of the Lists at the Leaves of the Virtual Tree

Let us consider a virtual tree V . Up to now, from the discretization, each list of the leaves of V must contain surfels which have the same size wP . But in the sequel (after multi-scale boolean set operations for example), it may happen

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F x O

...

z y Bounding box

Corresponding tree :

Corresponding virtual tree

Cracine the root ,

its 4 sons, . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

...

Fig. 4. The quad-tree of virtual surfels. We continue the construction of the tree until the size of the surfels is reached.

z

Fig. 5. The vitual tree according to the bounding box and the object composed of 4 voxels does not contain useless branches.

s= 1 d=3

s= 1 d=4

s= 1 d=3

s= 1 d=4

s= 1 d=3

s= 1 d=4

s= 1 d=3

s= 1 d=4

Fig. 6. Virtual tree and lists of surfels corresponding to the example of Figure 5.

that such a list has to contain surfels with diﬀerent sizes w ≤ wP , with w = wP , wP /2, wP /4, · · · , 1 . So, each leaf is a list of structures composed of a boolean quad-tree of surfels, both with the integer depth according the z axis (see Figure 6). In the following, we simply call these lists lists of surfels. We have just deﬁned a tree data structure which can represent a discretized surface according to a given direction. So, we know all the surfels which are parallel to the plane Oxy . We are going to explain how to perform boolean set operations on surfaces discretized at various scales using such structures, thus obtaining a multi-scale surface.

4

Boolean Set Operations on Tree Surfaces Representation

Before doing boolean set operations on the trees representing the surfaces, we make the assumption that the considered bounding boxes of the surfaces are the same. It is not really a drawback since the virtual tree does not contain useless branches (see section 3.1). Furthermore, the representations of the surfaces follow to the same axis of the coordinate space and we remind that the width of the surfels of the surfaces we consider are always powers of 2. Since our data structure is not translation invariant, we also assume that the diﬀerent objects are correctly located relatively to each other in the bounding boxes.

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The method is more or less the same whatever the considered boolean set operation, so we only explain the process for the union (therefore, we can replace the word union by intersection or diﬀerence in the following). 4.1

Cover of the Virtual Quad-Trees

We denote by A1 and A2 the virtual trees representing two surfaces S1 and S2 . We simultaneously apply a depth-ﬁrst exploration for the two trees. So, during the recover, we always consider the same part of the plane Oxy . We have 3 cases. If none of the trees is a leaf, we continue the cover with the ﬁrst children of A1 and A2 , the second ones and so on. If both of the trees are leaves, we compute the union of the corresponding respective lists L1 and L2 (see Section 4.3). If only one of the trees is a leaf (for example A2 ). Then the result of the union of A1 and A2 is the union of the children of A1 and the corresponding children of the surfels trees of the list of surfels of A2 (if a surfel tree is a node with value v, it is transformed into a tree with four leaves which have value v), and so on. In practice, we replace A1 by a leaf of which the list of surfels is constructed recursively from the lists of the children of A2 (see Figure 7). There remains to compute the union of two lists of the leaves of the virtual quad-trees. First, we introduce a useful tool. 4.2

The Booltrees

In the following, in order to perform the boolean set operations we need to know, during the cover of a list of surfels, if we enter or get out of the surface when we meet a quad-tree of surfels. So, we use a quad-tree that we call booltree and that represents a boolean matrix, for each list of surfels. At the beginning of the cover of a list of surfels, the booltree B is just a node with boolean value equal to 0. When we meet a quad-tree of surfels, for each surfel in the tree (i.e. a leaf with value equal to 1), we update B by reversing the boolean values of the corresponding part of the booltree. Then, we know if a surfel is opening (we enter into the surface), closing (we get out), partly opening or closing. For example, if a part of the booltree passes from 0 to 1, the surfel is opening for this part. Moreover, a booltree indicates the parts of the space which are in the interior or the exterior of the surface: between two quad-trees of surfels with depth d1 and d2 , the parts of the booltree with value equal to 1 are in the interior of the surface between d1 and d2 . For a better understanding, let us consider on Figure 8 the example with the list L. The booltree B is initially a matrix of 0 and is represented by a white square. The ﬁrst tree of surfels t1 is a node with a value equal to 1. Then, all the booltree is reversed and now B is a matrix of 1 (the square is gray). So, we know that the unique surfel in t1 is an opening one (we pass from 0 to 1 in the booltree). The next tree of surfels t2 is not a leaf. So, the booltree is partially transformed according to the parts of the tree that have a value equal to 1. Since these parts become 0, we know that the two corresponding surfels are closing

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d=1 d=2 d=3 Initial empty booltree

After the last tree

Fig. 8. Evolution of a booltree during the cover of a list.

ones. The last tree t3 contains two closing surfels. Since there are no more trees in L, the booltree is made of 0 again. We are ready to perform the boolean set operations on lists of surfels. 4.3

Boolean Set Operations on Two Lists of Surfels

Let us consider two non-empty lists of surfels L1 and L2 , in the same level in the virtual surfels tree. If one of the two lists is null, the result of the union is the other list. Initially, for the two lists, the respective booltrees B1 and B2 of L1 and L2 are completely made of 0. Let us consider the ﬁrst elements e1 of depth d1 and e2 of depth d2 of L1 and L2 . We also construct a booltree B representing L = L1 ∪ L2 , initially made of 0. There are three diﬀerent cases. First, if d1 < d2 , we update B1 using e1 . Then, we compute the union B∪ of B1 and B2 . If B is not equal to B∪ , for each part of B which is diﬀerent to B∪ , we have a surfel of L. Therefore, we can construct the quad-tree of surfels at the depth d1 . If B∪ = B, we do not construct a quad-tree. Then, B is now equal to B∪ , L1 becomes the next element of L1 , and we do the process again with the new values. The case d2 < d1 is similar. If d1 = d2 , we update B1 using e1 and B2 using e2 , we compute the union B∪ of B1 and B2 , we compare B∪ and B, and we construct the quad-tree of surfels, if it exists, at the depth d1 . Then, we update B with B∪ , L1 and L2 , and we do the process again with the new values. This procedure ends when L1 and L2 are both null. If at one step, one of the lists becomes null, the result list is equal to the other one. As an example, we will consider the union of the objects O1 and O2 of Figure 9. The lists L1 and L2 correspond to the objects and are at the same place in

Multi-scale Discrete Surfaces O1

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Fig. 9. Union of two objects represented by their lists of surfels using the corresponding booltrees.

the respective virtual quad-trees representing the objects. In this Figure are represented the evolution of the booltrees B1 and B2 , according to the depths, and the union of the booltrees B at each depth. Initially, the booltrees are white. At depth 2, we meet the ﬁrst surfel of L1 . The booltree B1 becomes grey, B2 remains the same (since there is no surfel at this depth in L2 ), and so B, which is the union of B1 and B2 at depth 2, becomes grey. From the comparison of the initial B and B at depth 2, we conclude that we have to construct a surfel of L of which the quad-tree is a node with value 1 and d = 2. At depth 3, we meet a surfel of L2 , and we consequently update B2 . The booltree B1 is unchanged. The union of B1 and B2 at depth 3 is equal to B at depth 2, so we do not have to create a surfel at this level. At depth 4, B1 becomes empty and B2 is unchanged. The comparison of the union of these booltrees and the last B shows that we have to create 3 surfels corresponding to the places of B that have changed at level 4. Finaly, at depth 5, B1 and B2 becomes white and so we have to build the last surfel, as shown on Figure 9. Then, we have obtained the list L which is the union of L1 and L2 . 4.4

Results of the Boolean Set Operations

Figure 10 shows an example of the result of the set operations. The surfels of the lists, which are orthogonal to the z axis, are shown. The bounding box has a width equal to 32, and this is the result of the diﬀerence between the surface of a voxel of width 16 and a sphere made of surfels of width 1.

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Fig. 10. Surfels of the lists of the leaves of a virtual tree. This object has been obtained by the diﬀerence of a voxel of width 16 and a sphere composed of voxels of width 1.

As we can see, the upper face of the cube has been subdivided into smaller surfels, but the lower one is unchanged.

5

Reconstruction of the Whole Surface

To build and to display the whole surface from the quad-tree structure V , we once again use booltrees. Note that there exist more than one possibility to reconstruct the surface depending on the chosen size of surfels. First, there are four orientations of the lateral surfels, one by edge of the surfels of the lists of surfels. We denote these ones by N for North, S for south, E for east and W for west. We explain the way we use to build the N surfels, and it is similar for the other orientations. First, we easily construct an array CB × CB denoted by A. Each cell of A contains a list of integers that is constructed from the recover of each list of surfels of V . They represent the intervals of the space which are in the interior of the surface and are obtained from the booltrees of the lists. If we consider a list of surfels L of width 1, and if the list at the north of L contains a list composed of 2 elements, 0 and (CB − 1), then the whole north of L is in the interior of the surface and there are no N surfel to construct for L. We construct the N surfels from A and the virtual tree, in such a way that the size of all surfels is as large as possible for the considered list (see the right part of Figure 11). To do this, we treat the lists at the leaves of V one by one. Let us consider a list of surfels L and the corresponding booltree B. Let us denote by Cmax the maximal width of the surfels of L (depending on the place of L in V ), by e the ﬁrst element of L and by d the depth of e. We consider L by successive intervals of length Cmax , and we begin at the depth p = d/Cmax × d. So, the ﬁrst interval we consider is I = [p, p + Cmax ]. We denote by w the width of the surfels we consider. Initially, w = Cmax . Let us introduce the procedure. If L is empty or contains only one element, we stop here the process for the current list. If d = p, we update B using e, L becomes the element after e in L, and d is the depth of the new ﬁrst element of L. We consider several cases for I. If, by considering A, the whole north of L is not completely full from p to p + w (i.e. is not in the interior of the surface)

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if all the following properties are satisﬁed: – B is not a leaf, – in the successive transformations of B using the possible elements of L that have a depth from p to p + w − 1, in B the north parts are always true, – the north of the list is completely free (by considering A), then we create a N lateral surfel with width w. Else if one of the following cases is true: 1. d < p + w, 2. B is not a leaf, 3. the north of L is not completely full or completely free, then, we subdivide. By subdividing, we mean that we do the process again for each n ∈ {1, 2, 3, 4} and: w = w/2, I = [p + i, p + i + w], with i = 0 and i = w, the nth child of B and the nth children of the elements of L. Indeed, if we verify one of these hypotheses, we can not put a surfel of the initial width w, and we try to put surfels with inferior widths. Else if B is a leaf. If the value of B is true and the north of the list is completely free (by considering A), then we create a N lateral surfel with width w. We update B thanks to the elements of L which have a depth strictly less than p + w, we consequently update L, and we continue with the next interval. Let us consider, on Figure 11, the reconstruction of the foreground faces of the object represented by the list of surfels L, with Cmax = 4. Since the second element of the list is at the depth 3, we cannot put a surfel of size 4. Then we subdivide L into 4 lists, l1 , l2 , l3 and l4 , numerated like indicated on the Figure. For each sub-list, Cmax = 2. From l1 and l3 , we can put any surfel because the foreground of these lists are hidden by respectively l2 and l4 . The list l2 is composed of 3 elements, at depths 0, 3 and 4. We update the initial booltree using the ﬁrst surfel and the booltree is now made of 1. We also update l2 . We can put a foreground surfel between 0 and 2 because the foreground side is completely free. After that we consider the next interval, between 2 and 4. The ﬁrst element of l2 is at depth 3, so we cannot put a surfel of width 2. Then we subdivide again, and we consider the sub-lists of l2 ll1 , ll2 , ll3 and ll4 . By considering ll1 , we put only a surfel of width 1 between 3 and 4, and from the lists ll2 and ll4 , we put three surfels of width 1 between 2 and 4. From the list l4 , we put two foreground surfels of width 2.

6

Final Results

Figure 12 shows the result of the reconstruction for the previous example of Figure 10.

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3

foreground faces

d=4 d=0

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Fig. 11. Reconstruction of the foreground faces from a list of surfels.

Fig. 12. Final result of the diﬀerence between a big voxel and a sphere, after the reconstruction of the lateral faces.

On Figure 13, we can see the result obtained by the diﬀerence between a wall of surfels of width 8 and a thorus made of surfels of width 1 in a bounding box of width 64. As we can see, only the surfels of the wall “in contact” with the thorus are subdivided in smaller surfels. Moreover, for each list composing the surface, we put the surfels with the maximal possible width. This allows us not to subdivide too much the bigger surfels of the surfaces.

7

Conclusion and Perspectives

In this paper, we presented a method to discretize surfaces represented by polyhedra. This method is easy to implement and is derived from the z-buﬀer algorithm. Next, we proposed a structure to represent the surfaces obtained by the discretization method. The quad-tree nature of this structure allows us to represent multi-scale surfaces. Then, we designed a process to perform easily boolean set operations of these tree structures and using booltree. From the discrete nature of the surfaces, there exists no class of surfaces that causes diﬃculties. Furthermore, since we consider only the surfaces of the objects, the data sizes are quite small and we can compute the set operations very quickly. We also proposed a method to restitute the whole surface, and some experimental results were shown. There are many perspectives to this work. Indeed, we want to be able to polyhedrize the obtained discrete multi-scale surface. As in the case of polyhedrization of a discrete surface presented in [1], we have to construct a graph of surfels to represent the surface. So, from the structure that represents a multiscale surface, we will construct the corresponding multi-scale graph of surface.

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Fig. 13. Result of the diﬀerence between a wall and a thorus. Some surfels of the wall have been subdivided into smaller ones.

We also will deﬁne a topology over the multi-scale surfaces. Then, we should be able to deﬁne a polyhedrization method analogue to the one exposed in [1]. Then, we will have a complete method to perform the boolean set operations on surfaces represented by polyhedrons, using a discrete step. After that, we will be able to use programs to perform a real-time navigation on the ﬁnal polyhedrized surfaces.

References 1. J. BURGUET, R. MALGOUYRES, Strong Thinning and Polyhedric Approximation of a Discrete Surface, Proceedings of DGCI’2000, Uppsala, Sweden, Lecture Notes in Computer Science, vol 1953, Springer, pp222-234, 2000. 2. R.A. FINKEL, J.L. BENTLEY, Quad trees: A data structure for retrieval on composite keys, Acta Informatica, Vol. 4, pp. 1-9, 1974. 3. J.D. FOLEY, A. VAN DAM, S.K. FEINER, J.F. HUGHES, Computer Graphics: introduction and practice (second edition in C), Addison-Wesley. 4. R.A. GOLDSTEIN, R.NAGEL, 3-D Visual Simulation, Simulation, 16(1), pp. 2531, 1971. 5. T. HENDERSON, C. HANSEN, CAD-Based Computer Vision, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(10), pp. 1181-1193, 1989. 6. G.T. HERMAN, Discrete Multidimensional Jordan Surfaces, CVGIP: Graphical Models and Image Processing, 55(5), pp. 507-515, 1992. 7. G.T. HERMAN, Boundaries in Digital Spaces: Basic Theory, in L.N. KANAL and A. ROSENFELD Ed., Topological Algorithms for Digital Image Processing, Vol. 19 of Machine Intelligence an Pattern Recognition, pp. 233-262, 1996. 8. G.T. HERMAN, Finitary 1-Simply Connected Digital Spaces, GMIP, 1(60), 1998. 9. A. KLINGER, C.R. DYER, Experiments on picture representation using regular decomposition, Computer Graphics and Image Processing, Vol. 5, pp. 68-105, 1976. 10. A. RAPPOPORT, S. SPITZ, Interactive Boolean Operations for Conceptual Design of 3-D Solids SIGGRAPH 97 Conference Proceedings, Annual Conference Series, pp. 269-278, 1997.

Invertible Minkowski Sum of Polygons Kokichi Sugihara Department of Mathematical Informatics, University of Tokyo Hongo, Bunkyo-ku, Tokyo 113-0033, Japan [email protected]

Abstract. The paper gives a new formulation of the Minkowski sum of polygons. In the conventional Minkowski sum, the inverse operation is not well deﬁned unless the polygons are restricted to be convex. In the proposed formulation, on the other hand, the set of polygons is extended to the set of “hyperpolygons” and the Minkowski sum forms a commutative group. Consequently, every polygon has its unique inverse, and the sum and the inverse operations can be taken freely. An example of a physical interpretation of the hyperpolygon is also given.

1

Introduction

The Minkowski sum is one of the most fundamental operations for geometric objects, and has a wide range of applications in data compression [4], generation of collision-free paths [10], and others. The Minkowski sum is also useful for constructing eﬃcient geometric algorithms [6]. Algorithms for computing the Minkowski sum have also been studied extensively; they include algorithms for restricted classes of polygons [8,13], for general polygons [6,12], and for higher dimensional ﬁgures [1,4,9]. However, there is a big open problem on the “inverse” of the Minkowski sum, because no clear deﬁnition of the inverse is known yet for a general class of ﬁgures. An inverse-like operation called the Minkowski diﬀerence is known, but it behaves as the inverse only for strongly restricted cases such as the case of convex ﬁgures [3]. Guibas et al. [6] studied objects which are obtained by the inverselike operation and pointed out that the boundaries of those objects change their directions partially. Ghosh [3,4] tried to deﬁne the inverse for nonconvex ﬁgures, but his deﬁnition is valid for a certain restricted class only. Sugihara et al. [14] solved this problem partially. They introduced a class of closed curves whose tangent direction is continuous and monotone, and deﬁned an algebraic operation in this class. This operation is a generalization of the Minkowski sum in that if we replace these closed curves with the regions bounded by the curves, the operation coincides with the Minkowski sum. Moreover, the operation is invertible in this class, and thus the inverse of the Minkowski sum is deﬁned for a much wider class than the class of convex ﬁgures. In this paper, we apply a similar idea to polygons, and construct an algebraic system in which a generalization of the Minkowski sum and its inverse is well deﬁned. In this new system, the set of polygons is extended to a set of more A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 350–359, 2002. c Springer-Verlag Berlin Heidelberg 2002

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general objects, which we call “hyperpolygons”. The extension from polygons to hyperpolygons has the same structure as the extension from integers to rationals. After we review conventional Minkowski algebra in Section 2, we introduce a new representation of the boundary of polygons in Section 3. Next, we deﬁne the new Minkowski sum on this representation in Section 4, and extend it to a larger invertible world in Section 5. In Section 6, we give an example of an interpretation of a physical meaning of a hyperpolygon, and make some concluding remarks in Section 7.

2

Conventional Minkowski Sum

We ﬁx an (x, y) Cartesian coordinate system Σ, and represent a point by its radial vector with respect to the origin of Σ. We call a set of points a ﬁgure. For two ﬁgures A and B, the new ﬁgure deﬁned by A ⊕ B = {a + b | a ∈ A, b ∈ B}

(1)

is called the Minkowski sum of A and B, where a + b represents the sum of the radial vectors a and b. Let A and B be the convex ﬁgures in Fig. 1(a) and (b). Then, the Minkowski sum of A and B is the ﬁgure C shown in Fig. 1(c). The Minkowski sum can be intuitively understood in the following way. Consider the ﬁgures A and B in Fig. 1(a) and (b) again. Let us choose and ﬁx an arbitrary point in B, as the dotted point in Fig. 1(b), and call it the reference point. As shown in Fig. 1(d), the Minkowski sum is the union of the ﬁgure A and the region swept by B when the reference point of B moves along the boundary of A. Next suppose that we are given two ﬁgures B and C. We want to ﬁnd the ﬁgure X that satisﬁes X ⊕ B = C. (2) This is the inverse problem of the Minkowski sum. The inverse problem can be partially solved by another operation called the Minkowski diﬀerence. The Minkowski diﬀerence of two ﬁgures C and B, denoted C B, is deﬁned as C B = {c − b | c ∈ C}. (3) b∈B

For the ﬁgures B and C in Fig. 1(b) and (c), respectively, C B coincides with the ﬁgure in Fig. 1(a), and hence in this particular case, the solution of eq. (2) is given by X = C B. However, the Minkowski diﬀerence does not give the complete solution of eq. (2). Actually the solution of eq. (2) is not necessarily unique. For example, let A be the ﬁgure in Fig. 1 (e). Then, we get A ⊕ B = C, and hence X = A is also a solution of eq. (2). Furthermore, eq. (2) does not necessarily admit a solution.

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Fig. 1. Minkowski sum of two polygons.

3

Angle-Parametric Polygonal Curves

While the conventional Minkowski sum is deﬁned on polygons, we consider a class of closed polygonal curves, a typical example of which is the boundary curve of a polygon. Let P be a closed polygonal curve with n vertices v0 , v1 , · · · , vn−1 , vn = v0 and n edges e1 , e2 , · · · , en such that edge ei is a line segment connecting vi−1 to vi . We consider that the edge ei has the direction from vi−1 to vi . Examples of closed polygonal curves are shown in Fig. 2. The polygonal curve in Fig. 2(a) is considered the boundary of a polygon surrounding counterclockwise, while the one in Fig. 2(b) has a self-intersection and hence does not correspond to a polygon in an ordinary sense. 0- 3 e3 0- 4

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Fig. 2. Polygon.

For our convenience of the notation, let us assume that we travel along P around in either direction endlessly, and name the vertices and the edges accordingly, i.e., vi+kn = vi and ei+kn = ei for k = ±1, ±2, ±3, · · ·. For each i = 0, ±1, ±2, · · ·, let φi be the angle required when we turn from the direction of ei to the direction of ei+1 counterclockwise. We call φi the left-turn

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angle at vi . Since we measure the angle counterclockwise, we have 0 < φi < π if P bends to the left at vi as in Fig. 3(a), while we have π < φi < 2π if P bends to the right at vi as in Fig. 3(b). We call the former vertex a convex vertex and the latter vertex a reﬂex vertex.

o/ i

o/i

v e

i +1

vi e

ei

vi

ei

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v (a)

i+1

(b)

Fig. 3. Left-turn angle

Let us choose and ﬁx an arbitrary edge, say es , and let θs be the angle of es with respect to the positive x direction measured counterclockwise with the convention that 0 ≤ θs < 2π. Next, for each i = 0, ±1, ±2, · · ·, we deﬁne θs + φs + φs+1 + · · · + φi−1 if i > s, θi = (4) θs − φs−1 − φs−2 − · · · − φi if i < s. In other words, θi basically represents the direction of the edge ei , but the ambiguity of a multiple of 2π is resolved by accumulating the left-turn angles starting with θs , as shown in Fig. 2(a). For θ ∈ R, let f (θ) be the set of points deﬁned by {vi } if θi < θ < θi+1 , f (θ) = (5) {tvi−1 + (1 − t)vi | 0 < t < 1} if θ = θi . That is, if θ = θi , f (θ) represents the set of points on the open line segment ei , and otherwise f (θ) represents the set of a single point representing the vertex vi satisfying θi < θ < θi+1 . We call f (θ) the angle-parametric polygonal curve. The function f (θ) is periodic. If P is the counterclockwise boundary of a convex polygon, the total left turn we make while we travel along P around is equal to 2π, and hence f (θ) = f (θ + 2π) (6) On the other hand, if P is the counterclockwise boundary of a polygon with k reﬂex vertices, the total left turn we make while we travel along P around is equal to (k + 1)2π, and hence we get f (θ) = f (θ + (k + 1)2π).

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We call the smallest positive integer m satisfying f (θ) = f (θ + 2mπ) the winding number of the polygonal curve. If P is the boundary of a convex polygon, its angle-parametric representation f (θ) is unique. However, for a general polygonal curve P, the angle parametric representation f (θ) depends on the choice of the starting edge es . For example, for the polygonal curve in Fig. 2(a), if we choose e1 , e2 , e3 , or e4 as es , we get the same angle-parametric representation, while if we choose e5 or e6 as es , the corresponding angle-parametric representation is diﬀerent from the former. In general, two edges ei and ei+k (k > 0) give the same angle-parametric representation if and only if we do not face the positive x direction when we turn from the direction of ei to the direction of ei+k counterclockwise. Hence, if the angleparametric representation f (θ) of the closed polygonal curve P has the winding number m, there are m diﬀerent angle-parametric representations for P. We consider them diﬀerent angle-parametric polygonal curves. In order to distinguish between the closed polygonal curve P and its angle-parametric representation f (θ), we call P the geometric polygonal curve. In what follows, f (θ) represents the set of points at the particular value θ of the parameter, and the function itself is represented by f or {f (θ)}. Let Π + be the set of all angle-parametric polygonal curves. Next, we introduce a binary operation to Π + , which corresponds the Minkowski sum.

4

New Minkowski Sum

For {f (θ)}, {g(θ)} ∈ Π + , we introduce a new operation ⊕ in the following way. First, for each value of the parameter θ, we deﬁne f (θ) ⊕ g(θ) = {a + b | a ∈ f (θ), b ∈ g(θ)},

(8)

where a + b denotes the sum of the radial vectors a and b. That is, if both f (θ) and g(θ) consist of a single point, f (θ)⊕g(θ) also consists of a single point; if one of f (θ) and g(θ) consists of a point and the other is an edge, f (θ) ⊕ g(θ) is the line segment obtained by translating the edge by the radial vector representing the point; if both f (θ) and g(θ) are edges, f (θ) ⊕ g(θ) is an open line segment connecting the sum of the start points to the sum of the end points. For the convenience in the subsequent discussion, we introduce the following notation. For each value of θ, we consider f (θ) the line segment; that is, if f (θ) is a point, we consider it the line segment with length 0. Let f s (θ) and f e (θ) be the start point and the end point of the line segment f (θ). Hence, if f (θ) is a point, we have f s (θ) = f e (θ). Next, we deﬁne {f (θ)} ⊕ {g(θ)} as {f (θ)} ⊕ {g(θ)} = {f (θ) ⊕ g(θ)},

(9)

that is, {f (θ)} ⊕ {g(θ)} is a function of the angle parameter θ, and its value at each θ is speciﬁed by eq. (8). As θ increases, almost everywhere {f (θ)} ⊕ {g(θ)} stays a constant singleton representing a vertex, and at some discrete values of

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θ it represents an edge connecting the vertex immediately before and the vertex immediately after. Moreover, {f (θ)} ⊕ {g(θ)} is periodic. Indeed its winding number is the least common multiple of the winding numbers of {f (θ)} and {g(θ)}. Hence, {f (θ)} ⊕ {g(θ)} also belongs to Π + . We call {f (θ)} ⊕ {g(θ)} the Minkowski sum of {f (θ)} and {g(θ)}. For example, let {f (θ)} and {g(θ)} be the angle-parametric representations of the polygon A in Fig. 1(a) and the polygon B in Fig. 1(b), respectively. Then, {f (θ)} ⊕ {g(θ)} coincides with the angle-parametric representation of the polygon C in Fig. 1(c). Fig. 4 shows another example. The polygonal curves in Fig. 4(a) and (b) both have winding number 2, and hence they have two diﬀerent angle-parametric representations. Let f1 and f2 be the two angle-parametric representations of the polygonal curve in Fig. 4(a), and g1 and g2 be the two angle-parametric representations of the polygonal curve in Fig. 4(b). Since f1 ⊕ g1 = f2 ⊕ g2 and f1 ⊕ g2 = f2 ⊕ g1 , there are essentially two diﬀerent Minkowski sums. They are shown in Fig. 4(c) and (d).

(a)

(b)

(c)

(d)

Fig. 4. Minkowski sums of two polygons with winding number 2.

In this case also we can consider the polygons bounded by the outermost part of the polygonal curves. The conventional Minkowski sum of the polygons in Fig. 4(a) and (b) corresponds to the union of the polygons in Fig. 4(c) and (d).

5

Extension to the Invertible World

As shown in eqs. (8) and (9), the Minkowski sum is deﬁned in terms of the sum of radial vectors. Consequently, the algebraic structure of the vector space is inherited in Π + . In particular, the following hold for any f, g, h ∈ Π + . Property 1. f ⊕ g = g ⊕ f (commutativity). Property 2. (f ⊕ g) ⊕ h = f ⊕ (g ⊕ h) (associativity). Property 3. f ⊕ g = f ⊕ h implies g = h (injectivity).

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There is a big theorem in algebra which says that, if Properties 1, 2 and 3 are satisﬁed, the set Π + can be augmented in such a way that all the elements have their inverses in it [2]. A well-known example is the extension of the set of natural numbers with the multiplication operation to the set of positive rational numbers. More precisely, we can augment Π + to a larger set Π in the following way. Let Π + × Π + be the set of all ordered pairs of the elements of Π + . For any (f1 , f2 ), (g1 , g2 ) ∈ Π + × Π + , we deﬁne (f1 , f2 ) ⊕ (g1 , g2 ) by (f1 , f2 ) ⊕ (g1 , g2 ) = (f1 ⊕ g1 , f2 ⊕ g2 ).

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[In the case of the natural numbers, eq. (10) corresponds to (f1 /f2 ) · (g1 /g2 ) = (f1 · g1 )/(f2 · g2 ).] Next, we write (f1 , f2 ) ∼ (g1 , g2 ) if and only if f1 ⊕ g2 = f2 ⊕ g1 . [In the case of the natural nubmers, f1 /f2 ∼ g1 /g2 if and only if f1 · g2 = f2 · g1 .] Then, the binary relation ∼ is an equivalence relation in Π + × Π + . Finally, we deﬁne Π as the set all equivalence classes, that is, the quotient of Π + × Π + by ∼: Π = Π + × Π +/ ∼ .

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[In the case of the natural numbers, f1 /f2 is considered a rational number, and f1 /f2 and g1 /g2 are considered identical if f1 /f2 ∼ g1 /g2 .] Elements of our new world Π can be considered more concretely in the following way. Let e(θ) be the origin for any θ. Then {e(θ)} (or e for short) is considered the “angle-parametric closed curve” consisting of the single point at the origin. Hence es (θ) = 0 and ee (θ) = 0 for any θ. e acts as the unit in Π, because for any f ∈ Π + , we get f (θ) ⊕ e(θ) is the line segment from f s (θ) + es (θ) = f s (θ) to f e (θ) + ee (θ) = f e (θ). For any f ∈ Π + , the inverse f −1 of f in Π is considered the angle-parametric closed curve for which the line segment f −1 (θ) at each value of θ starts at −f s (θ) and ends at −f e (θ). In other words, we get {f (θ)}−1 = {−f (θ)}. More generally, for any f, g ∈ Π + , the equivalence class containing (f, g) can be considered the angle-parametric closed curve whose value at each θ is the line segment from f s (θ) − g s (θ) to f e (θ) − g e (θ). Namely, the equivalence class containing (f, g) can be written {f (θ) − g(θ)}. The algebraic structure of the extension from Π + to Π is the same as that of the extension from the natural numbers to the positive rationals. Similar extension was also applied when the world of functions was augmented to the world of hyperfunctions [11]. From that analogy, we call elements of Π − Π + hypercurves or hyperpolygons.

6

Interpretation of Hyper Curves

Let us consider a numerically controlled polish machine as shown in Fig. 5. In this ﬁgure, A is a rail, B is a support plate, C is an arm, D is a polish disc, and E, F , G constitute the material to be polished. The upper part of Fig. 5 is the

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A

P

G

E

F

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C

A

Fig. 5. Numerically controlled polish machine.

top view (the projection on the xy plane) while the lower part is the side view (the projection on the xz plane). The rail A is ﬁxed to the outside world, on which the support plate B slides in the y direction. The arm C is held by B and can slide in the x direction. The polish disc is attached to the arm at P and can rotate around P. The material consists of mutually parallel two plates E and F , and they are connected by a small rod G. The material is ﬁxed to the outside world, and the upper surface of the plate F is to be polished by D. Note that the arm C cannot enter the material region E, because C and E are on the same z level, as shown in the side view of Fig. 5. If the arm C enters the material region E, they collide and broke each other. Hence the arm C can move only outside E. We are interested in which part of the surface of E can be polished by this machine. To answer this question, we usually apply the Minkowski algebra in the following way. Let R(C) represent the polygon obtained when we rotate the polygon C by 180 degrees around the point P, and let {fR(C) (θ)}, {fD (θ)} and {fE (θ)} be the angle-parametric polygonal curves representing the shapes of R(C), D and E, respectively. We assume that the point P is chosen as the origin of the coordinate system to represent fR(C) and fD . Then the Minkowski sum g = fE ⊕ fR(C) represents the region which the reference point P of the arm C cannot enter. Next, let {h(θ)} be the Minkowski diﬀerence; h is expressed by −1 −1 h = g ⊕ fD = (fE ⊕ fR(C) ) ⊕ fD .

(12)

The region represented by h corresponds to the region which the polish disc cannot reach. Hence, the intersection of this region with the material region E is the region that cannot be polished by this machine. This is the conventional

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way of ﬁnding the unpolishable region of the material surface (although we used our new notations). Note that the above computation can be done only after we are given the shape of the material E. In our new algebra, on the other hand, we can compute the Minkowski sums in any order, because the algebra forms a group and the uniqueness of the ﬁnal result is guaranteed. Now, let us rewrite eq. (12) into −1 h = (fR(C) ⊕ fD ) ⊕ fE .

(13)

Let {w(θ)} denote the ﬁrst part of the right-hand side of the equation: w = −1 fR(C) ⊕ fD . w is a hypercurve. If we compute according to eq. (12), the intermediate result g and the ﬁnal result h are both in Π + . On the other hand, if we compute according to eq. (13), the intermediate result w goes out from Π + , and hence the computation becomes invalid in the conventional algebra. This diﬀerence is important, because w can be computed even if we are not given the actual shape of the material. Actually, w depends only on the polish machine, and hence we can interpret that the hypercurve w represents the ability of the polish machine. This example shows the following two important points. First, the hypercurve has its own physical meaning, just as the “ability of the polish machine” in the above example. Secondly, the hypercurve can save the computational cost, because we need to compute the intermediate result w only once, and can apply it to any material shape, whereas in the conventional method we have to compute the intermediate result g every time a new material E is given.

7

Concluding Remarks

We have reformulated the Minkowski algebra in such a way that the new algebra forms a group and consequently the sum and its inverse can always be taken freely. In this new algebra, the conventional polygonal curves are extended to more general geometric objects, which we name the “hypercurves”. We also gave a physical interpretation of the hypercurve, and discussed the practical use of this new concept. There still remains many related problems to be solved. They include (i) eﬃcient algorithms for this algebra, (ii) optimal representation of a given polygonal region by an angle-parametric polygonal curve, (iii) physical interpretation of hypercurves generated by nonconvex curves, and (iv) extension to higher dimensions. Acknowledgements. This work is supported by the Grant-in-Aid for Scientiﬁc Research of the Japanese Ministry of Education, Science, Sports and Culture.

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References 1. H. Bekker and J.B.T.M. Roerdink: An eﬃcient algorithm to calculate the Minkowski sum of convex 3D polyhedra. Lecture Notes in Computer Science 2073, pp. 619–628, 2001. ´ ements de Math´ematique, Alg´ebre 1. Hermann, Paris, 1964. 2. N. Bourbaki: El´ 3. P. K. Ghosh: An algebra of polygons through the notion of negative shapes. CVGIP: Image Understanding, vol. 54 (1991), pp. 119–144. 4. P. K. Ghosh: Vision, geometry, and Minkowski operators. Contemporary Math., vol. 119 (1991), pp. 63–83. 5. P.K. Ghosh and R.M. Haralick: Mathematical morphological operations of boundary-represented geometric objects. J. Math. Imaging and Vision, vol. 6 (1996), pp. 199-222. 6. L. J. Guibas, L. Ramshaw and J. Stolﬁ: A kinetic framework for computational geometry. Proc. 24th IEEE Symp. Foundation of Computer Sciences, (1983), pp. 100–111. 7. S. Har-Peled, T. M. Chan, B. Aronov, D. Halperin and J. Snoeyink: The complexity of a single face of a Minkowski sum. Proc. 7th Canadian Conf. Comput. Geom., (1995), pp. 91–96. 8. A. Hermandez-Barrera: Computing the Minkowski sum of monotone polygons. IEICE Transactions on Information and Systems, vol. E80-D (1997) pp. 218–222. 9. M.-S. Kim and K. Sugihara: Minkowski sums of axis-parallel surfaces of revolution deﬁned by slope-monotone closed curves. IEICE Transactions on Information and Systems, vol. E84-D (2001), pp. 1540–1547. 10. D. Leven and M. Sharir: Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagram. Discrete and Comput. Geom., vol. 2 (1987), pp. 9–31. 11. J. Mikusinski: Operational Calculus. Pergamon Press, London, 1957. 12. D. Mount and R. Silverman: Combinatorial and computational aspects of Minkowski decompositions. Contemporary Math., vol. 119 (1991), pp. 107–124. 13. G. D. Ramkumar: An algorithm to compute the Minkowski sum outer-face of two simple polygons. Proc. 12th ACM Symp. Comput. Geom., (1996), pp. 234–241. 14. K. Sugihara, T. Imai and T. Hataguchi: An algebra for slope-monotone closed curves. Int. J. Shape Modeling, vol. 3 (1997), pp. 167–183.

Thinning Grayscale Well-Composed Images: A New Approach for Topological Coherent Image Segmentation Jocelyn Marchadier, Didier Arqu`es, and Sylvain Michelin Universit´e de Marne-la-Vall´ee, Equipe Image, Institut Gaspard Monge, 5, Boulevard Descartes, 77454 Champs sur Marne Cedex [email protected]

Abstract. Usual approaches for constructing topological maps on discrete structures are based on cellular complexes topology. This paper aims to construct a coherent topological map deﬁned on a square grid from a watershed transformation. We propose a deﬁnition of well-composed grayscale images based on the well-composed set theory and the cross-section topology. Properties of two diﬀerent thinning algorithms are studied within this scope, and we show how to obtain a thin crest network. We derive an eﬃcient algorithm that permits the construction of a meaningful topological map. Finally, we demonstrate the usefulness of this algorithm for multilevel image segmentation. Keywords. Topological map, thinning, well-composed images.

1

Introduction

A digital image may be seen as the digitalization of a piecewise continuous function. The discontinuities of the underlying continuous function are of primary importance in many shape recognition processes, as they usually describe the shapes of objects appearing on an image. A piecewise continuous function may be represented by a topological map, which can be viewed as a partition of the plane into three sets of points: a ﬁnite set S of points, a ﬁnite set A of disconnected Jordan arcs having elements of S as extremities, and a set of connected domains, the faces, which boundaries are unions of elements of S and A. The deﬁnition and the extraction of a topological map from a digital grayscale image coherent with the one describing the underlying continuous function would be a major achievment. Although such a structure can been deﬁned straightforwardly on an hexagonal grid, the square grid poses consistency problems. The notion of discontinuity is lost once a function is digitalized. Segmentation may be viewed as a process that tries to catch the discontinuities of an underlying continuous function. Two main approaches to segmentation may be distinghished. The ﬁrst approach consists of approximating the discrete function by a piecewise continuous function, and is usually referred to as a region oriented segmentation. The second approach tries to directly catch the discontinuities of an underlying continuous function using an heuristic provided by a A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 360–371, 2002. c Springer-Verlag Berlin Heidelberg 2002

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gradient operator deﬁned on discrete functions, and is referred as contour oriented segmentation. Recent works [4,7] aim to develop a coherent topological structure describing a digital image from the information provided by regions. The diﬀerent structures described use a discrete topology based on the decomposition of the support domain of an image into three kind of elements of diﬀerent dimensions, i.e. surface elements, associated with the discrete points of the support, edge elements, which are the edges seperating two surface elements, and vertices of the so deﬁned grid [6,10]. On one hand, such a partition has nice topological properties, but on the other hand, it suﬀers from many practical drawbacks, such as the amount of memory needed to store the entire partition, and diﬃculties faced when trying to construct the partition from the contour information. Alternatively, a topological partition of an image may be directly deﬁned by a digital topology only involving points on the square grid. In order to face the connectivity paradox, several neighborhood systems are usually used together. This is done either by considering diﬀerent adjacent relations for points belonging to a set and its complement [11], or by assigning diﬀerent neighborhoods to each point of ZZ 2 in a data independent manner, which can be formally stated using the framework proposed in [9] (this framework can also be used in the cellular complex approach [10]). Watersheds or more generally graytone skeletons can be used in this context to retrieve a vertex/arc network (crests lines) and faces (catchment basins) from a discrete topographic surface such that the modulus of the gradient viewed as a relief, which is exactly the seeked partition of the image. However, many consistency problems are encountered on a square grid. Approaches that work by suppressing points from a potential crests network (grayscale thinning) or by adding points to connected sets of points [1,3] do not usually guarantee that the extracted crests network is thin (ﬁgure 1). Thick conﬁgurations of crests pose

Fig. 1. Thick irreductible conﬁguration of points [1]

obvious problems when one is trying to link points from the resulting crests network in order to obtain digital curves and vertices of the topological partition. Approaches that work by linking potential crest points [16,17], constructing a raster graph, do not usually guarantee that the faces deﬁned by the cycles of the graph are composed of a unique connected component (ﬁgure 2).

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Fig. 2. Connected components assiciated with a cycle of a raster graph

Latecki and al. proposed to face the problem of thickness of skeletons on digital binary images by forbidding some conﬁgurations of points [15]. They propose a thinning operator that preserves the properties of the so called wellcomposed binary images, resulting in a thin skeleton. He also demonstrated a Jordan theorem that is veriﬁed on well-composed sets of points. He extended the property of well-composedness to multicolor images. In this contribution, we ﬁrst recall some classical notions of digital topology (section 2) and some properties of well-composed sets (section 3). We redeﬁne grayscale well-composed images using the cross section topology formalism [2,3, 16], adapt a grayscale thinning to well-composed graylevel images, and proove some of its properties (section 4). We derive an algorithm which construct a topological partition from an ultimate thinning of a well-composed image, and ﬁnally present an application (section 5).

2

Digital Topology: Basic Notions

A discrete image I is a function from ZZ 2 to a set E. When E = {0, 1}, I is said to be a binary digital image. When E = {0, . . . , k}, I is said to be a grayscale image. Two point p1 = (x1 , y1 ) ∈ ZZ 2 and p2 = (x2 , y2 ) ∈ ZZ 2 are – 4-adjacent if and only if d4 (p1 , p2 ) = |x1 − x2 | + |y1 − y2 | = 1. – 8-adjacent if and only if d8 (p1 , p2 ) = max(|x1 − x2 |, |y1 − y2 |) = 1. The n-neighborhood Γn (p) of a point p is the set of all the points n-adjacent to p, with n = 4 or n = 8. A n-connected path is an ordered set C = {p1 , . . . , pm } such that for all the points pi,1
3

Well-Composed Sets of Points

A coherent topological structure of a digital image should respect an equivalent of the Jordan curve theorem by which the complement of a simple closed curve

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is a set composed of two connected components. One of the major drawbacks of digital topology is known as the connectivity paradox. If the same neighborhood system is used for studying the connectivity of a set and its complement, the Jordan curve theorem does not have its digital counterpart on a squared grid. In order to solve this problem, two neighborhood systems (8 and 4-neighborhood) are used together, deﬁning the connectivity of respectively a set and its complement [11]. Latecki and al. [15] proposed a diﬀerent approach. By forbidenning some local conﬁgurations of the studied sets of points, one can use a coherent 4-neighborhood system. In this section, we present the results that they obtained which are usefull for the rest of this article. Readers interested in the demonstrations of those results may refer to [13,15]. A subset S of ZZ 2 is weakly well-composed iﬀ each 8-connected component of S is a 4-connected component of S. A subset S of ZZ 2 is well-composed iﬀ S and S are weakly well-composed. The ﬁgure 3 illustrates the concept of wellcomposedness. On the left hand side of the drawing, the set of gray points form a weakly well-composed set of points, while the right hand side drawing shows a set of points which is well-composed. In order for a set of points to be wellcomposed, conﬁgurations like the one surrounded by a bold square are forbidden. Thus, well-composed sets can be characterized by local conﬁgurations.

Fig. 3.

This leads to the statement of a new Jordan theorem on well composed set: Theorem 1. The complement of a well-composed simple closed curve is formed by two well-composed connected components. Classical thinning algorithms do not usually conserve the property of wellcomposedness. However, by slightly modifying the deﬁnition of simple points, we obtain a thinning that conserves well-composedness. A point p is simple for a well-composed set S ⊂ ZZ 2 iﬀ there exists a bijection between 4-connected components of S ∪ {p} and 4-connected components of S \ {p}, and between 4-connected components of S ∪ {p} and 4-connected components of S \ {p}. Simple points for well-composed sets can be characterized by a few local conﬁgurations. One can show that a point p is simple for a well-composed set S iﬀ there exists exactly one 4-connected component of S 4-adjacent to p and one 4-connected component of S 4-adjacent to p. A thinning of a well-composed set S is a set S obtained from S by iteratively deleting simple points of S. By construction, a thinning of a well-composed set is a well-composed set.

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A set S is irreductible iﬀ no point of S is simple. Let the n-interior of a set S of points be the set n − int(S) = {p ∈ S, Γn (p) ⊂ S}. An irreductible thinning of a well-composed set is thin, which can be formally stated as: Theorem 2. The 8-connected components of the 4-interior of an irreductible well-composed set are reduced to one point. The proof of this theorem found in [13] holds for the earlier deﬁnition of simple points. This can be used to enumerate all the irreductible local conﬁgurations (ﬁgure 4). On this ﬁgure, white lines show how to link points from an irreductible well-composed set of points in order to obtain a vertex/arc network deﬁning a coherent topological partition.

P

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(2)

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Fig. 4. Local irreductible conﬁgurations for well-composed sets

4

Well-Composed Grayscale Images

Meyer [16] proposed to characterize crest points on a grayscale image by studying a familly of binary images, which are obtained by thresholding the original image by all possible threshold values. Crest points are the points that change the homotopy of one of the binary images. More recently, cross-section topology has been introduced by Bertrand and al. [2,3]. It has led to the development of an eﬃcient watershed algorithm [5]. The deﬁnition of well-composed multicolor images has also been proposed [12]. In this section, we use the deﬁnition of well-composed grayscale images [14] , which is compatible with the cross section topology. We then study the properties of thinnings of well-composed grayscale images. Let I be a grayscale digital image, deﬁned as a function from ZZ 2 to E = {1, . . . , m}. A c-section of I is the set of points Fc = {(x, y) ∈ ZZ 2 , I(x, y) ≥ c}. A c-cut of I is the set of points Ec = {(x, y) ∈ ZZ 2 , I(x, y) = c} A grayscale digital image I is said to be well-composed iﬀ all its c-sections are well-composed sets. The critical conﬁgurations depicted on ﬁgure 5 are forbidden for well-composed grayscale images.

Thinning Grayscale Well-Composed Images 2

3

4

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Fig. 5. Forbiden conﬁgurations for well-composed grayscale images. Dark points are points having highest values of I.

A point p ∈ ZZ 2 is said to be destructible iﬀ it is simple for the section Fc , with c = I(p). We deﬁne the two sets Γ + (p) = {p ∈ Γ8 (p), I(p ) ≥ I(p)} and Γ − (p) = {p ∈ Γ8 (p), I(p ) < I(p)}. A destructible point p veriﬁes that both Γ + (p) and Γ − (p) are composed of a unique 4-connected component, with a point 4-adjacent to p. The destruction of a point p is the operation consisting of substracting 1 from I(p). An image I is a thinning of a grayscale well-composed image I iﬀ it can be obtained from I by iteratively destructing destructible points from I. Theorem 3. Let I be a well-composed grayscale image, and I be a thinning of I. Then I is a well-composed grayscale image. This can be demonstrated by observing that if I is a thinning of I, then a c-section of I is, consequently to the deﬁnition of a destructible point, a thinning of the c-section of I, independently of the value of c. All the c-sections of I are well-composed. As a thinning of a well-composed set is well-composed, all the c-sections of I are well-composed, thus I is well-composed. ✷ An graylevel well-composed image I is irreductible iﬀ no point of I is destructible. We deﬁne a plateau of a well-composed image I as a 4-connected set S such that ∀(p, p ) ∈ S 2 , I(p) = I(p ). A regional minimum S is a plateau such that ∀p ∈ S, Γ4 (p) ∩ S = ∅, ∀p ∈ Γ4 (p) ∩ S, I(p ) > I(p). Theorem 4. The set composed of all the points belonging to regional minima of a well-composed image I is well-composed. Let p and p be two 8-adjacent but not 4-adjacent points in regional minima of I. Suppose that Γ4 (p)∩Γ4 (p ) is not included in a regional minimum of I. Then I is not well-composed, since this construction is forbidden for well-composed images. This leads to a contradiction. ✷ Theorem 5. There exists a bijection between regional minima of a wellcomposed image I and regional minima of a thinning of I. We consider a thinning I of a well-composed image I. The points belonging to a regional minimum of a well-composed set I are not destructible. If a point

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of a regional minimum of I is destructible, then it is 4-adjacent to at least one point with a lower value of I, and thus, it is not a point of a regional minimum. The regional minima of I are then subsets of regional minima of I of I. Let’s suppose that a regional minimum S of I contains several regional minima of I. Let c be the value of the highest regional minimum S of I such that S ⊂ S . The set Fc+1 ∩ S contains at least two 4-connected components bounded by points adjacent to S (Jordan theorem). Thus, Fc+1 contains more connected components than the set Fc+1 , where Fc+1 is the (c+1)-section of I . I is not a thining of I, which contradicts the hypotheses. ✷ The complementary sets of regional minima are arbitrarely thick. This is partly due to conﬁgurations like the one depicted on the ﬁgure 6. On this ﬁgure,

Fig. 6. Regional quasi-minimum

the set composed of the points marked by a circle can be used to deﬁne an interseting face of a topological partition. We deﬁne a regional quasi-minimum as a 4-connected component of the set Q = {p ∈ ZZ 2 , Γ − (p) = ∅} deﬁned on I. Note that a regional minimum is a regional quasi-minimum. The set Q deﬁned on an irreductible well-composed image is not generally well-composed. For example, on the ﬁgure 7, Q is composed of all the points marked by a circle, and Q is not well-composed. This is due to the presence of peaks, i.e. points without upper neighbors. This leads to the following devlopments.

Fig. 7.

A leveling of an image I [3] is a graylevel image obtained from I by iteratively destructing destructible points and peaks from I.

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Theorem 6. Let I be a graylevel well-composed image, and let I be a leveling of I. Then I is well-composed. The destruction of a destructible point from a graylevel well-composed image I will result in a thinning of I which is well-composed (c.f. theorem 3). Now consider a peak p from a well-composed image I. Let p one of its neighbors such that ∀p ∈ Γ8 (p) I(p ) ≤ I(p ). As p is a peak, we have I(p) > I(p ). Let I the image obtained by destructing p from I. Then I (p) ≥ I (p ), and the ordering of the other points from Γ8 (p) is not changed. Thus, we do not create a forbiden conﬁguration for well-composed graylevel images, and I is well-composed. As the two operations are conserving well-composedness, their iterative application on a well-composed image will yield in a well-composed image. ✷ By applying the leveling transformation on a well-composed graylevel image until stability, we obtain a well-composed irreductible graylevel image such that none of its point is a peak. On such images, regional quasi-minima are wellcomposed. Theorem 7. Let I be an irreductible well-composed graylevel image such that no point of I is a peak. The set Q = {p ∈ ZZ 2 , Γ − (p) = ∅} deﬁned on I is well-composed. Let I be a well-composed grayscale image. Let p and p be two points 8-adjacent, but not 4-adjacent, belonging to the set Q = {p ∈ ZZ 2 , Γ − (p) = ∅}, such that the two points p1 and p2 4-adjacent to p and p are not belonging to Q. Necessarly, I(p) = I(p ), I(p1 ) ≥ I(p) and I(p2 ) ≥ I(p). Moreover, as I is well-composed, I(p1 ) = I(p) or I(p2 ) = I(p). Let’s suppose, without loss of generality, that I(p1 ) = I(p), which corresponds to the ﬁgure 8.

P5

P3

P1

P’

P6

P4

P

P2

Fig. 8.

I(p3 ) ≥ I(p) by hypothese. I(p3 ) = I(p) otherwise p3 is either destructible or I is not well-composed. I(p5 ) ≥ I(p3 ) and I(p6 ) < I(p3 ) otherwise p3 is destructible again. Then, p4 is destructible, which leads to a contradiction. ✷ We can now expose the following result, which states that each cut of an irreductible graylevel image without peak is thin. Theorem 8. Let I be an irreductible well-composed graylevel image from ZZ 2 to a set E, such that no point of I is a peak, and the set Q = {p ∈ ZZ 2 , Γ − (p) = ∅} deﬁned on I. For all c ∈ E, the 8-connected components of the 4-interior of the set Ec \ Q are composed of at most one point, where Ec is the c-cut of I.

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Let c ∈ E. Consider a point p with Γ − (p) = ∅ and I(p) = c. If {p ∈ Γ8 (p), I(p ) > I(p)} = ∅, then the c-cut of I respect one of the local conﬁgurations of the irreductible well-composed sets depicted on ﬁgure 4. We now consider two points p and p belonging to the same c-cut, 8-adjacent to each other, such that p is adjacent to at least one point of lower value and one point of upper value. Thus, one of the local conﬁgurations depicted on ﬁgure 9 holds.

P’

P’ D

D

P

P’

D

P

P

D P P’

Fig. 9.

The point noted D is destructible on these conﬁgurations and consequently I is not irreductible. ✷

5

Constructing a Topological Partition and Application

The topological map deﬁned on the digital plane is composed of: – a set S of points which are vertices of the map, – a set A of arcs, which are 4-connected disconnected digital curves, of which the extremities are elements of S, – a set F of faces, which are 4-connected sets of points, of which the boundaries are elements of S and A. Such a structure can be built from an irreductible well-composed grayscale image without peak, obtained by a leveling transformation of a well-composed grayscale image. The faces of the topological partition are the regional quasi-minima. The vertex/arc network can be retrieved by linking points that have at least one lower neighbor in the following way: – Points belonging to the same c-cut are linked together using one of the conﬁgurations of the ﬁgure 4. – Points belonging to diﬀerent c-cuts are linked according to one of the conﬁgurations of the ﬁgure 10.

Fig. 10.

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We can prove, by reasoning on local conﬁgurations, that the cycles of the graph constructed as such correspond to regional quasi-minima. The ﬁgure 11 represents an irreductible well-composed image without peaks and its constructed vertex/arc network.

Fig. 11.

Diﬀerent structures can be used to store this topological map, such as a combinatorial map [7] or a raster graph [17]. Note that both structures can be obtained from an irreductible well-composed image without peak with a single pass algorithm. Moreover, the raster graph structure can be obtained with a parallel algorithm, as the linking process involves only local conﬁgurations. The correspondance between the so deﬁned map and the map that can be constructed from the linear discontinuities of the underlying piecewise continuous function is achieved when the topology of the interesting set of points is preserved under digitalization. Note also that the digitalization of a continuous curve is a digital well-composed curve [13]. That motivates the use of the 4-neighborhood connectivity. Figure 12 demonstrates the usefulness of this results for image segmentation. The ﬁrst image of the ﬁgure is the original image. The second image is the irreductible well-composed image without peak that has been constructed from the modulus of a gradient of the original image. An algorithm, not detailed in this article, has been applied on the image of the modulus, turning it into a well-composed graylevel image. Thus, the second image is well-composed. The third image is the extracted crest network, where some arcs have been removed using a threshold strategy that was found to be stable and very fast on diﬀerent kinds of images. On all tested images, the crest network was found to be a thin set of points, although some thick conﬁgurations can be constructed. However, the linking is realized without ambiguity, and the topological map obtained in such a way have found to be usefull for shape recognition applications.

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Fig. 12.

The deﬁnition of destructible points presented in this article is not well suited for non well-composed images. For such images, destructible points are deﬁned with the classical 4/8-neighborhood system [3]. The thinning and leveling transformations deﬁned with that system are resulting in images with drawbacks such that thick conﬁgurations of points (when the 8-neighborhood system is used to characterized crest points) or 8-connected isolated points.

6

Conclusion

In this contribution, we have used the cross section topology formalism in order to deﬁne a thinning operator that is internal to well-composed gray level images. Moreover, we have proposed a way to construct a topological map from the resulting thin image, and have shown that the obtained map is coherent in the sense that: – a bijection exists between cycles of the map and 4-connected regions of the thin image, – each Jordan arc of the map is a 4-connected digital curve. The map can be constructed from a gray level image very eﬃciently, using previously published algorithms.

References 1. Arcelli C., Pattern Thinning by Contour Tracing, Computer Graphics and Image Processing, Vol. 17 (1981) 130–144 2. Bertrand B., Everat J.C., Couprie M., Topological approach to image segmentation, SPIE Vision Geometry V Proceedings, Vol. 2826 (1996) 3. Bertrand B., Everat J.C., Couprie M., Image segmentation through operators based upon topology, Journal of Electronic Imaging, Vol. 6(4) (1997) 395–405 4. Braquelaire J.-P., Brun L., Image Segmentation with Topological Maps and Interpixel Representation, Journal of Visual Communication and Image Representation, Vol. 9(1) (1998) 62–79

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5. Couprie, M., Bertrand G., Topological Grayscale Watershed Transformation, SPIE Vision Geometry V Proceedings, Vol. 3168 (1997) 136–146 6. Fiorio C., Approche interpixel en analyse d’images, une topologie et des algorithmes de segmentation, PhD Dissertation, Universit´e de Montpellier, France, (1995) 198 pages. 7. Fiorio C., A topologically Consistent Representation for Image Analysis: the Frontiers Topological Graph, DGCI’96, Lectures Notes in Computer Sciences, no. 1176, (1996) 151–162 8. Gangnet M., Herv´e J.-C., Pudet T., Van Tong J.-M., Incremental Computation of Planar Maps, SIGGRAPH Proc., Computer Graphics, Vol. 23(3) (1989) 345–354 9. Khalimsky E., Kopperman R., Meyer R., Computer Graphics and Connected Topologies on Finite Ordered Sets, Topology and its Applications, Vol. 36 1–17 10. Kovalevsky V. A., Finite Topology as Applied to Image Analysis, Computer Vision, Graphics and Image Processing, Vol. 46 141–161 11. Kong T. Y., Rosenfeld A., Digital Topology, Introduction and Survey, Computer Vision, Graphics, and Image Processing, Vol. 48 (1989) 357–393 12. Latecki L., Multicolor Well-Composed pictures, Pattern Recognition Letters, Vol. 16 (1995) 425–431 13. Latecki L., Discrete Representation of Spatial Objects in Computer Vision, Computational Imaging and Vision Vol. 11, Kluwer Academic Publishers (1999) 216 pages 14. Latecki L., Well-Composed Sets, Advances in Imaging and Electron Physics Vol. 112, Academic Press (2000) 95–163 15. Latecki L., Eckhardt U., Rosenfeld A., Well-Composed Sets, Computer Vision and Image Understanding, Vol. 61 (1995) 70–83 16. Meyer F., Skeletons and Perceptual Graphs, Signal Processing, Vol. 16 (1989) 335–363 17. Pierrot Deseilligny M., Stamon G., Suen C., Veinerization: A New Shape Description for Flexible Skeletonization, IEEE Trans. on PAMI, Vol. 20(5) (1998) 505– 521

An Incremental Linear Time Algorithm for Digital Line and Plane Recognition Using a Linear Incremental Feasibility Problem Lilian Buzer LLAIC, Universit´e Clermont 1, IUT d´epartement Informatique, B.P. 86, 63172 AUBIERE cedex, FRANCE, [email protected]

Abstract. We present a new linear incremental method for digital hyperplane1 recognition. The ﬁrst linear incremental algorithm was given for 8-connected planar lines in [DR95]. Our method recognizes any subset of line in the plane or plane in the space. We present the Megiddo linear programming (LP) algorithm in linear time and describe its adaptation to our problem. Then we explain its improvement toward a linear incremental method. Keywords: digital line recognition, digital plane recognition, feasibility problem, incremental, linear time. Conference Topic: Models for Discrete Geometry. Type of Presentation: oral presentation.

1

Introduction

We study digital line and plane recognition. The seminal deﬁnition of these objects was given by Reveill`es in [Rev91]. A set of points P of Zd is a digital hyperplane if it veriﬁes: d∗

∃(ai )1≤i≤d ∈ Z , ∃γ ∈ Z, ∀x ∈ P, we have: γ ≤

d

ai .xi < γ + ||a||∞

(1)

i=1

where ||.||∞ denotes inﬁnite norm equal to Sup{|ai |1≤i≤d }. We present a new algorithm which recognizes any set of points. Our technique is based on LP algorithm, more precisely the linear Megiddo method which is described in Section 2. In Section 3, we transform our recognition problem into a feasibility problem solvable by Megiddo algorithm. The improvement to linear incremental complexity is then presented in Section 4. 1

By extension of the notion of an euclidian hyperplane in a d-dimensional vectorspace. We remind that this object refers to a (d − 1)-dimensional aﬃne subspace.

A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 372–381, 2002. c Springer-Verlag Berlin Heidelberg 2002

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2 2.1

373

Megiddo Algorithm in R2 and R3 Preliminaries

Our digital line and plane recognition algorithm requires to solve LP problems in two and three dimensions. For this, we will use Megiddo linear-time algorithm. History. Working in the ﬁeld of computational geometry, Megiddo gave the ﬁrst deterministic algorithm for LP whose running time is linear in the number of constraints when the dimension is ﬁxed. The decimation technique was ﬁrst introduced in [Meg83] for the two and three-dimensional cases. Later, in [Meg84] d he extended his method to develop an O(22 .n) time algorithm for LP in Rd . The factor in d was improved by Clarkson to 3.d2 (see [Cla86]). In recent years, no progress has been made on this front, nevertheless new developments occured in randomized and parallel algorithms for linear programming. Numerous simpler and more practical randomized algorithms have been given (see [Cla98,Sei91]). An introduction can be read in [BKOS00] and a comprehensive summary of this ﬁeld can be found in [AS98]. We will hereafter present Megiddo’s technique. Note that randomized methods are unusable in the incremental approach. Summary of the prune and search technique of linear programming. We know that a limited number of constraints are tight at an optimal solution of a LP problem. This method tries to eliminate input constraints that do not aﬀect the optimum value. At each step, a constant fraction of n constraints is eliminated from the current set in O(n) time. Therefore after a logarithmic number of steps, the size of the problem becomes constant. By using any strongly polynomial LP algorithm, we solve our remaining set of constraints in constant time. Because of this decimation, the global cost remains bounded by the cost of the ﬁrst pruning step. Problem posing. We want to ﬁnd the optimum value of a d-dimensional LP problem of n constraints. We can always transform the gradient function into (0, . . . , 0, 1) by rotation. For presentation convenience, we only consider constraints that have a strictly positive coeﬃcient associated to xd (wlog). We want to solve: Minimize xd (2) d−1 So that xd ≥ Σj=1 aij .xj + bi (i = 1, . . . , n) Deletion criterion. The core of the technique consists in coupling constraints. Under the previous assumption, if we take two non-parallel constraints, there exists a vertical hyperplane passing through their intersection, that divides the space into two half spaces. Such a hyperplane is called a separating line (SL) or a separating plane (SP). If optimal solutions are located in a certain half space, then we may discard one of the two constraints.

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The Two-Dimensional Case

We give an overview of Megiddo two-dimensional algorithm described in [Meg83]. A comprehensive description of this method is given in [PS85] and in [Ede87] . We hereafter describe its inner loop in four steps (see Fig. 1). The algorithm steps. 1. Coupling: we create couples of constraints (except for one at most) and their associated SL. Under the assumption of 2.1, if two inequalities are parallel, one of them is redundant and can be immediately suppressed. 2. Selection of a test line : vertical SL have horizontal coordinates. We compute in linear time the median of these values. We select the test line to be the particular SL that corresponds to this median. 3. Testing a line: we want to know on which side of the test line are the optima located. For this, we ﬁrst compute the optimum of the LP problem restricted to this line. We determine the right and left slopes given by the constraints passing through this point. By convex properties of the feasibility polyhedron, if a decreasing slope exists, optima will be located on its side. Else this point is a minimum and the problem is solved. 4. Pruning: As the test line is a median for the SL set, we deduce the optima location relative to one half of the SL. We then apply the deletion criteria to each couple of constraints associated to these SL. After this, we iterate as long as the number of constraints is above a ﬁxed constant.

Separating line Test line

1. Coupling.

2. Selection of a test line.

Suppressed constraint

3. Testing a line.

4. Prunning.

Fig. 1. Steps of Megiddo two-dimensional algorithm.

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Oz Ox

P1 M

P2

Oy

I

P Fig. 3. Point location. Fig. 2. Testing a plane.

Global complexity. One quarter of the inequalities are rejected from the current set. Each of the four steps has a linear time complexity in the number of constraints. This implies that the runtime T (n) of our algorithm satisﬁes T (n) = O(n) + T ( 34 .n). Therefore this algorithm solves a linear program in two variables and n constraints in O(n) time. 2.3

The Three-Dimensional Case

We keep the technique described in the two-dimensional method. We test particular planes which enable us to drop a constant fraction of inequalities. After this, we iterate. We precisely describe steps 2 and 3 in the next sections. Step 2: Selection of testing planes. As the separating planes (SP) are all vertical, we represent them by lines in the Oxy plane. Our problem also becomes a two-dimensional search problem. Megiddo describes a technique to solve it eﬀectively in [Meg83] and in [Meg84]. Readers can refer to [Ede87]. Suppose we have two planar lines of opposite slopes, let I denote their intersection. If we know the position of a point relative to the vertical and horizontal lines passing trough I, we can determine the location of this point relative to at least one of the two given lines (see Fig. 3). The trick consists in ﬁnding in linear time the median of all present slopes. To shorten explanations, we consider the median direction to be vertical. We can also couple lines (except for one of them at most) and create couples of opposite slopes. So we use the previous remark and we obtain a set of vertical lines. We compute their median line and test it. We know the optimum position relative to one half of the vertical lines. We select the horizontal lines coupled with these vertical lines, take their median and test it. We also determine the optima position relative to 18 of SP and we can also 1 delete 16 of constraints. All these steps are shown in Fig. 4. Step 3: Testing a plane. We want to know on which side of a plane P lies the optima (see Fig. 2). We ﬁrst solve our LP problem restricted to this

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plane and obtain a minimum called M . If this problem is unbounded the threedimensional problem is unbounded too. Now, we have to test both sides of P . Let C denote the set of constraints passing through M . Because of the ﬁnite number of inequalities, there is a neighbouring ball of radius r around M where no other constraint can interfere, so the local decrease around M only depends on constraints in C. Without loss of generality, we can assume that P is Oxz and M is the origin. Let P1 and P2 denote the solutions of the two following systems: Minimize z, under C Minimize z, under C (3) with yi = r with yi = −r If one solution is better than M , it indicates the optima location. If both are worse, M is a solution of our problem. Because of the convexity of the feasibility polyhedron, we cannot have two better solutions. These two LP problems are linear in the number of constraints as we saw in the previous section. Remark 1. By deﬁnition of C, we do not have to determine a value for r. We can also choose r equal to 1. Oz

Oy

Oy

O

Ox

Ox

1. Coupling and creation of SPs.

3. Test of the horizontal median line.

2. Test of the vertical median line.

4. Optima location relative to two SPs.

Fig. 4. Solving the search problem.

3

Digital Line and Plane Recognition

We know we have to solve diophantine equations of the form: γ ≤ N.P (x1 , . . . , xd ) < γ + N ∞ . Each point P is linked to two inequalities. We

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cannot solve integer programming with LP method. I hereafter describe my rewriting of the three-dimensional problem that allows to use LP. As the normal vector is nonzero, the following transform is always possible: d

Ni .xi γ γ ≤ < +1

N ∞

N

N

∞ ∞ i=1

(4)

We now have a set of linear inequalities with d variables at most. For instance, in the three-dimensional case, we can solve this system using threedimensional Megiddo algorithm. We will hereafter only consider the case when

N (u, v, w) ∞ = w. We are not concerned with negative values of w because N is valid iﬀ −N is. We have: γ γ γ ≤ N.P (x, y, z) < γ + N ∞ ≤ u .x + v .y + z < w +1 ⇔ wu w v w (5)

N (u, v, w) ∞ = w | w | ≤ 1, | w | ≤ 1 Linear programming consists in optimizing linear function. Nevertheless, our recognition problem only requires to solve a feasibility problem: Find

(a, b, h)   a.xi + b.yi − h ≥ −zi So that a.xi + b.yi − h < −zi + 1  |a| ≤ 1, |b| ≤ 1

(i = 1, . . . , n)

(6)

Remark 2. This transform is equivalent to a linear separability problem. Let S be a set of points, we want to ﬁnd a plane passing between S and a virtual set of forbidden points S + ez . If such a plane exists, the convex hull of S and the convex hull of S +ez do not intersect, see [PS85] for details. The vertical distance between these two convex hulls is in ]0, 1[. This implies that S is lying in a band of vertical thickness less than 1. By deﬁnition, S is a subset of a digital plane. Strict and large inequalities in a same system may cause problems. Let P denote the polyhedron deﬁned with inequalities of (6) transformed into large inequalities and P the polyhedron associated to (6). The diﬀerence between P and P is equal to the superior border of P denoted Γ sup . From this observation, (6) is equivalent to this problem: (∗)

(a, b, h) ∈ / Γ sup   a.xi + b.yi − h ≥ −zi So that a.xi + b.yi − h ≤ −zi + 1  |a| ≤ 1, |b| ≤ 1 Find

(i = 1, . . . , n)

(7)

Remark 3. (∗) requires just a modiﬁcation in the algorithm. In the twodimensional problem, when we cut by a vertical line, we only have to keep the highest and lowest feasible points on the cut and check if they are diﬀerent. Otherwise, no solution lies on this cut. The three-dimensional problem calls the two-dimensional algorithm and uses its improvement.

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The Incremental Problem

We are able to recognize digital lines and planes in linear time using Megiddo’s technique. Nevertheless the incremental algorithm is quadratic. I adapt our feasibility problem to create a linear time incremental method. The core of this method uses the following lemma: Lemma 1. Let H be a separating hyperplane associated to a couple of constraints. If the set of feasible solutions lying on H is included in a hyperplane of H then one of the two constraints is inactive. Remark 4. An inactive constraint supports no face of the polyhedron and its suppression does not aﬀect the polyhedron. 4.1

The Starting Point

To apply Megiddo’s technique in Zd (d = 2 or 3 in our study), we must suppose that the ith coeﬃcient of each possible normal is equal to its inﬁnite norm. In other words, we want to know if all the possible hyperplanes can be written as a function of the same d − 1 coordinates. The two-dimensional case is quite easy. Suppose the ﬁrst inserted point M is the origin. If the next point P (x, y) veriﬁes |y| < |x|, (resp. >), all the possible normals N (u, v) will verify |u| > |v| (resp. <). In the three-dimensional case, we do not have the same property. Therefore we have to run a recognition for each direction. Test line

Suppressed constraints.

1

2

3

1. Lack of feasible points on the test line after the insertion of three new constraints.

2. Pruning phase.

Fig. 5. Three added constraints before the pruning step.

4.2

The Two-Dimensional Case

We now work with constraints oriented in any direction. We couple inequalities of the same form z ≥ ai .x + bi .x + ci or z ≤ ai .x + bi .x + ci for instance. At

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most two inequalities are not treated. We begin as in Megiddo algorithm and stop before the pruning step. We are cutting by a vertical line and we can have three diﬀerent cases: 1. We have no feasible point on the test line, the condition of Lemma 1 is fulﬁlled. We can thus go on with Megiddo’s method. 2. There is only one point on the test line. As the line is vertical, this point lies on the superior border of the polyhedron. According to Remark 3, it cannot be a solution for the global problem (7). This case is equivalent to case 1. 3. There is a segment of feasible points lying on the test line. The point of lowest ordinate is a solution. We do not enter the pruning phase and store new constraints in a waiting set. All the core of the incremental method is here. At once, Megiddo algorithm is frozen. For each new constraint we only update the segment of feasible points. While the result is a segment, its point of lowest ordinate is a solution, otherwise we are in case 1 or 2. We wait for new constraints to reject all the feasible points on the test line (see Fig. 5). After this, we can unfreeze Megiddo algorithm. As Lemma 1 is veriﬁed, the pruning step suppresses inactive constraints among the previously coupled constraints. Before the next iteration, we add all the inequalities from the waiting set to the current set of constraints. 4.3

The Three-Dimensional Case

We use the same retarded Megiddo approach (see Fig. 6). We freeze Megiddo algorithm when we want to test SPs. To cut with the ﬁrst SP, we apply the two-dimensional linear incremental technique. If we have a feasible point, we enter a three-dimensional frozen phase (new constraints are not coupled and they are collected in a waiting set). Adding new constraints in the two-dimensional problem, we obtain new feasible points until there is no more solution. We then test the plane and determine on which side a feasible region may exist. We also cut with a second SP and launch a two-dimensional linear incremental problem with inequalities from coupled constraints and from the constraints of the waiting set. We are in the frozen phase and so next constraints will be added to the waiting set. New constraints bring new solutions. When there are no more feasible points on the cut, we test it. We now know that the ﬁnal polyhedron lies at most by one edge on each cut. It implies that all the constraints that have to be suppressed verify Lemma 1. Therefore the pruning phase will only delete inactive constraints. We can also leave the frozen phase and enter the following step. Before the next iteration, the remaining constraints are united with the constraints of the waiting set. 4.4

Complexity of the Incremental Algorithm

We summarize all the diﬀerent steps of a three-dimensional sequence in Fig. 7. As O(m+k +k ) ∼ O(m)+O(k +k ), the complexity is equivalent to the linear time Megiddo algorithm complexity. More precisely, when we have less than a ﬁxed

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L. Buzer

Fig. 6. Frozen phase of the three-dimensional incremental algorithm.

number of constraints, we use a strongly polynomial LP method (SPM) to obtain a feasible solution in constant time. For instance, we suppose that k frozen phases (FP) are executed. We denote by (ai )1≤i≤k the number of constraints added in each FP (it can be zero) and by a0 the number of constraints at the beginning. i−1 Let ni = j=0 aj and mi denote the number of previously entered constraints and the number of present (coupled) constraints when we enter the ith FP. We denote by α.n + β the decimation function. It allows to bound the number of kept constraints when we apply the pruning technique to n constraints. We then deduce according to the previous deﬁnitions: T (nj ) ≤ T (nj−1 ) + O(mj−1 + aj ) + aj .ΓSP M

(8)

As we deﬁne mj+1 ≤ (α.mj + β) + aj , we establish: mj+1 ≤

j

ai .αj−i +

i=0

β 1−α

(9)

Using (8) and (9), we ﬁnally obtain: T (n) =

k j=1

O(

j−1

(ai .α

i=0 k−1

T (n) = O(

i=0

5

j−1−i

k β + )) + O(ai ) 1−α i=1

ai ) + O(n) = O(n) 1−α

(10)

(11)

Conclusion

We have presented a new linear incremental algorithm for line and plane recognition. It may suﬀer from an important linearity coeﬃcient and from a quite diﬃcult implementation. Nevertheless it provides not only linear incremental complexity, but also a robust and exact method for recognition of any set of points.

An Incremental Linear Time Algorithm

381

feasible point m constraints

Step 1 Step 2 Step 3

Coupling Projecting Angular median Second coupling Selecting cuts First cut

O(m) O(m) O(m) O(m) O(m) O(m) 15.m 16

feasible point

Entering k new constraints O(k) No more feasible points on the cut Testing

O(m+k)

Second cut

O(m+k)

Entering k’ O(k’) new constraints No more feasible points on the cut Testing

O(m+k+k’)

Frozen phase

+ k + k constraints

Pruning

O(m)

Step 4

Fig. 7. Modiﬁed Megiddo sequence in three dimensions.

Two-dimensional implementation was done and tested. Finding new feasible points is a low cost operation (cost of computing the intersection of two lines). The time-consuming part is the Megiddo pruning sequence (the four steps), but we cannot forget that it allows to limit the number of constraints and so to achieve linear incremental complexity. We are now studying and optimizing a speciﬁc three-dimensional implementation using rational arithmetics.

References [AS98]

Pankaj K. Agarwal and Micha Sharir (1998). Eﬃcient algorithms for geometric optimization. ACM Comput. Surv., vol. 30, pp. 412-458. [BKOS00] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf (2000). Computational Geometry: Algorithms and Applications (2nd ed.). SpringerVerlag. 2 [Cla86] K.L. Clarkson (1986). Linear Programming in O(n.23.d ) time. Inform. Process. Lett., vol. 22, pp. 21-24. [Cla98] K.L. Clarkson (1998). A Las Vegas algorithm for linear programming when the dimension is small. In Proc. 29th Annu. IEEE Sympos. Found. Comput. Sci., pp. 452-456. [DR95] I. Debled-Rennesson, J.-P. Reveill`es (1995). A linear algorithm for segmentation of digital curves. International Journal of Pattern Recognition and Artiﬁcial Intelligence, vol. 9, no. 6, pp. 635-662. [Ede87] H. Edelsbrunner (1987). Algorithms in Combinatorial Geometry. SpringerVerlag, New York. [Meg83] N. Megiddo (1983). Linear-time algorithms for linear programming in R3 and related problems. SIAM J. Comput., vol. 12 , pp. 759-776. [Meg84] N. Megiddo (1984). Linear programming in linear time when the dimension is ﬁxed. J. ACM, vol. 31 , pp. 114-127. [PS85] F.P. Preparata and M.I. Shamos (1985). Computational Geometry: an Introduction. Springer-Verlag, New York. [Rev91] J.P. Reveill`es (1991). G´eometrie discr`ete, calculs en nombre entiers et algorithmique. Th`ese d’´etat, Universit´e Louis Pasteur, Strasbourg. [Sei91] R. Seidel (1991). Small-Dimensional Linear Programming and Convex Hulls Made Easy. Discrete and Computational Geometry, vol. 6, pp. 423-434.

Reconstruction of Animated Models from Images Using Constrained Deformable Surfaces J. Starck, A. Hilton, and J. Illingworth Centre for Vision, Speech and Signal Processing University of Surrey Guildford, GU2 7XH, UK [email protected]

Abstract. In this paper we present a technique to generate animated 3D models from a limited set of images. We adopt a model-based approach to shape recovery to obtain a speciﬁc model parameterisation for animation. A generic animatable 3D mesh model is ﬁrst manually posed to match the pose in the images. Shape information from multiple 2D image silhouettes is integrated by reconstructing the occupied volume of an object. The mesh is then deformed to match the reconstructed volume using a robust coarse to ﬁne matching technique. We formulate an irregularly parameterised generic mesh as a shape constrained deformable surface to preserve the prior shape information and speciﬁc parameterisation in the model. Our interest lies in the generation of animated human models in a multi-camera studio. We assess the technique using artiﬁcial test cases and present results for a dancer in a studio.

1

Introduction

We introduce a novel model-based technique to recover shape from a limited set of calibrated images in order to generate animated models of people in a multi-camera studio. Applications include advertising, computer games, video conferencing, user-interface agents, virtual environments and clothing retail. Active sensing systems such as laser scanning or structured light projection are used to capture highly accurate body surface data through high-cost specialist scanning booths. In the studio setting, passive image based techniques such as stereo reconstruction [7] and shape from silhouette [3,10] have been adopted. These techniques produce static surface models and do not recover the necessary structure to animate the model. The reconstruction of animated human models has been addressed through model-based shape from silhouette [5] and model-based shape from stereo [4]. The model-based approach deforms a generic model to ﬁt the available shape information, the reconstructed model can then be animated using the pre-deﬁned animation structure of the generic model. Our approach extends current methods of shape from silhouette [15] from 2D silhouette matching to 3D volume matching using the approximate reconstructed volume of an object from multiple image silhouettes, the “visual-hull”. Matching of articulated models in 2D can be problematic as only small diﬀerences in shape A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 382–391, 2002. c Springer-Verlag Berlin Heidelberg 2002

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and pose can lead to large variations in the apparent image silhouette. We make the assumption that where ambiguities occur in a single 2D view, shape information is available from additional views to resolve the ambiguity in matching. We therefore integrate the shape information in multiple images by matching to the 3D visual-hull. The visual-hull provides only a convex approximation to the shape of an object, we therefore formulate the generic model as a shape constrained deformable surface in matching, to preserve the prior shape information in the model. Model-based shape from volume has been introduced previously for segmentation from volumetric medical image data using anatomical templates. Terzopoulos et. al. [16] ﬁrst introduced deformable models as a method of shape recovery. This has proved eﬀective in medical image analysis, exploiting prior shape knowledge in the shape of the model for segmentation while also allowing for signiﬁcant variation in the ﬁnal segmented shape [11]. Shape constraints have been included in models using global shape parameters as wells as probability distributions on shape variables to capture expected variations in shape [14]. The shape of the human body can be parameterised [17], however we expect signiﬁcant variability in shape with clothing and consider only a locally constrained shape. Our formulation for a shape constrained surface follows the work of Montagnat and Delingette [12], who made use of the speciﬁc parameterisation of 2-simplex meshes to formulate a constrained model. Here we extend the concept of shape constrained models to irregularly parameterised triangular meshes, where we attempt to preserve both the shape and the initial mesh parameterisation that may be optimised for model animation. We also combine the constrained deformable surface model with a coarse to ﬁne matching technique to achieve robust matching of the model to the volume. This paper describes a novel technique for model-based shape reconstruction of animated models from the approximate volume of an object, using a novel shape constrained deformable model and robust coarse to ﬁne matching of the model to volume surface. Section 2 introduces volume reconstruction from multiple view image silhouettes. Section 3 formulates a shape constrained deformable surface model. In Section 4 we deﬁne the assignment of the model to the surface of the reconstructed volume and give the algorithm for coarse to ﬁne matching of the model to the volume surface. Finally in Section 5 we present results for generating animated human models, ﬁrst in artiﬁcial test cases demonstrating the reconstruction of scanned human data-sets and secondly in a real test case of a dancer in a studio.

2

Volume Reconstruction

A volumetric representation of an object can be obtained from multiple binary image silhouettes by a technique termed “Volume Intersection”, introduced by Martin and Aggarwal [9]. Each silhouette together with the camera view point forms a generalised cone in space that contains the object. The intersection of multiple silhouette cones provides a bounding volume, termed the “visual hull”,

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on the object that approximates the object shape according to the position and number of cameras. Volume intersection is generally performed by dividing the scene into a set of 3D voxels. The voxels are then labelled as either occupied or vacant according to the intersection with the silhouette cones. A voxel is occupied if its projection overlaps all silhouettes and conversely it is vacant if the projection does not overlap any single silhouette. Here we wish to extract the set of surface voxels to generate a surface model. The volume of interest is divided into N×N×N voxels with (N+1)×(N+1)×(N+1) corners. The projection of each corner is tested for overlap with each image silhouette, a corner being marked as non-overlapping if the projection lies outside any silhouette and over-lapping otherwise. A voxel is deﬁned as occupied if it has at least one over-lapping corner. The surface set of voxels is extracted as the set of occupied voxels 6-connected to the non-occupied voxels. An octree representation of the surface voxels is constructed for eﬃcient retrieval of closest surface voxels in model to voxel matching. The normal for each surface voxel is also calculated using a central diﬀerence operator on the occupied voxels. Accurate camera calibration and silhouette extraction is critical for the reconstruction of detailed 3D shape. Calibration errors and silhouette noise typically lead to incorrect labelling of voxels as unoccupied. Our target environment is the virtual production studio and so we are free to use pre-calibrated ﬁxed cameras and a blue-screen background for silhouette extraction.

Fig. 1. Volume reconstruction for 2, 4, 6 and 8 simulated cameras spaced at equal angles around the subject.

It is important to note that volume intersection does not uniquely determine the shape of an object, only the visual hull can be recovered. The visual hull is the

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385

maximal volume that generates the original object silhouettes. Futhermore with only a limited number of viewpoints the apparent occupied volume increases, leading to protrusions around the object as shown in Figure 1. Here we make use of prior information on the human shape by seeking to match a generic human surface model to the surface of the occupied volume.

3

Constrained Surface Model

Standardised methods of representing 3D human models are provided by the VRML Humanoid Animation Working Group (H-Anim) [18]. We adopt a HAnim 1.2 seamless model consisting of a single triangular mesh deﬁning the body shape, attached to an underlying skeleton structure with 17 joints to synthesise the gross movements of the body and texture map(s) attached to change appearance. We make the assumption that the model is ﬁrst posed to match the subject in the images either by manual positioning or selection of joint positions in the images [15]. The model mesh is then deformed to ﬁt the extracted set of surface voxels. Deformable surface models were introduced by Terzopoulos et. al. [16] as a method of recovering shape from images. The problem is formulated as an energy minimisation task across the model surface where the potential energy derived from data ﬁtting is minimised along with the internal energy of the model, Equation (1). The deformation of the model in ﬁtting is then regularised by the changes to the internal energy. The most prominent regularising functional treats the surface as a thin-plate material under tension, producing a smooth model that ﬁts the data with a minimum area and minimum distortion. Etotal = Edata + αEinternal

(1)

Fig. 2. Facet based local frame. Central vertex locations x0 , x1 , x2 are deﬁned by barycentric coordinates u, v, h in outer frame x3 , x4 , x5

We seek to regularise data ﬁtting by minimising the deformation of a triangular mesh from the default shape and default parameterisation. Local mesh shape has been approximated using quadratic surface patches either in vertex

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based or facet based local frames [8]. For a vertex based scheme on an irregular mesh the default position of the central vertex, deﬁning the default shape and parameterisation, is not well deﬁned with respect to the neighbourhood of the vertex. We therefore consider a facet based local patch as introduced by Kobbelt et. al. [8] and shown in Figure (2). The default vertex positions of the central face x0 , x1 , x2 can be deﬁned by the barycentric coordinates in the frame of the surrounding vertices x3 , x4 , x5 of the patch. We minimise the deviation of the mesh vertices from the default vertex positions reconstructed from the default barycentric coordinates in each facet based patch, preserving the local shape and relative parameterisation of the mesh. Equation (2) gives the internal energy of the shape constrained deformable surface where xi is a vertex in the 0 mesh with valence Ni , and (u0if , vif , h0if ) are the default barycentric coordinates of the vertex in the f th facet based frame. Einternal =

0 xi − x(u0if , vif , h0if )2 i

f

Ni

(2)

Figure (3) demonstrates the eﬀect of the internal energy on an irregular mesh with added noise and no data ﬁtting term. The mesh recovers the original shape of the model under the shape constraint, even in the presence of severe distortions.

Fig. 3. Restoration of an irregular mesh (a) original shape (b) shape + noise (c) 50 itertions (d) 100 iterations (e) 150 iterations

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4

387

Model to Volume Surface Matching

Our task is to match the shape constrained active surface model to the surface set of voxels from volume intersection, in order to deform the model to match the apparent shape of a subject in a studio. Kakadiaris et. al. [6] addressed a comparable problem in ﬁtting deformable models to 2D images of body segments, making using of an “all-neighbours” assignment method for model to data matching. Here we use a similar assignment approach. A simple nearest neighbour type assignment would lead to distortions of the surface model according to the initial alignment of the model with the surface voxels. We therefore assign each vertex of the active surface model to all the surface voxels and use the robust point matching technique introduced by Rangarajan et. al. [13] to weight the assignment as the surface deforms. The energy function we seek to minimise in data-ﬁtting is the least-squared error between the model and the set of surface voxels. The energy function is given by Equation (3) where xi spans the set of I model vertices and y k spans the set of K surface voxels. Edata = mik y k − xi 2 (3) i

k

The all-neighbour assignment is deﬁned by a matrix M of continuous match variables mik subject to the following constraints (4): a voxel y k can match one model vertex mik = 1, partially match several vertices i mik ≤ 1 or may not match a vertex mik = 0, and a model vertex xi must be assigned to the voxels k mik = 1. The inequality constraint is transformed to an equality constraint by augmenting the match matrix by the slack variables mI+1,k . 0≤m ≤1 i=I+1 ik mik = 1 i=1 k=K m ik = 1 k=1

(4)

We explicitly constrain the model to match the surface voxels. If the vertices were free to not match the voxels, then the model to voxel assignment could be lost where the restoring forces from the internal energy of the model become comparable to the assignment forces in matching. It is important to note that this approach is therefore only appropriate where the majority of the surface area is reconstructed by volume intersection. We rely on the shape constraint energy of the model to prevent excessive distortions in the presence of occlusions. Rangarajan et. al. [13] make use of deterministic annealing approach to the assignment by adding an entropy term to the energy function. The temperature T of the entropy term deﬁnes the degree of fuzziness in the assignment (5). Edata =

i

k

mik y k − xi 2 + T

i

k

mik (lg(mik ) − 1)

(5)

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The ﬁnal energy equation for the shape constrained deformable model is then given by Equation (6). Etotal = α

0 xi − x(u0if , vif , h0if )2 i

+

i

Ni

f

k

mik y k − xi 2 + T

i

mik (lg(mik ) − 1)

(6)

k

For a ﬁxed temperature T and ﬁxed model conﬁguration xi the match parameters mik can be derived by diﬀerentiating the energy function with respect mik (7). The Sinkhorn balancing technique of alternating row and column normalisation of the match matrix M can then be performed to satisfy the match constraints (4) [1]. Equation (7) shows that a vertex assignment is therefore weighted according to the relative distance to a surface voxel y k − xi with the temperature T deﬁning the eﬀective region of matching in space. As the temperature tends to zero this method approaches nearest neighbour matching. dE dmik

= y k − xi 2 + T lg(mik ) = 0 mik = exp −

(7)

y −xi 2 k T

Similarly for a ﬁxed assignment we can derive the gradient of the energy function with respect to the mesh vertices xi , Equation (8), giving a gradient descent solution for the evolution of the mesh. We seek to ﬁt the active surface model to the surface voxels by alternately updating the assignment to the surface voxels mik using Equation (7) and Sinkhorn balancing of the match matrix M to satisfy the constraints in Equation (4), then updating the vertex positions of the model xi using Equation (8). The temperature is reduced as the model is updated to reﬁne the assignment, giving a coarse to ﬁne approach to matching. dxt dE 0 = − t = −α xi − x(u0if , vif , h0if ) + mik (y k − xi ) dt dx i i f

(8)

k

We start at an initial temperature Tinit that gives a matching region larger than the greatest expected error, typically we set Tinit = 0.4m2 . The match parameters are initialised according to Equation (7) and the slack matches mI+1,k set to the minimum match mik for each voxel k. We then iteratively update the model and reduce the temperature until the ﬁnal temperature Tf inal is reached. Tf inal is set in the order of the voxel size for a close ﬁt to the voxel surface. At a set temperature and deﬁned matches, the model is updated using explicit Euler integration steps until convergence. The convergence criteria is set as the maximum component of the √ gradient falling within the current region of matching, dE that is max { dx } ≤ T. i The number of surface voxels is potentially large, depending on the voxel size chosen in volume intersection. In practice we therefore restrict the number of voxels used in matching for each vertex to a closest sub-set. This also leads to

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more eﬃcient storage of the match matrix M . We use the octree representation of the surface voxels to eﬃciently retrieve the closest voxels. We also only match a vertex to those voxels whose surface normal lies in the same half-plane as the vertex normal in order to avoid incorrect surface matches.

5

Results

The technique was tested using virtual views generated for human 3D scanned data-sets, provided by Cyberware, in a simulated studio. A varying number of cameras were positioned in the studio at equal spacing and 3m radius around each data-set. An ideal pin-hole camera model was used with NTSC resolution and typical intrinsic parameters. Images of the data-sets were then rendered in OpenGL and used to test the shape reconstruction. Volume intersection was performed using a 1cm voxel size. Both the internal and data ﬁtting energy terms represent geometric errors at the mesh vertices and we give equal weight to both terms in matching, α = 1. We ﬁnd this provides adequate data ﬁtting without the need for ﬁne tuning of the parameter α, although we note that this may be case speciﬁc and that this requires further investigation.

Fig. 4. Error distribution (/m) in ﬁtting to voxel surface from 2,4,6,8 simulated cameras.

Figure (4) shows the distribution of errors between the deformed active surface model and the original scanned data-set across the surface of the model, for the surface voxels reconstructed from 2,4,6 and 8 cameras as shown in Figure 1. The Metro tool [2] has been used here to measure and display the errors. We can see that the shape constrained surface leads to feasible shape solutions even for a limited number of camera views in the presence of large deviations in the reconstructed volume from the true body shape. The shape constraint of the active surface model leads to a smooth approximation of the body shape. The model cannot recover the ﬁne detail and concavities in the original data-set that are not reconstructed in the visual hull. The illusion of additional surface shape

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information would be obtained by texture mapping the smooth model. Figure (5) shows the reconstructed shape for a dancer in a 6 camera studio.

6

Conclusions

We have presented a method for 3D reconstruction of animated models from a limited number of camera images. We introduce a model-based approach to shape from silhouette that preserves the prior shape information in the model, regularising data ﬁtting, and the parameterisation of the model for animation. A generic animated model is ﬁrst manually posed. The model is then matched to multiple image silhouettes simultaneously by ﬁtting to the reconstructed visual-hull of an object. We formulate an irregular triangular mesh model as a shape constrained deformable surface. The mesh is deformed to match the reconstructed volume using a robust coarse to ﬁne matching technique. This has been applied to reconstruct smooth animated models of humans. Further work is required to test this approach for a variety of subjects wearing diﬀerent clothing in a studio. The task also remains to apply colour texture from the images to generate photo-realistic models.

Fig. 5. Reconstructed model for a dancer in a studio showing 3 of the 6 captured image silhouettes and the reconstructed visual-hull

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References 1. H. Chui and A. Rangarajan. A New Algorithm for Non-Rigid Point Matching. In IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume 2, pages 44–51, 2000. 2. P. Cignoni, C. Rocchini, and R. Scopigno. Metro: measuring error on simpliﬁed surfaces. Computer Graphics Forum, 17(2):167–174, 1998. 3. L. Davis, E. Borovikov, R. Cutler, D. Harwood, and T. Horprasert. Multiperspective analysis of human action. In Proceedings of the 3rd International Workshop on Cooperative Distributed Vision, Kyoto, Japan, 1999. 4. P. Fua, R. Plankers, C. Miccio, and D. Thalmann. From Image Synthesis to Image Analysis: Using Human Animation Models to Guide Feature Extraction. EPFLLIG Computer Graphics Lab, Switzerland, 1998. 5. A. Hilton, D. Beresford, T. Gentils, R. Smith, and W. Sun. Virtual people: Capturing human models to populate virtual worlds. In IEEE International Conference on Computer Animation, May 1999. 6. I.A. Kakadiaris and D. Metaxas. Three-dimensional human body model acquisition from multiple views. International Journal of Computer Vision, 30(3):191–218, 1998. 7. T. Kanade, P. Rander, and P.J. Narayanan. Virtualized reality: Constructing virtual worlds from real scenes. IEEE Multimedia, 4(1):34–47, 1997. 8. L. Kobbelt, S. Campagna, J. Vorsatz, and H.P. Seidel. Interactive Multi-Resolution Modeling on Arbitrary Meshes. In SIGGRAPH Conference Proceedings, pages 105– 114, August 1998. 9. W.N. Martin and J.K. Aggarwal. Volumetric descriptions of objects from multiple views. IEE Transactions on Pattern Analysis and Machine Intelligence, 5(2):150– 158, 1983. 10. W. Matusik, C. Buehler, R. Raskar, S.J. Gortler, and L. McMillan. Image-based visual hulls. In SIGGRAPH Conference Proceedings. ACM SIGGRAPH, 2000. 11. T. McInerney and D. Terzopoulos. Deformable models in medical image analysis: a survey. Medical Image Analysis, 1(2):91–108, 1996. 12. J. Montagnat and H. Delingette. Volumetric Medical Images Segmentation using Shape Constrained Deformable Models. In First Joint Conference CVRMedMRCAS, volume Lecture Notes in Computer Science 1205, pages 13–22, March 1997. 13. A. Rangarajan, H. Chui, E. Mjolsness, S. Pappu, L. Davachi, P. Goldman-Rakic, and J. Duncan. A robust point matching algorithm for autoradiograph alignment. Medical Image Analysis, 4(1):379–398, 1997. 14. L.H. Staib and J.S. Duncan. Model-Based Deformable Surface Finding for Medical Images. IEEE Transactions on Medical Imaging, 15(5):720–731, 1996. 15. J. Starck, A. Hilton, and J. Illingworth. Human Shape Estimation in a MultiCamera Studio. In BMVC, volume 2, pages 573–582, September 2001. 16. D. Terzopoulos, A. Witkin, and M. Kass. Symmetry-seeking models and 3D object reconstruction. International Journal of Computer Vision, 1:211–221, 1987. 17. D. Thalmann, J. Shen, and E. Chauvineau. Fast human body deformations for animation and VR applications. In Proceedings of Computer Graphics International, pages 166–174. IEEE Computer Society Press, June 1996. 18. VRML Humanoid Animation Working Group, http://www.h-anim.org/. H-ANIM 1.1 Speciﬁcation, 1999.

Reconstruction of Binary Matrices from Absorbed Projections E. Balogh1 , A. Kuba1 , A. Del Lungo2 , and M. Nivat3 1

3

Department of Applied Informatics, University of Szeged, ´ ad t´er 2, H-6720 Szeged, Hungary Arp´ {bmse,kuba}@inf.u-szeged.hu 2 Department of Mathematics, University of Siena, Via del Capitano 15, 53100 Siena, Italy [email protected] Laboratoire d’Informatique Algorithmique: Fondements et Applications, Universit´e Paris 7 Denis-Diderot, Paris, France [email protected]

Abstract. A generalization of the classical discrete tomography problem is considered: Reconstruct binary matrices from their absorbed row and column sums. We show that this reconstruction problem can be linked to a 3SAT√ problem if the absorption is characterized with the constant β = ln( 1+2 5 ). Keywords: discrete tomography, reconstruction, absorption

1

Introduction

Let A = (aij )m×n be a binary matrix and let be β ≥ 1. Then we can deﬁne the absorbed row and column sums of A Rβ (A) and Sβ (A), respectively, as Rβ (A) = R = (r1 , . . . , rm )

where

ri =

n

aij β −j ,

i = 1, . . . , m, (1)

aij β −i ,

j = 1, . . . , n.

j=1

and Sβ (A) = S = (s1 , . . . , sn )

where

sj =

m

(2)

i=1

Then the reconstruction problem of binary matrices with absorption knowing the projections along horizontal and vertical lines can be posed as Reconstruction DA2D(β). Instance: β ≥ 1, m, n, R ∈ Nm , and S ∈ Nn Task: Construct a binary matrix A with size m × n such that Rβ (A) = R

and

Sβ (A) = S .

(3)

A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 392–403, 2002. c Springer-Verlag Berlin Heidelberg 2002

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If β = 1 then we have the classical reconstruction problem of binary matrices without absorption (as summaries see e.g. [1,2]). Other β values are suitable to describe the following model of the emission discrete tomography. Let us suppose that the discrete object represented by the binary matrix A is in an absorbing material having absorption coeﬃcient µ. If we measure the horizontal and vertical projections of A, then we have the absorbed row and column sums, i.e., Rβ (A) and Sβ (A), where β = eµ . Some more explanation to the motivation of this problem see [3,4] Select, for example, the mathematically interesting case β = β0 where β0−1 = β0−2 + β0−3

(4)

giving a solution β0

√ 1+ 5 = . 2

(5)

In this paper we discuss the problem of reconstruction of binary matrices from their row and column sums in the case of absorption characterized with β0 . A necessary and suﬃcient condition of uniqueness in this class is published in [3,4]. In this paper we are going to connect this kind of reconstruction problem with 3SAT. The SAT and the diﬀerent reconstruction problems have been connected already in [5,6].

2

β0 -Representation

Consider the row and column sums of the binary matrix A in the case of β = β0 : ri =

n

aij β0−j , i = 1, . . . , m, and sj =

j=1

m

aij β0−i , j = 1, . . . , n.

(6)

i=1

Using the terminology of numeration systems we can say that the ﬁnite (binary) word ai1 · · · ain is a (ﬁnite) representation in base β0 (or a ﬁnite β0 representation) of ri for each i = 1, . . . , m, and, similarly, a1j · · · anj is a β0 representation of sj for each j = 1, . . . , n. The equations (6) mean also that the row and column sums of A are nonnegative real numbers having a ﬁnite β0 representation with n and m binary digits, respectively (including the eventually ending zeros). Let Bk denote the set of nonnegative real numbers having a β0 -representation with k binary digits (k > 1), formally, Bk =

k i=1

Then

ai β0−i | ai ∈ {0, 1}

.

(7)

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ri ∈ Bn , i = 1, . . . , m,

and

sj ∈ Bm , j = 1, . . . , n,

(8)

are necessary conditions for the existence of a matrix A with Rβ0 (A) = (r1 , . . . , rm ) 2.1

and

Sβ0 (A) = (s1 , . . . , sn ) .

(9)

Switching in β0 -Representations

The β0 -representation is generally nonunique, because there are binary words with the same length representing the same number. For example, on the base of (4) it is easy to check the following equality between the 3-digit-length β0 representations 100 = 011 .

(10)

As direct consequences of (10), it is easy to see that 100 10x3 00 10x3 0x5 00 10x3 0x5 0x7 00

= = = = ...

011 01x3 11 01x3 1x5 11 01x3 1x5 1x7 11

(11)

where x3 , x5 , x7 , · · · denote the positions where both β0 -representations have the same (but otherwise arbitrary) binary digit. (That is, such kind of transformation 1(0x)k−1 00 → 0(1x)k−1 11 (k ≥ 1) between the subwords of the β0 representations can be performed without changing the represented value and without changing the values in the positions indicated by x’s.) The transformations described by (10) and (11) are called switchings. It is proved that any ﬁnite β0 -representation of a number can be get from its any other β0 -representation by switchings. Lemma 1. [3] Let a1 · · · ak and b1 · · · bk be diﬀerent, k-digit-length β0 representations of the same number. Then b1 · · · bk can be get from a1 · · · ak by a ﬁnite number of switchings. Consequence. If a1 · · · ak and b1 · · · bk are diﬀerent, k-digit-length β0 -representations of the same number, then there are positions i, i + 1, i + 2 (1 ≤ i ≤ k − 2) such that there is a switching between a1 · · · ak and b1 · · · bk on these positions. 2.2

β0 -Expansion

The k-digit-length β0 -expansion is a particular k-digit-length β0 -representation that can be computed by the “greedy algorithm”: Let r ∈ Bk , then its β0 expansion a1 · · · ak is determined as r0 := r , ai := β0 · ri−1 ,

ri := {β0 · ri−1 },

i = 1, . . . , k ,

(12)

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where . and {.} denote the integer and fractional, respectively, part of the argument. It is clear that the k-digit-length β0 -expansion of any number r ∈ Bk is uniquely determined (it is not the case with the k-digit-length β0 -representations as we saw it in the previous subsection). The ﬁnite β0 -expansion is characterized by the following property . Proposition 1. [3] Let a1 , . . . , ak ∈ {0, 1} (k ≥ 1). The word a1 · · · ak is the β0 -expansion of a number r ∈ Bk if and only if it has the form a1 · · · ak = T U V , where T = 0 · · · 0,

T = 1 · · · 1,

or

T =λ,

(13)

(λ denotes the empty symbol), U = U1 · · · Uu ,

u ≥ 0,

such that

Ui = 10 · · · 0,

i = 1, . . . , u, (14)

and each Ui contains at least one 0, V = 1

or

V = λ

(15)

and at least one of T , U , and V is not the empty symbol λ.

3

β0 -Representation and 3SAT Clauses

We are going to describe the β0 -representation by 3SAT expressions, that is, by Boolean expressions in conjunctive normal form with at most three literals in each clause. Let r be a real number having a k-digit long β0 -representation, a1 · · · ak . Let z1 , . . . , zk be Boolean variables and L be a Boolean function of z1 , . . . , zk , that is, L = L(z1 , . . . , zk ). We say that the Boolean values a1 , . . . , ak satisfy L if L(z1 = a1 , . . . , zk = ak ) is true. Now we are going to give the set of clauses, denoted by K, by which all k-digit length β0 -representations of any r ∈ Bk can be described for any k > 1. Let a1 · · · ak the k-digit-length β0 -expansion of r. Then, by Proposition 1, a1 · · · ak = T U V , where T , U , and V are given by (13), and (14), respectively. Accordingly, K = TT ∪ UU ∪ V V ,

(16)

where T T , U U , and V V denote the subsets of clauses describing the corresponding parts T , U , and V . First, consider the non-constant part of the β0 -representations, U = U1 · · · Uu (u ≥ 0). On the base of Lemma 1 we know that all β0 -representations of any r ∈ Bk can be generated from the β0 -expansion of r by elementary switchings. Accordingly, the clauses U U have to describe the set of binary words generated from Uk by elementary switchings (see Fig. 1). The elementary switchings done in U can be classiﬁed into two classes according to the places of switchings:

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(i) The switchings done in the positions of one Ui . (ii) The switchings done in the positions of Ui and Ui+1 , i.e. the last 1 of Ui “overﬂows” into the ﬁrst position of Ui+1 as a consequence of switchings. There can be such a switching if the length of Ui is even and the length of Ui+1 is not less than 3 (see the β0 -representations in Fig. 1 indicated by arrows). There are two consequences of overﬂowing switchings: We have diﬀerent clauses for Ui having even or odd length li and the sets of clauses of Ui , i = 1, · · · u, are not completely independent. The clauses of U U are given with the help of the Boolean variables γj , δj , ϕj , ψj , and χj , j = w1 , w1 + 1, · · · , wu + lu − 1, i.e. for all the variables of U U . For each j exactly one of these variables has value 1 (see the clauses of P OSIT ION S later). For this reason each binary word satisfying the clauses described by these variables can be represented in a 1-to-1 correspondence by a word of the alphabet {γ, δ, ϕ, ψ, χ}, indicating which variable has value 1 on that position. For example, z1 z2 z3 = ψγδ means that γ1 = 0, γ2 = 1, γ3 = 0, δ1 = 0, δ2 = 0, δ3 = 1, ϕ1 = 0, ϕ2 = 0, ϕ3 = 0, ψ1 = 1, ψ2 = 0, ψ3 = 0, χ1 = 0, χ2 = 0, χ3 = 0. The variables γj , δj , ϕj , ψj , and χj describing the clauses of U U will be transformed to 0’s and 1’s as follows: ϕj ⇒ aj = 0, δj ⇒ aj = 1, ψj ⇒ aj = 0, γj ⇒ aj = 1, χj ⇒ aj = 1 .

(17)

Continuing the previous example, then ψγδ = 011. Ui Ui+1 | 100000 | 10000 | | 011000 | 10000 | | 010110 | 10000 |

corresponding representations | δϕϕϕϕϕ | δϕϕϕϕ | | ψγδϕϕϕ | δϕϕϕϕ | | ψγψγδϕ | δϕϕϕϕ |

| 100000 | 01100 | | 011000 | 01100 | | 010110 | 01100 | | 010101 | 11100 |

| δϕϕϕϕϕ | ψγδϕϕ | | ψγδϕϕϕ | ψγδϕϕ | | ψγψγδϕ | ψγδϕϕ | | ψγψγψγ | χγδϕϕ | ←−

| 100000 | 01011 | | 011000 | 01011 | | 010110 | 01011 | | 010101 | 11011 |

| δϕϕϕϕϕ | ψγψγδ | | ψγδϕϕϕ | ψγψγδ | | ψγψγδϕ | ψγψγδ | | ψγψγψγ | χγψγδ | ←−

Fig. 1. All β0 -representations of Ui Ui+1 generated by elementary switchings and the corresponding representations with the variables γ, δ, ϕ, ψ, and χ (when li = 6 and li+1 = 5). The positions of Ui and Ui+1 are separated by vertical lines. The “overﬂowing” 1’s are indicated by χ in the rows with arrows.

Let B(Ui ) denote the set of (binary) sequences of Ui . Clearly, B(Ui ) = {01}b 1{0}c ,

(18)

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where b and c are nonnegative integers such that b + 1 + c = li . Then the binary sequences of Ui Ui+1 , B(Ui Ui+1 ), can be given as B(Ui )B(Ui+1 ), if li is odd B(Ui Ui+1 ) = (19) B(Ui )B(Ui+1 ) ∪ {01}li /2 1B (0) (Ui+1 ), if li is even, where B (0) (Ui+1 ) denotes the set of subsequences created from those sequences of B(Ui+1 ), where the ﬁrst element is 0, by omitting just this ﬁrst 0. For example, if li = 6 and li+1 = 5 then B(Ui ) = {100000, 011000, 010110}, B(Ui+1 ) = {10000, 01100, 01011}, and B (0) (Ui+1 ) = {1100, 1011}. We can describe these sequences with the letters γ, δ, ϕ, ψ, and χ as follows. Corresponding to (18) and (19) B(Ui ) = {ψγ}a δ{ϕ}b , B(Ui Ui+1 ) =

(20)

B(Ui )B(Ui+1 ), if li is odd B(Ui )B(Ui+1 ) ∪ {ψγ}li /2 χB (ψ) (Ui+1 ), if li is even.

(21)

According to (17) ψ, ϕ denote 0, γ, δ, and χ denote 1. B and B (ψ) are deﬁned in these sequences analogously to (18) and (19). Examples of generated in this way and the corresponding β0 -representations are in Fig. 1. The following sets of clauses will deﬁne a subword Ui .

DELT A =

wi +l i −2

(δj ⇒ ϕj+1 ) ∧

j=wi [

wi +l i −1

(δj ⇒ γj−1 ) ∧

j=wi +1

li

2 ]

δwi +2j−1 ∧ (ϕwi +1 ⇒ δwi ) .

j=1

The position of δ is crucial, because knowing this position all the elements succeeding δ can be computed as it is described in the ﬁrst part of this rule and all elements preceding δ can be computed as it is described in the second part. δ cannot be on an even position in the subword Ui . The last part of DELT A expresses that if there is a ϕ in the second position then there is a δ in the ﬁrst one. P HI =

wi +l i −2

(ϕj ⇒ ϕj+1 ) ∧ ϕwi .

j=wi +1

In other words, ϕ can be followed only by ϕ and ϕ cannot stand on the ﬁrst position of the subword. GAM M AP SI =

wi +l i −1 j=wi +2

(γj ⇒ ψj−1 ) ∧

wi +l i −2 j=wi

(ψj ⇒ γj+1 ) .

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The only predecessor of γ is ψ and the only successor of ψ is γ. CHI =

wi +l i −1

χj ∧ (χwi ⇒ γwi +1 ) .

j=wi +1

χ can stand only on the ﬁrst position. GAM M A =

wi +l i −1

(γj ⇒ ϕj+1 ) .

j=wi

γ cannot be followed by ϕ.

[

P OSIT ION S =

li

2 ]

(ϕwi +2j ∨ γwi +2j ) ∧ (δwi ∨ ψwi ∨ χwi ) ∧

j=1 [

li

2 ]

(δwi+2j−1 ∨ ψwi+2j−1 ∨ ϕwi+2j−1 ) .

j=1

On an even position in a subword can be ϕ or γ, on the ﬁrst position in the subword can stand δ, ψ, or χ, and on odd positions in the subword can stand δ, ψ, or ϕ. These are the only clauses containing 3 variables. EV EN = (γwi +li −1 ⇒ χwi +li ) . Actually wi + li = wi+1 , the ﬁrst element of the subword Ui+1 . This means that a subword with even length can inﬂuence the next subword. In this case the ﬁrst element is a χ followed by γ. ODD = χwi +li ∧ ψwi +li −1 . A subword with odd length cannot inﬂuence the next subword, this means that the ﬁrst element of the next subword cannot be χ and the last element cannot be ψ. DISJ =

li

(Aj ⇒ Bj ), for symbols A, B ∈ {ϕ, ψ, γ, δ, χ}, where A = B .

j=1

The clauses mean that exactly one of the variables ϕ, ψ, γ, δ, and χ has the value 1, for each j = 1, · · · , li .

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The clauses for a subword Ui . Knowing the length of the subword Ui we can construct a corresponding 3SAT expression:   DELT A ∧ P HI ∧ GAM M AP SI ∧ CHI      if li is odd ∧GAM M A ∧ P OSIT ION S ∧ ODD, Ki = (22)    DELT A ∧ P HI ∧ GAM M AP SI ∧ CHI    ∧GAM M A ∧ P OSIT ION S ∧ EV EN, if li is even The clauses describing U U . Let Γ = (γ1 , · · · , γk ), ∆ = (δ1 , · · · , δk ), Φ = (ϕ1 , · · · , ϕk ), Ψ = (ψ1 , · · · , ψk ), and X = (χ1 , · · · , χk ) be the vectors of Boolean variables. Then U U = U U (r; Γ, ∆, Φ, ψ, X) is deﬁned as follows: UU =

u

Ki .

i=1

The clauses describing T T and V V . In these clauses the same variables are as used in U U . Since the subwords corresponding to T and V have constant values in each β0 -representation of the same r ∈ Bk , the clauses describing these parts are  γ1 = · · · = γlt = 0, δ1 = · · · = δlt = 0,      ϕ1 = · · · = ϕlt = 0, ψ1 = · · · = ψlt = 0,      χ1 = · · · = χlt = 1,       T T = γ1 = · · · = γlt = 0, δ1 = · · · = δlt = 1,   ϕ1 = · · · = ϕlt = 0, ψ1 = · · · = ψlt = 0,      χ1 = · · · = χlt = 0,         φ, and

if T = 0 · · · 0;

(23) if T = 1 · · · 1;

if T = λ ,

γk = 0, δk = 1, ϕk = 0, ψk = 0, χk = 0, if V = 1; VV = φ, ifV = λ ,

(24)

The clauses describing K. As we saw T T , U U , V V , and so K are deﬁned with the help of r, Γ , ∆, Φ, Ψ , and X, i.e., K = K(r; Γ, ∆, Φ, Ψ, X) . K is given by (16) explicitly.

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Theorem 1. Let r ∈ Bk and a1 , . . . , ak be a binary word. a1 , . . . , ak is a β0 -representations of r if and only if there are vectors Γ, ∆, Φ, Ψ , and X of Boolean values such that a1 , . . . , ak is transformed by these vectors by (17) and K(r; Γ, ∆, Φ, Ψ, X) is true. Proof. Let a1 · · · ak be a k-digit-length β0 -representation of r. The corresponding word of γ, δ, ϕ, ψ, and χ is uniquely determined on the base of the forms (20) and (21). It is easy to check that all clauses of K (i.e. T T , V V , DELT A, · · · , DISJ) are satisﬁed by any word given by (20) and (21). In order to prove the other direction, consider an arbitrary word W satisfying the clauses of K. W has the uniquely determined structure T U V , where T same as (13), V same as (15) and U is a word of γ, ψ, ϕ, δ, and, χ. We have to show that U is a sequence of subsequences Ui , each of them satisfying (20) and (21). Knowing r we can determine the lengths li and positions of all Ui , i = 1, · · · u. Now we identify the subsequence Ui with length li starting from the end of U. 1. li is odd. According to P OSIT ION S, in the li th position can be δ, ψ, or ϕ. a. In the li position there is a δ. Now we have to prove that before δ there are only pairs of ψγ. From DELT A it follows that in the position li − 1 there is a γ. Let γ the position 2j, before δ. From GAM M AP SI it follows that in the position 2j − 1 there is a ψ. From P OSIT ION S it follows that in the position 2j − 2 there can be ϕ or γ. If in the position 2j − 2 is a ϕ, then according to P HI in the position 2j − 1 should be ϕ which is a contradiction (from DISJ) . This means, that in the position 2j − 2 is a γ, and let j = j − 1. This step has to be repeated till j > 1. If j = 1, i.e. in the second position is γ, then from P OSIT ION S we have that in the ﬁrst position can be δ, ψ, or χ. If in the ﬁrst position is δ then from DELT A follows that in the second position should be ϕ which is a contradiction. Conform to the equations (20) and (21), in the ﬁrst position can be ψ or χ, in this last case there is an overﬂow. b. In the li position there is a ψ This in contradiction with ODD. c. In the li position there is a ϕ. From P OSIT ION S it follows that in the previous position can be ϕ or γ. If it is γ, then from GAM M A it follows that in the li th position cannot be ϕ which is a contradiction. This means, that in the position li − 1 is ϕ. If li − 2 = 1 then in this position is δ (from DELT A). If li − 2 > 1 then from P OSIT ION S it follows that in the position li − 2 can be δ, ψ, or ϕ. If in the position i − 2 is δ then similar to Case a. we can prove that Ui satisﬁes (20) and (21). If in the position li − 2 is ϕ then similar to Case c. we can prove that Ui satisﬁes (20) and (21). If in the position li − 2 is ψ then from GAM M AP SI follows that in the position li − 1 is γ and this is in contradiction with DISJ. 2. li is even. According to P OSIT ION S in the position li can be ϕ or γ. If it is ϕ then using a similar deduction as in Case C. we can prove that Ui satisﬁes (20) and (21). If in the position li is γ, then conform EV EN in the

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next position is χ and conform CHI in the position li + 2 is γ, which means that Ui satisﬁes (20) and (21).

4

The Reconstruction Algorithm

In order to solve the reconstruction problem DA2D(β0 ) we express the β0 representations of the absorbed row and column sums with 3SAT clauses. (h) (h) (h) Boolean variables Γ (h) = (γij )m×n , ∆(h) = (δij )m×n , Φ(h) = (ϕij )m×n , (h)

(h)

Ψ (h) = (ψij )m×n , and X (h) = (χij )m×n are for describing relations of column (v)

(v)

sums (h stands for horizontal), and Γ (v) = (γij )m×n , ∆(v) = (δij )m×n , Φ(v) = (v)

(v)

(v)

(ϕij )m×n , Ψ (v) = (ψij )m×n , and X (v) = (χij )m×n for describing relations of (h)

(h)

(h)

(v)

(v)

column sums (v stands for vertical), Let, furthermore, Γi· = (γi1 ), · · · , γin ) (v) (v) (V ) be the ith row of Γ (h) , i = 1, · · · , m and Γ·j = (γ1j ), · · · , γmj )T be the jth (h)

(h)

(h)

(h)

(v)

(v)

column of Γ (v) , j = 1, · · · , n. ∆i· , Φi· , Ψi· , Xi· , ∆·j , Φ·j , Ψ·j , and X·j , be deﬁned similarly. The clauses describing the rows and columns. Now we can describe a whole row of the discrete set to be reconstructed by the following subset of clauses: (h)

(h)

(h)

(h)

(h)

K (h) (ri ; Γi· , ∆i· , Φi· , Ψi· , Xi· ) = T T ∧ U U ∧ V V, i = 1, · · · , m , where T T , U U , and V V are deﬁned in the previous section. All clauses describing the absorbed row sums are given by L(h) = L(h) (R, Γ (h) , ∆(h) , Φ(h) , Ψ (h) , X (h) ) m (h) (h) (h) (h) (h) = K (h) (ri ; Γi· , ∆i· , Φi· , Ψi· , Xi· ) .

(25)

i=1

Similarly, the columns can be described by (v)

(v)

(v)

(v)

(v)

K (v) (sj ; Γ·j , ∆·j , Φ·j , Ψ·j , X·j ) = T T ∧ U U ∧ V V, j = 1, · · · , n , and L(v) = L(v) (R, Γ (v) , ∆(v) , Φ(v) , Ψ (v) , X (v) ) m (v) (v) (v) (v) (v) = K (v) (sj ; Γ·j , ∆·j , Φ·j , Ψ·j , X·j ) . i=1

(26)

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The clauses describing the binary matrix. The last step is to deﬁne the connections between the Boolean matrices CON N = (

(h)

i1,j1

∧(

i1,j1

∧(

i1,j1

∧(

i1,j1

(v)

ϕi1,j1 ⇒ γi1,j1 ) ∧ (

(h)

i1,j1

(h)

(v)

(h) γi1,j1

(v) ϕi1,j1 )

ψi1,j1 ⇒ γi1,j1 ) ∧ (

i1,j1

(h)

⇒

(v)

∧(

i1,j1

δi1,j1 ⇒ ψi1,j1 ) ∧ (

i1,j1

(v)

ϕi1,j1 ⇒ δi1,j1 ) ∧ (

(h)

i1,j1

(h)

(v)

ψi1,j1 ⇒ δi1,j1 ) ∧ (

i1,j1

(h) γi1,j1 (h)

⇒

(v) ψi1,j1 )

∧(

(v)

i1,j1

χi1,j1 ⇒ ϕi1,j1 ) ∧ (

i1,j1

(v)

ϕi1,j1 ⇒ χi1,j1 ) ∧ (h)

(v)

(h)

(v)

ψi1,j1 ⇒ χi1,j1 ) ∧ δi1,j1 ⇒ ϕi1,j1 ) ∧ (h)

(v)

χi1,j1 ⇒ ψi1,j1 ) .

The 3SAT expression describing the whole discrete set is: L(h) ∧ L(v) ∧ CON N .

(27)

That is, in order to solve the reconstruction problem DA2D(β0 ) we have to do the following steps: 1. 2. 3. 4.

5

Determine the β0 -expansions of ri , i = 1, · · · , m, and j = 1, · · · n. On the base of β0 -expansions give the 3SAT expression (27). Solve the 3SAT problem using an eﬃcient SAT solver (e.g. CSAT, see [7]). If there is a solution of the 3SAT problem, give the binary matrix solution on the base of (17).

Discussion

A method is given to solve the reconstruction problem DA2D(β0 ), i.e., to reconstruct a binary matrix from it absorbed row and column sums, when the absorption can be represented by the special value β0 . It is shown that the problem DA2D(β0 ) can be transformed to a 3SAT expression such that if there is a solution of the 3SAT expression then it gives also a solution of the reconstruction problem (see Section 4). It is a natural question that how this method can be extended to other values of β. We believe that this idea is speciﬁc and cannot be generalised directly to all possible values of β. However, it is relative easy to show that very similar results are true for β’s having the property β −1 = β −2 + β −3 + · · · + β −l , where l ≥ 3. Then the switchings can be described by similar relations as in (11), β-representations can be given similarly as in Section 3, and so the reconstruction problem can be reduced to a 3SAT problem in such cases.

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Acknowledgements. This work was supported by the grant OTKA T 032241.

References 1. Brualdi, R.A.: Matrices of zeros and ones with ﬁxed row and column sums. Linear Algebra and Its Applications 33 (1980) 159-231. 2. Herman, G.T., Kuba, A. (Eds.): Discrete Tomography: Foundations, Algorithms and Applications. Birkh¨ auser, Boston (1999). 3. Kuba, A., Nivat, M.: A Suﬃcient condition for non-uniqueness in binary tomography with absorption, Technical Report, University of Szeged (2001). 4. Kuba, A., Nivat, M.: Reconstruction of discrete sets with absorption, accepted for publication in Linear Algebra and its Applications (2001). 5. M. Chrobak, C. D¨ urr, Reconstructing hv-convex polyominoes from orthogonal projections, Information Processing Letters 69 (1999) 283–289. 6. Y. Boufkhad, O. Dubois, and M. Nivat, Reconstructing (h,v)-convex twodimensional patterns of objects from approximate horizontal and vertical projections, to appear in Theoretical Computer Science. 7. O. Dubois, P. Andr´e, Y. Boufkhad, and J. Carlier, SAT versus UNSAT, in Second DIMACS Implementation Challenge, D. Johnson and M. A. Trick, eds., DIMACS Series in Discrete Mathematics and Theoretical Computer Science, AMS, 1993.

A Simpliﬁed Recognition Algorithm of Digital Planes Pieces Mohammed Mostefa Mesmoudi Department of Computer Science and Information Science (DISI) Science University of Genoa Via Dodecaneso, 35 -16146 Genoa (Italy) [email protected]

Abstract. Debled proposed an eﬃcient algorithm for the recognition of rectangular digital planes pieces. However, uses, in some cases called strongly exterior cases, some validity criteria which are only suﬃcient but not necessary. In this paper we give necessary and suﬃcient conditions (including strongly exterior cases) to recognize pieces of digital planes. We build up a simpliﬁed form of Debled’s algorithm. Furthermore, our approach is independent from the rectangular form of pieces considered by Debled.

1

Introduction

Let ν be a 18-connected bounded convex subset of Z 3 which injectively projects on a subset Π of the plane Oxy. Let ω be a positive integer, if there exist integers a, b, c, µ in Z such that ν is the set of solutions of the double Diophantine inequality µ ≤ ax + by + cz < µ + ω (1) (x, y) ∈ Π then we say that ν is a piece of a digital plane P (a, b, c, µ, ω) with characteristics a, b, c and lower bound µ. The number ω is called the arithmetic thickness of the plane. When ω = sup(|a|, |b|, |c|), the plane P (a, b, c, µ, ω) is called a na¨ıve plane and denoted by P (a, b, c, µ) [1]. Let us consider a family {νt }t∈[µ,µ+ω−1] of real parallel planes νt deﬁned by ax + by + cz = t. The digital plane ν is the intersection of the above family with Z 3 that can be geometrically represented by a set of voxels as shown in Figure 1. Let us suppose that the greatest common divisor of a, b, c is 1. We say that ν is recognized if it contains suﬃciently points to compute all the characteristics a, b, c, µ, ω. To this aim, it is suﬃcient that ν possesses four aﬃnely independent points that generate the bounding real planes νµ and νµ+ω−1 . The problem of studying pieces of digital na¨ıve planes can be reduced to the case where 0 ≤ a ≤ b ≤ c. Indeed, the general case can be obtained from the former case by rotations and symmetries [3]. There is a wide literature on the problem of recognizing digital plane pieces. Kim and Rosenfeld showed in [6] that a digital surface is a piece of a na¨ıve digital plane if and only A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 404–416, 2002. c Springer-Verlag Berlin Heidelberg 2002

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if there exists a face of the convex hull of the surface such that the distance between the points of the surface and the plane that supports the face is less than 1. They proposed an algorithm based on this property of complexity O(p4 ), where p is the number of points on the surface. In 1991, Kim and Stojmenovi´c [7] improved this algorithm to obtain another algorithm of complexity O(p2 logp). In the same year, Stojmenovi´c and To˜si´c [11] presented two other algorithms, the ﬁrst with complexity O(plogp) based on the construction of two convex hulls and the second with complexity O(p) and based on linear programming in 3D. These algorithms have a low complexity but they are not incremental which is a drawback in application. Furthermore, the construction of the convex hull in 3D is a delicate and expensive operation. In [13,12], Veelaert, relying on a generalization of a regularity property of digital straight lines introduced by Hung [5], developed a simple algorithm of complexity O(p2 ) which is satisfactory for small sets (p ≤ 100).

ν

y O x Π

Fig. 1. The shape of a possible digital plane piece

The arithmetic deﬁnition of digital planes led by relation (1) was introduced for the ﬁrst time by Reveilles in 1991, [10]. In 1995, Debled proposed, in her PhD thesis [3], an eﬃcient algorithm for the recognition of rectangular pieces of digital planes. This algorithm has more advantages than the other algorithms quoted above. It uses a simple and intuitive geometric properties of digital planes. It has a quadratic complexity and is incremental. It consists in sweeping a piece by sections that are parallel to a co-ordinate plane. At the beginning, one ﬁxes, for instance, y = 0 and let x vary. In this case, Debled’s algorithm tries to recognize digital straight lines until x reaches its maximum value in Π. Then, y is incremented by 1 and x varies again. In this case, Debled’s algorithm tries to recognize pieces of digital planes. At each step the algorithm tries to compute the characteristics of the new plane. Three cases may occur – If the added point M (x, y) satisﬁes relation (1), then the same characteristics of the last recognized piece are kept. – If the added point M satisﬁes one of the following relations ax + by + cz = µ − 1

or

ax + by + cz = µ + c

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where (a, b, c) is the normal vector of the last recognized piece, the point M is said to be 1-exterior to the last piece. Debled conjectured that the new piece is recognized in a new plane. By means of other conjectures, it is possible to construct the new characteristics. All these conjectures have been checked by numerous examples. Mathematical proofs of these conjectures have been presented in [2,9,8]. The complexity of Debled’s algorithm corresponding to this step is at most linear in the number of points on the piece, see [3] p.180. – The third possibility is that M satisﬁes one of the following inequalities ax + by + cz < µ − 1

or

ax + by + cz > µ + c

In this case we say that M is strongly k-exterior, where k is equal to µ − (ax + by + cz) or (ax + by + cz) − µ − c + 1 respectively. The new piece belongs to a digital plane if M is not too distant from the piece. Debled gave three validity criteria to be checked for the new piece to conclude its ﬂatness. These criteria are suﬃcient but not necessary, see [3] p. 172. In this paper we present necessary and suﬃcient conditions to recognize pieces of digital planes. We obtain a new algorithm that simpliﬁes and generalizes Debled algorithm. Its complexity is at most quadratic since we exploit, for each added point, only the part of Debled’s algorithm used for 1-exterior case. Next section introduces some fundamental notions of digital planes that will be used throughout the paper. Section 3 surveys our fundamental results on 1exterior case. Section 4 addresses the problem of recognition in strongly exterior case and two recognition theorems are presented. Section 5 describes our algorithm that simpliﬁes Debled’s one, whereas Section 6 explains it by an example.

2

Background

Let us begin by giving the deﬁnitions of some notions related to digital planes that we will use in this paper. Let M (x, y, z) be a point of a digital plane P (a, b, c, µ), the quantity r(M ) = ax + by + cz is called the remainder of M with respect to P . The Diophantine inequality(1) allows us to deﬁne P (a, b, c, µ) as the set of points (x, y, z) ∈ Z 3 such that µ−ax−by ≤ z < µ−ax−by +1. Since the c c plane projects injectively on Oxy, we can represent P (a, b, c, µ) in Oxy by level lines corresponding to values of z. We can also represent the plane P (a, b, c, µ) by the remainder of its points. In Figure 2, we combine the remainder and level lines representations to represent P (9, 13, 21, 0) on plane Oxy. The real plane deﬁned by r(a, b, c)(M ) = k is called the plane of index k. The plane of index µ is called lower leaning plane of P and the plane of index µ + c − 1 is called upper leaning plane of P . We denote them by (P i) and (P s) respectively. A piece of plane is a convex subset of voxels of a na¨ıve digital plane. A piece of a digital plane is said to be recognized if it possesses four leaning points that satisfy one of the following two cases: – Three upper (resp. lower) leaning points and one lower (resp. upper) leaning point. This conﬁguration is referred to as CAS3.1, see ﬁgure 2(a).

A Simpliﬁed Recognition Algorithm of Digital Planes Pieces

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– Two upper leaning points and two lower leaning points. This conﬁguration is refereed to as CAS2.2, see ﬁgure 2(b).

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Fig. 2. (a) CAS3.1. The remainder and level lines representation of a piece of P (9, 13, 21, 0) that contains three lower leaning points and one upper leaning point. (b) CAS2.2. The remainder and level lines representation of a piece of P (3, 5, 22, 0) that contains two lower leaning points and two upper leaning points

When the added point M is k-exterior, many geometric constructions are built and key voxels are extracted. These geometric constructions depend on leaning polygons and their positions with respect to M . These leaning polygons are deﬁned as follows: – If r(M ) < µ, we call (P S) the convex hull of the upper leaning points in the piece and we call it the upper leaning polygon. We deﬁne, in this case, the convex polygon of pivots CV P to be the upper leaning polygon P S. In the same way, we call P I the lower leaning polygon. In this case, we deﬁne the convex polygon of antipodes CV A to be the lower leaning polygon P I. Thus, we have CV P = P S and CV A = P I. – If r(M ) ≥ µ + c, the upper leaning polygon P S is called the convex polygon of antipodes CV A and the lower leaning polygon P I is called the convex polygon of pivots CV P . We have CV A = P S and CV P = P I. In ﬁgure 3(a) we give an example of the convex polygons of pivots and antipodes for a recognized piece in the plane P (5, 6, 7, −1). The polygonal line of pivot vectors L is constructed depending on the added k-exterior point and its associated convex polygon of pivots. All constructions are realized on the projections in the plane Oxy. Four cases are possible: 1. The CV P is reduced to one point. In this case, the polygonal line of pivot vectors L is reduced to this point. 2. The CV P is formed by points that are not all collinear. In this case, the polygonal line L of pivot vectors is composed by points of the CV P such that their projections in the plane Oxy are located on the part of the boundary of the convex hull of the CV P projection that disappears when the point M is added, see Figure 3(b). 3. The CV P is composed by collinear points in the plane Oxy:

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Fig. 3. The point M is 1-exterior of remainder −2. In (a) the parallelogram in bold is the convex of pivot points CV P and the dashed triangle is the convex polygon of antipodes CV A. In (b) U ∪ V form the polygonal line of pivot vectors L, point N is an antipode for both pivot vectors U and V , It is separating for the vector U . Point N is an antipode for the vector U only.

• If the projection of M is collinear with the projected points of the CV P , then the polygonal line L of pivot vectors is reduced to the nearest point of the CV P to M . • If the projection M is not collinear with the projected points of the CV P , then L is equal to all points of the CV P . An antipode A of a pivot vector V associated to a k-exterior point M is a summit of the CV A that has the maximal distance, among all points of the CV A, from the line directed by V and containing M . Furthermore, if the end points of V are separated by the line M A, then A is called a separating antipode.

3

The 1-Exterior Case

Let S be a recognized piece of the plane P (a, b, c, µ) and M an added point to S. If the remainder of M with respect to P (a, b, c, µ) is between µ and µ + c − 1, then S = S ∪ {M } is still recognized in the same plane. If the point M is 1exterior, Debled’s conjectures [3,4] assure the existence of a new plane in which S is recognized. Furthermore, following the shape of the polygonal line of pivot vectors, these conjectures give methods to construct the new characteristics A ≤ B ≤ C such that C is the smallest for all possible planes that contain S . These conjectures have been mathematically proven in recent works [2,8,9]. In fact, we prove in [8] the following key theorem that assures the existence of planes that contain S . We gave also a method to construct the smallest characteristics. Theorem([8]). Let S be a recognized piece of the plane P (a, b, c, µ) and M (x0 , y0 , z0 ) ∈ Z 3 be a point such that S = S ∪{M } is convex. If M is 1-exterior to S, then there exists (A, B, C, µ ) ∈ Z 4 with A ∧ B ∧ C = 1 such that S = S ∪ {M } is a piece of the plane P (A, B, C, µ ). The proof of this theorem provides two necessary and suﬃcient conditions on the

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choice of points that allow the construction of the new base. These conditions are: β2 (x2 − x0 ) − α2 (y2 − y0 ) = c,

(2)

where (x2 , y2 ), (x1 , y1 ) and (α2 , β2 ) are respectively the projections on Oxy of a pivot point M2 , an antipode M1 and a vector V2 formed by M2 and another pivot point. dm − (µ + c − 1) +

h2 )m hm ≤ ≤ dm − (µ − 1) ∀m ∈ S, H H

(3)

where )m is the sign of β2 (x − x0 ) − α2 (y − y0 ), the numbers H, h2 et hm are the Euclidean distances between the points M1 , M2 and m respectively and the real line directed by V2 and containing M .

4

The Strongly Exterior Case

Let us now suppose that M is strongly exterior to the piece S ⊂ P (a, b, c, µ). This implies that there exists an integer P ≥ 2 such that M satisﬁes one of the following equalities: ax0 + by0 + cz0 = µ − P (i.e, M is located under the piece S) or ax0 + by0 + cz0 = µ + c − 1 + P (i.e, M is located over the piece S). An axial rotation of the plane by π reverses the positions of its points. Points located under the plane get over it and vice versa. The characteristics of the plane remain unchanged. This fact allows us to reduce our study to one case, say M being located under the plane. All relations we will obtain are still valid in the case where M is located over the plane. The following lemma is a direct consequence of convexity. Lemma 1. Let S be a piece of a discrete plane P (a, b, c, µ) and M (x0 , y0 , z0 ) a point such that S = S ∪ {M } is convex. If M is strongly P -exterior then P ≤ c. Proof. Suppose that the remainder of M is r(M ) = µ − P and let consider the point M (x0 , y0 , z0 +1). The remainder of M is r(M ) = µ−P +c. If r(M ) < µ, then M is disconnected from S and S is not convex. Hence, r(M ) ≥ µ which gives P ≤ c. 4.1

Recognition Theorems

The piece S is bounded by two real leaning planes of indices µ and µ + c − 1. When the added point M is exterior to S, located for instance under S, then, in order to geometrically contain S = S ∪ {M } in another digital plane, one should lean the plane of index µ in a suitable direction until M is reached. This latter plane becomes the lower leaning plane of the ”eventual” digital plane we are looking for. Then let us consider another plane located over S, tangent to S and parallel to the new lower leaning plane. The distance between these two planes should be less than or equal to 1; otherwise we don’t get a digital plane.

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The operation of leaning planes can be done by revolving on some parallel straight lines all planes of indices between µ and µ + c − 1. If the line, on which pivots the plane of index µ + c − 1, is too distant from M , then we may lose some upper leaning points (i.e., points of CV P ) located between this line and M after the rotation, see Figure 4(a). Thus, this line should be chosen as close as possible to the point M . In the same way, the rotation of the lower leaning plane should not exclude the lower leaning points. Therefore, the line on which we pivot this plane should be chosen as far as possible from M so that all points of the CV A will be located between this line and M , see Figure 4(b). This explains the importance of polygonal lines of pivots and antipodes deﬁned above. M’2

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Fig. 4. Proﬁle view: Axes of rotation are shown by (), the result is delimited by the dashed lines. In the 3D-space points, as M, M ”2 , are not piled up. In (a) the rotation excludes leaning points M1 , M2 and eventual internal points as M ”1 and M ”2 . In (b) the rotation does not exclude any point of the piece. In (c) the rotation R1 excludes point M0 , while rotation R1 includes all points of S, its axe contains M1 . Leaning point M4 becomes an internal point.

After this operation M become naturally a lower leaning point of the new digital plane that contains S . In practice, this operation does not always give a recognized piece of a digital plane. This is due to the fact that when M is too distant from S, the rotation of the above upper (resp. lower) leaning plane may excludes some points (i.e., their remainders are beyond the bounds given by relation 1). To palliate to this problem one can choose the lines, on which the leaning planes pivot, outside the piece. But the result we obtain is that either the new digital plane is not na¨ıve (the distance between the resulting leaning planes is greater than 1) or the piece is not recognized (the number of the new leaning points in the piece is strictly inferior to 4). Thus, to get a satisfactory solution we have to carefully handle the operation of leaning planes. We proceed as follows: We rotate the upper leaning plane, by a suitable angle, as explained bellow on the nearest line Lp to M . We rotate the lower leaning plane on the most far line La from M which is parallel to Lp (in the sense that it is directed by the same vector of Lp ) and contains a leaning lower point in S. These lines strongly depend on the polygonal line of pivots L and the polygonal line of antipodes L . Since each of L and L contains no more than two independent vectors [3,9] then Lp and La are chosen in one of the following conﬁgurations: 1. If the polygonal line of pivots L is not reduced to a point and if there exists a separating antipode M1 , then Lp is the line containing vector V2 of L which

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is separated by M1 . We take La as the line directed by V2 and containing M1 . 2. If L is reduced to one point, then, since S is recognized, four leaning points exist that deﬁne the normal vector of S and its thickness. These points deﬁne at least two independent vectors. Since the leaning points on L are collinear, then other leaning points (antipodes) exist on the CVA that deﬁne a transverse direction to L. Among antipodes on this transverse direction we take the line La which contains the most far antipode from M . Then we deﬁne line Lp to be parallel to La and to contain the nearest pivot point to M (which is, in this case, equal to L). 3. If L is not reduced to a point and there is no separating antipode, then we proceed in the same way as in the previous case to deﬁne La and Lp . The angle of rotation is chosen such that (i)

if M is reached without excluding any point of S (i.e., all points of S still satisfy a relation of type (1) after the rotation), then this rotation, say R1 , gives a new na¨ıve digital plane in which the piece S is recognized. (ii) if the previous rotation excludes a point only on Lp , since all possible points on Lp should have the same remainder, then all these points are excluded. Thus M is too distant. To keep the points of Lp in the new plane we have to stop rotation R1 before reaching M . In this case, Lp and La are still the pivots and antipodes lines of the new plane and there is no way to reach M by a rotation. We note that, in this case, M becomes less strongly exterior but never 1-exterior because in this later case the piece becomes recognizable which is impossible. (iii) if the previous rotation R1 excludes a point M2 of Lp (or La ) after having reached a point M0 of remainder µ + c − 2 (or µ + 1), then we reduce the angle of rotation until M0 is reached. Let R1 be the corresponding rotation. Algorithmically, this can be done by moving line La to the next line La that is parallel to La , contains antipodes, and its distance to M is bigger than La ’s one, see Figure 4(c). The leaning points of La and Lp always remain leaning points of the new digital plane, the other leaning points become internal points in general, and M0 becomes a new leaning point. Thus, we get enough leaning points so that the piece S still be recognized in the new digital plane. • If point M becomes 1-exterior for the new characteristics, then Debled algorithm allows us to recognize the piece S in another digital plane. • If point M remains strongly exterior, then we have to apply a new rotation. But in this case, point M2 will be excluded and we are in situation (ii). Therefore, piece S ∪ {M } can not be recognized in a digital plane. (iv) If rotation R1 excludes a point M0 of remainder µ + c − 2 (or µ + 1) and points of Lp and La remain included by R1 then we have to stop the rotation before excluding M0 . This corresponds to an extremal position and piece S ∪ {M } can not be recognized.

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All these cases imply that strongly exterior cases can be recognized only through 1-exterior cases. This proves our ﬁrst theorem of recognition. However, before doing any rotation, there is an analytic way to check what kind of situation we may have. This analytic way is given by condition 5 in the second theorem of recognition. Theorem 1. Let S be a recognized piece of a digital plane P (a, b, c, µ) and M ∈ Z 3 be a point such that S = S ∪ {M } is convex. Suppose that M is strongly P exterior ( P ≥ 2). Then there exists at most one digital na¨ıve plane in which S is recognized and admits M as a strongly q-exterior for some integer 2 ≤ q < P . When a such number q exists, then S cannot be recognized in a na¨ıve digital plane. Theorem 2. Let S be a recognized piece of a digital plane P (a, b, c, µ) and M (x0 , y0 , z0 ) ∈ Z 3 be a point such that S = S ∪ {M } is convex. Suppose that M is strongly P -exterior ( P ≥ 2). Let M2 (x2 , y2 , z2 ) be a point on the polygonal line of pivots and V2 (α2 , β2 , γ2 ) a vector based on M2 and located either on the polygonal line of pivots if it is not reduced to a point or parallel to the polygonal line of antipodes. Let V1 (α1 , β1 , γ1 ) be a vector linking M to an antipode M1 suﬃciently distant from M in order for it to exceed at least the polygonal line of antipodes. Let A = β1 γ2 − β2 γ1 , B = α2 γ1 − α1 γ2 , C = α1 β2 − α2 β1 and k = A ∧ B ∧ C). B C Then piece S belongs to a digital plane P ( A k , k , k , µ ) such that M is a leaning point (q = 0) or 1-exterior (q = 1) if and only if M1 is arbitrarily chosen on a line, among at most three parallel lines, directed by V2 such that the following relations are satisﬁed: 0 ≤ A ≤ B ≤ C,

(4)

P β2 (x2 − x0 ) − α2 (y2 − y0 ) − C(P − 1) = kc(1 − q),

(5)

dm − µ − c + P k(1 − q)c )m hm dm − µ + P + ≤ ≤ ; ∀m ∈ S, P |V2 |H H P

(6)

with q = 0 or 1 and H (resp. hm ) being the height of point M1 (resp. m) to the real line passing by M and directed by V2 . The number of lines that may contain M1 is 1 if k < P and at most 3 if k = P . Remarks. 1. In the algorithm that we propose, only relations (4) and (5) are needed explicitly, relation (6) serves mathematically to prove Debled’s conjectures and explains how to choose M1 between all antipodes. 2. We note that if in theorem 2 we assume that P can take the value 1, relations (4),(5) and (6) become equivalent to those quoted in the theorem of the previous section. Following lemma 2 below, the number k = A ∧ B ∧ C divides P which is assumed to be 1 here. This fact is also true for the 1-exterior case. Thus theorem 2 includes the 1-exterior case.

A Simpliﬁed Recognition Algorithm of Digital Planes Pieces

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Proof Idea. Let us suppose that the added point M is located under plane P (a, b, c, µ). The Point M satisﬁes the relation ax0 + by0 + cz0 = µ − P . Let M1 −−−→ be a lower leaning point of P (a, b, c, µ) and let V1 = M M1 =: (α1 , β1 , γ1 ) and (A, B, C) =: V1 ∧ V2 . Vector V1 satisﬁes the relation aα1 + bβ1 + c(γ1 − Pc ) = 0. Suppose that V2 is linearly independent from V1 (α1 , β1 , γ1 − Pc ). The vectorial product V1 ∧ V2 = (A , B , C ) is a rational multiple, say λ, of N (a, b, c). The numbers A , B , C are given by: A = A + β2

P = λa; c

B = B − α2

P = λb; c

C = C = λc.

We can always assume (by considering −V2 in place of V2 ) that 0 ≤ A . Since 0 ≤ a ≤ b ≤ c, then 0 ≤ A ≤ B ≤ C which is equivalent to 0 ≤ A + β2 Pc ≤ B − α2 Pc ≤ C. If we increase the length of V1 , then the numbers A , B , C also increase, and so does the distance between each other. Note that the quantities β2 Pc and α2 Pc do not change when we increase the length of V1 . Let us take V1 suﬃciently long so that 0≤A≤B≤C

(4)

By dividing A, B, C by their greater common divisor, we obtain the characteristics of the eventual new plane which may contain the piece S = S ∪ {M }. Lemma 2. If the numbers α2 , β2 and γ2 are prime together (α2 ∧ β2 ∧ γ2 = 1), then P GCD(A, B, C) = A ∧ B ∧ C divides number P . Let us search conditions under which there exists µ such that the point M is q-exterior to S ⊂ P (A, B, C, µ ), with q ∈ {0, 1}. This means that for all m (x, y, z) ∈ S we have: µ ≤ Ax + By + Cz < µ + C Ax0 + By0 + Cz0 = µ − P + q where q = 0 or 1 For all m(x, y, z) ∈ S ⊂ P (a, b, c, µ) we have µ ≤ ax + by + cz ≤ µ + c − 1. The added point M satisﬁes the relation ax0 + by0 + cz0 = µ − P . Then µ − P ≤ ax + by + cz ≤ µ + c − 1, for all points of S . By multiplying all members with λ we obtain: λ(µ − P ) − P (

β2 α2 β2 α2 x− y) ≤ Ax + By + Cz ≤ λ(µ − 1) − P ( x − y) + C. c c c c

Let us take k = P GCD(A, B, C). By dividing the previous inequality with k we obtain: 2 y) 2 y) λ”(µ − P ) − P (β2 x−α ≤ A”x + B”y + C”z ≤ λ”(µ − 1) − P (β2 x−α + C”, c c B C λ C” where P = Pk , A” = A , B” = , C” = etλ” = = . k k k k c 2 y) Let us take µ(x, y) = λ”(µ − P ) − P (β2 x−α . Then for all points in S we c have: µ(x, y) ≤ A”x + B”y + C”z ≤ µ(x, y) + C” + λ (P − 1). Using the facts that M is P -exterior and should become q-exterior and that M2 (one of the end points of vector V2 ) is an upper leaning point of pieces S and S , we obtain the condition

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P [β2 (x2 − x0 ) − α2 (y2 − y0 )] − C(P − 1) = kc(1 − q).

(5)

This condition expresses the position of points M2 , M2 and M1 with respect to M . Now let us search in what region we can choose V1 so that surface S will be a recognized piece of the digital plane of parameters (A”, B”, C”, µ(x0 , y0 ) + q) with M a q-exterior, where q = 0 or 1. The former double inequality can be 0 )−α2 (y−y0 ) transformed to get dm −µ−c+P + kc(1−q) ≤ β2 (x−x ≤ dm −µ+P . This P PC β2 α1 −α2 β1 P expression can be expressed in terms of heights by: dm −µ−c+P P

+

k(1−q)c |V2 |H

≤

m h m H

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dm −µ+P P

(6)

with dm ∈ [µ−P, µ+c−1]. It is the condition that all points m(x, y, z) ∈ S should satisfy so that S will be a recognized piece in the digital plane of characteristics (A”, B”, C”, µ = µ(x0 , y0 )). By studying the sign of the ﬁrst member of relation (5), we can show that this relation can be satisﬁed at most by three lines directed by V2 and containing antipodes.

5

Simpliﬁed Recognition Algorithm

The algorithm that we describe in this section directly derives from the discussion and the proofs of the two previous theorems. It uses only the part of Debled’s algorithm corresponding to 1-exterior case with small modiﬁcations to recognize the general case of rectangular pieces. The complexity of the simpliﬁed algorithm decreases to become at most quadratic in the number of points in the piece. Following the necessary and suﬃcient conditions quoted in theorem 2, the simpliﬁed algorithm becomes completely decidable where the validity criteria given by Debled fail in the strongly exterior case. These two advantages are added to the other advantages of Debled’s algorithm. The algorithm begin by sweeping the piece to be recognized following sections parallel to one co-ordinate plane, Oxz for instance, by successively adding voxels. At the beginning we initialize y = 0 and we let x vary in its interval of deﬁnition. At each added voxel the algorithm tries to recognize a piece of a digital straight line. When all values of x are considered we increment y by 1 and we let x sweep again over all its possible values. At each step the algorithm tries to recognize a piece in a digital plane and compute its characteristics. Three cases are possible: 1. If the added point M satisﬁes the double inequality (1) for the plane constructed before adding M , then we keep the same characteristics and the updated piece is still recognized in this plane with the same leaning points with a possible addition of M . 2. If the added point is 1-exterior to the plane, then we apply Debled’s algorithm to recognize the new piece. This step consists of computing polygonal pivots and antipodes lines to determine the vector V2 which satisﬁes relation (2), and also search an adequate antipode M1 that satisﬁes (3)). 3. In the third case, if the added point is strongly exterior, we search the polygonal line of pivots and antipodes and then check relation (5). If this relation

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is not satisﬁed for any point M2 on the polygonal line of pivots, then the piece is not recognizable. If relation (5) is satisﬁed for some q diﬀerent from 0 and 1 then the piece is not recognizable. If relation (5) is satisﬁed for q = 0, then we apply 1-exterior part of Debled’s algorithm. In this case the piece is recognized with M as a leaning point. If relation (5) is satisﬁed for q = 1, then we again apply Debled’s algorithm for the 1-exterior case at most three times to decide the recognition of the piece. Note that in this case, the choice of the antipode M1 is modiﬁed from the choice given by Debled. In the ﬁrst running of the algorithm, point M1 is taken as in Debled’s algorithm. If M becomes 1-exterior, then we apply Debled’s algorithm once more and the piece is recognized. At this point, if after one run there are some points which are excluded, then we choose the antipode M1 on the following line proposed by theorem 2; if the point M becomes 1-exterior, then we are done. If there are some points which are excluded then we take M1 on the last line proposed by the theorem and the point becomes 1-exterior. Usually, only two lines are suﬃcient.

6

Example

In Figure 5(a), we represent a recognized piece S of the plane P (18, 21, 23, 0). The added point M (3, 5, −7) is 2-exterior of remainder r(M ) = 2. The nearest pivot point to M is M2 (4, 2, −4). The polygonal line of antipodes contains only two points M1 (0, 0, 0) and M1 (8, 3, −9). The vector V2 is then (8, 3, −9) and V1 = (−3, −5, 7). The vectorial product of V1 and V2 gives (A, B, C) = (24, 29, 31). With this choice relation (6) is not satisﬁed. Take for instance the point m = 2 (y−y0 ) (9, 0, −7). The quantity hHm = β2 (x−x0 )−α = 58 C 31 = 1, 86 is greater than dm −µ+P = 3/2 = 1, 5. Let us take the antipode located on the line directed by P V2 and that comes just after V1 . Let M ”1 (5, −1, −3) be a such antipode. The −−−−→ point M ”1 is located outside the piece S. In this case vector V1 is M ”1 M = (2, −6, 4). The vectorial product V1 with V2 is (42, 50, 54) = 2(21, 25, 27). Note that P GCD = k = 2 divides P = 2. The quantity P β2 (x2 − x0 ) − α2 (y2 − y0 ) − C(P − 1) is equal to 2 3(4 − 3) − 8(2 − 5)) − 54(2 − 1) = 54 − 54 = 0. Moreover, for q = 1 we obtain kc(1 − q) = 0. Thus, relation (5) is satisﬁed for q = 1. We can check that all points of S satisfy relation (6). For m(9, 0, −7), which did not satisfy (6) for the ﬁrst basis, we get dm −µ+P hm 58 = 3/2 The value of µ = µ(x0 , y0 ) + q H = 54 = 1, 07 which is small to P is −1 + 1 = 0. Piece S is recognized in the plane P (21, 25, 27, 0), see Figure 5(b). Thus, we return back to 1-exterior case. The polygonal line of pivots is reduced to the point M2 (4, 2, −4). The polygonal line of antipodes contains two −−−−→ points M1 (0, 0, 0) and M1 (9, 0, −7). Vector V2 is then equal to M1 M1 (9, 0, −7) −−−→ and V1 = M1 M = (−3, −5, 7). The vectorial product of V1 with V2 gives (35, 42, 45). Piece S = S ∪ {M } becomes recognized in the plane P (35, 42, 45, 0) with at l;east three lower leaning points M, M1 , M1 and at least one upper leaning point M2 , see Figure 5(c).

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References [1] E. Andr`es. Cercles discrets et rotations discr` etes. PhD thesis, Louis Pasteur University, December 1992. [2] Y. Boukhatem. Sur la reconnaissance des plans discrets. Master Thesis, Mostaganem University, April 2001. [3] I. Debled-Rennesson. Etude et Reconnaissance des Droites et Plans Discrets. PhD thesis, Louis Pasteur University, October 1995. [4] I. Debled-Rennesson and J.P. Reveilles. Incremental algorithm for recognizing pieces of digital planes. In Spie’s Internat. Symp. on Optical Science, Engeneering and Instrumentation, Technical conference Vision Geometry 5, Denver, USA, Aug 1996. [5] S. H. Hung. On the straightness of digital arcs. In IEEE Transactions on Pattern Analysis and Machine Intelligence, volume PAMI 7, pages 203–215, 1985. [6] C.E. Kim and A. Rosenfeld. Convex digital solids. In IEEE Transactions on Pattern Analysis and Machine Intelligence, volume PAMI 6, pages 639–645, 1984. [7] C.E. Kim and I. Stajmenovi´c. On the recognition digital planes in threedimensional space. In North-Holland, editor, Pattern Recognition Letters, volume 12, pages 665–669, 1991. [8] M. M. Mesmoudi and I. Debled-Rennesson. Contribution ` a la reconnaissance des plans discrets. Technical report, Done at Loria Laboartory INRIA-Lorraine, April 2000. [9] M.M. Mesmoudi and I. Debled-Rennesson. Sur la reconnaissance des plans discrets. In Colloque d’Analyse et Application. Mostaganem University, Oct 1999. [10] J.P. Reveilles. G´eom´etrie discr`ete, calculs en nombre entiers et algorithmique. Doctorat d’Etat thesis, Louis Pasteur University, Strasbourg, 1991. [11] I. Stojmenovi´c and R. Tosi´c. Digitaziation schemes and the recognition of digital straight lines, hyperplanes and ﬂats in arbitrary dimensions. In Vision Geometry, Contemporay Mathematics series, volume 119, pages 197–212. AMS, Providence, RI, 1991. [12] P. Veelaert. Digital planarity of rectangular surface segments. In IEEE Transactions on Pattern Analysis and Machine Intelligence, volume 16, pages 647–652, 1994. [13] P. Veerlaert. On the ﬂatness of digital hyperplanes, volume 3. Journal of Mathematical Imaging and Vision, 1993.

Ridgelet Transform Based on Reveill` es Discrete Lines Philippe Carr´e and Eric Andres Laboratoire IRCOM-SIC, bˆ at. SP2MI, av. Marie et Pierre Curie BP 30179 - 86960 Chasseneuil-Futuroscope C´edex - FRANCE

Abstract. In this paper we present a new discrete implementation of ridgelet transforms based on Reveill`es discrete 2D lines. Ridgelet transforms are particular invertible wavelet transforms. Our approach uses the arithmetical thickness parameter of Reveill`es lines to adapt the Ridgelet transform to speciﬁc applications. We illustrate this with a denoising and a compression algorithm. The broader aim of this paper is to show how results of discrete analytical geometry can be sucessfully used in image analysis.

1

Introduction

Image analysis is traditionally aimed at understanding digital signals obtained by sensors (in our case cameras). Digital information is considered as sampled continuous information and the theoretical background for it is signal theory. This is sometimes referred to as “digital geometry” in opposition to “discrete geometry” for computer graphics. These last ten years, since J-P. Reveill`es has introduced it [1], discrete analytical geometry has made an important progress in deﬁning and studying classes of discrete objects and transformations. This greatly enhanced our understanding of the links between the discrete world Zn and the continuous world Rn . In the same time, a new discrete signal decomposition has been developed in image analysis: the wavelet representation. This new representation has many applications such as denoising, compression, analysis, etc. One of the aims of this paper is to apply this new insight in discrete geometry to image analysis and more speciﬁcally to a particular wavelet transform: the ridgelet transform. Wavelets are very good at representing point singularities ; however they are signiﬁcantly less eﬃcient when it comes to linear singularities. Because edges are a extremely common phenomena in natural images, an eﬃcient multiresolution representation of images with edges would be quite advantageous in a number of applications. A team of Stanford has recently developed an alternative system of multiresolution analysis speciﬁcally designed to eﬃciently represent edges in images [2]. Their attempt was to design a new system, called ridgelet transform, in the continuous domain so that an image could be approximated within a certain margin error with signiﬁcantly fewer coeﬃcients than would be required A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 417–427, 2002. c Springer-Verlag Berlin Heidelberg 2002

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after a wavelet decomposition. However, most of the work done with ridgelets has been theoretical in nature and discussed in the context of continuous functions. The important bridge to digital implementation is tenuous at best. To our knowledge, we can ﬁnd in the literature only two solutions for the digital ridgelet decomposition [3], [4] (notice that the study proposed by Gu´edon et al is similar [5]).This paper presents a new approach that aims at representing linear singularities with a discrete ridgelet transform based on Reveill`es discrete lines. In this article, we propose a new approach of the ridgelet transform based on several types of Reveill`es discrete lines deﬁnitions in the Fourier domain. Our decomposition has an exact inverse reconstruction process and the redundancy of our Ridgelet representation can be adjusted with the arithmetical thickness of the Reveilles discrete lines. To illustrate this new decomposition, we propose a method of restoration of noised images which uses a wavelet undecimated method deﬁned in [6].

2 2.1

The Ridgelet Transform The Wavelet Transform

The discrete wavelet transform (DWT) stems from the multiresolution analysis and ﬁlter bank theory [7]. The multiresolution analysis is a decreasing sequence of closed subspace {Vj }j∈Z that approximates L2 () (f ∈ L2 () if ∞ 2 f (x) dx < ∞). A function s ∈ L2 () is projected, at each step l, onto −∞ the subset Vl . This projection is deﬁned as the scalar product, noted cl , of s with a scaling function, noted φ that is dilated and translated: cl (k) = s(x), 2−l/2 φ 2−l x − k = s(x), φl,k (x)

(1)

k is the translation parameter and l is the dilatation parameter with k, l ∈ Z. At each step (if l is growing), the signal is smoothed. The lost informations can be restored using the complementary subspace Wl+1 of Vl+1 in Vl . This subspace is generated by a wavelet function ψ with integer translation and dyadic dilatation; the projection of s on Wl is deﬁned as the scalar product, noted dl : dl (k) = s(x), 2−l/2 ψ 2−l x − k = s(x), ψl,k (x) (2) Then, the analysis is deﬁned as : 1 1 cl (k) = √ h(n − 2k)cl−1 (k), dl (k) = √ g(n − 2k)cl−1 (k) 2 n 2 n with cl the coarse approximation, dl the decimated wavelet coeﬃcients at scale l and c0 the original signal, the sequence {h (k) , k ∈ Z} is the impulse response

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of a low-pass ﬁlter and the sequence {g (k) , k ∈ Z} is the impulse response of a high-pass ﬁlter. Notice that with conditions required on the ﬁlters, we get an exact restoration. Mallat’s multiresolution analysis is connected with so called ”pyramidal” algorithms in image processing [8]. Because of decimation after ﬁltering, the Mallat’s decomposition is completely time variant. A way to obtain a time-invariant system is to compute all the integer shifts of the signal. Since the decomposition is not decimated, ﬁlters are dilated between each projection. This algorithm presents many advantages, particularly a knowledge of all wavelets’ coeﬃcients: coeﬃcients removed during the downsampling are not necessary for a perfect reconstruction, but they may contain information useful for the denoising. 2.2

Continuous Theory of Ridgelet Transform

A substantial foundation for Ridgelet analysis is documented in the Ph.D. thesis of Cand`es [2]. We brieﬂy review the ridgelet transform and illustrate its connections with the radon and wavelet transforms in the continuous domain. The continuous ridgelet transform of s ∈ L2 2 is deﬁned by : r(a, b, θ) = ψa,b,θ (x)s(x)dx R2

with ψa,b,θ (x) the ridgelet 2-D function deﬁned from a wavelet 1-D function ψ as: x1 cos θ + x2 sin θ − b −1/2 ψa,b,θ (x) = a ψ a b is the translation parameter, a is the dilatation parameter and θ is the direction parameter. The function is oriented at the angle θ and is constant along lines x1 cos θ + x2 sin θ = cst. Transverse to these ridges it is a wavelet. In comparison, the analysis continuous 2-D wavelet function are tensor products of 1-D wavelet ψa,b : ψa,b (x) = ψa1 ,b1 (x1 )ψa2 ,b2 (x2 ) The Radon transform seems to be similar to the 2-D wavelet transform but the translation parameters (b1 , b2 ) are replaced by the line parameters (b, θ). Then, the wavelets are adapted to analyse isolated point discontinuities, while the ridgelets are adapted to analyse discontinuities along lines. A basic tool for calculating ridgelet coeﬃcients is to view ridgelet analysis as a form of wavelet analysis in the Radon domain: in 2-D, points and lines are related via the radon transform, thus the wavelet and ridgelet transforms are linked via the Radon transform. The Radon transform of s is deﬁned as: Rs(θ, t) = s(x)δ(x1 cos θ + x2 sin θ − t)dx1 dx2 R2

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where δ is the Dirac distribution. The ridgelet coeﬃcients r(a, b, θ) of s are given by the 1-D wavelet transform to the projections of the Radon transform where the direction θ is constant and x is varying: r(a, b, θ) = ψa,b (x)Rs(θ, x)dx R

Notice that the Radon transform can be obtained by applying the 1-D inverse Fourier transform to the 2-D Fourier transform restricted to radial lines going through the origin (this is exactly what we are going to do in the discrete Fourier domain with help of discrete Reveill`es lines): s (ω cos θ, ω sin θ) = e−jωx Rs(θ, x)dx R

with s (ω) the 2-D Fourier transform of s. This is the projection-slice formula which is used in image reconstruction from projection methods. We deduce that the Radon transform can be obtained by applying the 1-D inverse Fourier transform to the 2-D Fourier transform restricted to radial lines going through the origin. These relations are shown in ﬁgure 1.

Fig. 1. Relation between transforms

2.3

Discrete Ridgelet Transform

As we have seen, a basic strategy for calculating the continuous ridgelet transform is ﬁrst to compute the Radon transform Rs(θ, t) and secondly, to apply a 1-D wavelet transform to the slices Rs(θ, .). The discrete procedure uses the same principle. As presented in the ﬁrst section, the discrete wavelet decomposition is easy to implement, is stable and invertible, and can be associated to a discrete orthogonal representation. The discretization of the Radon transform is more diﬃcult to achieve. The majority of methods proposed in the literature have been devised to approximate the continuous formula. But, none of them were speciﬁcally designed to be invertible transforms for discrete images and can not be used for the discrete

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Ridgelet transform. Recently, some articles studied the implementation of the digital Ridgelet transform. Two approaches have been developed: – Spatial strategy for digital Radon transform: the Radon transform is deﬁned as summations of image pixels over a certain set of lines. Those lines are deﬁned in a ﬁnite geometry in a similar way as the line for the continuous Radon transform in the Euclidean geometry. Rs(p, q, b) = s(x, y)δ (b + px − qy) with (p, q) direction of projection x

y

In [5] an inverse transform based on erosion and dilatation operations is proposed. Vetterli et al. proposed in [3] an orthonormal ridgelet transform. – Fourier strategy for digital Radon transform: the projection-slice formula suggests that approximate Radon transforms for digital data can be based on discrete Fast Fourier transforms (FFT). This is a widely used approach in the literature of medical imaging and synthetic aperture radar imaging. The Fourier-domain computation of an approximate digital radon transform is deﬁned as: 1. Compute the 2-D FFT of f 2. Extract Fourier coeﬃcients which fall lines Lθ going through the origin. 3. Compute the 1-D FFT on each line Lθ (deﬁned for each value of the angular parameter). In this strategy too, discrete lines must be deﬁned. In [4], Starck et Al proposed to use an interpolation scheme which substitutes the sampled value of the Fourier transform obtained on the square lattice with sampled value of s on a polar lattice. In this paper, we propose to deﬁne the lines Lθ with the discrete geometry in the Fourier domain. This solution allows us to have diﬀerent Ridgelet decompositions according to the arithmetical thickness of the discrete Reveill`es lines. Our transformation is redundant but the repetition of information depends on the type of the discrete lines used and can be adapted with the application. Moreover we obtain an exact reconstruction.

3 3.1

Digital Radon Transform Based on Reveill` es Discrete 2D Lines Deﬁnition of Discrete Lines

The discrete lines that are used in our application are not classical discrete lines such as, for instance, Bresenham lines nor the classical Reveill`es lines. These lines are not suitable for our purpose because they do not provide a central symmetry in the Fourier domain. Without central symmetry, the inverse Fourier transform would produce imaginary values during the Radon transform. Central symmetry is obtained easily by using closed Reveill`es discrete lines deﬁned as follows: 2 Lω (p,q) = (x, y) ∈ Z ||px + qy| ≤ ω/2

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with (p, q) ∈ Z2 the direction of the line (direction of Radon projection) and ω the arithmetical thickness. The parameter ω deﬁnes the connectivity of the discrete lines. The closed discrete lines have many interesting properties. One of the most important ones is that each type of closed discrete line is directly linked to a distance: for instance √ 2 2 p2 + q 2 p +q 2 L(p,q) = (x, y) ∈ Z |px + qy| ≤ 2 is equal to M ∈ Z2 d2 M, L(p,q) ≤ 12 where L(p,q) : px + qy = 0 is the Euclidean line of direction (p, q) and d2 the Euclidean distance [9].

Fig. 2. Redundancy on the cover of the Fourier lattice by (a) closed na¨ıve lines (b) supercover lines

3.2

Closed Reveill` es Discrete Lines for Digital Radon Transform

Our Digital Radon transform is deﬁned by: Rω s(p, q, b) =

K k=0

k

s (fk )e2πj K b with fk =

k f1 such that pf1k + qf2k ≤ ω/2 k f2

and K the length of a line segment of Lω p,q We must deﬁne the set of discrete directions (p, q) in order to provide a complete representation. The set of line segments must cover all the square lattice in Fourier domain. For this, we deﬁne the direction (p, q) according to pairs of symmetric points from the boundary of the 2-D Discrete Fourier Spectra.

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2 Proposition 1. Let a square lattice be deﬁned as ΩN = [−N, N ] × [−N, N ]. Let us consider the set of directions (pm , qm ) with, for 0 ≤ m ≤ 2N, (pm , qm ) = (N, m − N ) and for 2N + 1 ≤ m ≤ 4N − 2, (pm , qm ) = (m − 3N + 1, N ). The set of all the closed lines deﬁned by |pm f1 + qm f2 | ≤ ωm /2 with ωm ≥ 2 sup (|pm | , |qm |) provides a complete cover of the lattice ΩN .

The proof of this proposition is obvious because of a well known result in discrete analytical geometry that states that a closed discrete line of direction (p, q) is connected if and only if ω ≥ sup (|p| , |q|) [1]. For thinner (non connected) discrete lines, with values of ω < sup (|p| , |q|) , it is possible but not certain that 2 we also achieve a complete cover of the lattice ΩN depending on the value of ω compared to N . However, for our applications, we preferred working with connected discrete lines. Figure 2 illustrates the cover of the Fourier lattice (on the ﬁrst octant) by two diﬀerent types of discrete lines. The grey value of the pixels represents the redundancy in the projection (number of times a pixels belongs to a discrete line). One isolated line is drawn to shown the illustrate the arithmetical thickness of each type of line. Three diﬀerent types of closed discrete lines have been tested: – closed naive discrete lines: ω = sup (|p| , |q|). These lines are the thinnest connected closed discrete lines. They are 8-connected. They provide therefore the smallest redundancy as we can see on ﬁgure 2(a). Closed naive discrete lines sup(|p|,|q|) are related to the distance d1 : L(p,q) = M ∈ Z2 d1 M, L(p,q) ≤ 12 where d1 (A, B) = |xA − xB | + |yA − yB |; – supercover lines: ω = |p| + |q|. These lines are the thickest connected closed discrete lines that have been considered in our applications. They are the thinnest closed lines that are 4-connected and that cover the Euclidean line they approximate. They provide of course an important redundancy as we can see on ﬁgure 2(b). Supercover lines are related to the distance |p|+|q| d∞ : L(p,q) = M ∈ Z2 d∞ M, L(p,q) ≤ 12 . The supercover lines have an important theoretical importance. – closed Pythagorean lines: ω = p2 + q 2 . These lines are 8-connected and oﬀer a medium redundancy, in between the naive and supercover lines. The lines are related to the Euclidean distance d2 : √ 2 2 1 p +q L(p,q) = M ∈ Z2 d2 M, L(p,q) ≤ . 2 These lines possess the property of having a number of pixels per period close to its length. This means, in practice, that if pixels of the discrete line would hold energy, this energy would be distributed evenly along the line in the same way independently of the slope of the line.

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Discrete Ridgelet Transform

Now, to obtain the Ridgelet transform, we simply apply the 1-D wavelet transform on each discrete Radon coeﬃcients Rω s(p, q, b) obtained on the line segment Lω p,q . This transform is easily invertible. The reconstruction procedure works as follows: 1. Compute the inverse 1-D wavelet transform followed by the inverse 1-D FFT transform for each set Rω s(pm , qm , .) with m ∈ [0, 4N − 2] 2. Substitute the sampled value of f on the lattice where the points fall on lines

Lω p,q with the sampled value of f on the square lattice. The precedent procedure permits one to obtain an exact reconstruction if the set of M = 4N − 2 lines provides a complete cover of the square lattice.

Fig. 3. (a) Noisy image “object”(b) denoised by ridgelet decomposition with pythagoricean discrete lines, ω = q 2 + p2 (c) noisy image woman (d) denoised by ridgelet decomposition with na¨ive lines, ω = max(|p| , |q|) (e) denoised by ridgelet decomposition with supercover lines, ω = |p| + |q| .

Now with our invertible discrete Radon transform, we can obtain an invertible discrete Ridgelet transform by taking the discrete wavelet transform on

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each Radon projection sequence {Rω s(pm , qm , k)}b∈[0,K−1] where the direction (pm , qm ) is ﬁxed. This wavelet transform can be decimated or undecimated and the wavelet base can be adapted according to the application, as for the classical wavelet decomposition. Notice that our strategy generalizes and uniﬁes the methods proposed in the litterature that use particular forms of discrete lines (see section on discrete ridgelet transform).

4

Illustration and Discussion

To illustrate the diﬀerent applications that can be achieved with the new discrete ridgelet transform based on closed Reveill`es discrete lines, we have developed two examples: a denoising and a compression algorithm. The procedure of denoise by Ridgelet transform consists simply in thresholding the Ridgelet coeﬃcients and computing the inverse Ridgelet transform. The thresholding is performed with help of an undecimated method developed for the wavelet decomposition [6]. The redundancy of the wavelet decomposition, associated with this method, reduces artifacts which appear after thresholding [6].

Fig. 4. (a) Original image (b) noisy image (c) denoising with na¨ıve lines (d) denoising with pythagoricean lines (e) denoising with supercover lines

We present in the ﬁgure two results of our denoising method. With the ﬁrst example, we can see that this method can reconstruct very noisy images. Because of the adaptation of this decomposition to linear singularities, the edges of

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objects are preserved and the noise seems to be removed. The second example illustrates the results for diﬀerent deﬁnition of the lines Lω p,q . As for the ﬁrst image, the features are generally correctly reconstructed and the noise is smoothed. But if we study more precisely the result on the woman’s hat, we see that the denoising is better for ω = |p| + |q|, supercover lines, than for ω = max(|p| , |q|), naive discrete lines. The ﬁrst choice of arithmetical thickness ω introduces more redundancy into the decomposition. Due to this redundancy we obtain an average value during the reconstruction process that reduces the artifacts. In order to illustrate more precisely the result of the denoising algorithm with diﬀerent type of discrete closed lines we have generated an artiﬁcial image (Figure 4 (a)) and added important white noise (Figure 4 (b)). To show the eﬀect of the noise we have added a vertical slice of each image (at the left of (a) and right of (b)). Figures 4 (c), (d) and (e) are the results obtained with the denoising algorithm for the three deﬁnitions of closed discrete lines. As we can see, for a more redundant decomposition (supercover discrete lines, ﬁgure 4 (e)) the denoising is better than for a lesser redundant decomposition (4 (c)).

Fig. 5. (a) Original image (b) image compressed at 70% with na¨ıve discrete lines (c) image compressed at 70% with supercover lines

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Contrary to the denoising algorithm problematic, for an eﬃcient compression algorithm, redundancy is of course not interesting (more redundancy means more information and thus less compression). In our example, Figure 5, in order to obtain a compression of the image, we have selected in the ridgelet decomposition, the 30% most important (highest) coeﬃcients. This leads to a 70% compression rate. Of course, this is not a very sophisticated procedure and postprocessing would be applied in real applications. This illustrates however how the arithmetical thickness of the discrete lines employed in our ridgelet transform inﬂuences the quality of the compressed image. As expected, the lower redundancy representation (naive discrete lines) preserves all the features of the original image after thresholding (Figure 5 (b)). On the other hand, with the higher redundancy representation (supercover lines) we loose features and the image is globally of lower quality. This work can be extended in several directions. One of the more theoretical discrete geometry question that is the question of the smallest value of ω for which we one could obtain a full cover of the Fourier lattice. This is still an open and it seems diﬃcult arithmetical problem. We are also considering extending our denoising and compression algorithms with more sophisticated ﬁlters and parameters. It is clear that, for instance, the quality of the result of compression algorithm, where we have performed a simple thresholding, can be increased.

References 1. Reveill`es, J.P.: G´eom´etrie discr`ete, calcul en nombres entiers et algorithmique. Habilitation, Universit´e Louis Pasteur de Strasbourg (1991) 2. Cand`es, E.: Ridgelets: Theory and Applications. PhD thesis, Stanford (1998) 3. Do, M., Vetterli, M.: Discrete ridgelet transforms for image representation. Submitted to IEEE Trans. on Image Processing (2001) 4. Starck, J.L., Cand`es, E.J., Donoho, D.L.: The curvelet transform for image denoising. Technical report, Department of Statistics, Stanford (2000) 5. Normand, N., Guedon, P.: Transformee mojette : une transformee redondante pour l’image. Compte rendu de l’Acad´emie des Sciences (1998) 6. Carr´e, P., Leman, H., Marque, C., Fernandez, C.: Denoising the EHG signal with an undecimated wavelet transform. IEEE Trans. on Biomedical Engineering 45 (1998) 1104–1113 7. Mallat, S.: A theory for multiresolution signal decomposition: the wavelet transform. IEEE Trans. on PAMI 11 (1989) 674–693 8. Burt, P., Adelson, E.: The laplacien pyramid as a compact image code. IEEE Trans. Comm. 31 (1983) 482–550 9. Andr`es, E.: Mod´elisation analytique discr`ete d’objets g´eom´etriques. Habilitation, Universit´e de Poitiers (2000)

A Discrete Radiosity Method R´emy Malgouyres LLAIC, Universit´e Clermont 1, IUT departement Informatique, B.P. 86, 63172 AUBIERE cedex, France. [email protected] Abstract. We present a completely new principle of computation of radiosity values in a 3D scene. The method is based on a voxel approximation of the objects, and all occlusion calculations involve only integer arithmetics operation. The method is proved to converge. Some experimental results are presented. Keywords: Discrete Graphical Models, Voxel, Global Illumination, Radiosity.

Introduction Radiosity ([SP94]) is a technique which has proved being eﬃcient (in spite of its large complexity) and accurate to simulate several physical processes involving exchange of energy. Its ﬁelds of application include weather forecast, heat transfer simulation and light propagation, and in particular 3D rendering in computer graphics. However, radiosity remains so expensive that it cannot yet be extensively used in the movie industry and similar applications. Moreover, the basic radiosity algorithms rely on the so-called ideal diﬀuse hypothesis, which represents a limitation of the range of applications, and often implies a complicated combination of radiosity with other techniques (such as ray-tracing) in order to obtain a correct simulation. On the other hand, integer only geometric computations are everyday improved and several techniques have been introduced for the use of discrete geometric models in the ﬁeld of modeling or computer graphics ([YCK92], [SC95], [ANF97], [BM00]). Though discrete ray-tracing has not become very widely used, it remains an interesting discrete geometric modelisation and a good ﬁrst attempt to use discrete geometric models for 3D rendering. The purpose of this paper is to introduce a completely new radiosity method, which is based on a voxel representation of objects, and whose occlusion calculations involve only integer arithmetical operations. The objects of the scene are ﬁrst approximated by a set of voxels, which are stored in an octree data structure. Then the radiosity computations are performed in the discrete voxel space. Finally, some classical techniques such as z−buﬀer or ray-tracing are used for display of the results. The paper is organized as follows: ﬁrst we remind the reader the required notions about classical radiosity, then we set the basics of discretization. Afterwards, we explain how we can approximate the so-called diﬀuse illumination A. Braquelaire, J.-O. Lachaud, and A. Vialard (Eds.): DGCI 2002, LNCS 2301, pp. 428–438, 2002. c Springer-Verlag Berlin Heidelberg 2002

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equation by a discrete equation, and propose a numerical solution which is proved to converge towards a solution of our discrete equation. Then comes a section about implementation and complexity of the method, and ﬁnally we present some experimental results.

1

Basic Notions for Radiosity

In this section we recall the basics of the radiosity method. We refer to [SP94] for more details on classical radiosity methods. 1.1

The Data of a 3D Scene

We assume that we are given a 3D scene, which consists of a set of polyhedra (also called objects), or surfaces approximated by polyhedra, each polyhedron P of the scene being provided with a reﬂectance coeﬃcient ρP ∈ [0, 1[. The physical deﬁnition of this reﬂectance coeﬃcient is intuitively that ρP is the ratio of the total power of the outgoing light at a given point of the polyhedron and of the power of the incoming light at the same point. Note that the hypothesis that ρP < 1 means that no energy is created while light is reﬂected by the object. For convenience, given a point x of the object P , we denote ρ(x) = ρP . In this paper, we adopt the simplifying hypothesis that the reﬂectance ρP does not depend on the point on the polyhedron P , which means that the objects are made of a uniform material: they are not textured. This is not an intrinsic limitation of the presented method, but rather a hypothesis that we made for the ﬁrst investigations. There is also another hypothesis, which is that the reﬂectance does not depend on the incoming and outgoing direction of the light. This hypothesis is known as the ideal diﬀuse reﬂectors hypothesis, and is classically the basic hypothesis in radiosity methods. Intuitively, an ideal reﬂector is an object such that the light emitted from a point x of this object has the same properties in all the directions of the half-space limited by the tangent plane at point x and which contains the outgoing normal vector at x. A light source is a particular object P which is provided with a positive real number EP called the exitance of P . Intuitively, the exitance is the total power of light leaving the object per unit of area. As the reﬂectance, the exitance is assumed to depend neither on the point of the object, nor on the direction of emission. Given a point x of the object P , we denote E(x) = EP . 1.2

The Diﬀuse Illumination Equation

Here we describe a continuous equation which expresses the power of light leaving a point x (radiosity at point x) of an object P as a function of the power of light of all other points in the scene. Let us denote by B(x) the radiosity at point x. Given a point y in the scene, we denote by V (x, y) the number equal to 1 if y is visible form x in the scene (i.e. if no object of the scene intersects the straight

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line segment [x, y]), and equal to 0 otherwise. The function V thus deﬁned is called the visibility function. We denote by θ(x, y) the angle between the vector − → and the normal vector − → xy n at point x if this angle is less than π2 , and equal to π 2 otherwise (so that cos θ(x, y) = 0 if y is not in the half space limited by the tangent plane at x and containing the outgoing normal vector). The diﬀuse illumination equation at point x, which corresponds to the physical ideal diﬀuse reﬂection model, is the following (see [SP94]): cos θ(x, y) cos θ(y, x) B(x) = E(x) + ρ(x) B(y) V (x, y) dy (1) π r2 y∈scene where r denotes the distance between x and y. The problem of simulating light propagation and reﬂections in the scene, which consists in computing the radiosity B(x) at each point x of the scene, reduces to solving the diﬀuse illumination equation. However, this integral equation cannot be solved analytically except in some very particular cases which are of no use in the ﬁeld of computer graphics. Therefore, we have to ﬁnd an approximate numerical solution. A classical way to do this, called the radiosity method, is do break down the objects of the scene into a ﬁnite number of patches, and solving a discrete version of the diﬀuse illumination equation, the solution of this discrete version of the equation consisting in solving a (huge) matrix equation. However, in this solution, the coeﬃcients of the involved matrix depend on the so-called form factors, which are double integrals over the surface of couples of patches. The computation of these form factors constitute the main part of the runtime of radiosity programs, and is either very slow, or quite rough. The idea of this paper is to discretize the scene to obtain voxels, and, roughly speaking, to consider each voxel as a single point in order to replace the computation of integrals by computation of sums.

2 2.1

Discretization of a 3D Scene Basic Notions of Discrete Geometry

A voxel v = (i, j, k) is a point of Z3 , i.e. a point with integer coordinates. Classically, such a voxel v = (i, j, k) can be seen as a unit cube centered at the point (i, j, k), and whose edges are parallel to coordinates axis. Given two voxels v = (i, j, k) and v = (i , j , k ), we say that v and v are 26−adjacent if max(|i − i |, |j − j |, |k − k |) = 1. We say that v and v are 18−adjacent if they are 26−adjacent and have at least one coordinate in common. Finally, v and v are said to be 6−adjacent if they are 26−adjacent and diﬀer only by one of their coordinates. Given v a voxel and n ∈ {6, 18, 26}, we call n−neighborhood of v, and we denote by Nn (v) the set of all voxels which are n−adjacent to v. Let n ∈ {6, 18, 26}. Using the notion of n−adjacency, we can deﬁne n−connectivity as follows. Let X ⊂ Z3 be a set of voxels. An n−path in X

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form v0 to vp is a ﬁnite sequence (v0 , . . . , vp ) of elements of X such that for i = 1, . . . , p the voxels vi is n−adjacent to vi−1 . Now let v and v be two elements of x. We say that v and v are n−connected in X if there exists an n−path in X from v to v . The relation “to be n−connected in X” is an equivalence relation, and we call n−connected components of X the equivalence classes of this equivalence relation. The set X is called n−connected if it has exactly one n−connected component. Given X ⊂ Z3 , we denote by X the complement Z3 \X of X in Z3 . The set X is said to be n−separating if X has exactly two n−connected components. 2.2

Discretizing a Polyhedron

The main problem is the following: given a closed polyhedron P , with an interior and an exterior, how to generate a list of voxels which approximate the polyhedron P (in the sense for instance that the Hausdorf distance between the obtained set of voxels and P is less than 1) , and such that the obtained set of voxels is 6−separating. There is no solution to this problem if P is an arbitrary polyhedron (just think of pinched, thin or highly curved polyhedra), but, as we shall see from experimental results, we can ﬁnd practically acceptable solutions for thick enough polyhedra. We use a discretization scheme similar to the one described in [BM02]: for each face of the polyhedron, we use a polygon ﬁlling algorithm ([FVD96]) to go over the pixels of the projection of the face onto (say) the z = 0 plane, and for each of these pixels we sample the height z of the face over this pixel. 2.3

Our Discrete Data-Structure

In order to represent the discrete scene obtained after discretization, we have chosen an octree data structure, for the following reasons. First it is much less memory consuming than a boolean matrix representation of discrete objects, and second it is compatible with a hierachical version of the method, i.e. a version of the method in which some parts of the scene are discretized more roughly than others. As we shall see in the conclusion, such a hierachical method is the only way to obtain a technique which can seriously be compared to the most recent and advanced versions of classical radiosity methods. Since octrees have been very widely used in computer graphics, and more generally in computer imagery since long ago (see [S85] and [W95] among many others), we do not describe them here into many details. We just mention that an octree is a tree structure in which each node has less than 9 children, each child of a node representing an eighth portion of the space represented by the node. The root of the tree thus represents the whole matrix. A node is a leaf of the tree either if the portion of space it represents is only composed of 1’s (object leaf case), or if this portion of space is only composed of 0’s (complement leaf case). Thus, it is possible to represent an n × n × n boolean matrix by a tree with depth (at most) log n. Then, determining if a given element of the matrix is 1 can be done in O(log n).

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In our case, since 1’s correspond to voxels approximating surfaces, we are likely to ﬁnd many wide portions of space composed only of 0’s, and thus many low depth complement leaves. This is another argument for the use of the octree data structure. Indeed, as was previously pointed out by authors using octrees for ray-tracing or discrete ray-tracing ([SC95], [YCK92]), and as we shall see below, the existence of low depth complement leaves enables us to speed up the computation of the intersection between a ray (i.e. a half-line) and the objects of the scene.

3 3.1

Discretizing and Solving the Diﬀuse Illumination Equation Transforming the Integral Equation

Now we are going to explain how, by transforming a bit the diﬀuse illumination equation (Equation 1), and by using sums to approximate integrals, we can obtain a discrete equation which can be numerically solved. The ﬁrst term on the right side of the diﬀuse illumination equation, due to emitance, can easily be handled, henceforth we concentrate on the second term, which is an integral representing the light leaving the point x which is due to reﬂection of light arriving from other points y of the scene. We denote by SR (x) the sphere centered at the point x with radius R, with R ∈ R+ . Moreover, given σ ∈ SR (x), we denote by y(x, σ) the ﬁrst point of the scene met when going over the ray (i.e. half line) having x as extremity and containing σ. cos θ(y,x) B(y) cos θ(x,y) V (x, y) dy π r2 y∈scene d(y(x,σ)) = σ∈SR B(y(x, σ)) cos θ(x, y) cos θ(y, x) π||x−y(x,σ)|| 2 dσ = σ∈SR B(y(x, σ)) cos θ(x, σ) πR 2 d(y(x,σ)) dσ Note that we can identify the terms cos θ(y, x) π||x−y(x,σ)|| 2 and πR2 because both can be recognized as an element of solid angle viewed from the point x.

3.2

The Discrete Sphere Method

Now we explain how we approximate the latter integral over a sphere with radius R by a sum over the voxels of a discrete sphere. The idea is simply that the integral of a function can be approximated by sum over small patches of the area of the patch multiplied by the value of the function at some point of the patch. We use a similar idea to the so-called Hemicube method (see [SP94] for instance) in order to discretize the set of directions in space. We consider a discrete sphere ΣR with radius R, with R ∈ R, centered at a voxel x = (a, b, c), i.e. the set of all voxels at distance less than or equal to R from x and having a 6−neighbor at distance more than R from x. In our method, we shall construct as an initialization the voxels of such a discrete sphere. We can use a straightforward construction algorithm, since the time for constructing the

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sphere will anyway be very small as compared to the overall radiosity method runtime. Then we consider, for each voxel v ∈ ΣR , the set F (v) of all the faces of the voxel v which are shared by a voxel with distance greater than R from x. All these faces constitute the frontier of the discrete ball with radius R. For each face f ∈ F (v), we consider the solid angle A(f ) formed by f viewed from x. The solid angle A(f ) can be approximated as follows: assume for instance that f is a face shared by v = (a + i, b + j, c + k) and the voxel v = (a + i, b + j, c + k + 1). Then k + 0.5 A(f )

3 . 2 π ∗ (i + j 2 + (k + 0.5)2 ) 2 Now, consider SR the continuous sphere centered at x with radius R over which we want to compute the integral as above. Consider, for each v ∈ ΣR and each f ∈ F (v), the patch p(f ) which is the central projection (with center x) of f on the sphere SR . We have: dσ B(y(x, σ)) cos θ(x, σ) πR 2 σ∈SR dσ = v∈ΣR ,f ∈F (v) σ∈p(f ) B(y(x, σ)) cos θ(x, σ) πR 2

v∈ΣR ,f ∈F (v) B(y(x, v)) cos θ(x, v)A(f ) The latter approximation is obtained by considering the integrand B(y(x, σ)) cos θ as constant on the patch p(f ); the obtained integral is B(y(x, v)) cos θ multiplied by the solid angle of p(f ) viewed from x, this solid angle being equal to A(f ). Now, y(x, v) can be approximated by the ﬁrst voxel I(x, v) encountered by going over a discrete line from the voxel x through the voxel v. Finally, the diﬀuse illumination equation of Section 1.2 is approximated by the following discrete linear equation for each voxel x of the discrete scene: B(x) = E(x) + ρ(x) B(I(x, v)) cos θ(x, v)A(f ) (2) v∈ΣR ,f ∈F (v)

3.3

Numerical Solution of the Discrete Equation

The Proposed Algorithm Lemma 1. We have

lim

R→+∞

cos θ(x, v)A(f ) = 1.

v∈ΣR ,f ∈F (v)

An immediate consequence of this lemma is that our linear system (Equation 2) satisﬁes the formal properties under which the Jacobi relaxation and Gauss-Seidel relaxation both converge to a solution of the system (see [SP94] for the similar use of Gauss-Seidel relaxation in classical radiosity). Now we explain the iterative scheme. We set B0 (x) = E(x) for each voxel x of the scene. Then we inductively deﬁne Bi (x) for i ≥ 1 by: Bi (x) = E(x) + ρ(x) Bi−1 (I(x, v)) cos θ(x, v)A(f ) (3) v∈ΣR ,f ∈F (v)

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When i tends to inﬁnity, the numbers Bi for all voxels exponencially converge to a solution of the discrete Equation 2.

4 4.1

Implementation and Complexity Going over a Discrete Line

We remind the reader that the voxels of the surfaces are encoded in an octree. The problem is, given a voxel x and an integer vector v, to compute the ﬁrst voxel I(x, v) encountered by following a discrete line from x in the direction of v. First, which discrete line should we choose, since there are several ? Here, the good arithmetical properties of the chosen line are not really important. The main problem is to perform a fast computation of a 6−connected discrete line which is close to the continuous half line from x directed by v. To do so, we introduce a current voxel M initialized to x. In fact, we must initialize M to a voxel close to x on the considered discrete line in such a way that M is not, as is x, a surface voxel, i.e. the leave L(M ) of the octee corresponding to M is a complement leave. The choice of an initial M is a bit tricky. Then we iterate the following procedure until the leaf L(M ) is an object leave. We assume for instance that the coordinates of the vector v are all positive. 1. Find the limits xmax , ymax , zmax of the cube corresponding to the leaf L(M ) of the octree ; 2. Determine, using integers, which limit will be crossed ﬁrst from x by following the line in the direction v ; 3. Assume that, say, the limit xmax is crossed ﬁrst. Compute a new voxel M having xmax + 1 as ﬁrst coordinate on the discrete line. To insure 6−connectedness, we choose M such that M.y ≤ ymax and M.z ≤ zmax ; 4. Find the new leaf L(M ) in the octree data structure. Let W be the width of the voxels matrix. Since the time for searching a leaf in the octree is at most log W , the time for going over the discrete line is at most W log W . In fact, the required time is generally much less because, when the complement leave L(M ) corresponding to M represents a large cube C, ﬁrst the depth of L(M ) is less than log W , and second, we jump directly to the exit of C without considering intermediate voxels. We see here one advantage of the octree data structure: going over a discrete line is fast. 4.2

The Two Main Steps of the Method

The purpose of the method is to compute the numbers Bi (x) for each voxel x of the discretized scene, and for i suﬃciently large so that Bi (x) is a correct approximation of the solution of Equation 2. Fortunately, the convergence is exponential with respect to i, and it turns out that, in practice, we have an accurate enough estimation for i = 5, . . . , 10. As the classical radiosity techniques, our method consists in two main steps :

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1. Computation, for all voxels x of the scene and for all voxels v of the discrete sphere ΣR centered at the point x, of the ﬁrst voxel I(x, v) encountered by following the discrete 6−connected line issued from x through v. The address of the corresponding leave of the octree data structure may be stored. 2. Computation, using the inductive deﬁnition of Bi (x) (Equation 3) and the I(x, v) computed during the ﬁrst step, of the numbers Bi ’s. The First Step. Note that the result of the ﬁrst step depends only on the geometry of the scene, and not on the material properties of the objects (numbers ρ(x)) nor on the illumination conditions (light sources, emitance). This ﬁrst step is analogous to the comptation of the form factors in classical radiosity methods, and doesn’t need to be performed again in case of a change in the illumination conditions. Note the important diﬀerence of our method with respect to other radiosity methods, that the ﬁrst step consists in integer only computations. The complexity of the method is the time L for going over the longest discrete line segment included in the complement of the scene, multiplied by the number |ΣR | of voxels of the considered discrete sphere (which is O(R2 )), multiplied by the number N of voxels of the scene. The Second Step. Now we come to the second step which, after the ﬁrst step, is quite straightforward. The second step is called the propagation step. Let us mention that all calculations concerning discrete spheres, including computation of the solid angles A(f ), can be performed once for all, independently from the voxel x. Note that we stored the normal vector at each voxel (which is used for computing cos θ(x, v)) while discretizing the scene, and that this normal vector is computed by continuous techniques (such as the Phong method). An interesting point is that we can, instead of duplicating the variables Bi in order to store Bi−1 and Bi as Equation 3 suggests, we can use the previously computed Bi (y) instead of Bi−1 (y) in order to compute Bi (x). The obtained method can also be proved to converge, and, practically speaking, it converges even faster. The complexity of this step is the number i of iterations (typically 5 or 6), multiplied by |ΣR |, multiplied by the number of voxels of the scene. Therefore, as we can also observe from experiments, the complexity of the second step is lower than the complexity of the ﬁrst step. Hence the overall complexity of the method is LN |ΣR | + iN |ΣR |. 4.3

About the Space Complexity

The memory cost of the method, as described above, mainly corresponds to the cost of storage of the adresses of I(x, v) for all x and v, which is N |ΣR |. This can be managed by storing the octree data structure (hence voxel information) in the RAM, while storing the addresses I(x, v)’s on a disk. Indeed, by implementing carefully the algorithm, we can write once for all each address I(x, v) (maybe write blocks of a few MB) during the ﬁrst step, and then, during the second step, read these addresses as many times as the number i of propagations, always in

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(a) 25 hours, viewpoint 1.

(b) 72 hours, viewpoint 2. Fig. 1. Color plates: the living-room scene.

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the same order as the addresses were written. Thus, the disk write/read cost is low, and the memory cost aﬀordable.

5

Experimental Results

First let us describe the computer with which the experiments were made: the processor is 1.2 GHz, we needed 500Mo of RAM, and a 50Go IDE disk. We present 2 experiments with the same 3D scene: – The ﬁrst experiment uses a 315*315*190 matrix of voxels, the surfaces being approximates by about 1 million voxels; the radius of the used discrete sphere is 30, so that the cardinality of the discrete sphere (which is the number of directions in space taken into account) is 9194. The amount of memory used is less than 120Mo in RAM to store the octree data structure, about 18Go were written on the disk. The runtime is about 25 hours. – The second experiment (Figure 1(a)) uses a 420*420*250 matrix of voxels; the radius of the used discrete sphere is 38, so that the cardinality of the discrete sphere is around 15000. The amount of RAM to encode the octree data structure is about 200Mo, and less than 50Go were written on the disk. The runtime is about 72 hours.

Conclusion We have presented a completely new simulation technique for lighting in a 3D scene made of ideal diﬀuse reﬂectors. This method is based on a space voxelization, integer only arithmetic, and give promizing results. However, this is the ﬁrst paper on this method and a huge amount of work remains to be done, including, ﬁrst the evaluation of the method as a simulation technique and comparison with classical radiosity methods; second the generalization to an adaptative voxel space by working with higher resolution arround highly curved objects (see [BM02]); third the generalization to specular reﬂectors and transparency; and fourth arithmetical optimization.

References [ANF97] [BM00]

[BM02]

E. Andres, Ph. Nehlig, J. Francon, Tunnel-Free Supercover 3D Polygons and Polyhedra, Eurographics ’97, Budapest, Computer Graphics Forum, ed. Blackwell Publishers, vol. 16, 3, pp. C3-C13, 1997. J. Burguet, R. Malgouyres, Strong Thinning and Polyhedrization of the surface of a Voxel Object, Proceedings of DGCI’2000, Uppsala, Sweden, Lecture Notes in Computer Science vol 1953, Springer, pp 222-234, December 2000. J. Burguet, R. Malgouyres, Multiscale Discrete Surfaces, Proceedings of DGCI’2002, Bordeaux, France, Lecture Notes in Computer Science, to appear, April 2002.

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[FVD96] [S85] [SP94] [SC95] [W95] [YCK92]

J.D. Foley, A. Van Dam, S.K. Feiner and J.F. Hughes, Computer Graphics: introduction and practice (second edition in C), Addison-Wesley. J. Sandor: Octree Data Structures and Perspective Imagery, C&G Vol. 9, No. 4, pp.393-405, 1985 F.X. Sillon and C. Puech, Radiosity & Golbal Illumination, Morgan Kaufmann Publishers, San Francisco, California, 1994. N. Stolte and R. Caubet. Discrete Ray-Tracing of Huge Voxel Spaces, Eurographics 95, pages 383-394, Maastricht, August 1995. Blackwell. K. Y. Whang et al, Octree-R: an adaptive octree for eﬃcient ray tracing, IEEE TVCG, Vol. 1, No. 4, pp. 343-349, 1995 R. Yagel, D. Cohen, and A. Kaufman, Discrete Ray Tracing, IEEE Computer Graphics and Applications, September 1992, 19-28.

Author Index

Agnus V., 155 Amat J., 255 Andres E., 313, 417 Arqu`es D., 360 Attali D., 57

Kenmochi Y., 301 K¨ othe U., 22 Kong T.Y., 81 Kropatsch W.G., 1, 92 Kuba A., 392

Balogh E., 392 Bertrand G., 102, 301 Billhardt H., 165 Boissonnat J.D., 197 Borgefors G., 244 Bretto A., 124 Bruckstein A.M., 145 Brun L., 92 Burguet J., 338 Buzer L., 372

Lindblad J., 267 Lohou C., 102

Carr´e P., 417 Chastel S., 124 Ciria J.C., 45 Coeurjolly D., 326 Colantoni P., 124 Cortadellas J., 255 Couprie M., 301 Crespo J., 165 Damiand G., 220 De Floriani L., 69 Del Lungo A., 392 Dom´ınguez E., 45 Franc´es A.R., 45 Frigola M., 255 Frosini A., 136 Gau C.J., 81 H´etroy F., 57 Herman G.T., 279 Hilton A., 382 Illingworth J., 382 Jonker P.P., 187

Malandain G., 197 Malgouyres R., 338, 428 Maojo V., 165 Marchadier J., 360 Mesmoudi M.M., 69, 404 Michelin S., 360 Morando F., 69 Mu˜ noz A., 165 Nivat M., 392 Nystr¨ om I., 267 Puppo E., 69 Resch P., 220 Ronse C., 155 Ros L., 209 Sanandr´es J.A., 165 Simi G., 136 Sintorn I.M., 244 Soille P., 175 Song Y., 114 Starck J., 382 Sugihara K., 209, 350 Thomas F., 209 Th¨ urmer G., 34 Veelaert P., 289 Zhang A., 114 ˇ c J., 232 Zuni´

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Discrete Geometry for Computer Imagery: 7th International Workshop, DGCI '97, Montpellier, France, December 3-5, 1997, Proceedings

Discrete Geometry for Computer Imagery: 11th International Conference, DGCI 2003, Naples, Italy, November 19-21, 2003, Proceedings

Discrete Geometry for Computer Imagery: 13th International Conference, DGCI 2006, Szeged, Hungary, October 25-27, 2006, Proceedings

Discrete Geometry for Computer Imagery: 9th International Conference, DGCI 2000 Uppsala, Sweden, December 13-15, 2000 Proceedings

Computational Science - ICCS 2002: International Conference, Amsterdam, The Netherlands, April 21-24, 2002. Proceedings

Coordination Models and Languages: 5th International Conference, COORDINATION 2002, YORK, UK, April 8-11, 2002 Proceedings

Distributed Computing: 16th International Conference, DISC 2002. Toulouse, France, October 28-30, 2002, Proceedings

Embedded Software: Second International Conference, EMSOFT 2002, Grenoble, France, October 7-9, 2002. Proceedings

Object-Oriented Information Systems: 8th International Conference, OOIS 2002, Montpellier, France, September 2-5, 2002, Proceedings

Electronic Government: First International Conference, EGOV 2002, Aix-en-Provence, France, September 2-5, 2002. Proceedings

Computer Performance Evaluation: Modelling Techniques and Tools. 12th International Conference, TOOLS 2002 London, UK, April 14-17, 2002 Proceedings

Discrete and Computational Geometry: Japanese Conference, JCDCG 2002, Tokyo, Japan, December 6-9, 2002, Revised Papers

Discrete Geometry for Computer Imagery: 15th IAPR International Conference, DGCI 2009, Montréal, Canada, September 30 - October 2, 2009, Proceedings. (Lecture Notes in Computer Science)

Genetic Programming: 5th European Conference, EuroGP 2002, Kinsale, Ireland, April 3-5, 2002. Proceedings

Discrete Geometry for Computer Imagery: 15th IAPR International Conference, DGCI 2009, Montréal, Canada, September 30 - October 2, 2009, Proceedings. (Lecture Notes in Computer Science)

Diagrammatic Representation and Inference: Second International Conference, Diagrams 2002 Callaway Gardens, GA, USA, April 18-20, 2002 Proceedings

Logic Programming: 18th International Conference, ICLP 2002, Copenhagen, Denmark, July 29 - August 1, 2002 Proceedings

Pervasive Computing: First International Conference, Pervasive 2002, Zürich, Switzerland, August 26-28, 2002. Proceedings

Infrastructure Security: International Conference, InfraSec 2002 Bristol, UK, October 1-3, 2002 Proceedings

Data Warehousing and Knowledge Discovery: 4th International Conference, DaWaK 2002, Aix-en-Provence, France, September 4-6, 2002. Proceedings

Discrete Geometry for Computer Imagery: 10th International Conference, DGCI 2002, Bordeaux, France, April 3-5, 2002. Proceedings

Discrete Geometry for Computer Imagery - DGCI 2011

Discrete Geometry for Computer Imagery: 14th IAPR International Conference, DGCI 2008, Lyon, France, April 16-18, 2008, Proceedings

Discrete Geometry for Computer Imagery: 12th International Conference, DGCI 2005, Poitiers, France, April 11-13, 2005, Proceedings

Discrete Geometry for Computer Imagery: 16th IAPR International Conference, DGCI 2011, Nancy, France, April 6-8, 2011, Proceedings

Discrete Geometry for Computer Imagery: 8th International Conference, DGCI'99, Marne-la-Vallee, France, March 17-19, 1999 Proceedings

Discrete Geometry for Computer Imagery: 6th International Workshop, DGCI'96, Lyon, France, November 13 - 15, 1996, Proceedings

Discrete Geometry for Computer Imagery: 14th IAPR International Conference, DGCI 2008, Lyon, France, April 16-18, 2008, Proceedings (Lecture Notes in Computer Science)

Discrete Geometry for Computer Imagery: 12th International Conference, DGCI 2005, Poitiers, France, April 11-13, 2005, Proceedings (Lecture Notes in Computer Science)

Discrete Geometry for Computer Imagery: 14th IAPR International Conference, DGCI 2008, Lyon, France, April 16-18, 2008, Proceedings (Lecture Notes in Computer Science)

Discrete Geometry for Computer Imagery: 7th International Workshop, DGCI '97, Montpellier, France, December 3-5, 1997, Proceedings

Discrete Geometry for Computer Imagery: 11th International Conference, DGCI 2003, Naples, Italy, November 19-21, 2003, Proceedings

Discrete Geometry for Computer Imagery: 13th International Conference, DGCI 2006, Szeged, Hungary, October 25-27, 2006, Proceedings

Discrete Geometry for Computer Imagery: 9th International Conference, DGCI 2000 Uppsala, Sweden, December 13-15, 2000 Proceedings

Computational Science - ICCS 2002: International Conference, Amsterdam, The Netherlands, April 21-24, 2002. Proceedings

Computational Science - ICCS 2002: International Conference, Amsterdam, The Netherlands, April 21-24, 2002. Proceedings

Coordination Models and Languages: 5th International Conference, COORDINATION 2002, YORK, UK, April 8-11, 2002 Proceedings

Distributed Computing: 16th International Conference, DISC 2002. Toulouse, France, October 28-30, 2002, Proceedings

Embedded Software: Second International Conference, EMSOFT 2002, Grenoble, France, October 7-9, 2002. Proceedings

Object-Oriented Information Systems: 8th International Conference, OOIS 2002, Montpellier, France, September 2-5, 2002, Proceedings

Electronic Government: First International Conference, EGOV 2002, Aix-en-Provence, France, September 2-5, 2002. Proceedings

Computer Performance Evaluation: Modelling Techniques and Tools. 12th International Conference, TOOLS 2002 London, UK, April 14-17, 2002 Proceedings

Discrete and Computational Geometry: Japanese Conference, JCDCG 2002, Tokyo, Japan, December 6-9, 2002, Revised Papers

Discrete Geometry for Computer Imagery: 15th IAPR International Conference, DGCI 2009, Montréal, Canada, September 30 - October 2, 2009, Proceedings. (Lecture Notes in Computer Science)

Genetic Programming: 5th European Conference, EuroGP 2002, Kinsale, Ireland, April 3-5, 2002. Proceedings

Discrete Geometry for Computer Imagery: 15th IAPR International Conference, DGCI 2009, Montréal, Canada, September 30 - October 2, 2009, Proceedings. (Lecture Notes in Computer Science)

Diagrammatic Representation and Inference: Second International Conference, Diagrams 2002 Callaway Gardens, GA, USA, April 18-20, 2002 Proceedings

Logic Programming: 18th International Conference, ICLP 2002, Copenhagen, Denmark, July 29 - August 1, 2002 Proceedings

Pervasive Computing: First International Conference, Pervasive 2002, Zürich, Switzerland, August 26-28, 2002. Proceedings

Infrastructure Security: International Conference, InfraSec 2002 Bristol, UK, October 1-3, 2002 Proceedings

Data Warehousing and Knowledge Discovery: 4th International Conference, DaWaK 2002, Aix-en-Provence, France, September 4-6, 2002. Proceedings

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