Improved signal and image interpolation in biomedical applications: (MRI)

Improved Signal and Image Interpolation in Biomedical Applications: The Case of Magnetic Resonance Imaging (MRI) Carlo Ciulla Lane College, USA

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Published in the United States of America by Information Science Reference (an imprint of IGI Global) 701 E. Chocolate Avenue, Suite 200 Hershey PA 17033 Tel: 717-533-8845 Fax: 717-533-8661 E-mail: [email protected] Web site: http://www.igi-global.com and in the United Kingdom by Information Science Reference (an imprint of IGI Global) 3 Henrietta Street Covent Garden London WC2E 8LU Tel: 44 20 7240 0856 Fax: 44 20 7379 0609 Web site: http://www.eurospanbookstore.com Copyright © 2009 by IGI Global. All rights reserved. No part of this publication may be reproduced, stored or distributed in any form or by any means, electronic or mechanical, including photocopying, without written permission from the publisher. Product or company names used in this set are for identi.cation purposes only . Inclusion of the names of the products or companies does not indicate a claim of ownership by IGI Global of the trademark or registered trademark. Library of Congress Cataloging-in-Publication Data Ciulla, Carlo, 1964Improved signal and image interpolation in biomedical applications : the case of magnetic resonance imaging (MRI) / by Carlo Ciulla. p. ; cm. Includes bibliographical references and index. Summary: "This book presents novel concepts supported through mathematics to create unique theories related to interpolation"--Provided by publisher. ISBN 978-1-60566-202-2 (h/c) 1. Magnetic resonance imaging--Digital techniques. 2. Magnetic resonance imaging--Mathematical models. 3. Interpolation. I. Title. [DNLM: 1. Magnetic Resonance Imaging--methods. 2. Algorithms. 3. Image Enhancement--methods. WN 185 C581i 2009] RC78.7.N83C53 2009 616.07'548--dc22 2008038153 British Cataloguing in Publication Data A Cataloguing in Publication record for this book is available from the British Library. All work contributed to this book is original material. The views expressed in this book are those of the authors, but not necessarily of the publisher. If a library purchased a print copy of this publication, please go to http://www.igi-global.com/agreement for information on activating the library's complimentary electronic access to this publication.

To my sweet and marvelous son, Roman Alexander Ciulla

Table of Contents

Foreword......................................................................................................................................................... x Preface ..........................................................................................................................................................xii Acknowledgment.........................................................................................................................................xix Preamble: On the Philisophical Basis Underlying the Unifying Theory...............................................xxi Chapter I Magnetic Resonance Imaging and the Signal-Image Processing Techniques Developed Under the Umbrella of the Unifying Theory............................................................................................... 1 Introduction...................................................................................................................................................... 1 How the Review of the Literature Proceeds Throughout the Book.................................................................. 2 The Magnetic Resonance Imaging Database.................................................................................................. 3 Classic Interpolation...................................................................................................................................... 10 SRE-Based Interpolation............................................................................................................................... 11 Fast Fourier Transform and Spectral Power Analysis.................................................................................. 11 Non-Linear Separability of the Spectral Power Evolutions of Functional MRI Images: An Arti.cial Neural Network Experiment . ............................................................................................... 12 The Novel Theory........................................................................................................................................... 15 The Significance of the Unifying Theory........................................................................................................ 16 Implications for Signal and Image Processing.............................................................................................. 17 Motion Correction in Functional Magnetic Resonance Imaging.................................................................. 18 Summary........................................................................................................................................................ 18 References...................................................................................................................................................... 19

Section I On the Process Driven Through Deduction, Which Starts from the Intuition, Formulates a Conception, Assesses the Truth Foreseen In the Intuition, and Finally Arrives to the Derivation of the Notion Chapter II The Intuition................................................................................................................................................. 23 Introduction.................................................................................................................................................... 23 Definition I..................................................................................................................................................... 23 Definition II.................................................................................................................................................... 23

De.nition III .................................................................................................................................................. 24 Definition IV................................................................................................................................................... 24 Definition V.................................................................................................................................................... 24 Definition VI................................................................................................................................................... 26 Definition VII................................................................................................................................................. 26 Observation I................................................................................................................................................. 27 Theorem......................................................................................................................................................... 27 Summary........................................................................................................................................................ 29 References...................................................................................................................................................... 30 Chapter III The Conception of the Intensity-Curvature Functional........................................................................... 31 Introduction.................................................................................................................................................... 31 Image Energy................................................................................................................................................. 32 Definition I..................................................................................................................................................... 32 Lemma I......................................................................................................................................................... 33 Lemma II........................................................................................................................................................ 33 Deduction I..................................................................................................................................................... 34 Lemma III....................................................................................................................................................... 34 The Energy of the Interpolated Signal (Image) is not Equal to Zero at Equilibrium.................................... 36 The Energy of the Original Signal (Image) At Equilibrium........................................................................... 38 Summary........................................................................................................................................................ 39 Reference........................................................................................................................................................ 39 Chapter IV The Conception of the Sub-Pixel Ef.cacy Region .................................................................................... 40 Introduction.................................................................................................................................................... 40 Sub-Pixel Efficacy Region.............................................................................................................................. 40 Summary........................................................................................................................................................ 46 References...................................................................................................................................................... 47 Chapter V Assessment of the Truth Foreseen in the Intuition................................................................................... 48 Introduction.................................................................................................................................................... 48 The Assessment.............................................................................................................................................. 49 Summary........................................................................................................................................................ 50 Reference........................................................................................................................................................ 51 Chapter VI The Notion.................................................................................................................................................... 52 Introduction.................................................................................................................................................... 52 The Curvature of the Interpolation Function................................................................................................. 52 Summary........................................................................................................................................................ 56 Reference........................................................................................................................................................ 56

Section II Sub-Pixel Efficacy Region of the Bivariate Linear Interpolation Function Chapter VII The Theoretical Approach to the Improvement of the Interpolation Error: Bivariate Linear Interpolation Function................................................................................................... 58 Goals and Organization of the Forthcoming Text......................................................................................... 58 Scope.............................................................................................................................................................. 59 Deduction of the Intensity-Curvature Functional.......................................................................................... 60 Definition of Intensity-Curvature Functional................................................................................................ 61 Calculation of the First Order Partial Derivatives of the Intensity-Curvature Functional.......................... 62 Sub-Pixel Efficacy Region.............................................................................................................................. 62 Analysis of the Intensity-Curvature Functional............................................................................................. 63 Calculation of the Second Order Partial Derivatives of the Intensity-Curvature Functional....................... 64 The Role of the Curvature in the Sub-Pixel Efficacy Region......................................................................... 65 A Conceptual Pathway to Attempt Interpolation as Independent From the Resolution................................ 66 Assigning the Role of Energy to the Relationships Between Neighboring Pixel Intensity Values................. 67 Bivariate Linear Interpolation Function: Attempting to Obtain Resilient Interpolation.............................. 68 Summary........................................................................................................................................................ 70 References...................................................................................................................................................... 70 Chapter VIII The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function........... 72 Introduction.................................................................................................................................................... 72 Validation Paradigm...................................................................................................................................... 72 Simulations..................................................................................................................................................... 74 On the Interpretation of the Results............................................................................................................... 77 Results with Real Image Data........................................................................................................................ 79 On the Improvement of the Bivariate Linear Function................................................................................ 138 Summary...................................................................................................................................................... 170 References.................................................................................................................................................... 170 Section III Sub-Pixel Efficacy Region of the Trivariate Linear Interpolation Function Chapter IX Interpolation Procedures........................................................................................................................... 172 Introduction.................................................................................................................................................. 172 Classification and Applications of the Interpolation Procedures................................................................ 172 Error Bounds Existing in Literature and the Indirect Measures of the Interpolation Error of the Proposed Theory..................................................................................................................................... 175 Summary...................................................................................................................................................... 176 References.................................................................................................................................................... 176 Chapter X The Extension of the Theory to the Trivariate Linear Interpolation Function.................................... 180 Organization of the Forthcoming Text......................................................................................................... 180 Scope............................................................................................................................................................ 180 Methodology................................................................................................................................................ 181

Definition of Intensity-Curvature Functional.............................................................................................. 181 Study of the Intensity-Curvature Functional................................................................................................ 182 Summary...................................................................................................................................................... 186 Reference...................................................................................................................................................... 187 Chapter XI The Results of the Sub-Pixel Efficacy Region Based Trivariate Linear Interpolation Function...................................................................................................................................................... 188 Introduction.................................................................................................................................................. 188 Validation Procedure................................................................................................................................... 188 Analysis of the Rmse Ratio Obtained with MRI Data.................................................................................. 189 Results with Functional Magnetic Resonance Volumes............................................................................... 193 Summary...................................................................................................................................................... 205 Reference...................................................................................................................................................... 205 Chapter XII Equating the Two Intensity-Curvature Terms, Before and After Interpolation, Attempting to Obtain Resilient Interpolation: Trivariate Linear Interpolation Function................. 206 Introduction.................................................................................................................................................. 206 The Mathematical Procedure....................................................................................................................... 207 Summary...................................................................................................................................................... 212 Reference...................................................................................................................................................... 212 Section IV Improvement of One Dimensional B-Spline Functions Chapter XIII On the Literature of B-Spline Interpolation Functions.......................................................................... 214 The Message of the Chapter........................................................................................................................ 214 Hierarchy and Formal Definition................................................................................................................ 214 Literature: Methodologies and Applications............................................................................................... 215 The Intent of the Unifying Theory in Relashionship to the Improvement of One-Dimensional B-Splines.................................................................................................................... 218 Summary...................................................................................................................................................... 218 References.................................................................................................................................................... 219 Chapter XIV The Extension of Theory and Methodology to B-Splines....................................................................... 223 Organization of the Forthcoming Text......................................................................................................... 223 Quadratic and Cubic B-Splines................................................................................................................... 223 Calculation of the Intensity-Curvature Terms Before and After Interpolation for the Quadratic B-Spline................................................................................................................................. 224 Calculation of the Intensity-Curvature Functional ΔE of the Quadratic B-Spline...................................... 226 Study of the Intensity-Curvature Functional ΔE of the Quadratic B-Spline................................................ 226 The Sub-Pixel Efficacy Region of the Quadratic B-Spline........................................................................... 226 Solution of a 4th Degree Polynomial............................................................................................................ 227 The Derivation of the Novel Re-Sampling Locations of the Quadratic B-Spline........................................ 228

Calculation of the Intensity-Curvature Terms Before and After Interpolation for the Cubic B-Spline................................................................................................................................................... 229 Calculation of the Intensity-Curvature Functional ΔE of the Cubic B-Spline............................................. 230 Study of the Intensity-Curvature Functional ΔE of the Cubic B-Spline....................................................... 230 The Sub-Pixel Efficacy Region of the Cubic B-Spline................................................................................. 231 The Derivation of the Novel Re-Sampling Locations of the Cubic B-Spline............................................... 232 Equating the Intensity-Curvature Terms for Quadratic and Cubic B-Splines: The Deduction................... 233 Summary...................................................................................................................................................... 237 References.................................................................................................................................................... 238 Chapter XV The Results of the Sub-Pixel Efficacy Region Based B-Spline Interpolation Functions..................... 239 Introduction.................................................................................................................................................. 239 Data............................................................................................................................................................. 239 Results with the Quadratic B-Spline............................................................................................................ 240 Results with the Cubic B-Spline................................................................................................................... 295 Summary...................................................................................................................................................... 337 Reference...................................................................................................................................................... 337 Chapter XVI On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region...................... 338 Introduction.................................................................................................................................................. 338 Efficiency, Approximation and Efficacy....................................................................................................... 338 Frameworks Existing in Literature.............................................................................................................. 341 Summary of the Properties of the Sre-Based Interpolation Functions........................................................ 341 Characteristics of the Methodology for the Improvement of the Interpolation Error................................. 342 Fourier Properties of the Sub-Pixel Efficacy Region................................................................................... 343 Summary...................................................................................................................................................... 344 References.................................................................................................................................................... 344 Section V Extension to Lagrange and Sinc Interpolation Functions Chapter XVII The Main Innovation Determined By the Sub-Pixel Efficacy Region................................................... 348 Introduction.................................................................................................................................................. 348 Literature..................................................................................................................................................... 348 Message to the Reader................................................................................................................................. 350 Summary...................................................................................................................................................... 351 References.................................................................................................................................................... 351 Chapter XVIII The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions..................................... 353 Organization of the Forthcoming Text......................................................................................................... 353 Lagrange Interpolation Function................................................................................................................ 353 Calculation of the Lagrange Intensity-Curvature Terms Before and After Interpolation........................... 354 Calculation of the Lagrange Intensity-Curvature Functional ΔE............................................................... 355

Study of the Lagrange Intensity-Curvature Functional ΔE......................................................................... 356 The Sub-Pixel Efficacy Region of the Lagrange Interpolation Function..................................................... 356 The Derivation of the Novel Re-Sampling Locations of the Lagrange Interpolation Function.................. 358 Characterization of the Interpolation Error Improvement and Interpolation Error Bounds...................... 360 The Intensity-Curvature Functional of Sinc Interpolation.......................................................................... 360 The Sub-Pixel Efficacy Region of Sinc Interpolation................................................................................... 362 The Novel Re-Sampling Locations of Sinc Interpolation............................................................................. 363 Equating the Intensity-Curvature Terms for Lagrange and Sinc Interpolation Functions: The Deduction......................................................................................................................................... 365 Summary...................................................................................................................................................... 370 References.................................................................................................................................................... 370 Chapter XIX The Results of the Sub-Pixel Efficacy Region Based Lagrange and Sinc Interpolation Functions............................................................................................................................. 371 Introduction.................................................................................................................................................. 371 Validation Methodology and MRI Dataset.................................................................................................. 371 Lagrange Interpolation: Results with T1, T2 and Functional MRI Data.................................................... 372 Sinc Interpolation: Results with T1, T2 and Functional MRI Data............................................................. 420 Summary...................................................................................................................................................... 470 Reference...................................................................................................................................................... 470 Chapter XX On the Implications of the Sub-Pixel Efficacy Region and the Bridging Concept of the Unifying Theory................................................................................................................................... 471 Introduction.................................................................................................................................................. 471 Novel Conceptions and Limitations of the Unifying Theory........................................................................ 472 On the Estimation of Unknown Signals....................................................................................................... 473 On the Interpolation Error Representation.................................................................................................. 480 The Concept Bridging Classic Interpolation with the Unifying Theory...................................................... 481 The Basic Formulas of the Unifying Theory................................................................................................ 482 Influence of the Resolution on Interpolation Error and Its Improvement.................................................... 484 An Investigation on the Influence of the Scale Parameter and a Performance Comparison Across Classic and Sre-Based Interpolation Functions.......................................................................... 493 Summary...................................................................................................................................................... 503 References.................................................................................................................................................... 508 Appendix..................................................................................................................................................... 512 About the Author....................................................................................................................................... 612 Index............................................................................................................................................................ 613



Foreword

I am pleased to introduce to the scientific community and specifically to the body of undergraduate and graduate students in disciplines that relate to signal (image) processing, applied mathematics, biomedical imaging, biomedical engineering imaging technology, and more generally: applied computational engineering the works of Dr. Carlo Ciulla in signal (image) interpolation. These works propose a novel scientific approach to an ancient problem, interpolation. I had the possibility to meet with Carlo at Lane College during the summer of 2008 and had the opportunity to discuss his works and to see manifestly the proposition Carlo had to readapt the math related to signal interpolation with his own thinking and beliefs. The science in this book is made following the style most common to academia which is that of developing novel concepts supported through mathematics and to converge to a unifying theory. I had the possibility to see through his way of thinking of science and noticed that his approach is made through deduction versus hypothesis testing. The school of thought that drives the development of the science in this book is addressing the undoubtedly challenging task of teaching students how to do research. The school of thought and the science proposed in this book are therefore correlating with each other and tracing the pathway that Dr. Carlo Ciulla follows along while employing his general view on how teaching should be conducted. The pathway to teaching is well accepted here at Lane College from his students and colleagues. The task addressed while employing signal (image) interpolation relates strictly to sampling theory wherefore analog/digital conversion is necessary to make a continuous signal a discrete one. In the quest for the validation of the mathematical assertions made through the unifying theory, this book addresses the specific case of Magnetic Resonance Imaging (MRI). Nevertheless, while proposing innovation the true underlying task that the manuscript addresses, is certainly related to handling algebra and calculus. This fact is quite relevant to students interested in deepening their understanding of these disciplines while viewing the wider panorama displayed through a diagnostic radiology application as relevant as MRI is in modern days. The benefits provided through the book can also be appreciated from audiences in the interests, but not limited to applied computational engineering. Dr. Reda Abraham, PhD Lane College, USA Professor Reda Abraham was awarded the Bachelor of Science in physics and pure mathematics (1968), University of Assiut in Egypt, Master of Science in physics, Ain Shams University, Cairo Egypt (1977) and the PhD in physics, Zagazig University, Egypt (1986) with close collaboration with the University of London. Professor Abraham presented at the Italian Physical Society and was later appointed postdoctoral researcher at Varian Application Lab (Germany) in 1988. He was assistant lecturer from 1977 to 1981 and from 1984 to 1986 at Zagazig University, graduate assistant at the University of London in between the years 1981 and 1984. Since 1986 he was appointed faculty positions at Zagazig University, Brookdale Community College and the Board

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of Education of Vocational Schools in New Jersey, USA. In 2000, Abraham was appointed assistant professor of physics at Lane College, Jackson, TN. In 2007, Abraham received an additional faculty training at the University of Oregon. Professor Abraham has taught a variety of courses in physics with calculus, electronics, classical mechanics, thermal physics, mathematical physics, electromagnetism, quantum mechanics, physical science, and college algebra. He has authored 5 journal publications and supervised the theses and dissertations of three graduate students. In the year 2008, Abraham was awarded “Professor of the Year” at Lane College.

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Preface

OVERVIEW OF THE SUBJECT MATTER “If I am anything, which I highly doubt, I have made myself so by hard work.” Isaac Newton This book presents a novel theoretical framework for the improvement of the interpolation error from which is derived a unified methodology applied to several interpolation paradigms: two and three-dimensional linear, one-dimensional quadratic and cubic B-Splines, Lagrange, and Sinc. Advances are made to derive a framework that is unifying in its purpose and that thus achieves improvement of the approximation properties of the interpolation function regardless to its dimensionality or degree. The framework proposes a mathematical formulation, which consequentially to its developmental approach formulates interpolation error improvement as dependent from the joint information content of node intensity and the second order derivative of the interpolation function. From the theory herein developed, the mathematical formulation is derived by two main intuitions called the Intensity-Curvature Functional and the Sub-pixel Efficacy Region, and these are such to set the grounds for the improvement of the interpolation error. From the theory it descends that given a location where the interpolation function ought to estimate the value of the unknown signal, a novel sub-pixel (intra-node) re-sampling location is determined which varies locally pixel-by-pixel (node-by-node). The novel theory asserts and demonstrates by a-posteriori knowledge, that the interpolation error improvement can be achieved based on the joint information content of the node intensity and the second order derivative of the interpolation function. Consequentially to the formulation, re-sampling is performed locally at locations that are variable depending on the signal intensity at the neighborhood and local curvature of the interpolator. The book initiates presenting two mathematical intuitions and given that absolute truth cannot be reached by them, the intuitions serve the purpose to derive novel conceptions of interpolation error improvement. These conceptions are determined during a process driven through deduction and thus a theoretical framework is presented. The boundary of the truth determined through the mathematical intuitions in explaining interpolation error improvement are empirically tested. The purpose of the theory is that of unifying under the same approach interpolators of diverse degree and dimensionality such to determine novel interpolation functions so called: SRE-based interpolation functions, which possess improved approximation characteristics. The bridging concept between SRE-based and classic interpolation functions is the local curvature of the function as expressed by the second order derivative. There are extended mathematical descriptions of the theory with detailed formulations to conceptualize the steps of the approach. Also, improved interpolation functions are validated experimentally by a motion correction paradigm. The motion correction paradigm shows both the error reduction and the resulting image quality after improvement. Spectral power analyses are performed and results are presented to show that local re-sampling changes the band-pass characteristics of the interpolation function. Validation is then extended to the estimation of signals at unknown locations to simulate the estimation beyond the Nyquist’s frequency in

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common experimental settings. The book cites the most current relevant and compelling literature in the field of signal-image interpolation and explains the need for advancing to the novel theory it proposes. Also, the book literally explains scope and methodological approach and details the mathematical characterization. Illustrations are given for the validation method and benefits and limitations of the theory are shown quantitatively and qualitatively. Outlines are given of the implications in image and signal processing with particular attention to biomedical imaging applications such as anatomical and functional Magnetic Resonance Imaging (MRI). Given the level of details that the book provides to the reader while furnishing proofs to the mathematics and also because of the inclusion of the source code, the intended audience is within academic level among students interested in exploring the innovative approach that the treatise proposes, and within disciplines like signal-image processing, medical imaging and applied mathematics. It is known that students appreciate and benefit from detailed explanations (Hopcroft & Ullman, 1969). The alternative approach that the book presents can be viewed with the intent of (i) devising novel mathematics for signal-image interpolation applications, and (ii) implement them in software. Thus the purpose of the book is targeted to complementing current textbooks for the exploration of the alternative approach that is proposed. At the aim of further research and exercise and because of the unifying challenge of this work, students can undertake the application of the methodology to interpolation paradigms that are not presented in the book. While the book is appropriate for the junior-level undergraduate, particularly in image processing and biomedical imaging courses within computer science and biomedical engineering programs, it can also be a compelling source of research material for graduate courses in the same majors. This book is also dedicated to whoever believes in science as combined effort of disciplines and which ultimate goal is that of the quest for truth. As within the course of the process of acquiring knowledge, doubt arises it is of necessity to remind that knowledge ultimate goal is that of improving understanding. Intuitions lead to the development of general conceptions, derived in the light of a process determined through deduction, which subsequently became a unifying theory. As absolute truth cannot be derived from intuitions, conceptions must undergo testing and thus by means of the process of empirically validating the theory, the boundaries of the truth were derived.

THE CHALLENGES Titles of distinguished books in the field of signal-image processing are available in literature (Agarwal & Wong, 1993; Castleman, 1996; De Boor, 1978; Sonka et al., 1999). Mathematical formulations of the problem of signal interpolation with in-depth coverage are provided (Agarwal & Wong, 1993; De Boor, 1978) and they convey currently adopted and recognized protocols that are currently being studied in colleges at undergraduate and graduate level. These authors have contributed enormously to the effort of the research community in the development of signal processing applications that employ interpolation within the context of quite large and diverse web of tasks. For a comprehensive review of on interpolation procedures and their applications, the reader is referred recent surveys (Amidror, 2002; Meijering, 2002). The knowledge provided through these two books (Agarwal & Wong, 1993; De Boor, 1978) is highly specific as much as remarkable and focuses on signal-image interpolation with several approaches dedicated to the improvement of the approximation characteristics of the interpolation functions. Whereas the other two books (Castleman, 1996; Sonka et al., 1999) have presented works that are compelling because of their comprehensive coverage of topics related to signal-image processing. These other authors devote specific as well as remarkable focus to the solution of problems in computer vision and medical imaging contexts. The challenges that the book addresses are herein listed: (i) derivation through deduction, (ii) unifying theoretical approach, (iii) extendibility of theory and methodology.

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Derivation through deduction. Theory and methodology are derived through deduction, thus through a process of posing the question and answering to it with the appropriate mathematical tool. It descends that the book introduces novel mathematics and the correspondent novel conceptions for which an appropriate name has been given. Unifying theoretical approach. In the book it is presented a unifying methodology for the improvement of the interpolation error and thus the improvement of the approximation characteristics of the interpolators. This is done through a unique and novel approach that is applied to interpolation functions regardless to their degree or dimensionality. Extendibility of theory and methodology. The book sets the grounds for a unifying theory under which interpolation functions of diverse degree and dimensionality are embraced under the same methodological approach. It is possible to extend the theory to other interpolation paradigms and consequentially the book serves the purpose to allowing the reader to produce knowledge that can go beyond the one presented herein.

SEARCHING FOR A SOLUTION This book presents a novel approach while in the quest to finding the solution to the problem of devising a unifying theory and methodological approach for the improvement of the interpolation error which is extendible to diverse mathematical functions regardless to their degree and dimensionality. Along the quest for the solution, the book delivers a message to the reader. The message is that: “there exists a unifying approach that regardless of the dimensionality or the degree of the interpolation function achieves interpolation error improvement”. The book and the approach is original and unique, thus both the theory and the methodology that are presented are absent from other signal processing text books. Some of the distinguished features of the book are listed here to follow. • • • • • •

The mathematics are novel and original. The methodology is derived through deduction. Novel terminology is introduced specifically within the developmental effort. The validation paradigm is consistently applied to all of the interpolation functions. Results are discussed within the context of the most relevant literature in signal-image interpolation published in scholarly written papers that appeared in leading journals concerned with signal (image) interpolation. Consistent application of theory and methodology across interpolation functions determines a unifying approach to the improvement of the interpolation error.

Once the solution is found the book provides to the reader with knowledge. Hereto follow it is anticipated an explanation of the meaning of the basic conception: the Sub-pixel Efficacy Region (SRE) which allowed the deduction of the mathematics that are the constituents of the unifying theory. Hence in explaining the meaning of the spatial set of points named SRE is due considering their effect on the interpolation function. Given a classic interpolation function, the values of the independent variable are re-organized within their domain through the projection determined by means of the SRE such to obtain novel values of the independent variable called novel re-sampling locations. Re-calculation of the classic interpolation function at the novel re-sampling location consists of the SRE-based interpolation function. Thus, the effect of the SRE on the classic interpolation function is that of changing the function in that the model to the data changes because of the re-organization of the independent variable within its domain.

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ORGANIZATION OF THE BOOK The book is organized in 20 chapters and a brief description of each chapter is given in the following. A prelude of philosophical nature precedes the chapters of the book. Chapter I provides the reader with an introduction aimed to acknowledge the basic issue of interpolation for signal and image processing. The Magnetic Resonance Imaging (MRI) database is introduced. It is presented as an outline on how the review of the literature progresses throughout the book along with an introduction to classic and SRE-based interpolation, and the most relevant signal processing techniques developed for the specific purpose to undertake the research presented in the book. The unifying theory is introduced parallel to a discussion on its significance and its implications for signal and image processing and for motion correction in functional MRI. Chapter II opens Section I of the book and introduces to the reader the intuition that has set the roots for the development of the work presented in the book. The intuition consists of a math process that starts with definitions, yields to observations, derives the definition of the Sub-pixel Efficacy Region (SRE) and on its basis a claim is made that the interpolation function assumes most accurate approximation within the spatial extent of the SRE. This claim is deemed to be a truth foreseen in the intuition and will be analyzed in the following chapters with the purpose of deriving the true notion. Chapter III provides the reader with one conception named the Intensity-Curvature Functional, which consists of a math formulation derived on the basis of two intensity-curvature terms. The two intensity curvature terms consist of the measure of the energy level that the signal (image) possess before interpolation (in its original form) and after interpolation. It is explained that the Intensity-Curvature Functional is well suited to measure the change of energy level of the signal (image) caused by the model interpolation function. Chapter IV formulates another conception: the Sub-pixel Efficacy Region. As the Sub-pixel Efficacy Region was an intuition in chapterII; it becomes a conception in chapter IV. Along this process the concept of Intensity-Curvature Functional is refined in that it becomes the measure of the energy level change determined by the interpolation function for the particular re-sampling location where the signal is calculated. Chapter V assesses the truth foreseen in the intuition presented in chapter II. The definition of the SRE given in chapter II is evaluated on the basis of the results of chapter IV. The purpose is to ascertain how big is the degree of match between the theoretical definition of SRE given in chapter II and the conception of SRE derived in chapter IV which embeds practical implications that derive from the conception of the IntensityCurvature Functional. The deduction carried out in chapter IV show that the SRE consists of the extremes (either minimum or maximum) of the Intensity-Curvature Functional. The degree of match is found to be the concept of curvature of the interpolation function and this concept ascends to the role of the truth foreseen in the intuition as well as it embeds practical implications for the estimation of signals of unknown nature. Chapter VI concludes Section I of the book deriving the notion on the basis of the evaluation of the truth foreseen in the intuition. Chapter IV asserts that the Sub-pixel Efficacy Region is that spatial set of intranodal points where the energy level change of the signal is minimum or maximum. Chapter VI clarifies that the Sub-Pixel Efficacy Region targets the spatial set of intra-nodal points for which the local second order partial derivatives are smaller than the increment or decrement of the function measured on the tangent to the first order derivative. The notion is therefore that the curvature of the model interpolation function is quite relevant to the extent of improving the interpolation error. Chapter VII opens Section II of the book while introducing, studying, and solving the problem of the improvement of the bivariate linear interpolation function by the use of the SRE. The two intensity-curvature terms, calculated at the grid point and at the generic intra-pixel location (x, y) respectively, are employed to formulate the Intensity-Curvature Functional (∆E). The solution of the polynomial system of the first order derivatives of ∆E furnishes a sub-pixel set of points located in the domain of existence of the signal: the so called SRE. The chapter also explains that on the basis of the theory proposed in the book re-sampling can vary on a pixel-by-pixel basis at the image space locations called novel re-sampling locations. The chapter concludes presenting a math formulation that attempts resilient interpolation. The validation of such formulation is left to the willing audience of this book.

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Chapter VIII presents results of the performance of the SRE-based bivariate linear interpolation function starting with a detailed description of the validation paradigm. The validation paradigm is employed on Magnetic Resonance Imaging (MRI) data within the context of three different modalities: T1-MRI, T2-MRI, and functional MRI. Additionally, the validation paradigm is employed on signals of known nature such as trigonometric functions. Quantitative and qualitative analyses are presented of the reduction of the interpolation error and of the improved approximation properties of the SRE-based bivariate linear interpolation function. Chapter IX opens Section III of the book and reports on relevant literature that classifies interpolation procedures on the basis of the image processing task to perform and on the basis of the information content employed at the aim to reconstruct the continuous signal. The chapter does inform the reader as to what is the relevance of the interpolation error bounds characterization forms existing in literature, and how they relate to the mathematical foundations of the theory presented in the book. Chapter X outlines, studies, and solves the problem of the improvement of the trivariate linear interpolation function by means of the Sub-pixel Efficacy Region. This is carried out with detailed description leading to a comprehensive coverage of the mathematics. Chapter XI presents the results of the Sub-pixel Efficacy Region based trivariate linear interpolation function. The chapter focuses on functional MRI data and the presentation is given quantitatively through the root-mean-square-error (RMSE) of the two classes of trivariate linear interpolation paradigms: (i) classic and (ii) SRE-based. Also, they are presented quantitative results obtained through the use of the signal processing tool called spectral power evolution. This signal processing tool complements the spectral information provided through the Fast Fourier Transform (FFT) showing the differences existing in terms of frequency components between the three types of images: (i) original, (ii) processed with classic interpolation, and (iii) processed with SRE-based interpolation. Chapter XII follows the path traced by the last section of chapter VII in attempting resilient interpolation for the case of the trivariate linear function. The chapter introduces the novel concept of signal intensity value resilient to interpolation and this concept is accompanied with the corresponding math formulation. This concept has theoretical implications in the meaning that the signal resilient to interpolation would be that one reconstructed through the model function under the constraint that the intensity-curvature content does not change either with or without interpolation. The reconstructed signal attempts to be the one that could not been sampled because of the limitations imposed by the Nyquist’s theorem. The validation of this chapter is intended for the willing audience of this book available to undertake this research path. It might be possible to determine resilient interpolation. Chapter XIII opens up Section IV of the book, informs the reader of the quite extensive literature existing for the piecewise polynomial interpolation functions and devotes particular attention to the quadratic and cubic B-Splines. This chapter reinforces on the two conceptions that were object of experimentation and studies in the previous chapters. These conceptions are: (i) interpolation error improvement can be formulated as dependent on the joint information content of the node intensity and the second order derivative of the interpolation function, and (ii) re-sampling is an issue of local relevance, which therefore depends on the neighboring signal intensity values. Chapter XIV reports meticulously on the mathematics employed to devise the one dimensional quadratic and cubic SRE-based interpolation functions. The methodological approach undertaken descends from the novel theory as an extension of the unifying approach that has already characterized the SRE-based linear interpolation paradigms. The chapter reinforces on the fact that the second order derivative of the function incorporates the curvature information content, which is joined by the signal intensity values to form the mathematical tools that are able to set the grounds for the improvement of the B-Spline functions’ approximation properties. Consistently with chapters VII and XII, this chapter presents also the math formulation that continues to outline the attempt to device resilient interpolation. Chapter XV presents results for quadratic and cubic B-Splines SRE-based interpolation functions obtained through the experimentation conducted on the basis of the motion correction validation paradigm employed consistently throughout the book. Data employed consists of T1-MRI, T2-MRI, and functional MRI. Qualita-

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tive and quantitative investigation is conducted through Fast Fourier Transform (FFT) analysis complemented with the analysis of the spectral power evolutions of the resulting images and with the analysis of the error images obtained after processing. The analysis embraces the two classes of interpolators: (i) classic and (ii) SRE-based. This investigative method reveals the difference between classic and SRE-based B-Spline interpolation in terms of interpolation error and spectral frequency content. Chapter XVI concludes Section IV of the book. There is specific reference to compelling literature and mention to the dependency of the mathematical formulation on the joint information content of node intensity and curvature of the interpolation function. It is finally reinforced that the mathematics derived through the unifying theory are capable to determine a common ground for the improvement of the interpolation error for functions of diverse degree and dimensionality. Emphasis is given to the Fourier properties of the SREbased interpolation functions which are featured with the capability to retain the spectral components of the original images. Chapter XVII opens Section V of the book and introduces to the reader with the intent to improve the performance of two one-dimensional interpolation functions (Lagrange and Sinc) that have been shown through Fourier analysis to present excellent pass-band characteristics. The chapter reports due reference to compelling literature and also emphasizes on the main innovation introduced through the SRE-based interpolation functions. Chapter XVIII presents the Sub-pixel Efficacy Region of one-dimensional cubic form of Lagrange and one-dimensional Sinc interpolation functions. Following the approach of the unifying theory, the deduction of the mathematics yields, consistently with what seen for the other functions treated in this book, to the determination of the novel re-sampling locations for the two interpolation functions. Additionally, the chapter provides reference to the characterization of the interpolation error and interpolation error improvement bounds, which in their formulation descend from the unifying approach that the theory presents throughout the book. Consistently with chapters VII, XII, and XIV, the chapter concludes the presentation of the math formulation that attempts to device resilient interpolation. The validation of this math is left to the willingness of the reader. Chapter XIX presents results of the application of the Sub-pixel Efficacy Region to the cubic form of Lagrange interpolation and Sinc interpolation. In this chapter specific focus is given to the potentiality of these two SRE-based interpolation functions in improving the interpolation error and also in preserving spectral components of the images. Quantitative and qualitative evaluations are presented employing the following three modalities of MRI data: T1, T2, and T2* (functional MRI). Chapter XX reports concluding remarks recalling to the reader of the intent of the book. Characteristics of the methodology outlined in the unifying theory are referenced to the current literature pointing out to the flexibility of the mathematics and their capability to group interpolators under same methodological approach regardless to their degree and dimensionality. Among the implications of the SRE are: (i) preservation of spectral frequencies into the interpolated images and (ii) reduction of the smoothing effect of the interpolation function. It is shown that the SRE-based interpolation functions can outperform classic interpolation functions in the estimation of signals at locations that are not sampled because of the Nyquist’s theorem constraint. Also, quantitative evaluation is conducted across the SRE-based polynomial interpolation functions and the Sinc function to elucidate which one performs the best in terms of interpolation error. Finally, qualitative evaluation is conducted in order to ascertain the influence of an important parameter on (i) the calculation of the novel re-sampling locations and (ii) the capability to preserve the spectral content of the original images. This parameter is used to scale the convolutions of the polynomial interpolation functions and the sums of cosine and sine functions of the Sinc interpolation function. Due to the uniqueness of the theory and methodology and the originality of the mathematics that the book presents, it would not be appropriate to consider the present book as competing with any other in the field of signal-image processing. The book is intended as vehicle to add to the current state-of-the-art knowledge in signal-image interpolation and would be more suitable for complementariness rather than being an alternative to current text books. The keyword for this book is thus complement and not competitiveness. Furthermore,

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while in the process of validating the theory and methodology, this book devotes specific focus to the case of Magnetic Resonance Imaging (MRI) and thus adds to the existing literature a source of reference specifically oriented to MRI applications.

REFERENCES Agarwal, R. P., & Wong, P. J. Y. (1993). Error inequalities in polynomial interpolation and their applications. Mathematics and its applications. Dordrecht, The Netherlands: Kluwer Academic Publishers. Amidror, I. (2002). Scattered data interpolation methods for electronic imaging systems: A Survey. Journal of Electronic Imaging, 11(2), 157-176. Castleman, K. R. (1996). Digital image processing. Englewood Cliffs, NJ: Prentice Hall. De Boor, C. (1978). A practical guide to splines. Applied mathematical sciences. New York, NY: SpringerVerlag. Hopcroft, J. E., & Ullman J. D. (1969). Formal languages and their relation to automata (p. 242). Reading, MA: Addison-Wesley. Meijering, E. (2002). A chronology of interpolation: From ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90(3), 319-342. Sonka, M., Hlavac, V., & Boyle, R. (1999). Image Processing, Analysis and Machine Vision. (2nd ed.). Pacific Grove, CA: Brooks/Cole.

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Acknowledgment

The author wishes to express exceptional and sincere gratitude to Professor Fadi P. Deek (New Jersey Institute of Technology) for the great, vigorous and continuous support provided during the publications of the three papers (Ciulla & Deek, 2005a, 2005b, 2006) and also during the preparation of the proposal of this book. Professor Deek suggestions were magnificent to the extent to set key and remarkable milestones in the development of this work. The author also wishes to express exceptional and sincere gratitude to the entire IGI Global publishing team that since the inception of the book idea made possible the development of the book. Particular mention is due to Medhi Khosrow-Pour, to Julia Mosemann for the constant, vigorous and effective support and also the invaluable suggestions provided during the developmental effort of the book, to Kristin Klinger for the invaluable suggestion in proposing the title for the book and for proposing to expand the book from its original size, also to Kristin Roth, Jessica Thompson, Corrina Chandler, Lindsay Johnston, Donna Lattanzio and Jan Travers for the development process of the book. The author is very grateful to Professor Reda Abraham (Lane College) for the foreword written for this book with expertise, tact and politeness, to raise the interest of the readers on the content of the book. Professor Reda Abraham is acknowledged for his invaluable insights provided in his text. Naturally, my appreciation and gratitude is expressed also for the excellent insights that the anonymous reviewers of the draft of the book have provided along with invaluable suggestions aimed to improve both quality and organization of the final manuscript. I was truly honored by the remarkable level of appreciation of my work expressed by the reviewers. Also, sincere gratitude is extended to the anonymous reviewers of the book proposal submitted to IGI Global, and to the anonymous reviewer that has helped improving the presentation of the style of the draft of the book proposal prior to submission to IGI Global. Due recognition is to be given to ICGST - International Congress Global Science Technology and the International Journal on Graphics Vision and Image Processing for the possibility to publish the three above mentioned manuscripts and also to the anonymous reviewers for the excellent suggestions given in order to improve the quality of the three manuscripts. The author is very grateful to all the contributors of Magnetic Resonance Images that were provided for the development of this book. Dr. Daniel Marcus for making available the T1-MRI data set, and all the contributors to the Open Access Series of Imaging Studies (OASIS) (www.oasis-brains.org), and the authors and co-authors of the key publications relevant to the development of OASIS. Dr. Rand S. Swenson (www. Dartmouth.edu/~rswenson/Atlas) for making available the T2-MRI data set. Dr. Pierre Kornprobst and Dr. Ronald Peeters (www-sop.inria.fr/odyssee) and also the Department of Radiology KULeuven for making available the functional MRI data set named fMRI-TS1 in this book. Dr. Will Penny (www.fil.ion.ucl.ac.uk) for making available the functional MRI data set named fMRI-snffM00587 in this book, and also the authors of the key manuscript relevant to these data (Buchel & Friston, 1997). The author is very grateful also to the developers of the software ImageJ (http://rsb.info.nih.gov/ij/) which made possible to calculate Fast Fourier Transform (FFT) maps, histograms of the FFT maps and the error

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images presented in this book. Also, ImageJ made possible to visualize the maps of the novel re-sampling locations and the MRI images shown in the book. Also due recognition is due to the authors of the software MRIcro (http://www.sph.sc.edu/comd/rorden/ mricro.html) which made possible to prepare the images in analyze format. Great recognition is due to Wikipedia, the free encyclopedia (www.wikipedia.org) for the invaluable knowledge that was provided in order to solve the cubic equation, the quartic equation and also to employ the Newton’s Iterative Method (Newton - Raphson, Newton - Fourier) for the solution of the equation that leads to the Sub-pixel Efficacy Region of the Sinc interpolation function.

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Preamble: On the Philosophical Basis Underlying the Unifying Theory

The process of deriving knowledge from scientific investigation has its main goal in improving understanding of phenomena. To develop a theory, an approach that is customarily used in science is that of stating hypotheses and then move into experimentation to seek for empirical confirmation. Such mechanism of hypothesis testing requires collection of data and reconstruction of the reality data portrait through the modulation of empirical results, and comparison of the degree of match with the vision of the reality that has been formulated a-priori. Naturally, it is quite difficult to foresee the reality as it will itself manifest and thus it is difficult to formulate a-priori hypotheses that make a model of it. Therefore such methodological approach is normally based on a-priori knowledge that has been collected through other studies and experimentations which leave room for testing further conceptions such to add to the picture of reality other pieces of evidence. Many disciplines benefit of this type of approach which allowed several majors to build the foundations of knowledge and understanding. Conversely, another approach to determine the boundary of reality as manifestation of the phenomena is that of collecting data prior to the formulation of any hypotheses and thus to leave full freedom of expression to data such that the hypotheses can later be tested on them and a theory can be developed (De Laszlo, 1959/1990). Such approach was suggested, among others, by the eminent psychologist C. G. Jung, whom by means of the practice of his profession, which was based on the process of acquisition of data from its patients and their interpretation, developed the root foundations of modern psychology. Therefore, it is through the process of data collection that reality can manifest itself and allow answering research questions. It is then possible through hypotheses testing and thus validation to collect knowledge and to build a theory such to improve the understanding. During the past century however, subsequently to the publication and the acceptance of the special theory of relativity developed by Albert Einstein (Lorentz et al., 1923/1952), a novel discipline called quantum theory came to light through the works of Werner Heisenberg whom described the uncertainty principle and Niels Bohr whom gave birth to the Copenhagen interpretation of Quantum Mechanics (Herbert, 1985). Albert Einstein was eminently and intellectually adversary of the uncertainty principle without the proper realization that it consisted of a considerable extension of his own works. These scientists showed to the scientific community that reality has its own ways to keep itself manifests to the observer. And on the basis of quantum theory it can also be conjectured that the reality we see is what the measurement instrument makes of it. That means, what the instrumentation allows us to measure. Thus the reality we measure consists of an approximation of what the instrumentation allow us to see under certain experimental settings. Above mentioned ways of producing scientific knowledge need certainly to be corroborated through the process which was most familiar and used by Albert Einstein, whom accompanied his quest for knowledge through deduction, starting from intuitions and driven through a process that through the use of mathematical methods poses questions to the findings and subsequently obtain answers from the results, and drive to its own determination a theory which can be such to explain phenomena of nature. Undoubtedly though, every theory has its own core property in the symmetry property. By means of the symmetry property a theory can be validated but can also lose validity when the experimentation furnishes the negation (Greene, 1999; Herbert, 1985). To cite two most eminent examples of the last century, while the special theory of relativity had its symmetry property in the Lorentz invariance between mass and velocity

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(Lorentz et al., 1923/1952), quantum theory had the symmetry property in the equivalence principle introduced by Bohr through which light can be seen at the same time as particle and as wave (Herbert, 1985). In its metaphysics, Aristotle, driven from a process of investigation aimed to discover the first principle, by means of, all the subsequent could be explained, had illuminated the paths to later comers to the proposition that ideas and thus the principle of senses are capable to furnish the answers as to how understanding can be formed through observation (Aristotle, trans. 1991). Among the most accurate intuitions followed by empirical validation made by Isaac Newton was the mathematical instrument called interpolation (Newton & Huygens, trans. 1934) which was later adopted in approximation theory. Newton, on the basis of a discrete representation of the curve, devised through simple equations a method that based on finite differences was able to estimate the value of a curve at locations where the true value was unknown. It is thus due to recall that above mentions to scientific methods of enquiry and prominent scientific discoveries fill the duty to supervise the novel theory presented in this book in terms of explaining what is the philosophical basis allowing its conception and development. And also to recall that through the clarification of the process of investigation undertaken in order to determine the theory, the statement of its symmetry property and the outline of its aims and scopes, the meaning of the theory can be understood. A metaphor will be presented in order to start the overview of the theory presented in this book. The potentiality of an interpolation function seen as the potentiality to estimate the true value of a signal on the basis of its discrete representative collection can be termed as equivalent to that of a model that attempts to reconstruct a hidden reality from a set of ordered data that is not chaotic. Thus, this potentiality is to be transformed into energy precisely through the process of estimation, and the energy following the potentiality is as high as most accurate is the estimation performed through the model in the representation of reality. Being approximation inherent to interpolation functions, in the light that they are prone to produce an approximation error, it is likely to imagine such error as inversely proportional to the potentiality and so to the energy of the interpolation model. Thus, the question that needs to be answered is that of finding the mean that can elevate the potentiality of the model to a higher energy level and thus reducing the approximation error and therefore improve the estimation. The mean that can raise the energy level of the interpolation function in this book is called the Sub-pixel Efficacy Region and it is one of the two starting intuitions that allowed the development of the theory. Such region, as seen in between adjacent nodes where a discrete sample of signal has been collected, is defined throughout the book as the spatial set of points at locations to be determined such to allow improvement of the interpolation approximation characteristics in terms of reduction of the interpolation error. The interpolation model can elevate its potentiality by means of the Sub-pixel Efficacy Region (SRE). The Sub-pixel Efficacy Region has been derived from the first intuition that this book presents and this is a mathematical formulation that groups together two properties. One that relates to the signal and this is the intensity as seen as the discrete set of values that makes the data, and the other one is the local curvature of the interpolation function as expressed through the second order derivative. This mathematical formulation is called in the book the Intensity-Curvature Functional (ΔE) and it provides with the basis of the symmetry property of the herein developed theory. Such property is thus the relationship between the signal intensity and the curvature of the interpolation function. The reason why this happens rests in the fact that the interpolation function assumes its characteristic local curvature on the basis of the sequel of discrete signal intensities collected in the sampled data. Hence, what binds together the two variables of signal intensity and local curvature is the energy of the interpolation paradigm: the potentiality capable to produce an approximation. In this book the potentiality is named intensity-curvature term and the mean that can raise such potentiality is called Sub-pixel Efficacy Region. Namely, to raise the potentiality to produce an approximation shall be seen as the act of raising the goodness of the estimate of the true value of signal. The intensity-curvature term is calculated in two conditions: (i) if the signal would not be interpolated and (ii) if the signal would be interpolated. The difference between intensity-curvature term and Intensity-Curvature Functional consists in that ΔE is the ratio of the two terms calculated in above conditions.

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Therefore, the symmetry property relates signal intensity and local curvature in the following terms. Signal intensity tends to be constant in the neighborhood wherefore local curvature approaches zero. It thus to be seen by the reader of this book that the key to the understanding and the interpretation of the content of the materials rests into a process which driven through deduction, starts from intuitions, and poses the question: what if, and by its answer, arrives to formulations allowing conceptions that build the theory. It is also true that intuitions which, in the light of Kantian philosophy constitutes absolute truth, and which derives from pure reason; in the modern scientific world have lost the meaning that was given to them by the German philosopher, and for them to be such to produce knowledge and thus improve understanding, empirical confirmation is needed (Kant, 1900/2003). It is thus presented in this book an extensive experimentation that was conducted through the use of different validation paradigms. Also, it is presented a large collection of results that allow building the evidence necessary to validate the theory herein introduced. One of the undoubtedly foremost goals to achieve by academia is that of unifying that is to determine an umbrella capable to embrace under the same set of mathematical equations the most of the phenomena as they are apt to explain nature and the reality that itself manifests. The aim and the scope of unifying the methodology is that of making available the method for the improvement of interpolation functions of diverse nature by means of a set of formulations that make possible to cover different cases which otherwise would have presented different pathways to solving the problem of improving the interpolation error. Due to the flexibility of the mathematics developed the theory has evolved, through extensions aimed to cover the case to diverse interpolators, into a framework that permits to explain the phenomenon of the improvement of the interpolation error through the same methodological approach. Another philosophical basis chosen to develop the theory is that of not allowing any a-priori assumptions on the nature of the relationships between interpolation functions as models, and the concept of sampling resolution, which is modulated through the sampling frequency of the instrument that collects the discrete samples of data. Not allowing any a-priori assumptions, contrary to what commonly used in scientific quests, which often bases its progress through hypotheses testing, allowed to reveal reality as itself wanted to be manifest through the mathematics that were used as instrument for its measurement. The reference in question is due to be recalled, as it is related to the commonly recognized evidence that literature has established since the last century. This evidence asserts that the interpolation error, which is the measure of the goodness of the approximation produced through the interpolator, is inversely proportional to the sampling resolution. This evidence as proposed and recalled in literature has been derived from an assumption and its demonstration. And the demonstration, which in the view of the Aristotelian philosophy is the first principle of an axiom, has provided the basic mean for a school of researchers to develop interpolation schemes and to describe their featuring characteristics such as those related to the approximation properties of the interpolator. The a-priori assumption of dependence of the interpolation error on the sampling resolution hasn’t been made in the present book. Instead the mathematical formulation that sets the basis of the unifying theory has been developed at the aim to trace a step of detachment of the interpolation approximation (which is the remarkable featuring characteristics of any interpolator) from the sampling resolution. Thus the reader is likely to note throughout the book that the mathematics is such to tend to the aim of setting the interpolation error improvement as independent from the resolution. Practically, to avoid relying on such basic assumption that many investigators have used and recognized, presents the advantage of showing the resulting truth as free from any imposed constraints and thus allows the reality to manifest itself as it is. The evidence produced through this work still confirms the existence of an effect of the sampling resolution on the approximation characteristics of the interpolation function, but such truth has been given the freedom to manifest itself freely from any constraints imposed by a-priori assumptions. As to what is the question that the book poses and answers, it shall be seen that such question is that of setting the grounds for the determination of a unifying theory for the improvement of the interpolation error.

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The reader may then ask as to if the theory proposed in the book embeds sufficient mathematical solidness. In this regard it shall be noticed that the formulation offered in the book brings to the actualization a framework that embraces diverse interpolators regardless to their dimensionality or degree. Thus, the mathematical solidness of the theory can be evaluated through the flexibility of the mathematical formulation. Another relevant question that is critical to the essence of the book is that of establishing a criterion for assessing the boundaries of the truth determined by means of the theory. It is possible to answer immediately to such question through the evidence furnished with the results and therefore through empirical validation (Barankin, 1956). This is certainly necessary, but the sufficiency can be reached through the evaluation of the requirements of the herein proposed theory to be solid: (i) the continuity of the interpolation function; (ii) the existence of the non-null second order derivatives of the interpolation function, which property is strictly related to the curvature and it is also the bridging concept between the classic interpolation functions and the novel class of interpolators offered in the book; (iii) the existence of non-null extremes of the Intensity-Curvature Functional; (iv) the capability of the Sub-pixel Efficacy Region (SRE) to effectively determine a successful projection onto a sub-nodal point where the approximation characteristics of the interpolator are improved (this property being related to the effectiveness of the relationship that links the intensity-curvature terms to the Intensity-Curvature Functional) and finally (v) the validation paradigm, which is such to show the capability to re-sample at the novel location found through the projection of the sampling location by means of the SRE intra-nodal point. As shown throughout the book all above requirements are satisfied through the mathematical formulation that determines the grounds for the theory to be solid in its essence. But assessing the boundaries of the truth determined by the theory also involves evaluating the results that are presented through several experimentations that were conducted. Nevertheless, under a strict definition that refers to the meaning of the word theory, a theory might be falsified. And this should not be misleading to the reader when shown with some of the results presented in this book. Therefore, on the basis of such statement it is possible that the conclusion that can be intuitively drawn is that the theoretical framework depicts a reality which boundaries of truth are not absolute. Consistently, the non absolute power of the theory is likely to be a consequence of the unifying challenge that is being attempted. The challenge of unifying is a difficult pursuit in each scientific discipline, but valuable because of its aims when it is given ground of solidness to the mathematics behind the theory that is being developed. But the opinion of the writer, shall also invite the reader to undertake an evaluation of the solidness of the theory that the book proposes and to assess on the boundaries of the truth determined. To facilitate such task it is possible to enlist and analyze how beneficial are the requirements of the theory to the extent of proving the theory solidness. This shall be done borrowing the Cartesian scientific method of investigation. As the Cartesian method teaches (Descartes, 1614/2004), the first step to undertake is that of never accepting anything that is free from any grounds of doubt, and in this respect the requirements of the theory as listed above are essential to the applicability of the framework to an interpolation function. That means when is undeniable for the interpolation function that its mathematical formulation is such to fulfill as true all of the criterions listed above then the theory stands solid for the aim it wants to achieve. At the second step of the Cartesian method is the task of dividing the difficulties in subsequently smaller problems each admitting solution. It is then confirmed the solidness of the theory only if all of the above requirements are fulfilled through the results. That means that the theory is solid if all requirements suffice to the improvement of the interpolation error. Nevertheless, in those cases when the theory does not find empirical confirmation it should be considered that the requirements are not sufficient. Thus, the boundaries of the truth portrayed through the manifested reality are of such particular nature that is not possible to explain them through the theory. As third step, Descartes places the emphasis in dividing each of the objective requirements, from the simplest to the most complex, and to assign knowledge to each of them. As the knowledge is meant to improve understanding the reader is invited to undertake an analysis of the meaning of all the requirements.

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The continuity is essential to any interpolation function to be such and this concept derives from the aim inherent to the interpolator as that of reconstructing a continuous signal from a discrete sample. Thus the knowledge here is inherent to the essence of the purpose. The existence of non-null second order derivative of the interpolation function (its local curvature) is linked to the symmetry property of the theory. Thus, the knowledge here is that of furnishing the basis for an improved understanding as to how an interpolation function can improve its approximation properties. This knowledge is related to the overall aim of the theory, which is that the interpolation error decreases when the curvature is taken into account and it is incorporated into the Intensity-Curvature Functional. The knowledge of the existence of a relationship between the signal intensity and the curvature of the interpolation function is also relevant to the attempt that the theory makes to determine an approach to improve the interpolation error. Through its mathematical formulation such approach detaches itself, at least partially, from the influence of the sampling resolution. This knowledge is relevant to the effect of inverse proportionality of the sampling resolution on the interpolation error. This fact has being extensively demonstrated in literature. The knowledge that descends from the non-null value of the extremes of the Intensity-Curvature Functional implies the existence of the set of spatial points called Sub-pixel Efficacy Region. Also, this knowledge is relevant to the extent that the extremes of the Intensity-Curvature Functional are capable to find the SRE location necessary to determine the projection at the intra-nodal point called novel re-sampling location. And the novel re-sampling location is the spatial location where the SRE-based paradigm of the interpolator assumes improved approximation characteristics. This is assertion is supported through the empirical evidence collected with the validation paradigms adopted in the book. The validation paradigm is the last of the requirements and the knowledge associated with it is that it provides the unifying theory with the means to validate itself. Once above is ascertained, then it shall be noted true that by satisfaction of each of the requirements of the theory, knowledge and therefore value is added to the overall solution the theory proposes, which is that is to improve the interpolation error. Finally the fourth step of the Cartesian method asks that nothing is omitted. Consistently with what stated above that means that if all of the requirements of the theory are fulfilled and therefore the empirical evidence has given its confirmation, it follows that the given interpolation function has improved its approximation properties. Due to mention that the philosophical bases relevant to the innovation that the unifying theory intends to bring to the literature are related to the process of fostering freedom while conceiving innovation. In fact, this book presents the source code at www.sourcecodewebsite.com that was developed to produce the results which constitute the direct validation of the mathematical formulation herein proposed. The software code is provided with the intent to serve the purpose of further development by the willing audience. Open Source (Deek & McHugh, 2008) provides with the philosophical bases at the root of the author’s intent. Along this line of thought, sections of the book presenting mathematical formulations of the process of re-sampling that is resilient to interpolation are testimony of conceptions that are novel in their nature and they are still waiting for empirical confirmation from the willing audience. Within this context the author is serving the purpose to foster innovation.

REFERENCES Aristotle. (1991). The metaphysics (J. H. McMahon, Trans.). New York, NY: Prometheus Books. Barankin, E. W. (1956). Toward an Objectivistic Theory of Probability. Proceeding of the Berkeley Symposium on Mathematical Statistics and Probability: Vol. 5. (p. 25). Berkeley: University of California Press.

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Deek, F. P., & McHugh, J. A. (2008). Open source. Technology and policy. New York, NY: Cambridge University Press. De Laszlo, V. S. (1990). The basic writings of C. G. Jung (R. F. C. Hull, Trans.). Princeton, NJ: Princeton University Press. (Original work published 1959) Descartes, R. (2004). Discourse on Method. New York, NY: Barnes & Noble. (Original work published 1614) Greene, B. (1999). The elegant universe. Superstrings, hidden dimensions, and the quest for the ultimate theory. New York, NY: W. W. Norton & Company. Herbert, N. (1985). Quantum Reality. New York, NY: Anchor Books. Kant, I. (2003). Critique of pure reason (J. M. D. Meiklejohn, Trans.). Mineaola, NY: Dover Publications. (Original work published 1900) Lorentz, H. A., Einstein, A., Minkowski, H., & Weyl, H. (1952). The principle of relativity: A collection of original memoirs on the special and general theory of relativity (W. Perrett & G. B. Jeffery, Trans.). NewYork, NY: Dover Publications. (Original work published 1923) Newton I., & Huygens, C. (1934). The motion of the moon’s nodes. In R. M. Aynard Hutchins (Ed.), Mathematical Principles of Natural Philosophy: Optics, Treatise on light (A. Motte, Trans.). (pp. 338-339). William Benton.



Chapter I

Magnetic Resonance Imaging and the Signal-Image Processing Techniques Developed Under the Umbrella of the Unifying Theory

INTRODUCTION Since its inception (Lauterbur, 1973; Mansfield & Grannell, 1973) Magnetic Resonance Imaging (MRI) has produced a beneficial and revolutionary trend in several biomedical imaging related applications. Diagnostic imaging, among others, is one of the most relevant. Following the discovery of MRI and parallel to the widespread use of computer technology, an immense research literature has been produced on signal-image processing techniques devoted to Magnetic Resonance Imaging. Several text books exist in literature that explain the concepts behind the MRI physics and the concepts that drive the need and the types of signal-image processing techniques that were and are being developed (Cho et al., 1993; Haacke et al., 1999; Jezzard et al., 2001; Liang & Lauterbur, 2000). Interpolation is indeed one of the most relevant signal processing applications related to MRI and also to any scientific discipline that requires the estimation of the signal at time-space locations where the signal is not known because of the limitations to sampling as imposed by the recording (imaging) equipment. This fact should provide with an immediate answer as to what is the relevance of the contents of this manuscript. The main objective of this book is to present a unifying theory, supported through empirical validation, for the improvement of the approximation properties of mathematical functions employed in signal-image interpolation to estimate the value of the signal at those space-time locations that are not sampled. Given the relevance of MRI, this book presents the validation of the unifying theory employing Magnetic Resonance Imaging signals (images). The forthcoming text of this chapter is intended to raise the debate about the content of the book. To fulfill this purpose, is presented in first instance a section that summarizes on how the literature is reviewed throughout the book and also the MRI database employed to validate the unifying theory. FolCopyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

lowing this presentation, the chapter introduces to the reader the concept behind the basic issue relating to signal-image interpolation, which is the preservation of the signal (image) energy after processing with interpolation. This concept is strictly related to the approximation properties of the interpolation functions and consequently to the interpolation error. This concept is widely accepted in literature and is viewed in this works with a different and innovative perspective. The discussion undertaken in this chapter then introduces the signal-image processing techniques that were developed under the umbrella of the unifying theory. These are: (i) the SRE-based interpolation functions, (ii) the spectral power evolutions, (iii) a three layered artificial neural network with the added capability to generate its internal architecture during the learning process such to adjust to the given pattern classification task. The chapter also introduces the very basic concept of the unifying theory which is the distinctive feature of the SRE-based interpolation functions: the local curvature of the function. The main implication of the theory is introduced as it relates to the possibility to calculate the re-sampling locations through a computational approach based on the combined information content of signal intensities and curvature (second order derivative) of the interpolation function. Finally the chapter gives a general overview on the implications of the unifying theory in signal-image processing and particularly in motion correction in functional MRI. The discussion highlights that the creation of signal-image artifacts, which are detrimental to signal reconstruction, can derive directly from the approximate nature of interpolation, but can be reduced through SRE-based interpolation.

How THE REVIEW OF THE LITERATURE PROCEEDS THROUGHOUT THE BOOK Chapter I acknowledges literature relevant to the basic issue in signal-image interpolation which is that of the preservation of the signal (image) energy and also discusses on the signal processing techniques related to MRI which are most likely to benefit from interpolation. Chapter IX classifies interpolation procedures, and within this process, the SRE-based interpolation functions are viewed within the larger framework provided in literature. Chapter XIII focuses on the literature that relates to B-Splines and provides the reader with information as to how the unifying theory achieves the improvement of the classic quadratic and cubic interpolation paradigms. Chapter XVI discusses on the properties of the unifying theory while referencing literature concerned with concepts like efficiency, approximation, efficacy and more generally pointing out to frameworks existing in literature that classify interpolation methodologies. Chapter XVII presents literature that relates to the Sinc interpolation function while delivering one of the main messages that the book furnishes to the reader: the combined use of intensity and curvature increases the likelihood to devise an improved interpolation paradigm. Chapter XX presents literature on interpolation while discussing: novel conceptions of the unifying theory and the limitations, the interpolation error representation and the basic concept that connects classic and SREbased interpolation paradigms. More generally, in relationship to the literature reviewed the above mentioned chapters determine an ongoing discussion within the book and this discussion starts with the proposition of the issues in MRI that most relate to interpolation. The discussion proceeds while providing to the reader with an opinion as to how SRE-based interpolation fits in the context of the present scientific literature. Moreover, while proceeding further, it is outlined the intent of the unifying theory as far as the improvement of one dimensional B-Splines is concerned. Subsequently, the literature is reviewed while the discussion



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

places its emphasis on the properties of the unifying theory. At the next level the literature is presented while delivering to the reader one of the most relevant messages of the book. And finally, the review of the literature is presented while mentioning the novel conceptions and the bridging concept between classic and SRE-based interpolation. In summary, it can be seen that the review of the literature proceeds in parallel with the outline of the most relevant topics covered in this book.

The Magnetic Resonance Imaging Database The math formulation derived from the theoretical conceptions presented in this book has been tested extensively on three different Magnetic Resonance Imaging (MRI) modalities. They were: T1-weighted, T2-weighted and T2*-weighted (functional Magnetic Resonance Imaging). This section presents the image database accompanied with information on the resolution (matrix size and voxel size) of each of the three dimensional MRI modalities. Each MRI was collected to image the human brain. Throughout the book the T1-weighted imaging modality is referred as to T1 MRI, the T2-weighted imaging modality is referred as to T2 MRI, and the T2*-weighted imaging modality is referred as to functional MRI. The T1 MRI was kindly provided by the Open Access Series of Imaging Studies (OASIS), www. oasis-brains.org, courtesy of Dr. Daniel Marcus (Buckner et al., 2004; Fotenos et al., 2005; Marcus et al., 2007; Morris, 1993; Rubin et al., 1998; Zhang et al., 2001). The T2 MRI was kindly provided by Dr. Rand S. Swenson (Dartmouth College, NH – USA), www.Dartmouth.edu/~rswenson/Atlas. The functional MRI was kindly provided by (i) INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee (Peeters et al., 2004) and the Department of Radiology KULeuven, courtesy of Dr. Pierre Kornprobst; and (ii) The Wellcome Trust Centre for Neuroimaging (University College London – UK), www.fil.ion. ucl.ac.uk (Buchel & Friston, 1997), courtesy of Dr. William Penny. The author of this book is greatly appreciative to all of the above contributors for the image database that has been generously provided and wishes to express the most sincere gratitude to all of them. T1 MRI resolution was 176 x 208 with 1.00 x 1.00 mm of pixel size (x and y directions of the coordinate system respectively), with 176 slices (z direction of the coordinate system) and an inter-slice resolution of 1.00 mm. Two T1 MRI volumes (Open Access Series of Imaging Studies - OASIS) were employed and from each of the two, 75 slices were employed for the experimentations. T2 MRI resolution was 888 x 912 with 1.0 x 1.0 mm of pixel size, with 20 slices and an inter-slice resolution of 1.00 mm. For all of the experimentations except those relevant to the bivariate liner interpolation functions, the T2 MRI (Dartmouth College, NH – USA) was down-sampled to 177 x 182 in plane resolution. Nineteen of the 20 slices were employed for all of the experimentations. Functional MRI resolution was 128 x 128 with 1.72 x 1.72 mm of pixel size, with 22 slices and an inter-slice resolution of 1.72 mm (INRIA Odyssee Lab - France). Twenty slices were employed for the experimentations. Also, functional MRI resolution was 53 x 63 with 3.00 x 3.00 mm of pixel size, with 46 slices and an inter-slice resolution of 3.00 mm (Wellcome Trust Centre for Neuroimaging - UK). Thirty slices were employed for the experimentations. This paragraph presents the Magnetic Resonance Imaging Database through the pictures reported hereto. In each of the slices of the T1 MRI, T2 MRI, functional MRI, appears a number as identifier.



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Figure 1. The T1-MRI-230 data set, slices 67-78. The number located in the upper left corner identifies the slice within the volume and it will appear in the pictures that present the results of the experiments and that are located in the subsequent chapters of the book. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org.

Figure 2. The T1-MRI-230 data set, slices 79-90. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. Data seen in this figure are human brain slices that are sorted in ascending order from bottom (neck) to top and consistently the number that identifies each slice increases too.

Figure 3. The T1-MRI-230 data set, slices 91-102. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. Each of the slices seen in this figure presents different details of the human cerebral cortex.

Figure 4. The T1-MRI-230 data set, slices 103-114. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. Data shown in this figure show how the human brain changes its shape while reaching the motor cortex.



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Figure 5. The T1-MRI-230 data set, slices 115-126. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. Details of the motor cortex are clearly distinguishable in this figure.

Figure 6. The T1-MRI-230 data set, slices 127138. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. In this figure, the cerebral cortex seen in the slices show reduced size because is gradually reaching the top of the head.

Figure 7. The T1-MRI-230 data set, slices 139141. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. The three last slices shown here continue to show gradually reduced size.

Figure 8. The T1-MRI-450 data set, slices 67-78. The number located in the upper left corner identifies the slice within the volume and it will appear in the pictures that present the results of the experiments and that are located in the subsequent chapters of the book. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org.



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Figure 9. The T1-MRI-450 data set, slices 79-90. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. Like wise for the T1-MRI-230 data set the numbering of the slices is in ascending order, showing details of the cerebral cortex while moving from bottom (neck) to top. The contrast of these images is however different with respect to the image contrast seen for the T1-MRI-230 data set.

Figure 10. The T1-MRI-450 data set, slices 91-102. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. The brain slices progress towards the human motor cortex.

Figure 12. The T1-MRI-450 data set, slices 115-126. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. Some of the sulci Figure 11. The T1-MRI-450 data set, slices 103-114. and the fissures of the human cortex can be visually Courtesy of: Open Access Series of Imaging Studies appreciated. (OASIS), www.oasis-brains.org. The figure shows clearly the differentiation between white matter (brighter part of the cortex) and grey matter.



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Figure 13. The T1-MRI-450 data set, slices 127138. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. In this figure the human brain cortex progress toward the top of the head.

Figure 14. The T1-MRI-450 data set, slices 139-141. Courtesy of: Open Access Series of Imaging Studies (OASIS), www.oasis-brains.org. In this figure are seen the last there slices of the MRI volume that were employed for the validation of the SRE-based interpolation functions.

Figure 15. The T2 MRI data set, slices 1-4. The number located in the upper left corner identifies the slice within the volume and it will appear in the pictures that present the results of the experiments and that are located in the subsequent chapters of the book. Courtesy of: R. S. Swenson, www. Dartmouth.edu/~rswenson/Atlas.

Figure 16. The T2 MRI data set, slices 5-8. Courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/ Atlas. The slices of this figure progress from front to rear and consistently is seen an increase of the number that identifies each slice.



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Figure 17. The T2 MRI data set, slices 9-12. Courtesy of: R. S. Swenson, www.Dartmouth. edu/~rswenson/Atlas. The figure allows appreciating different aspects of the human cortex and particularly the difference between imaged tissues types.

Figure 18. The T2 MRI data set, slices 13-16. Courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/ Atlas. As the slices progress from front to rear is seen the remarkable reduction of the size of the human cortex. Also, the cerebellum of the human brain is clearly shown.

Figure 19. The T2 MRI data set, slices 17-19. Courtesy of: R. S. Swenson, www.Dartmouth. edu/~rswenson/Atlas. As the slices progress toward the rear, the cerebellum gradually disappears from the sight leaving room for the rest of the human cortex located in the rear.

To allow understanding of the location of each brain slice within the volume, this number will appear in the pictures that present the results and such pictures are located in the subsequent chapters of the book. While the first T1 MRI dataset will be identified with the name T1-MRI-230 (slices 67-141), the second T1 MRI dataset will be identified with the name T1-MRI-450 (slices 67-141). In figures 1 through 14 are shown the two T1 MRI datasets: T1-MRI-230 and T1-MRI-450 (Open Access Series of Imaging Studies - OASIS) respectively. Figures 15 through 19 show the T2 MRI data set (Dartmouth College, NH – USA) (slices 1-19). Finally, figures 21 and 22 show the two functional MRI data sets. They will be identified throughout the book as fMRI-TS1 (slices 1-20 seen in figure 20) (INRIA Odyssee Lab – France) and fMRI-snffM00587 (slices 13-42, seen in figure 21) (Wellcome Trust Centre for Neuroimaging – UK).



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Figure 20. The fMRI-TS1 data set, slices 1-20. The number located in the upper left corner identifies the slice within the volume and it will appear in the pictures that present the results of the experiments and that are located in the subsequent chapters of the book. Courtesy of: INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee, also courtesy of the Department of Radiology KULeuven.

Figure 21. The fMRI-snffM00587 data set, slices 13-42. The number located below each fMRI image identifies the slice within the volume and it will appear in the pictures that present the results of the experiments and that are located in the subsequent chapters of the book. Courtesy of: The Wellcome Trust Centre for Neuroimaging (University College London – UK), www.fil.ion.ucl.ac.uk.



Magnetic Resonance Imaging and the Signal-Image Processing Techniques

CLASSIC INTERPOLATION Shannon’s Information Theory has lead to the introduction of the Sinc interpolation function as the ideal one. The 1-D ideal interpolation can be achieved in the spatial domain by convolution with the Sinc function h(x) = sin (πx)/ πx (Lehmann et al., 1999). It satisfies the zero crossing conditions: h(x) ≡ 0 | x | = 1, 2...



(1)

Thus, this process avoids smoothing and preserves high frequencies. Any interpolation function that does not satisfy (1) is defined as an approximator. Accuracy of the interpolation function can therefore be defined as the capability to generate a continuous signal using the information from a discrete one, thus achieving minimal approximation. The location of the re-sampled points with respect to the initial coordinate system affects accuracy of the interpolation functions (Parker et al., 1983). Similar findings were reported (Meijering et al., 2001; Ostuni et al., 1997), confirming that excellent accuracy is obtainable with Spline interpolation and that the high speed of the trivariate linear is desirable for some applications. Comparative evaluation of interpolation techniques was reported (Lehmann et al., 1999; Meijering et al., 2001), emphasizing accuracy and efficiency in terms of computational demand. Also, through the analysis of interpolation functions’ Fourier properties, it was shown that Gaussian interpolation also provided excellent results (Lehmann et al., 1999). A more quantitative evaluation was presented (Meijering et al., 2001) through an image normalized root- mean-square-error (RMSE), pointing out that excellent results in terms of accuracy are obtained with Spline interpolation, and with a satisfactory trade off in computational time. Two main issues of image processing applications were described (Joyeux & Penczeck, 2002) as being relevant to the detrimental effects of the interpolation paradigm: (i) the order of the interpolation function and (ii) the amount of sub-pixel shift. Also, signal-to-noise ratio (SNR) is a determinant of alignment accuracy. Since all interpolation functions produce an approximation, another issue to consider is that of the image’s energy. There exists a difference between the energy of re-sampled and original images. To avoid changing the image energy for a given interpolation process, the sum of all samples should be equal to one for any misplacement: k=- h (d + k) = 1 (2) It can also be shown that k=-

h (d + k) = -



(3)

where: Ш (x + d) = Σk=-∞ δ(x + k) is a sum of delta functions δ(x) each of which corresponds to the amplitude of the interpolation function h (x) at the position (x + k) of the function δ (Lehmann et al., 1999). The definition of image’s energy given in part I of this book is different from the one reported earlier (Lehmann et al., 1999) because it comprises of the joint intensity-curvature information content of the interpolation function.

10

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

SRE-BASED INTERPOLATION The view expressed in this book is intended to address the extent to which the interpolation function might overestimate or underestimate a true pixel value depending on its local concavity or convexity. From this view the reader is addressed with the following message: to assess how exact is the estimation of the interpolation function of the true signal value, it is relevant to consider the local concavity or convexity of the model interpolator. Ciulla and Deek (2005a) reported due mentions to research dedicated to the improvement of the approximation properties of the linear interpolation function (Blu et al., 2002, 2004) as well to a recent review of interpolation schemes (Meijering, 2002). With specific focus on the bivariate linear function, Ciulla and Deek (2005a) also reported the premise, the derived concepts, the aims and the practical implications of an original approach derived for the improvement of the interpolation error. Along the same line of thought, the aim of this book is that of tracing the path of the novel avenues to reach interpolation error improvement and within this goal the approach earlier presented is extended to several other interpolation functions. Such an approach is missing from the current literature. Ciulla and Deek (2005a) clarified the dependency of the SRE on the pixel intensity values and the local curvature of the interpolation function. The approach is conceptually equivalent to finding those subpixel locations for which the energy of the signal (image) is the same before and after interpolation. The perfect interpolation function is that one capable to produce zero error and it is obviously quite difficult to device (if not impossible, given the present knowledge provided in literature) because the signal remains unknown at the locations that cannot be sampled because of the Nyquist’s theorem constraints. The methodological approach outlined in this book for the improvement of the interpolation error is consistent across the following interpolation paradigms: (i) bivariate linear, (ii) trivariate linear, (iii) quadratic B-Spline, (iv) cubic B-Spline, (v) Lagrange and (vi) Sinc with odd numbers of nodes. This spectrum of functions has been chosen to cover the presentation of the quest for the solution to the problem of improved signal (image) interpolation. Such solution is novel in its nature. Therefore, they were chosen one, two and three dimensional interpolation paradigms, also interpolation paradigms having first, second and third degree as per the order of the function. Nevertheless, the book also covers the case of trigonometric interpolation by presenting the SRE-based Sinc function.

Fast Fourier Transform and Spectral Power Analy sis Re-sampling at locations that vary voxel-by-voxel might lead to the preservation of spectral characteristics to the extent that it attenuates the smoothing effect of the interpolator. To quantify differences in the frequency spectrum, the magnitude evolution obtained through the Fast Fourier Transform was calculated and a spectral power evolution was built as described herein. The intent in analyzing the spectral power evolutions was to quantify interpolation improvement based on the differences observed in the spectrum of classic and SRE-based interpolation functions (Ciulla & Deek, 2005b). The FFT was shifted in order to place the highest signal’s frequency at the center of the spectrum. For each frequency component (represented by a complex number) the magnitude was calculated and multiplied by the sign resulting from the ratio between the signs of the real and the imaginary part of

11

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

the FFT complex values, thus obtaining negative, zero valued and positive valued magnitudes. We call these magnitudes: “the resulting magnitudes”. The advantage of doing so consists in that the multiplication of the FFT magnitude by the sign resulting from the ratio between the signs of the real and the imaginary part of the FFT complex values incorporates both real and imaginary parts of the complex number into a single coefficient. This coefficient can either be zero, positive or negative. The resulting magnitudes were therefore sorted from the smallest negative to the biggest positive. The entire sorted range of resulting magnitudes was divided into 2000 intervals and a number growing from 1 to 2000 was assigned to each interval. The sum of the squares of the resulting magnitudes (the spectral energy) was calculated for each of the 2000 intervals. By plotting the energy for each of the 2000 intervals, it was therefore depicted the evolution of the spectral energy of the processed image. It is relevant to note that the plot of the spectral energy was versus the intervals. The value 2000, although arbitrary, was kept consistent across all of the images that were processed. The behavior of the plot of the spectral energy (spectral energy evolution) was analyzed for: (i) the original image, (ii) the image processed with the classic interpolation function and (iii) the image processed with the SRE-based interpolation function. The aim was to capture at each interval the differences in energy evolution between the original image and the images processed with either of the two interpolation paradigms (classis and SRE-based). Also, to analyze quantitatively the two evolutions such to determine which one of the two was matching more closely the evolution of the original image. Attention was focused to the intervals corresponding to the high frequencies. Intervals [979, 1023] were analyzed in the following manner. Differences between the classic and SRE-based interpolation functions were detected in the following terms. At each interval, the absolute value of the difference between the energy derived from the original image and the energy derived processing the image with the classic function was calculated and named abs(O_E - NOSRE_E). Also, at each interval the absolute value of the difference between the energy derived from the original image and the energy derived processing the image with the SRE-based function was calculated and named abs(O_E - SRE_E). If the difference abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) was positive, then the spectral energy obtained processing the image with the classic function was found closer to the spectral energy of the original image. If above difference was negative, then the spectral energy obtained processing the image with the SRE-based function was found closer to the spectral energy of the original image.

Non-Linear Separability of the Spectral Power Evolutions of functional MRI images: An Artificial Neural Network Ex periment Classification results for the three classes of spectral energy evolutions (original image, image processed with classic interpolation and image processed with SRE-based interpolation) relevant to functional MRI (fMRI-snffM00587, slice 13 through 42) were obtained through the implementation of a three layered perceptron named ANN98 and employing the backpropagation algorithm supported by the delta rule (Werbos, 1990). The objective of the experiment was to confirm existing difference between the original functional MR images and the functional MRI processed with classic and SRE-based interpolation functions. The differences were searched within the three classes of spectral power evolutions. Thus, 44 intervals were

12

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

selected from each evolution and the value of the spectral energy (sum of the squares of the resulting magnitudes) constituted the input to the 44 input neurons of ANN98. The intervals were those between 979 and 1023 (see the spectral power evolutions in Chapter XI) except for the interval 1000 which had the same spectral energy associated with it for all of the three classes. As anticipated above, the three classes for which the non-linearity of the differences was investigated were: (i) original functional MRI, (ii) functional MRI processed with classic interpolation function, and (iii) functional MRI processed with SRE-based interpolation function, (for a total of 3 output neurons). The total number of learning patterns was 90 and particularly 30 patterns for each of the three classes. The functional MRI volume (see the spectral power evolutions in Chapter XI) was processed with the misplacement: X = 0.49, Y = 0.49, Z = 0.49; with the two interpolation paradigms (classic and SRE-based). The mathematic implemented in ANN98 is herein reported. A neural network is a system composed by many computational units called neurons (nodes) distributed along cascaded layers. The neurons of the output layer are connected through internal variables called weights to the inputs of the hidden layers and so forth till the connections to the neurons of the input layer. Changing the weights of the connections alters the behavior of the whole system. The goal is to archive a desired input-output response at the end of learning process. Backpropagation is one of the most popular algorithms used in order to perform supervised learning in artificial neural networks. Indeed, backpropagation carries an exact and efficient method for calculating the derivatives of the cost function and this method is called: generalized delta rule (Werbos, 1990). The network adopted and developed in this work is a three layered Artificial Neural Network (ANN98). At the input layer the neurons are not computational as they are in the hidden and output layers where each neuron has a decision hyper plane associated with it, and the hyper plane is mathematically defined through the linear combinations determined by inputs and weights leading to that node. The input 0 ≤ I L (j ) ≤ 1 to the neuron is defined as: nL −1 ÷ L L −1 I L (j ) = ≈ ∆ ƒ w (i, j )∂O (j )◊ −  i =0 

L

(j )

L=1, 2; j=1...n

L



(4)

with O being the output of the neuron of the preceding layer, being the firing threshold of the neuron, wL (i, j) being the weighting connection between the i-th neuron of the (L-1)-th layer and the j-th neuron of the L-th layer, O 0 (j) being the input pattern to the network with: j = 1...n0 . Furthermore in the hidden and output layers (L = 1, 2), each neuron is characterized by a transfer function f . The backpropagation algorithm requires differentiability along the signal paths of the network, therefore ANN98 cannot work with functions such as hard-limiter or threshold logic. The sigmoid defined by equation (5) was herein adopted as the f transfer function:

[

]

f I L (j ) = O L (j ) =

1 1 + e −I

L (j )

L = 1, 2; j = 1...n L



(5)

13

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

During the learning process the algorithm calculates the partial derivatives of the cost function defined by equation (6) with respect to wL (i, j): N

S

E = 1/2 ∂ƒ ∂ƒ

[ (j )− O (j ) ] L

t =1 j =1

L

2

for L = 2

(6)

with N being the number of patterns included into the learning set, S being the number of classes to discriminate, ΩL (j) being the desired output for class discrimination. Since neurons are distributed along cascade layers, it is obvious that the relation existing between E and wL (i, j) is expressed by equations (4), (5) and (6). Above relationship requires an efficient method for calculating the derivatives. Therefore, the backpropagation algorithm uses the generalized delta rule defined through equation (7) at the p-th step of the learning process:

L p

(i, j ) = −

≈ ∂E ÷  ◊ (i, j )⁄ L=1, 2; ∆ ∂… ∆∂w pL ◊   … ⁄

(7)

with 0 ≤ ≤ 1 being the learning coefficient. Equation (7) indicates that if increasing a given weight would lead to more error, the delta rule adjusts that weight’s value downwards. Inversely, if increasing a weight leads to less error the delta rule adjusts its value upwards. Hence, after computing the derivatives, equation (7) is rewritten as: L

(i, j ) =

where L

L

∂

L

(j )∂O L (i )

L= 1, 2;



(j ) is the error term, which for the sigmoid nonlinearity (5) is given by:

(j ) = O L (j )∂[1 − O L (j ) ]∂[ L (j )− O L (j ) ] L = 2;

L

(8)

(j ) = O L (j )∂[1 − O L (j ) ]∂ ƒ

nL +1 k =1

L +1

(k )∂w L+1 (j, k )



L = 1;

(9)

(10)

To update the weights, backpropagation uses the following learning rule:

w pL (i, j ) = w pL−1 (i, j )+ with L p

L p

L p

(i, j )

L=1, 2;







(11)

(i, j ) being the correction term for the weighting connection of the computational neuron. Also,

(i, j ) cumulates all partial correction terms corresponding to each of the patterns of the learning

set. Therefore, each weight is adjusted only after all N patterns are processed (a process called batch learning). In order to improve convergence of the learning process, a smoothing factor ς is introduced so that equation (11) can be rewritten as follows:

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Magnetic Resonance Imaging and the Signal-Image Processing Techniques

w pL (i, j ) = w pL−1 (i, j )+

L p

(i, j )+

∂

L p -1

(i, j )

L=1, 2;

(12)

L L where both w p− 1 (i, j ) and p−1 (i, j ) are calculated at the preceding step of the learning process. In the present work the mathematics described above designed the three layered perceptron, which was coded and implemented in the C programming language and employed with the aim to solve the threeclass problem relevant to the proper classification of the three types of spectral energy evolutions. Due to mention is the issue of generating the proper neural network architecture. It is clear that the mathematical model associated with the neurons and defined by equation (4) as a decision hyper plane allows stating that the neural network architecture is designed at the aim of non-linearly separate the samples belonging to the learning set at the output layer once the learning task is completed. Training a neural network might be frustrating as normally many trials might be needed before the number of hyper planes and corresponding neurons are found such to suffice in the classification task for the given problem. In this work however a single hidden layer was enough to reveal the sufficient number of hidden neurons preventing the process to get stuck in a local minimum. Generally the learning task would be faster if the number of neurons to include into the network architecture were known a priori. This problem was addressed by this research and solved by allowing the user to interact with ANN98 learning process and thus to add neurons in the hidden layer during the learning process and this helped overcoming local minimums. ANN98 gives the options: (i) to train only those neurons added during learning thus leaving unchanged the network architecture built previous to adding the neuron; or (ii) to train the newly added neuron along with retraining a fixed number of previously added neurons. This computational engineering solution allowed the backpropagation algorithm to generate the neural network architecture appropriate for the given classification problem and this was achieved while performing the learning process. The learning process (inclusive of the three classes of spectral energy evolutions: original image, image processed with classic interpolation and image processed with SRE-based interpolation) employed the values of 0.089 and 0.098 for the learning rate and the smoothing factor respectively. At the end of the learning process 6 hidden neurons were generated and the network achieved convergence for a cumulative residual error of 0.0419. All patterns’ values (44 x 90) were normalized to the range [0, 1] before being processed with ANN98.

The Novel Theory As a recent development, adaptive linear interpolation, was devised to estimate missing pixels from images based on neighboring pixel intensity and Euclidean distances (Amidror, 2002; Xu et al., 2000). For a review of linear interpolation paradigms that use scattered data points and not a uniform grid the reader is referred to Amidror (2002). Within this context, the new theory herein proposed assumes relevance because determines re-sampling as directly related to the curvature (i.e. second order derivative) that the model interpolation function assumes on the basis of the sequel of discrete data (pixel or node intensity) of which an image or more generally a signal is composed. The concept that the novel theory proposes in order to improve the error inherent to re-sampling is that of finding an intra-pixel location where the approximation characteristics of the model interpolation function are improved, which corresponds to a decrease in interpolation. This concept has been initially

15

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

illustrated in figure 7 (pg. 16) of section 4.2 (Ciulla & Deek, 2005a), and in figure 9 (pg. 10) of section 4.3 (Ciulla & Deek, 2006). Within the above conception, the (xsre, ysre) location reported in equations (20.a) and (20.b) of Chapter VII assumes the role of the reference point that allows determining the projection of (xsre - x0, ysre - y0) onto (xsre - xr0, ysre - yr0) such to determine the novel location re-sampling location (xr0, yr0). At the novel re-sampling location the interpolation approximation might be improved. The idea that is brought to light through this novel conception is that the model interpolation function is as accurate in its approximation, as much as its curvature tends to the curvature of the signal that attempts to approximate. As the knowledge of the curvature of the signal to approximate remains unknown, the interpolation model tries to reconstruct the signal based on the knowledge provided by sequel of node intensities. Thus the key in improving the approximation of the interpolation function is that of re-sampling at the location where the curvature of the model is as close as possible to that of the signal to approximate. This is truly an ill posed problem and the reason for it is because the true curvature of the signal is unknown. However, the novel theory brings to light a key notion: the local curvature of the interpolation function is one viable and effective variable to take into account while attempting to improve the approximation properties of the interpolation function. As anticipated in the previous section, further implications of the theory will be elucidated with FFT analysis. The images resulting after processing with SRE-based interpolation will show the existence of frequencies that are absent in those images obtained with classic interpolation, and vice versa. Frequency ranges affected by the employment of the SRE will be examined quantitatively. This will be conducted through the calculation and the comparison of the spectral energy determined at given frequency magnitude ranges using classic interpolation and SRE-based interpolation schemes and also through the comparison with the spectral energy of the original image. This will be shown for bivariate and trivariate linear interpolation functions, quadratic and cubic B-Splines, Lagrange and Sinc interpolation functions. It must be noted however, that frequencies found in the images resulting from motion correction of the misplacement and subsequent interpolation are partially influenced by the paradigm itself. Shifting the image and bringing it back to the original position does make use of re-sampling twice, whereas re-sampling following interpolation is usually performed once at the aim to estimate the signal at unknown locations.

The Significance of the Unify ing Theory The Sub-pixel Efficacy Region consists of a set of points that allow calculation of novel re-sampling locations where the interpolation function is calculated with improved approximation properties. This is done with the same theoretical approach for all the interpolation functions treated in this book. The significance of the unifying theory herein presented consists of determining the improvement of the interpolation error through the exploitation of the joint information content of nodes’ intensities and curvature of the interpolation function. As a consequence of the SRE approach used by the theory it descends that re-sampling becomes an operation that is provided with the property of locality. This latter property given to re-sampling is however not novel. Naturally re-sampling of a two or three dimensional signal (image) occurs at a given misplacement that is constant across the grid of nodes. Based on the Sub-pixel Efficacy Region, it is possible instead

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Magnetic Resonance Imaging and the Signal-Image Processing Techniques

to calculate novel re-sampling locations that vary from node to node. The unifying theory elucidates that it is possible to calculate re-sampling locations that vary node-by-node, using the herein envisioned joint information content of nodes’ intensities and curvature of the interpolation function. The joint information content has been formulated mathematically by the Intensity-Curvature Functional and has lead to the discovery and the mathematical characterization of the Sub-pixel Efficacy Region. In summary, it is presented in this book a theory that has been elaborated through extensive mathematical formulation which shows existence of the spatial set of intra-nodal points called: the Sub-pixel Efficacy Region (SRE). The SRE is formulated as dependent on the relationships between the intensity values at the pixel to re-sample and its neighbors, and the local curvature of the interpolation function. Consequentially, the present work asserts that for given misplacement there exists a sub-pixel location where to re-sample and this location is not necessarily the same as the given misplacement. It is called novel re-sampling location. It is shown that at this location re-sampling provides an improved approximation and thus a decreased root-mean-square-error (RSME). Heretofore the aim of this book is that of tracing the path of the novel avenues to reach interpolation error improvement and within this goal it is proposed an approach that is missing in the current literature.

Implications for Signal and Image Processing Implications of the proposed theory and methodology are in improving the approximation of the interpolation paradigms within the context of signal and image processing. Image artifacts can be created as spurious intensities resulting from the approximation inherent to interpolation. These newly calculated signal values do not correspond to the true intensity values and lead to errors in subsequent image processing techniques. Interpolation paradigms might then produce artifacts that are detrimental, for instance in circumstances where re-sampling takes place at particular values of magnitude of the misplacement (Brock & Wiseman, 2001), that is at a given Euclidean distance between original grid points and misplacement. Given the misplacement where the signal is to be estimated, re-sampling the signal on the basis on the novel re-sampling locations obtained through the Sub-pixel Efficacy Region could prevent creation of spurious intensities. As the results will show, this happens because interpolation approximation is improved through the estimation of the signal at the novel re-sampling location. The estimation that takes place at the novel re-sampling location is not the same as the estimation that takes place at the given re-sampling location. Also, the novel re-sampling location differs from node to node within the grid because they are being determined by a functional that is dependent on the pixel intensity at the neighbourhood and on the curvature of the interpolation function. Therefore, to consider re-sampling as a local issue (i.e. node-by-node) proves beneficial in improving image quality after interpolation. Since the mathematical formulation is applicable to low resolution imaging and because maximizes information content extraction from bio-signals, it might prove beneficial for biomedical applications at a level that is quite interdisciplinary. Examples include: functional Magnetic Resonance Imaging (fMRI) where the interdependencies between bio-signals detected at different times is to be established, and in high resolution three-dimensional microscopy where the issue of resolution is highly relevant to the extraction genomic information. 17

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Motion Correction in functional Magnetic Resonance Imaging Among the various tasks in image processing, motion correction offers a viable option for investigating the effects of the approximated nature of the interpolation paradigms. Artifacts in images are intensity values resulting from signal processing techniques like motion correction algorithms, which make use of interpolators, and are thus prone to errors due to the approximate nature of the interpolation function. Therefore, due to the diversity in error sources, in the specific case of motion correction in functional MRI, a classification of the artifacts is desirable. Artifacts that are related to the spin excitatory history effect were discovered and reported (Friston et al., 1996). Artifacts that are attributable to the approximate nature of the interpolation paradigm and inherent to re-sampling of brain images were suggested (Grootoonk et al., 2000). Along this line of thought it was demonstrated that interpolation paradigms produce motion artifacts in motion free data, and they can also be detrimental in circumstances where the magnitude of the motion to be removed is small compared to the pixel size (Freire & Mangin, 2001). Two other types of artifacts have also been explored: activation-related and task-related artifacts (Freire et al., 2002). Activation related artifacts are types of false activations caused by the similarity measures of the brain images (e.g. least squares), which mishandles real brain activations and use them to estimate non existent motion. The above delineated artifacts were demonstrated by realigning motion free data and obtaining motion estimates (Freire & Mangin, 2001). Task-related artifacts, mostly, but not necessarily, visible at the edges of the brain images, might be due to the approximate nature of the interpolation. Detrimental effects of interpolation paradigms within the context of fMRI motion correction were characterized in literature (Freire & Mangin, 2001). It was consistently shown (Freire et al., 2002), that the magnitude of detrimental effects produced by the interpolation functions is dependent on the size of the motion. Though expressed in the implicit form of location of re-sampled points relative to the initial coordinate system, this phenomenon was already known (Parker, 1983). The Sub-pixel Efficacy Region can be used to reduce the possibility of creating image artifacts. This goal can be reached through re-sampling at the novel re-sampling locations. The novel re-sampling locations are dependent on the node intensity at the location to re-sample and the neighborhood node intensities and thus they vary across nodes. Furthermore the SRE is relevant to improving interpolation approximation at low resolutions such as in functional Magnetic Resonance Imaging.

SUMMARY The chapter opens up with an overview given to the reader as to how the literature review will evolve throughout the book. Also, it is presented the Magnetic Resonance Imaging database that has been employed in these works for the validation of the SRE-based interpolation functions. Classic and SRE-based interpolation paradigms are introduced to the reader and mention is given to the model interpolation functions that will be embraced under the umbrella of the unifying theory. It is also introduced the signal processing tool that will be used throughout the book to expand from the information provided by the Fast Fourier Transform (FFT) to the frequency properties of the images processed with classic and SRE-based interpolation paradigms. This signal processing tool is called spectral power evolution and provides valuable information through the display of the plot of the combined spectral energy differences between the three types of images: (i) original, (ii) processed with classic interpolation and (iii) processed with SRE-based interpolation.

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Magnetic Resonance Imaging and the Signal-Image Processing Techniques

It is presented an experiment conducted with a three layer artificial neural network (perceptron) which was employed to confirm the difference in the spectral characteristics of the functional MRI signal that was processed with classic and SRE-based trivariate linear interpolation paradigms. The reader is provided with a discussion aimed to summarize and to emphasize on the characteristics and implications of the novel theory, and the practical implications of the Sub-pixel Efficacy Region for both signal processing and motion correction in Magnetic Resonance Imaging. Clear mention is reported in the chapter to the so called image artifact that are those spurious pixel intensity created by image processing tools like interpolation and is recalled the possibility offered through the SRE to decrease their occurrence because of the improved approximation properties of the interpolation function. The discussion goes deep in details while citing the most compelling literature in order to classify image artifact based on the mathematical nature of the processing tool that causes them. Also, particular emphasis is given to those image artifacts that are inherent to the approximate nature of interpolation. This discussion is undertaken within the context of functional Magnetic Resonance Imaging (fMRI).

References Amidror, I. (2002). Scattered data interpolation methods for electronic imaging systems: A Survey. Journal of Electronic Imaging, 11(2), 157-176. Blu, T., Thevenaz, P., & Unser, M. (2002). How a simple shift can significantly improve the performance of linear interpolation. IEEE Proceedings International Conference on Image Processing 3, (pp. III377 III-380). Blu, T., Thevenaz, P., & Unser, M. (2004). Linear interpolation revitalized. IEEE Transactions on Image Processing, 13(5), 710-719. Brock, J. S., & Wiseman, J. R. (2001). Discrete-expansions for linear interpolation functions. Computer Physics Communications, 142(1-3), 206-213. Buchel, C., & Friston, K. (1997). Modulation of connectivity in visual pathways by attention: Cortical interactions evaluated with structural equation modelling and fMRI. Cerebral Cortex, 7, 768-778. Buckner, R. L., Head, D., Parker, J., Fotenos, A. F., Marcus, D., Morris, J. C., & Snyder, A. Z. (2004). A unified approach for morphometric and functional data analysis in young, old, and demented adults using automated atlas-based head size normalization: Reliability and validation against manual measurement of total intracranial volume. Neuroimage, 23(2), 724-738. Cho, Z. H., Jones, J. P., & Singh, M. (1993). Foundations of medical imaging. New York, NY: John Wiley & Sons. Ciulla, C., & Deek, F. P. (2005a). On the approximate nature of the bivariate linear interpolation function: A novel scheme based on intensity-curvature. ICGST - International Journal on Graphics, Vision and Image Processing, 5(7), 9-19. Ciulla, C., & Deek, F. P. (2005b). Novel schemes of trivariate linear and one-dimensional quadratic B-Spline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53. 19

Magnetic Resonance Imaging and the Signal-Image Processing Techniques

Ciulla, C., & Deek, F. P. (2006). Extension of the sub-pixel efficacy region to the Lagrange interpolation function. ICGST - International Journal on Graphics, Vision and Image Processing, 6(1), 1-12. Fotenos, A. F., Snyder, A. Z., Girton, L. E., Morris, J. C., & Buckner, R. L. (2005). Normative estimates of cross-sectional and longitudinal brain volume decline in aging and AD. Neurology, 64, 1032-1039. Freire, L., & Mangin, J. F. (2001). Motion correction algorithms may create spurious brain activations in the absence of subject motion. Neuroimage, 14(3), 709-722. Freire, L., Roche, A., & Mangin, J.F. (2002). What is the best similarity measure for motion correction in fMRI time series? IEEE Transactions on Medical Imaging, 21(5), 470-484. Friston, K. J., Williams, S., Howard, R., Frackowiak, R. S., & Turner, R. (1996). Movement-related effects in fMRI time series. Magnetic Resonance in Medicine, 35(3), 346-355. Grootoonk, S., Hutton, C., Ashburner, J., Howseman, A. M., Josephs, O., Rees, G., Friston, K. J., & Turner, R. (2000). Characterization and correction of interpolation effects in the realignment of fMRI time series. Neuroimage, 11(1), 49-57. Haacke, E. M., Brown, R. W., Thompson, M. R., & Venkatesan, R. (1999). Magnetic Resonance Imaging: Physical Principles and Sequence Design. New York, NY: John Wiley & Sons. Jezzard, P., Matthews, P. M., & Smith, S. M. (2001). Functional MRI an introduction to methods. Oxford, England: Oxford University Press. Joyeux, L., & Penczeck, P. A. (2002). Efficiency of 2D alignment methods. Ultramicroscopy, 92, 3346. Lauterbur, P. C. (1973). Image formation by induced local interactions: Examples employing nuclear magnetic resonance. Nature, 242, 190-191. Lehmann, T. M., Gonner, C., & Spitzer, K. (1999). Survey: Interpolation methods in medical image processing. IEEE Transactions on Medical Imaging, 18(11), 1049-1075. Liang, Z. P., & Lauterbur, P. C. (2000). Principles of Magnetic Resonance Imaging, A signal processing perspective. New York, NY: IEEE Press. Mansfield, P., & Grannell, P. K. (1973). NMR ‘diffraction’ in solids? Journal of Physics C: Solid State Physics, 6(22), L422-L426. Marcus, D. S., Wang, T. H., Parker, J., Csernansky, J. G., Morris, J. C., & Buckner, R. L. (2007). Open Access Series of Imaging Studies (OASIS): Cross-sectional MRI data in young, middle aged, nondemented, and demented older adults. Journal of Cognitive Neuroscience, 19(9), 1498-1507. Meijering, E. H. W., Niessen, W. J., & Viergever, M. A. (2001). Quantitative evaluation of convolutionbased methods for medical image interpolation. Medical Image Analysis, 5(2), 111-126. Meijering, E. (2002). A chronology of interpolation: From ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90(3), 319-342.

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Morris, J. C. (1993). The clinical dementia rating (CDR): current version and scoring rules. Neurology, 43(11), 2412b-2414b. Ostuni, J. L., Santha, A. K. S., Mattay, V. S., Weinberger, D. R., Levin, R. L., & Frank, J. A. (1997). Analysis of interpolation effects in the reslicing of functional MR Images. Journal of Computer Assisted Tomography, 21(5), 803-810. Parker, J. A., Kenyon, R. V., & Troxel, D. E. (1983). Comparison of interpolating methods for image resampling. IEEE Transactions on Medical Imaging, 2(1), 31-39. Peeters, R. R., Kornprobst, P., Nikolova, M., Sunaert, S., Vieville, T., Malandain, G., Deriche, R., Faugeras, O., & Van Hecke, P. (2004). The use of super-resolution techniques to reduce slice thickness in functional MRI. International Journal of Imaging Systems and Technology, 14(3), 131-138. Rubin, E. H., Storandt, M., Miller, J. P., Kinscherf, D. A., Grant, E. A., Morris, J. C., & Berg, L. A. (1998). A prospective study of cognitive function and onset of dementia in cognitively healthy elders. Archives of Neurology, 55(3), 395-401. Werbos, P. J. (1990). Backpropagation through time: what it does and how to do it. Proceedings of the IEEE, 78(10), 1550-1560. Xu, F., Liu, H., Wang, G., & Alford, B. A. (2000). Comparison of adaptive linear interpolation and conventional linear interpolation for digital radiography systems. Journal of Electronic Imaging, 9(1), 22-31. Zhang, Y., Brady, M., & Smith, S. (2001). Segmentation of brain MR images through a hidden markov random field model and the expectation maximization algorithm. IEEE Transactions on Medical Imaging, 20(1), 45-57.

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Section I

On the Process Driven through Deduction, Which Starts from the Intuition, Formulates a Conception, Assesses the Truth Foreseen in the Intuition, and Finally Arrives to the Derivation of the Notion

23

Chapter II

The Intuition

Introduction This chapter reports on the initial idea which gave birth to the investigation which subsequently became the unifying theory. The intuition herein illustrated consists of the realization that the following concept might have been supported by reasonable ground of truth after extensive study. The concept is that there exists a region of spatial extent within two nodes (1D), a pixel (2D), or a voxel (3D) where the interpolation function has best approximation properties. Naturally, the adjective best is to be interpreted with its relativity to the potentiality of the specific model interpolation function to determine approximation properties. Such potentiality according to the intuition resides in: (i) the sequel of discrete samples (e.g. the pixel intensities for the two-dimensional case), and (ii) the curvature of the model interpolator as expressed by its second order derivatives. The study in this chapter is initiated for the trivariate linear interpolation function and formalized through a set of definitions, an observation and a theorem.

Definition I Let V1 = (0, 0, 0), V2 = (1, 0, 0), V3 = (1, 1, 0), V4 = (0, 1, 0) be the quadruple of vertices of the rectangle α. Let α be lying on the plane π1 of equation C * z + D = 0 ║ to the XY plane of the absolute right handed coordinate system Ξ with origin in O. Let V5 = (0, 0, 1), V6 = (1, 0, 1), V7 = (1, 1, 1), V8 = (0, 1, 1) be the quadruple of vertices of the rectangle β. Let β be lying on the plane π2 of equation C * (z + ξ1) + D = 0 ║ to the same XY plane and with ξ1 being a constant. While Vi (i = 1…4) follows each other counterclock wise on α, Vi (i = 5…8) do it on β. These eight vertices are located at the boundary surface ∑ of the parallelepiped (voxel) as shown in figure 1.

Definition II Let V1 = (0, 0, 0), V4 = (0, 1, 0), V5 = (0, 0, 1), V8 = (0, 1, 1) be the quadruple of vertices of the rectangle δ. Let δ be lying on the plane π3 of equation A * x + D = 0 ║ to the ZY plane of the absolute right handed coordinate system Ξ with origin in O. Let V2 = (1, 0, 0), V3 = (1, 1, 0), V6 = (1, 0, 1), V7 = (1, 1, 1) be the Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Intuition

quadruple of vertices of the rectangle γ. Let γ be lying on the plane π4 of equation A * (x + ξ2) + D = 0 ║ to ZY; with ξ2 being a constant. While Vi (i = 1, 4, 5, 8) follows each other counter-clock wise on δ, Vi (i = 2, 3, 6, 7) do it on γ. Also these eight vertices are located at the boundary surface ∑ of the parallelepiped (voxel) as shown in figure 1.

Definition III Let V9 = (X, 0, 0), V10 = (1, Y, 0), V11 = (X, 1, 0), V12 = (0, Y, 0) be any points of the segments [V1, V2], [V2, V3], [V3, V4], [V4, V1] respectively and V13 = (X, 0, 1), V14 = (1, Y, 1), V15 = (X, 1, 1), V16 = (0, Y, 1) any points of the segments [V5, V6], [V6, V7], [V7, V8], [V8, V5] respectively. Also, let V17 = (1, 0, Z), V18 = (1, 1, Z) be any points of the segments [V4, V8], [V3, V7] respectively, and V19 = (0, 0, Z), V20 = (1, 0, Z) be any points of the segments [V1, V5], [V2, V6] respectively. Where X, Y and Z are in between the range [0, 1], and Vi (i = 9…20) is located at the boundary surface ∑ of the parallelepiped (voxel) as shown in figure 1. Figure 1 is given here for illustration purposes. However throughout the rest of the book, it is employed the convention that the origin of the coordinate system of the intra-nodal distance (sampling step in 1D), or the pixel (2D) or the voxel (3D) is at the corner and not at the center. The corner is the left one to the eye of the observer for an intra-nodal distance, the location (0, 0) in plane, or the location (0, 0, 0) in the three dimensional space. Therefore the value of the misplacement is never negative.

Definition IV Let h = h(x, y, z) be the trivariate linear interpolation function and let it be a continuous function. Castleman (1996) defines h(x, y, z) as: h(x, y, z) = f(0,0,0) + x [f(1,0,0) – f(0,0,0)] + y [f(0,1,0) – f(0,0,0)] + z [f(0,0,1) – f(0,0,0)] + x y [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] + z y [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] + x z [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] + x y z [f(1,1,1) - f(1,1,0) - f(0,1,1) - f(1,0,1) + f(0,0,1) + f(0,1,0) + f(1,0,0) - f(0,0,0)]. (1) Where: f(0,0,0), f(1,0,0), f(1,1,0), f(0,1,0) and f(0,0,1), f(1,0,1), f(1,1,1), f(0,1,1) respectively are the two quadruple of values at the vertices the two rectangles α and β of the voxel Ψ.

Definition V Let: ∂2h /δxδy = ∂2h /∂y∂x = [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] + z ωf ∂2h/∂x∂z = ∂2h/∂z∂x = [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] + y ωf ∂2h /∂y∂z = ∂2h /∂z∂y = [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] + x ωf be the second order derivatives of the function h with respect to xy, xz and yz, where:

24

The Intuition

Figure 1. Intuition: The Sub-pixel Efficacy Region. The Voxel (a), the hyperbolic paraboloid given by equation (1) for arbitrary values of nodes’ intensity (b) and the visualization of the Sub-Pixel Efficacy Region as seen by the intuition presented in this chapter. This picture is found in: Ciulla, C. (2002). Development and characterization of methodology and technology for the alignment of fMRI time series. Unpublished doctoral dissertation, New Jersey Institute of Technology - Newark.

(a)

(b)

continued on following page

25

The Intuition

Figure 1. continued

(c)

ωf = [f(1,1,1) - f(1,1,0) - f(0,1,1) - f(1,0,1) + f(0,0,1) + f(0,1,0) + f(1,0,0) - f(0,0,0)]. Also let be: θx = [f(1,0,0) - f(0,0,0)] θy = [f(0,1,0) - f(0,0,0)] θz = [f(0,0,1) - f(0,0,0)]

Definition VI Let: θxy = θyx = - [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] / ωf θxz = θzx = - [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] / ωf θyz = θzy = - [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] / ωf be the slopes of the functions ∂2h /δxδy, ∂2h /∂x∂z and ∂2h /∂y∂z respectively.

Definition VII The function h = h(X, Y, Z) is the hyperbolic paraboloid that exists for each (X, Y, Z) of the voxel Ψ and it estimates the signal at any intra-voxel location. Let Ω = { (X, Y, Z) : | ∂2h /δxδy |z=Z = | ∂2h /δyδx |z=Z ≤ Z θxy and | ∂2h /δxδz |y=Y = | ∂2h /δzδx |y=Y ≤ Y θxz and | ∂2h /δyδz |x=X = | ∂2h /δzδy |x=X ≤ X θyz } be the definition of the Sub-pixel Efficacy Region in Ψ. It is tentatively postulated that within this region

26

The Intuition

the function h embraces characteristics of best approximation, that is: the error in signal estimation is minimized.

Observation I Let: bxy = - [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] bxz = - [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] byz = - [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] From definition VII it follows that if (X, Y, Z) belongs to Ω, then: | ∂2h /δxδy |z=Z = | ∂2h /δyδx |z=Z ≤ Z θxy | ∂2h /δxδz |y=Y = | ∂2h /δzδx |y=Y ≤ Y θxz | ∂2h /δyδz |x=X = | ∂2h /δzδy |x=X ≤ X θyz This can be re-written as: | - bxy + Z ωf | ≤ Z (bxy / ωf ) | - bxz + Y ωf | ≤ Y (bxz / ωf ) | - byz + X ωf | ≤ Z (byz / ωf )



(2) (3) (4)

Thus, if (X, Y, Z) belongs to Ω, then from equations (2), (3) and (4), it follows that: Z ≤ ωf bxy / (ωf 2 - bxy) Z ≥ ωf bxy / (ωf 2 + bxy) Y ≤ ωf bxz / (ωf 2 - bxz) Y ≥ ωf bxz / (ωf 2 + bxz) X ≤ ωf byz / (ωf 2 - byz) X ≥ ωf byz / (ωf 2 + byz)







(5) (6) (7) (8) (9) (10)

For convenience, let: ωf bxy / (ωf 2 - bxy) = Z(1) ωf bxz / (ωf 2 - bxz) = Y(1) ωf byz / (ωf 2 - byz) = X(1)

ωf bxy / (ωf 2 + bxy) = Z(2) ωf bxz / (ωf 2 + bxz) = Y(2) ωf byz / (ωf 2 + byz) = X(2)

Theorem The function h embraces characteristics of best approximation if and only if (X, Y, Z) belongs to Ω. Proof:

27

The Intuition

Claim (i): if (X, Y, Z) belongs to Ω, then h embraces characteristics of best approximation. Given (X, Y, Z) into Ω, it follows from definition VII that: | ∂2h /δxδy |z=Z = | ∂2h /δyδx |z=Z ≤ Z θxy | ∂2h /δxδz |y=Y = | ∂2h /δzδx |y=Y ≤ Y θxz | ∂2h /δyδz |x=X = | ∂2h /δzδy |x=X ≤ X θyz Thus, h embraces characteristics of best approximation. Claim (ii): if h embraces characteristics of best approximation at the location (X, Y, Z), then (X, Y, Z) belongs to Ω. Given that h embraces characteristics of best approximation at the location (X, Y, Z), then from observation I it follows that: ∂2h /δxδy |z=Z ≤ Z θxy ∂2h /δxδy |z=Z ≥ - Z θxy ∂2h /δxδz |y=Y ≤ Y θxz ∂2h /δxδz |y=Y ≥ - Y θxz ∂2h /δyδz |x=X ≤ X θyz ∂2h /δyδz |x=X ≤ - X θyz



(11) (12) (13) (14) (15) (16)

Now assume the absurd that (X, Y, Z) does not belong to Ω, then from observation I it follows that: Z > ωf bxy / (ωf 2 - bxy) Z < ωf bxy / (ωf 2 + bxy) Y > ωf bxz / (ωf 2 - bxz) Y < ωf bxz / (ωf 2 + bxz) X > ωf byz / (ωf 2 - byz) X < ωf byz / (ωf 2 + byz) Therefore: Z θxy > Z(1) θxy Z θxy < Z(2) θxy Y θxz > Y (1) θxz Y θxz < Y (2) θxz X θyz > X (1) θyz X θyz < X (2) θyz





(17) (18) (19) (20) (21) (22)

and combining the definition given for Z(1), Z(2), Y(1), Y(2), X(1) and X(2) in observation I with equations (5) though (10), it follows that: | ∂2h /δxδy |z=Z = | ∂2h /δyδx |z=Z ≤ Z θxy | ∂2h /δxδz |y=Y = | ∂2h /δzδx |y=Y ≤ Y θxz

28

(23) (24)

The Intuition

| ∂2h /δyδz |x=X = | ∂2h /δzδy |x=X ≤ X θyz





(25)

are not true. This is because equations (17) through (22) do falsify equations (23) through (25). Thus the assumption that h embraces characteristics of best approximation at the location (X, Y, Z) is contradicted by equations (17) through (22). Therefore, since to assume that (X, Y, Z) does not belong to Ω when (11) through (16) are true, yields an absurd, then it must be true that (X, Y, Z) does belong to Ω (Ciulla, 2002).

SUMMARY From an intuition, it has been given a set of definitions among which the one for the Sub-pixel Efficacy Region (SRE). Observations have been made and a claim has been formulated such to asserts tentatively the truth that the interpolation function incorporates best approximation characteristics within the spatial extent of the SRE. Therefore, the definition of the Sub-pixel Efficacy Region given herein stands for the purpose of revealing the intuition. In the following chapters of the book the definition of the Sub-pixel Efficacy Region will be refined once the analysis of the truth foreseen in the intuition will allow determining the true notion. Worth noting that the first part of the book presents a progression of five chapters that introduces to the reader one key note of the book: the process of deduction as driven by the intuition. Within this context, this chapter has expressed in mathematically organized terms the concept of Sub-pixel Efficacy Region as envisioned by the first and original idea the writer had in order to undertake a mathematical process that was aimed to the improvement of the interpolation error. The interpolation error is related to the mismatch between the estimate of the interpolation function and true value of the signal. The interpolation error is the above mentioned mismatch only for signals of known nature (e.g. defined in the realm of the floating point numbers such as cos (x)). For the rest of the signals (i.e. MRI signals) the mismatch with the true value of the signal is not yet known because of the limitations of the sampling device. The term “Sub-pixel Efficacy Region” (SRE) was chosen to express the concept that there should exists into the intra-nodal distance (sampling step), or pixel (in two dimensions), or voxel (for three dimensions), a region where the interpolation function is capable to estimate the true signal with reduced approximation error. To undertake the investigation as to where this foreseen region should be located within the voxel, a mathematical statement was given in this chapter which asserts without a-priori knowledge that the best approximation properties of the interpolation function are in that region of the voxel where the second order derivative is smaller than the increment or decrement of the function h measured on the tangent to the first order derivative. Without empirical demonstration, such assertion is certainly in necessity of waiting for confirmation. However, as such, the intuition has brought to light two theoretical implications. One implication rests in the fact that the concept of curvature is now brought into play when considering the development of a methodology aimed to improve the interpolation error. The second implication is that if the curvature of the model interpolation function is as close a possible to zero then the interpolation error is minimized. In such cases the requirement that is imposed to the signal is that of

29

The Intuition

being quasi-constant. Such requirement makes the definition given in this chapter quite restrictive as it is true that in most biomedical applications the signal is scarcely constant at all. Therefore, the main role of this chapter is that of having introduced the concept of curvature and to have set the beginning of the process which driven through the deduction illustrated in the remaining four chapters of Section I of the book will assess the truth embedded into the intuition. Also, it will be possible to derive the subsequent notion and the conception of the two foremost mathematical formulations that allowed the development of the unifying theory presented in this book. The mathematical formulations are: (i) the Intensity-Curvature Functional and (ii) the Sub-pixel Efficacy Region.

References Castleman, K. R. (1996). Digital image processing. Englewood Cliffs, NJ: Prentice Hall. Ciulla, C. (2002). Development and characterization of methodology and technology for the alignment of fMRI time series. Unpublished doctoral dissertation, New Jersey Institute of Technology - Newark.

30

31

Chapter III

The Conception of the Intensity-Curvature Functional

Introduction The preceding chapter is to be viewed as a purely theoretical math intuition and the claim consists in that of the existence of a region within the voxel (the three dimensional pixel) where interpolation is most beneficial because it is meant to produce the least approximation of the true intensity value and this region has been named: “Sub-pixel Efficacy Region” (SRE). An energy function will be defined here as the ratio between the energy of the original image and the energy of the interpolated image. This ratio which is called the Intensity-Curvature Functional, is symbolized by the expression ∆E = Eo / EIN, and it is prone to be studied to reveal its behavior within the voxel, and it is prone to determine the boundary of the Sub-pixel Efficacy Region within the voxel. In this chapter the Intensity-Curvature Functional will be treated for what concerns the trivariate liner interpolation function such to present its original conception. An application of the Intensity-Curvature Functional for the improvement of the trivariate liner interpolation function was however previously reported (Ciulla & Deek, 2005). Eo is the intensity-curvature term before interpolation and shall be envisioned as the energy of the original signal (image) and EIN is the intensity-curvature term after interpolation and shall be envisioned as the energy of the interpolated signal (image). What is demonstrated in this chapter is that because of interpolation the signal (image) is subject to energy change and the Intensity-Curvature Functional is the measure of such energy change. This chapter presents a mathematical demonstration of this presupposition. ∆E is a measure of energy change which covers the cases: (i) interpolated and original signal (image) differs in their voxel intensity; or (ii) interpolated and original signal (image) differs in their local second order derivative (curvature); or (iii) interpolated and original signal (image) differs in their voxel intensity and also in their local second order derivative. This is logical to presuppose since the intensity correction inherent to interpolation determines a change in node (voxel) intensity h(x, y, z) with respect to the original value f(0, 0, 0). And also this change of intensity is strictly linked to the change in local curvature of the interpolation function h(x, y, z).

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Conception of the Intensity-Curvature Functional

IMAGE ENERGY The formulation of the interpolation function h(x, y, z) was given in definition IV of Chapter II. The curvature consists of the second order derivatives of h(x, y, z): ∂2 (h(x, y, z))/∂x∂y, ∂2(h (x, y, z))/∂y∂z, and ∂2(h (x, y, z))/∂z∂x. For a voxel, let the energy of the interpolated image be defined as: EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz)

(1)

where:

x/2 y/2 z/2

EIN (Ψxy) = ∫ ∫

∫ h(x, y, z) (∂2 (h(x, y, z)) /∂x∂y) dx dy dz



(2)



(3)



(4)

-x/2 -y/2 -z/2

x/2 y/2 z/2

EIN (Ψzx) = ∫ ∫

∫ h(x, y, z) (∂2 (h(x, y, z)) /∂z∂x) dx dy dz

-x/2 -y/2 -z/2

x/2 y/2 z/2



EIN (Ψyz) = ∫ ∫

∫ h(x, y, z) (∂2 (h(x, y, z)) /∂y∂z) dx dy dz

-x/2 -y/2 -z/2

For the original image, let the energy be defined as: Eo = Eo(x, y, z) = Eo (Ψxy) + Eo (Ψzx) + Eo (Ψyz)

(5)

where: x/2 y/2 z/2

Eo (Ψxy) = ∫ ∫

∫ f(0, 0, 0) (∂2 I /∂x∂y) dx dy dz

(6)

-x/2 -y/2 -z/2



x/2 y/2 z/2

Eo (Ψzx) = ∫ ∫

∫ f(0, 0, 0) (∂2 I /∂z∂x) dx dy dz

(7)

-x/2 -y/2 -z/2



x/2 y/2 z/2

Eo (Ψyz) = ∫ ∫

∫ f(0, 0, 0) (∂2 I /∂y∂z) dx dy dz

-x/2 -y/2 -z/2

Definition I Let h be a continuous function, as given by definition IV of Chapter II.

32

(8)

The Conception of the Intensity-Curvature Functional

Let: θxy = θyx = - [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] / ωf θxz = θzx = - [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] / ωf θyz = θzy = - [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] / ωf ωf = [f(1,1,1) - f(1,1,0) - f(0,1,1) - f(1,0,1) + f(0,0,1) + f(0,1,0) + f(1,0,0) - f(0,0,0)]

(9) (10) (11) (12)

It is deducted from definition I above and definition VI of Chapter II that: (∂2 I /∂x∂y) = (∂2 (h(x,y,z))/∂x∂y) (0 ,0, 0) = -θxy ωf (∂2 I /∂y∂z) = (∂2 (h(x,y,z))/∂y∂z) (0 ,0, 0) = -θyz ωf (∂2 I /∂x∂z) = (∂2 (h(x,y,z)) /∂x∂z) (0 ,0, 0) = -θxz ωf and they are expected to be constant within the original image. This is likely to be assumed since the total variation of velocity of the intensity I of the original image, that is: (∂2 I /∂x∂z) + (∂2 I /∂z∂x) + (∂2 I /∂y∂z) is likely to be uniform within the voxel Ψ, whereas the total variation of velocity of the intensity of the interpolated image, that is: (∂2 (h(x, y, z)) /∂x∂z) + (∂2 (h(x, y, z)) /∂z∂x) + (∂2 (h(x, y, z)) /∂y∂z) is not uniform because of the model interpolation function h(x, y, z).

Lemma I Let Hz (x, y, z) be the primitive of h(x, y, z) with respect to z. For the trivariate linear function h(x, y, z), this Lemma shows that the second order derivative of Hz (x, y, z) with respect to the variables x and y, is a function of the variable z and not a function of the variables x and y. This can be seen by being: Hz (x, y, z) = f(0,0,0) z + x z θx + y z θy + z2/2 θz - x y z θxy ωf - y z2/2 θzy ωf - x z2/2 θxz ωf + x y z2/2 ωf (∂2 (Hz (x, y, z)) /∂x∂y) = - z θxy ωf + z2/2 ωf Also, it follows that: Hz (x, y, z) is proportional to: z * h(x, y, z)

Lemma II For the trivariate linear function h(x, y, z) this Lemma shows that its primitive with respect to the variables y and x, is proportional to the product of the function h(x, y, z) times the term xy. The primitive of the function h(x, y, z) with respect to the variable y is:

33

The Conception of the Intensity-Curvature Functional

Hy (x, y, z) = f(0,0,0) y + x y θx + y2/2 θy + y z θz - x y2/2 θxy ωf - y2/2 z θzy ωf - x y z θxz ωf + x y2/2 z ωf The primitive of the function h(x, y, z) with respect to the variables y and x is: Hyx (x, y, z) = f(0,0,0) x y + x2/2 y θx + x y2/2 θy + x y z θz - x2/2 y2/2 θxy ωf - x y2/2 z θzy ωf - x2/2 y z θxz ωf + x2/2 y2/2 z ωf It follows that: Hyx (x, y, z) is proportional to: x * y * h(x, y, z).

Deduction I Learning from the knowledge provided by Lemma I and Lemma II it follows that the primitive of the function h(x, y, z) with respect to the variables x and z is: Hxz (x, y, z) = f(0,0,0) x z + x2/2 z θx + x y z θy + x z2/2 θz - x2/2 y z θxy ωf - x z2/2 y θzy ωf - x2/2 z2/2 θxz ωf + x2/2 y z2/2 z ωf and that the primitive of the function h(x, y, z) with respect to the variables x, y and z is: Hxyz (x, y, z) = f(0,0,0) x y z + x2/2 y z θx + x y2/2 z θy + x y z2/2 θz - x2/2 y2/2 z θxy ωf - x y2/2 z2/2 θzy ωf - x2/2 z2/2 y θxz ωf + x2/2 y2/2 z2/2 ωf

Lemma III For the trivariate linear function h(x, y, z) this Lemma calculates the primitive of Hxyz (x, y, z) while the integrals are calculated with respect to the variables x, y and z respectively.

x

Þx (Hxyz (x, y, z)) = ∫ f(0,0,0) x y z + x2/2 y z θx + x y2/2 z θy + x y z2/2 θz - x2/2 y2/2 z θxy ωf - x 0

y /2 z /2 θzy ωf - x2/2 z2/2 y θxz ωf + x2/2 y2/2 z2/2 ωf dx = 2

2

f(0,0,0) x2/2 y z + x3/6 y z θx + x2/2 y2/2 z θy + x2/2 y z2/2 θz - x3/6 y2/2 z θxy ωf - x2/2 y2/2 z2/2 θzy ωf - x3/6 z2/2 y θxz ωf + x3/6 y2/2 z2/2 ωf = [ ζx × Hxyz (x, y, z) ] * x

34

The Conception of the Intensity-Curvature Functional

Where: ζx = [ (1/2), (1/3), (1/2), (1/2), (1/3), (1/2), (1/3), (1/3) ] Þx (Hxyz (x, y, z)) = [ζx × Hxyz (x, y, z) ] = (1/2) f(0,0,0) x y z + (1/3) x2/2 y z θx + (1/2) x y2/2 z θy + (1/2) x y z2/2 θz - (1/3) x2/2 y2/2 z θxy ωf - (1/2) x y2/2 z2/2 θzy ωf - (1/3) x2/2 z2/2 y θxz ωf + (1/3) x2/2 y2/2 z2/2 ωf Þy (Hxyz (x, y, z)) = ∫ f(0,0,0) x y z + x2/2 y z θx + x y2/2 z θy + x y z2/2 θz - x2/2 y2/2 z θxy ωf - x 0

y /2 z /2 θzy ωf - x2/2 z2/2 y θxz ωf + x2/2 y2/2 z2/2 ωf dx = 2

2

f(0,0,0) x y2/2 z + x2/2 y2/2 z θx + x y3/6 z θy + x y2/2 z2/2 θz - x2/2 y3/6 z θxy ωf - xy3/6 z2/2 θzy ωf - x2/2 z2/2 y2/2 θxz ωf + x2/2 y3/6 z2/2 ωf = [ζy × Hxyz (x, y, z) ] * y Where: ζy = [ (1/2), (1/2), (1/3), (1/2), (1/3), (1/3), (1/2), (1/3) ] Þy (Hxyz (x, y, z)) = [ζy × Hxyz (x, y, z) ] = (1/2) f(0,0,0) x y z + (1/2) x2/2 y z θx + (1/3) x y2/2 z θy + (1/2) x y z2/2 θz - (1/3) x2/2 y2/2 z θxy ωf - (1/3) x y2/2 z2/2 θzy ωf - (1/2) x2/2 z2/2 y θxz ωf + (1/3) x2/2 y2/2 z2/2 ωf

z

Þz (Hxyz (x, y, z)) = ∫ f(0,0,0) x y z + x2/2 y z θx + x y2/2 z θy + x y z2/2 θz - x2/2 y2/2 z θxy ωf - x 0

y /2 z /2 θzy ωf - x2/2 z2/2 y θxz ωf + x2/2 y2/2 z2/2 ωf dx = 2

2

f(0,0,0) x y z2/2 + x2/2 y z2/2 θx + x y2/2 z2/2 θy + x y z3/6 θz - x2/2 y2/2 z2/2 θxy ωf - xy2/2 z3/6 θzy ωf - x2/2 z3/6 y θxz ωf + x2/2 y2/2 z3/6 ωf = [ζz × Hxyz (x, y, z) ] * z Where: ζz = [ (1/2), (1/2), (1/2), (1/3), (1/2), (1/3), (1/3), (1/3) ] Þz (Hxyz (x, y, z)) = [ζz × Hxyz (x, y, z) ] = (1/2) f(0,0,0) x y z + (1/2) x2/2 y z θx + (1/2) x y2/2 z θy + (1/3) x y z2/2 θz - (1/2) x2/2 y2/2 z θxy ωf - (1/3) x y2/2 z2/2 θzy ωf - (1/3) x2/2 z2/2 y θxz ωf + (1/3) x2/2 y2/2 z2/2 ωf 35

The Conception of the Intensity-Curvature Functional

The Energy of the Interpolated Signal (Image) Is Not Eq ual to Zero at Eq uilibrium Hereto follow the term equilibrium shall be defined as the state corresponding to the energy of the signal (image) obtained through the calculation of the sum of the intensity-curvature terms across the entire voxel: EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) It shall be seen that the energy measured by EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) at equilibrium within a voxel is not zero. The meaning of equilibrium shall be such to reveal the intensity-curvature terms EIN (Ψxy), EIN (Ψzx), EIN (Ψyz) to be a measure of the effect of interpolation on a given voxel. The effect of the model interpolation function is that of changing the characteristics of the signal. That means that the signal does not remain unchanged after interpolation. Consistently, this section reveals that calculating the intensity-curvature terms EIN (Ψxy), EIN (Ψzx), EIN (Ψyz) across the entire voxel, the model interpolation function changes the energy level of the signal (image) and therefore the characteristics of the signal. The non zero energy level demonstrated through the intensity-curvature terms in this section also allows inferring that the intensity-curvature terms EIN (Ψxy), EIN (Ψzx), EIN (Ψyz) are effective measures of the energy change determined by the specific model interpolation function.

The Intensity-Curvature Term EIN (Ψxy) EIN (Ψxy) = (∂2 (h(x, y, z)) /∂x∂y) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 – Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 +



Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { - Þz (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þz (Hxyz (x, y, z)) | -x/2 y/2 z/2 +

Þz (Hxyz (x, y, z)) | x/2 y/2 -z/2 - Þz (Hxyz (x, y, z)) | -x/2 y/2 -z/2 -



Þz (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þz (Hxyz (x, y, z)) | -x/2 -y/2 z/2 +



Þz (Hxyz (x, y, z)) | x/2 -y/2 -z/2 - Þz (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } (13)

The proof to the assertion herein stated is given in appendix I.

36

The Conception of the Intensity-Curvature Functional

The Intensity-Curvature Term EIN (Ψzx) EIN (Ψzx) = (∂2 (h(x, y, z)) /∂z∂x) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2





Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 -



Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 +



Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { - Þy (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þy (Hxyz (x, y, z)) | -x/2 y/2 z/2 +

Þy (Hxyz (x, y, z)) | x/2 -y/2 z/2 - Þy (Hxyz (x, y, z)) | -x/2 -y/2 z/2 Þy (Hxyz (x, y, z)) | x/2 y/2 -z/2 + Þy (Hxyz (x, y, z)) | -x/2 y/2 -z/2 + Þy (Hxyz (x, y, z)) | x/2 -y/2 -z/2 - Þy (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 }

(14)

The proof to the assertion herein stated is given in appendix I.

The Intensity-Curvature Term EIN (Ψyz) EIN (Ψyz) = (∂2 (h(x, y, z)) /∂y∂z) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | x/2 y/2 -z/2





Hxyz (x, y, z) | -x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 -



Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 +



Hxyz (x, y, z) | -x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { -Þx (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þx (Hxyz (x, y, z)) | x/2 y/2 -z/2 +

Þx (Hxyz (x, y, z)) | -x/2 y/2 z/2 - Þx (Hxyz (x, y, z)) | -x/2 y/2 -z/2 Þx (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þx (Hxyz (x, y, z)) | x/2 -y/2 -z/2 + Þx (Hxyz (x, y, z)) | -x/2 -y/2 z/2 - Þx (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 }

(15)

The proof to the assertion herein stated is given in appendix I.

37

The Conception of the Intensity-Curvature Functional

The Energy of the Interpolated Image EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) = { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x + y + z ] } * - { -2 f(0,0,0) x y z / 6 - 2 x/2 y2/8 z/2 θy - 2 x/2 y/2 z2/8 θz + 2 x/2 y2/8 z2/8 θzy ωf } + { ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x + y + z ] } * - { -2 f(0,0,0) x y z / 6 + 2 x/2 y2/8 z/2 θy + 2 x/2 y/2 z2/8 θz + 2 x/2 y2/8 z2/8 θzy ωf } + { ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x - y + z ] } * - { -2 f(0,0,0) x y z / 6 + 2 x/2 y2/8 z/2 θy - 2 x/2 y/2 z2/8 θz - 2 x/2 y2/8 z2/8 θzy ωf } + { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x - y + z ] } * -{ -2 f(0,0,0) x y z / 6 - 2 x/2 y2/8 z/2 θy + 2 x/2 y/2 z2/8 θz - 2 x/2 y2/8 z2/8 θzy ωf } ≠ 0

(16)

for (x, y, z) ≠ 0 The proof to the assertion herein stated is given in appendix I.

The Energy of the Original Signal (Image) at Eq uilibrium Equation (5) is solved here and the energy of the original signal (image) that is before any interpolation is calculated. Eo = Eo(x, y, z) = Eo (Ψxy) + Eo (Ψzx) + Eo (Ψyz) = [- f(0, 0, 0) x y z θxy ωf - f(0, 0, 0) x y z θyz ωf - f(0, 0, 0) x y z θxz ωf ]x/2 y/2 z/2 [- f(0, 0, 0) x y z θxy ωf - f(0, 0, 0) x y z θyz ωf - f(0, 0, 0) x y z θxz ωf ] -x/2 -y/2 -z/2 = [- f(0, 0, 0) x y z ωf (θxy + θyz + θxz)] x/2 y/2 z/2 [- f(0, 0, 0) x y z ωf (θxy + θyz + θxz)] -x/2 -y/2 -z/2 = - ( 1/ 4) * [ f(0, 0, 0) x y z ωf (θxy + θyz + θxz) ]

38

(17)

The Conception of the Intensity-Curvature Functional

This knowledge combined with the result of equation (16) allows to see that the Intensity-Curvature Functional ∆E = Eo / EIN is a measure of the change of energy that the signal (image) experiences because of interpolation. Equations (16) and (17) allow inferring that: (i) the sum EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) is evidence of the fact that the model interpolation function changes the characteristics of the signal (image) before and after interpolation and also (ii) that the intensity-curvature terms EIN (Ψxy), EIN (Ψzx), EIN (Ψyz) are effective measures of the energy change determined by the specific model interpolation function.

SUMMARY This chapter has presented the conception of the Intensity-Curvature Functional within the context of the trivariate linear interpolation function. The Intensity-Curvature Functional symbolized by the term ΔE has been envisioned as the ratio between two energy terms. One, called the intensity-curvature term before interpolation (symbolized by Eo) is the energy that the signal (image) has prior to any interpolation, and the other, called the intensity curvature term after interpolation (symbolized by EIN) is the energy level that the signal (image) has after the model interpolation function has re-sampled the signal intensity. It was demonstrated through a meticulous math process that the energy of the interpolated signal is not equal to zero at equilibrium. Equilibrium is associated with the state of the signal either before or after interpolation. From such demonstration it follows that the energy of the original signal (before interpolation) is not equal to the energy of the signal after interpolation. Thus grounds are set for the measurement of the change of energy of the signal (image) determined through the interpolation function and this change can be measured through the Intensity-Curvature Functional. ΔE is also apt to determine an indirect measure of the interpolation error. This is because the intensity-curvature term after interpolation is strictly related to the location within the voxel where re-sampling takes place. The idea that this chapter brings to the attention of the reader is that by changing the energy level of the signal, the model interpolation function is prone to an error and such error can be indirectly measured through the Intensity-Curvature Functional. It thus descends the next step of the process driven through deduction is that of finding the extremes (minimum and maximum) of the IntensityCurvature Functional.

Reference Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53.

39

40

Chapter IV

The Conception of the Sub-Pixel Ef.cacy Region Introduction This chapter presents the mathematical deduction of the Sub-pixel Efficacy Region (SRE) for the case of the trivariate liner interpolation function. The questions that the reader may have at this point are: (i) what is the need of defining the Sub-pixel efficacy Region once again? And (ii) why the Sub-pixel Efficacy Region is to be calculated on the basis of the Intensity-Curvature Functional? To reconnect with the study undertaken in Chapters II and III it is due to recall that the SRE is only an intuition before the beginning of this chapter. Therefore, the answer to the first question is immediate to the writer to the extent that the SRE mathematically characterized here consists of a conceptual refinement to what already anticipated through the intuition presented in Chapter II. This is necessary in order to move the first step towards the determination of the SRE-based interpolation functions and this shall be seen throughout the rest of the book. Nevertheless, this is also consequential to the process of deduction that started from the intuition. In this chapter, the Intensity-Curvature Functional is associated with the concept of metric employed to measure the intensity-curvature content of the interpolation function. Nevertheless, Chapter III has presented the conception of the Intensity-Curvature Functional and concluded that such functional is conceived and demonstrated to be the measure of change of the energy of the signal (image) that the interpolation function determines. The answer to the second question is therefore the following: to calculate the SRE on the basis of the Intensity-Curvature Functional is conceptually equivalent to a process that finds the intra-pixel locations where the change of image energy determined through the model interpolation function is either minimal or maximal. This answer naturally descends from the fact that the SRE is calculated from the solution of the polynomial system of equations consisting of the three first order partial derivatives of the Intensity-Curvature Functional.

Sub-pix el Efficacy Region The Sub-Pixel Efficacy Region is derived through the study of the Intensity-Curvature Functional, particularly through the process of finding the extremes of ΔE. Based on the results of Chapter III, the

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Conception of the Sub-Pixel Ef.cacy Region

Intensity-Curvature Functional is herein calculated: ΔE(x, y, z) = Eo / EIN



(1)

EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) = x y z

xyz

∫ ∫ ∫ h(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) dx dy dz + ∫ ∫ ∫ h (x, y, z) (∂2(h(x,y,z)) /∂z∂x) dx dy dz + 0 0 0 xyz

000

∫ ∫ ∫ h (x, y, z) (∂2(h(x,y,z)) /∂y∂z) dx dy dz

(2)

000

Eo = Eo(x, y, z) = Eo (Ψxy) + Eo (Ψzx) + Eo (Ψyz) = xyz

∫ ∫ ∫ f (0, 0, 0) (∂2 (h(x,y,z))/∂x∂y + ∂2 (h(x,y,z))/∂y∂z + ∂2 (h(x,y,z)) /∂x∂z) (0 ,0, 0) dx dy dz (3) 000 It is due to clarify that above integrals are calculated from (0, 0, 0) (the origin of the coordinate system of the voxel, assumed at the location of the node of intensity f(0, 0, 0)) to anywhere (x, y, z) in the voxel. For the rest of the interpolation functions treated in the book, the integrals of the intensity-curvature terms are calculated from the origin of the coordinate system in between two nodes (1D) or the pixel (2D) respectively, to anywhere within the intra-nodal distance or the pixel. The calculation shown in equations (2) and (3) is different from the one previously seen in Chapter III, where the integrals where calculated from (-x/2, -y/2, -z/2) to (x/2, y/2, z/2) such to quantify the energy level of the signal (image) obtained before and after interpolation. In this chapter the concept of energy is temporarily subsidized by the concept of Intensity-Curvature content embedded into the model interpolation function at a given re-sampling location (x, y, z) in the voxel. From this concept it is derived the measure of intensity-curvature of the interpolation function, which, herein and in the rest of the book will be called: the Intensity-Curvature Functional. Hereto is presented the calculation of the integrals and also for what concerns the deduction of the Sub-pixel Efficacy Region is presented the math process that derives the extremes of the ratio expressed by equation (1). Keeping in mind that (∂2 (h(x, y, z)) /∂x∂y) = - θxy ωf + z ωf is a function of the variable z only, and that (∂3 (h(x,y,z)) /∂x∂y∂z) = ωf and also by the benefits of Lemma III of Chapter III, let us proceed to write: xyz

EIN (Ψxy) = ∫ ∫ ∫ h(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) dx dy dz = 000 zy

∫ ∫ Hx(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) dy dz = 00

41

The Conception of the Sub-Pixel Efficacy Region

z

∫ Hxy(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) dz = [ Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) ]z 0



[ Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) ]0 -

z

∫ Hxyz(x, y, z) (∂3 (h(x,y,z)) /∂x∂y∂z) dz = 0

Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) - [ζz × Hxyz (x, y, z) ] * z (∂3 (h(x,y,z)) /∂x∂y∂z) = Hxyz(x, y, z) [ (∂2 (h(x,y,z)) /∂x∂y) - (z * ζz ×) (∂3 (h(x,y,z)) /∂x∂y∂z) ] = Hxyz (x, y, z) [- θxy ωf + z ωf - (z * ζz ×) ωf ] = Hxyz (x, y, z) [- θxy ωf + z (1 - ζz ×) ωf ]

(4)

xyz

Eo (Ψxy) =∫ ∫ ∫ f (0, 0, 0) (∂2 (h(x,y,z))/∂x∂y ) (0 ,0, 0) dx dy dz = 000 f (0, 0, 0) x y z (∂2 (h(x,y,z))/∂x∂y ) (0 ,0, 0) = - f (0, 0, 0) x y z θxy ωf

(5)

Keeping in mind that (∂2 (h(x, y, z)) /∂z∂x) = - θxz ωf + y ωf is a function of the variable y only, let us write: xyz

EIN (Ψzx) = ∫ ∫ ∫ h(x, y, z) (∂2 (h(x,y,z)) /∂z∂x) dx dy dz = 000 xy

∫ ∫ Hz(x, y, z) (∂2 (h(x,y,z)) /∂z∂x) dx dy = 00 y

∫ Hxz(x, y, z) (∂2 (h(x,y,z)) /∂z∂x) dy = [ Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂z∂x) ]y 0



[ Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂z∂x) ]0 -

y

∫ Hxyz(x, y, z) (∂3 (h(x,y,z)) /∂x∂y∂z) dz = 0

Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂z∂x) - [ζy × Hxyz (x, y, z) ] * y (∂3 (h(x,y,z)) /∂x∂y∂z) = Hxyz(x, y, z) [ (∂2 (h(x,y,z)) /∂z∂x) - (y * ζy ×) (∂3 (h(x,y,z)) /∂x∂y∂z) ] =

42

The Conception of the Sub-Pixel Efficacy Region

Hxyz (x, y, z) [- θxz ωf + y ωf - (y * ζy ×) ωf ] = Hxyz (x, y, z) [- θxz ωf + y (1 - ζy ×) ωf ]

(6)

xyz

Eo (Ψzx) =∫ ∫ ∫ f (0, 0, 0) (∂2 (h(x,y,z))/∂z∂x ) (0 ,0, 0) dx dy dz = 000 2 f (0, 0, 0) x y z (∂ (h(x,y,z))/∂z∂x ) (0 ,0, 0) = - f (0, 0, 0) x y z θzx ωf

(7)

Also, keeping in mind that (∂2 (h(x, y, z)) /∂y∂z) = - θzy ωf + x ωf is a variable of the sole x, let us write: xyz

EIN (Ψyz) = ∫ ∫ ∫ h(x, y, z) (∂2 (h(x,y,z)) /∂y∂z) dx dy dz = 000 xy

∫ ∫ Hz(x, y, z) (∂2 (h(x,y,z)) /∂y∂z) dx dy = 00 x

∫ Hyz(x, y, z) (∂2 (h(x,y,z)) /∂y∂z) dx = [ Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂y∂z) ]x 0

x

[ Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂y∂z) ]0 -

∫ Hxyz(x, y, z) (∂3 (h(x,y,z)) /∂x∂y∂z) dz = 0

Hxyz(x, y, z) (∂2 (h(x,y,z)) /∂y∂z) - [ ζx × Hxyz (x, y, z) ] * x (∂3 (h(x,y,z)) /∂x∂y∂z) = Hxyz(x, y, z) [ (∂2 (h(x,y,z)) /∂y∂z) - (x * ζx ×) (∂3 (h(x,y,z)) /∂x∂y∂z) ] = Hxyz (x, y, z) [- θzy ωf + x ωf - (x * ζx ×) ωf ] = Hxyz (x, y, z) [- θzy ωf + x (1 - ζx ×) ωf ]

(8)

xyz

Eo (Ψyz) =∫ ∫ ∫ f (0, 0, 0) (∂2 (h(x,y,z))/ ∂y∂z) (0 ,0, 0) dx dy dz = 000 f (0, 0, 0) x y z (∂2 (h(x,y,z))/ ∂y∂z) (0 ,0, 0) = - f (0, 0, 0) x y z θyz ωf

(9)

Summing up equations (4), (6) and (8) it can be written that: EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) = Hxyz (x, y, z) [ - θxy ωf + z (1 - ζz ×) ωf ] + Hxyz (x, y, z) [ - θxz ωf + y (1 - ζy ×) ωf ] + Hxyz (x, y, z) [ - θzy ωf + x (1 - ζx ×) ωf ] = ωf Hxyz (x, y, z) [z/2 - θxy + y/2 - θxz + x/2 - θyz ] =

43

The Conception of the Sub-Pixel Efficacy Region

ωf Hxyz (x, y, z) [x (1 - ζx ×) + y (1 - ζy ×) + z (1 - ζz ×) - θxy - θxz - θyz ] = - ωf Hxyz (x, y, z) [ θxy + θxz + θyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]

(10)

Summing up equations (5), (7) and (9) it can be written that: Eo = Eo(x, y, z) = Eo (Ψxy) + Eo (Ψzx) + Eo (Ψyz) = - f (0, 0, 0) x y z θxy ωf - f (0, 0, 0) x y z θzx ωf - f (0, 0, 0) x y z θyz ωf = - f (0, 0, 0) x y z ωf [θxy + θyz + θzx]

(11)

Therefore, the Intensity-Curvature Functional expressed by equation (1) is written as: ΔE(x, y, z) = Eo / EIN = { f (0, 0, 0) x y z [θxy + θyz + θzx ] / Hxyz (x, y, z) [ θxy + θxz + θyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }



(12)

Calculation of the First Order Partial Derivatives of Hxyz (x, y, z) (∂ Hxyz (x, y, z) / ∂x) = f (0, 0, 0) y z + x y z θx + y2/2 z θy + z2/2 y θz - x y2/2 z θxy ωf - y2/2 z2/2 θzy ωf - x y z2/2 θxz ωf + x y2/2 z2/2 ωf = y z [ f (0, 0, 0) + x θx + y/2 θy + z/2 θz - x y/2 θxy ωf - z/2 y/2 θzy ωf - x z/2 θxz ωf + x y/2 z/2 ωf ]



(13)

(∂ Hxyz (x, y, z) / ∂y) = f (0, 0, 0) x z + x2/2 z θx + x y z θy + x z2/2 θz - x2/2 y z θxy ωf - x y z2/2 θzy ωf - x2/2 z2/2 θxz ωf + x2/2 y z2/2 ωf = x z [ f (0, 0, 0) + x/2 θx + y θy + z/2 θz - x y/2 θxy ωf - z y/2 θzy ωf - x/2 z/2 θxz ωf + x/2 y z/2 ωf ]



(14)

(∂ Hxyz (x, y, z) / ∂z) = f (0, 0, 0) x y + x2/2 y θx + x y2/2 θy + x y z θz - x2/2 y2/2 θxy ωf - x y2/2 z θzy ωf - x2/2 y z θxz ωf + x2/2 y2/2 z ωf = x y [ f (0, 0, 0) + x/2 θx + y/2 θy + z θz - x/2 y/2 θxy ωf - z y/2 θzy ωf - x/2 z θxz ωf + x/2 y/2 z ωf ] (15) From equations (13), (14) and (15) it follows that:

44



The Conception of the Sub-Pixel Efficacy Region

(∂ Hxyz (x, y, z) / ∂x) = Hyz (x, y, z) (∂ Hxyz (x, y, z) / ∂y) = Hxz (x, y, z) (∂ Hxyz (x, y, z) / ∂z) = Hxy (x, y, z)

Study of the Intensity-Curvature Functional Hereto is the outcome of the study of the Intensity-Curvature Functional (ΔE). The entire math process that relates to the study of ΔE is reported in appendix II. The outcome consists of the solution of the polynomial system of equations derived from the three first order partial derivatives of ΔE with respect to the variables x, y and z. x = [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ]

(16)

y2 { (θy/2 - z θzy ωf/2) [ ( θxy ωf /2 - z ωf/2) + (-θxy ωf/4 + z ωf/4)] } + y { -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + (-f(0,0,0) - z θz) [(- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf )]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf )/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf ) + z θz (-θxy ωf/4 + z ωf/4) } + { f(0,0,0) [ θx/2 - z/2 θxz ωf ] + z θz (θx/2 - z/2 θxz ωf ) } = 0

(17)

Equation (17) furnishes the following two solutions: y1,2 = ( -b ± Δ1/2 ) / 2 a where: b = { -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + (-f(0,0,0) - z θz) [(- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf )]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf )/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf ) + z θz (-θxy ωf/4 + z ωf/4) } a = { (θy/2 - z θzy ωf/2) [ ( θxy ωf /2 - z ωf/2) + (-θxy ωf/4 + z ωf/4)] } Δ = b2 - 4 a c c = { f(0,0,0) [ θx/2 - z/2 θxz ωf ] + z θz (θx/2 - z/2 θxz ωf ) } z2 {(θz/2 - y/2 θzy ωf ) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (θxz/4 ωf - y/4 ωf ) (- θz + y/2 θzy ωf )} + z {[ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] y/4 ωf + (θx/2 - y/2 θxy ωf ) [- θz + y/2 θzy

45

The Conception of the Sub-Pixel Efficacy Region

ωf ] - θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (θxz/4 ωf - y/4 ωf ) [-f(0,0,0) - y/2 θy] } + {[- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] - (θx/2 - y/2 θxy ωf ) [-f(0,0,0) - y/2 θy ]} = 0





(18)

Equation (18) furnishes the following two solutions: z1,2 = ( -b ± Δ1/2 ) / 2 a where: b = {[ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] y/4 ωf + (θx/2 - y/2 θxy ωf ) [- θz + y/2 θzy ωf ] θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (θxz/4 ωf - y/4 ωf ) [-f(0,0,0) - y/2 θy] } a = {(θz/2 - y/2 θzy ωf ) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (θxz/4 ωf - y/4 ωf ) (- θz + y/2 θzy ωf )} Δ = b2 - 4 a c c = { [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] - (θx/2 - y/2 θxy ωf ) [-f(0,0,0) - y/2 θy ] }

The Deduction of the Sub-Pixel Ef.cacy Region The Sub-pixel Efficacy Region of the trivariate liner interpolation function (the case treated so far) is given by equations (16), (17) and (18) and these equations are the solution of the polynomial system resulting from the three first order partial derivatives of the Intensity-Curvature Functional ΔE.

SUMMARY The study undertaken in this chapter presents the conception of the Sub-pixel Efficacy Region in its revised form with respect to the form earlier presented in Chapter III. The study relevant to the SubPixel Efficacy Region is given for the trivariate linear interpolation function and this is consistent with the previous two chapters. To undertake the math process that derives the SRE, this chapter further refines the concept of Intensity-Curvature Functional. While in Chapter III, ΔE was calculated covering the entire voxel thus quantifying the energy change determined by the model interpolation function, this concept is herein subsidized by the concept of intensity-curvature content. The metric ΔE is thus calculated to measure the intensity-curvature content

46

The Conception of the Sub-Pixel Efficacy Region

of the interpolation function. This calculation is done by setting the origin of the coordinate system of the voxel at the grid’s corner and measuring the intensity-curvature content from the origin to the intra-voxel location where re-sampling takes place. The rationale behind this conceptual adjustment rests in the justification that the Intensity-Curvature Functional is set to measure the energy level change provoked by interpolation for the particular re-sampling location that is chosen for the model to be calculated at. Thus, the Intensity-Curvature Functional measures energy change and it does that by focusing on the intensity-curvature content of the model interpolation function that is measured in between the origin of the coordinate system of the voxel and the re-sampling location where the interpolation function is calculated. At the end of the meticulous math process, the Sub-pixel Efficacy Region of the trivariate linear interpolation function is found as the solution of the polynomial system resulting from the three first order partial derivatives of the ΔE. The idea that the Intensity-Curvature Functional is a measure of energy level change determined by the interpolation function can be viewed as an extension of the concepts that are relevant to signal (image) energy reported earlier (Ciulla & Deek, 2005b). Also, as it shall be seen throughout the book that for what pertains to the development of the herein proposed innovative theory, the Sub-pixel Efficacy Region is ultimately responsible for the adaptive behavior of the SRE-based interpolation functions that will be presented. The conception of the SRE (Ciulla & Deek, 2005a, 2005b, 2006) can be seen as precursor of later works on adaptive interpolation (Cha & Kim, 2006, 2007; Chand & Kapoor 2006; Goodman & Ong, 2005), which employ derivatives to devise improved versions of the interpolation functions.

References Cha, Y., & Kim, S. (2006). Edge-forming methods for image zooming. Journal of Mathematical Imaging and Vision, 25(3), 353-364. Cha, Y., & Kim, S. (2007). The error-amended sharp edge (EASE) scheme for image zooming. IEEE Transactions on Image Processing, 16(6), 1496-1505. Chand, A. K. B., & Kapoor, G. P. (2006). Spline coalescence hidden variable fractal interpolation functions. Journal of Applied Mathematics, 2006(Article ID 36829), 1-17. Ciulla, C., & Deek, F. P. (2005a). On the approximate nature of the bivariate linear interpolation function: A novel scheme based on intensity-curvature. ICGST - International Journal on Graphics, Vision and Image Processing, 5(7), 9-19. Ciulla, C., & Deek, F. P. (2005b). Novel schemes of trivariate linear and one-dimensional quadratic B-Spline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53. Ciulla, C., & Deek, F. P. (2006). Extension of the sub-pixel efficacy region to the Lagrange interpolation function. ICGST - International Journal on Graphics, Vision and Image Processing, 6(1), 1-12. Goodman, T. N. T., & Ong, B. H. (2005). Shape-preserving interpolation by splines using vector subdivision. Advances in Computational Mathematics, 22(1), 49-77.

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48

Chapter V

Assessment of the Truth Foreseen in the Intuition

Introduction This chapter continues the investigation conducted through deduction from the previous three chapters. Chapter II of the book has brought to light an intuition that has allowed the conception of the SubPixel Efficacy Region and its definition. Chapter III outlines the conception of the Intensity-Curvature functional as a measure of the change in energy level of the signal because of the effect of the model interpolation function on the signal (image). Chapter IV studies the Intensity-Curvature Functional (ΔE) and through the solution of the polynomial system obtained from the first order partial derivatives of ΔE, elucidates that a spatial set of points within the voxel is calculated from the Intensity-Curvature Functional and is assigned the name of Sub-pixel Efficacy Region (SRE). The next step is then to assess the truth foreseen in the intuition of Chapter II for what concerns the definition of the Sub-pixel Efficacy Region. The definition of the SRE given in Chapter II has to be re-evaluated on the basis of the results of Chapter IV. Conceptually, what Chapter II states is that there is a region within the voxel where the value of the second order partial derivative (curvature) of the model interpolation function is bounded by two constants. Also, Chapter II tentatively asserts that within this region the model interpolation function has optimal approximation characteristics. That means that the function admits minimal interpolation error with respect to the error obtainable in the spatial domain that is located outside the SRE. On the basis of the theoretical grounds furnished in Chapter III as far as the meaning of the IntensityCurvature Functional is concerned, Chapter IV derives the spatial set of points within the voxel where the extremes of ΔE are located and assigns to these points the name: Sub-pixel Efficacy Region. Formulas given for the SRE in Chapter IV (equations (16), (17) and (18)) do not have immediate correspondence to the definition of the SRE given in Chapter II. Therefore, it is due to investigate what is the degree of match between the two definitions of the SRE: (i) the one given in Chapter II, which is purely theoretical, and (ii) the other one deducted through the study of the Intensity-Curvature Functional in Chapter IV, which is practical. Through the evaluation of the degree of match between these two definitions of the SRE it is possible to assess on the truth that was foreseen in the intuition presented in Chapter II. Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

Assessment of the Truth Foreseen in the Intuition

THE ASSESSMENT It is presented here and in appendix III the mathematical process that through further study of equations (16), (17) and (18) of Chapter IV attempts to establish as to whether or not the SRE set of points found in Chapter IV has the theoretical property stated in Chapter II. This means to see as to if at the SRE spatial points of Chapter IV the local curvature of the model interpolation function is bounded by the constants given by the equations (5) through (10) of Chapter II. In order to undertake the investigative process and for convenience to the reader let us recall the definition of SRE given in Chapters II and IV.

Chapter II (x, y, z) belongs to the SRE if and only if the following is true:

z ≤ ωf bxy / (ωf 2 - bxy) z ≥ ωf bxy / (ωf 2 + bxy) 2 y ≤ ωf bxz / (ωf - bxz) y ≥ ωf bxz / (ωf 2 + bxz) x ≤ ωf byz / (ωf 2 - byz) x ≥ ωf byz / (ωf 2 + byz)



(1) (2) (3) (4) (5) (6)

Chapter IV (x, y, z) that belongs to the SRE have the following properties: x = [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ]

(7)

y1,2 = ( -b ± Δ1/2 ) / 2 a

(8)



where: b = { (-f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + (-f(0,0,0) - z θz) [(- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf )]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf )/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf ) + z θz (-θxy ωf/4 + z ωf/4) } a = { (θy/2 - z θzy ωf/2) [ ( θxy ωf /2 - z ωf/2) + (-θxy ωf/4 + z ωf/4)] } Δ = b2 - 4 a c

49

Assessment of the Truth Foreseen in the Intuition

c = { f(0,0,0) [ θx/2 - z/2 θxz ωf ] + z θz (θx/2 - z/2 θxz ωf ) } z1,2 = ( -b ± Δ1/2 ) / 2 a



(9)

where: b = {[ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] y/4 ωf + (θx/2 - y/2 θxy ωf ) [- θz + y/2 θzy ωf ] θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (θxz/4 ωf - y/4 ωf ) [-f(0,0,0) - y/2 θy] } a = {(θz/2 - y/2 θzy ωf ) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (θxz/4 ωf - y/4 ωf ) (- θz + y/2 θzy ωf )} Δ = b2 - 4 a c c = { [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] - (θx/2 - y/2 θxy ωf ) [-f(0,0,0) - y/2 θy ] } What follows from equations (8) and (9) is reported in appendix III. It is not readily possible to the understanding of the writer to see if equation (7) and the solutions of equations (8) and (9) obey to the requirements posed by equations (1) through (6). Equations (1) through (6) descend from the definition of Sub-pixel Efficacy Region given in Chapter II. In conclusion, the degree of match of the two definitions of Sub-pixel Efficacy Region given in Chapters II and IV respectively resides in the concept of curvature. This concept is seen in equations (1) through (6) and in the conception and definition of the Intensity-Curvature Functional. The truth foreseen in the intuition of Chapter II is that: the curvature of the model interpolation function embeds quite important properties that are relevant to the estimation of the unknown signals. The curvature of the interpolation function is therefore the notion derived from the process driven through deduction, which was undertaken in the previous four chapters. The curvature of the interpolation function will be object of discussion in the next chapter.

SUMMARY This chapter has presented the mathematical investigation aimed to assess the truth foreseen in the intuition of Chapter II. The definitions given for the Sub-pixel Efficacy Region in Chapters II and IV were under investigation such to determine as to what is the degree of match between them. For the set of intra-voxel points where the Intensity-Curvature functional admits its extremes, which correspond to the definition of SRE given in Chapter IV, the math process given in this chapter reveals that it is unclear as to if the definition of SRE given in Chapter II obeys to the definition of SRE given in Chapter IV. The chapter concludes stressing on the main concept realized through the intuition, which is that of the curvature of the interpolation function (Ciulla & Deek 2005). As it shall be seen in the remaining chapters, this concept is relevant to the extent of the material presented in the book because it plays a

50

Assessment of the Truth Foreseen in the Intuition

fundamental role in devising the methodology of interpolation error improvement through the unifying theory.

Reference Ciulla, C., & Deek, F. P. (2005). On the approximate nature of the bivariate linear interpolation function: A novel scheme based on intensity-curvature. ICGST - International Journal on Graphics, Vision and Image Processing, 5(7), 9-19.

51

52

Chapter VI

The Notion

Introduction The goal of this chapter is to discuss the practical implications of the notion. This chapter concludes the logical reasoning outlined through the process of deduction that begun in Chapter II while reporting on the intuition. The logical reasoning was necessary in order to arrive to a determination as to what is the meaning of the intuition. Along the way it was possible to introduce the novel conception of Intensity-Curvature Functional and to connect such conception to the initial intuition of SRE. At that point the intuition was assessed in its truth and thus it was manifest to reconsider the formulation of the SRE. The implication of doing so rests in that the formulation and the conceptualization of the SRE given in Chapter IV is useful for the remainder of the book in presenting the unifying theory for the improvement of the interpolation error. The process of deduction (logical reasoning) has finally arrived to the notion that the curvature of the model interpolation function is quite relevant to the extent of improving the approximation properties of the function. The realization of the notion was already anticipated in Chapter V and this chapter further extends on the notion through the illustrations of its practical implications. In other words the process of deduction has arrived to a beneficial conclusion and for this reason it proves itself to have been useful in order to derive the notion.

The curvature of the interpolation function This chapter concludes Section I of the book deriving the notion from the evaluation of the truth foreseen in the intuition. The presentation is consequential to what already seen in the previous four chapters and it is particularly focused on the trivariate liner interpolation function. The conception of the Intensity-Curvature Functional that was given in Chapter III has shown that it possible to give a mathematical formulation that is such to measure the energy level of the signal (image) before and after interpolation. The Intensity-Curvature Functional is thus capable to determine the energy level change of the signal (image) when is subject to re-sampling at any intra-voxel location through the model interpolation function. Based on this conception it was derived the empirical formulation of the Sub-pixel Efficacy Region (SRE), which deduction is subsequent to the study of the Intensity-

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The Notion

Curvature Functional. Such formulation asserts that the Sub-pixel Efficacy Region is that spatial set of intra-voxel points where the energy level change of the signal (image) is minimum or maximum. This descends from the fact that the set of points that belong to the SRE are derived through the solution of the polynomial system of first order partial derivatives of the Intensity-Curvature Functional (ΔE) and that is the task that corresponds to the identification of the extremes of ΔE. In Chapter II however, the definition given to the SRE was purely theoretical and was derived from the intuition. Definition VII seen in Chapter II was given within the context of the hyperbolic paraboloid function h = h(X, Y, Z) that exists for each (X, Y, Z) of the voxel Ψ and that is capable to estimate the signal at any intra-voxel location. The Sub-pixel Efficacy Region in Ψ was thus defined as that intravoxel set of points: Ω = { (X, Y, Z) : | ∂2h /δxδy |z=Z = | ∂2h /δyδx |z=Z ≤ Z θxy and | ∂2h /δxδz |y=Y = | ∂2h /δzδx |y=Y ≤ Y θxz and | ∂2h /δyδz |x=X = | ∂2h /δzδy |x=X ≤ X θyz }. Also, for convenience to the reader, it is herein recalled from definition V of Chapter II that: ∂2h /δxδy = ∂2h /∂y∂x = [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] + Z ωf ∂2h/∂x∂z = ∂2h/∂z∂x = [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] + Y ωf ∂2h /∂y∂z = ∂2h /∂z∂y = [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] + X ωf Above relationships recall us the knowledge that the geometrical meaning of the derivative of a function is that of the tangent to the function. Assuming the origin of the coordinate system of the voxel at the location (0, 0, 0), the local second order partial derivatives at (X, Y, Z) which are taken into account in the definition of Ω: | ∂2h /δxδy |z=Z = | ∂2h /δyδx |z=Z | ∂2h /δxδz |y=Y = | ∂2h /δzδx |y=Y | ∂2h /δyδz |x=X = | ∂2h /δzδy |x=X embed the meaning of the angles subtended by the tangent to the first order derivatives of h. Such tangent is measured from (0, 0, 0) to (X, Y, Z). Let us now recall the meaning of θxy θxz and θyz according to definition VI of Chapter II: θxy = θyx = - [f(0,0,0) – f(1,0,0) – f(0,1,0) + f(1,1,0)] / ωf θxz = θzx = - [f(0,0,0) – f(1,0,0) – f(0,0,1) + f(1,0,1)] / ωf θyz = θzy = - [f(0,0,0) – f(0,1,0) – f(0,0,1) + f(0,1,1)] / ωf θxy θxz and θyz are thus a measure of the slopes of the second order partial derivatives ∂2h /δxδy = ∂2h /∂y∂x, ∂2h/∂x∂z = ∂2h/∂z∂x, and ∂2h /∂y∂z = ∂2h /∂z∂y. Thus, the geometrical meaning of the values Z θxy,Y θxz and X θyz given in the definition of the Sub-Pixel Efficacy Region are those of: the increment or decrement of the function h measured on the tangent to the first order derivative. The concepts formulated above are illustrated in figure 1, and for simplicity the picture is one dimensional in the variable x.

53

The Notion

When seen within the context of the definition of the SRE, above knowledge yields the concept behind the definition that was given to the SRE (Ω) in Chapter II: The Sub-pixel Efficacy Region is that spatial set of intra-voxel points for which the local second order partial derivatives are smaller than the increment or decrement of the function h measured on the tangent to the first order derivative. Let us recall that the geometrical meaning of the second order derivative embeds the meaning of the angle subtended by the tangent to the first order derivative of h. To understand above concept let us analyze three cases of interest. Let the first case be when the angle subtended by the tangent to the first order derivative of h is small. In such a case the curvature of the function h (measured by the local second order derivative) is small because the function h tends to be flat. Conversely, let the second case be when the angle subtended by the tangent to the first order derivative of h is big. In such a case the curvature of the function h is big because the function h tends to be far from being flat. Finally, let the limit case be when the angle subtended by the tangent to the first order derivatives of h is zero. In such case the curvature is clearly zero and it indicates that the function h is constant. Intuitively, the Sub-Pixel Efficacy Region targets the spatial set of intra-voxel points for which the function h local curvature is limited: its value is not too big compared to the increment (or decrement) Figure 1. The increment of the function h measured on the tangent to the first order derivative. Increment of the function h measured on its tangent

y = h (x) h' (x)

h (x) X

x

is the tangent to the function h Increment X)

of the function h measured on the tangent

1

y = h' (x) h' (x)

1

The angle

h (x) subtended by the tangent

1 to

X 1 is

54

the tangent to the first order derivative h'

h' (x)

x

The Notion

of the function h measured on the tangent to the first order derivative. Conceptually this is equivalent to identifying those points in the voxel where the function’s curvature is small. The curvature of the model interpolation function is ultimately depending on the signal (image) intensity values at the grid corners in the neighborhood (eight values of pixel intensity for the case of the trivariate liner interpolation function). Setting as desirable that the curvature is as small as possible (preferably close to zero) is conceptually equivalent to desire a signal with almost no variation of intensity across the neighborhood. This requirement is fulfilled in regions of the signal that are quasi constant, or for instance, in regions of the image where the voxel-to-voxel gradient is quasi null. As it is fair to admit that these regions of quasi null curvature exist in a signal (image) as those regions of constant node-to-node intensity, it is due as well as quite realistic to admit that in most instances this is not likely to be true. Biological signals, MRI of the human brain among others, are examples of such instance. Definition VII given in Chapter II of the Sub-Pixel Efficacy Region is therefore conveying the message that a model interpolation function works best (its approximation produces the smallest error) when the signal is constant. This is quite limitative since signal’s behaviors in the nodes’ neighborhood can be generally scarcely constant. The truth foreseen in the intuition of Chapter II is thus applicable to particular, sometimes ideal cases, however extendible in their essence, because the truth furnishes a concept that is intermediate to the development of further investigation. Therefore the notion that this chapter wants to bring to the attention of the reader is that: the curvature of the model interpolation function is quite relevant to the extent of the improvement of the interpolation error. From this discussion it thus descends the necessity to formulate a definition of SRE that is wider and that adapts itself to those cases that are most likely to be found in real applications, cases for which the curvature of the model interpolation function varies across the neighborhood as well across neighborhoods. This necessity also provided with the justification as to why it is important to measure the change in curvature of the model interpolation function. Based on the sequel of discrete intensity values of the signal, it is important to measure the change in curvature of the model interpolation function, and yet more important to measure the adaptability of the curvature of the model interpolation function to the curvature of the true and unknown signal to estimate. This is certainly relevant to the extent of the minimization of the interpolation error. To measure the change of the curvature of the interpolation function and its degree of match with the true signal to estimate, this book introduces a mathematical formulation called: Intensity-Curvature Functional (ΔE), and it admits, as seen in Chapter III, that the Sub-pixel Efficacy Region (for the case of the trivariate linear interpolation function) is that set of intra-voxel points for which ∂ (ΔE(x, y, z)) /∂x = 0, ∂ (ΔE(x, y, z)) /∂y = 0 and ∂ (ΔE(x, y, z)) /∂z = 0. Such definition of the SRE allows: (i) to overcome the limitation introduced by the requirement given in Chapter II, that is of quasi-null curvature, (ii) to extend the domain of the SRE to a larger set of points (locations) within the voxel and (iii) to set for these SRE point the requirement that for them the first order partial derivatives of the Intensity-Curvature Functional are to be equal to zero. Therefore, the Sub-pixel Efficacy Region becomes: the spatial set of intra-voxel points where ΔE admits extremes (Ciulla & Deek, 2005).

55

The Notion

At this point it is reasonable and also due to clarify to the reader what is the meaning of finding the extremes of ΔE. This clarification shall be given in the subsequent chapters of the book, through the empirical validation of ΔE conducted for several model interpolation functions and the demonstration of its usefulness in finding the novel re-sampling locations where the interpolation error is reduced. The subsequent chapters of the book will thus undertake a comprehensive investigation of the properties and the usefulness of the Intensity-Curvature Functional as well as its relevance in determining the SRE and the novel re-sampling locations where the interpolation error is reduced.

SUMMARY The main concept realized through the intuition, which is that of the curvature of the interpolation function is recalled in this chapter along with the knowledge provided by the previous chapters and to this concept is assigned the role of the notion gained through the investigation driven through deduction which is presented in part I of the book. The curvature of the model interpolation function becomes thus the key to the model approach presented in this book at the aim to generate a unifying theory for the improvement of the interpolation error. Within this framework, with similar formulations and exactly the same concepts, it is possible to address the audience with the clear message that there exists a unifying approach to the improvement of the interpolation error, and such approach shall be used regardless to the degree or dimensionality of the model interpolation function. The reader shall see that the curvature as identified by the second order derivative of the interpolation function is used at the aim to derive each of the novel interpolation paradigms presented in this book. To these interpolation paradigms it has been assigned the name: SRE-based interpolation functions.

Reference Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53.

56

Section II

Sub-Pixel Efficacy Region of the Bivariate Linear Interpolation Function

58

Chapter VII

The Theoretical Approach to the Improvement of the Interpolation Error:

Bivariate Linear Interpolation Function

GOALS AND ORGANIZATION OF THE FORTHCOMING TEXt In the sections of this chapter the reader will be introduced to the sequence of mathematical processes which, starting from a model interpolation function yield to the corresponding SRE-based paradigm. Particularly, this chapter addresses the development of the SRE-based bivariate interpolation function. The mathematical procedure is consistently iterated in the rest of the book for all the other model functions that the unifying theory embraces. The first step of the procedure is that of the calculation of the intensity-curvature terms and through their ratio the Intensity-Curvature Functional is calculated for the model function. The second step is that of calculating the first order partial derivatives of the Intensity-Curvature Functional. Thirdly, the polynomial consisting of the first order partial derivatives is solved to obtain the Sub-pixel Efficacy Region. At this point, the formula of the unifying theory (equation [21]) sets the stage to obtain the novel re-sampling locations. Worth noting that this formula can be adapted to cover cases of one, two, and three dimensional interpolation functions and it is also consistently employed for linear quadratic cubic and trigonometric (Sinc) models. This shall be manifest throughout the remainder of the book. The remainder of this chapter discusses on the nature of the SRE, also makes a connection with Chapter XX of the book within the context of the relationship existing between resolution and interpolation error, and in the last section, the concept of resilient interpolation is introduced and the relevant math is illustrated for the case of the bivariate linear function. Due to mention the following introductory observations that are relevant to the specific case of the bivariate linear function treated in the present chapter. Even though the linearity of the function is manifest when one variable is kept constant, and thus the other variates linearly, the term bivariate linear is used instead of the more common term bi-linear. This simply fits with the mathematical notion for which the interpolation function is not linear in its nature, being a hyperbolic paraboloid (Amidror, 2002).

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The Theoretical Approach to the Improvement of the Interpolation Error

Efficacy is introduced in this chapter as the capability of the interpolation function to generate minimal approximation. This corresponds to keeping the pixel intensities of the re-sampled image as close as possible to the original. The issue of the optimal design of interpolation functions was recently addressed in the research literature concerned with non-parametric root mean square error (RMSE) based design of linear interpolation curves (Wang, 2001). Also, quantitative measures like RMSE can reveal accuracy of interpolation functions if the area under the interpolation curve is calculated as a sum of RMSE over the sampling intervals, rather than a sum of areas under the curve, each corresponding to a sampling interval (Wang, 2001). This evidence is definitely useful to the purpose of validating the SRE-based interpolation functions as it shall be seen in the remainder of the book.

Scope The basic aim of the theory is that of determining the mathematical characterization of the relationship existing between misplacement and pixel intensity at the re-sampling location and those neighboring pixel intensities. This is done in the Section II of the book for the bivariate linear interpolation function. This characterization should be dependent on the local properties of the interpolation function and independent from any constant shifts and also dependent on the local properties of the discrete signal. In this book the calculations establish a relationship between misplacement and pixel intensity and such relationship is used to derive a methodology to minimize the interpolation error. The curvature of the bivariate linear interpolation function as expressed by its second order derivatives is incorporated into an intensity-curvature term, named Eo when calculated at the grid point, and named EIN when calculated at the generic intra-pixel location. Either of the two intensity-curvature terms is defined as the integral of the interpolation function times the sum of the non-null second order derivatives (∂2 (h(x, y))/∂x∂y and ∂2(h (x, y))/∂y∂x). The intensity-curvature term becomes a measure of the particular curvature behavior of the interpolation function across the pixel. Given a function value (i.e. a signal intensity value), the function curvature, which is the local convexity or concavity, is a mean of differentiation between shapes and thus behaviors (Ciulla & Deek, 2005). The inclusion of the second order derivatives into the intensity-curvature term incorporates the curvature and therefore increases the information content of the behavior of the function in relationship to its neighbors, versus considering the intensity value alone. The intensity-curvature term is therefore computed at the grid point (0, 0) and at the generic intra-pixel location (x, y). Through the study of the ratio between the two terms, which that is by studying the extremes of Eo / EIN, it is possible to derive a mathematical characterization of locations where approximation properties and thus error of the interpolation function are either at their maximum or at their minimum. Such locations are designated: the Sub-pixel Efficacy Region (SRE) of the interpolation function. These are used to calculate the novel re-sampling locations where the signal is calculated to obtain approximation improvement.

59

The Theoretical Approach to the Improvement of the Interpolation Error

Deduction of the Intensity-Curvature Functional Definition of h(x y) Let h be a continuous function that takes the form: h(x, y) = f(0,0) + x (f(1,0) - f(0,0)) + y (f(0,1) - f(0,0)) + xy (f(1,1) + f(0,0) - f(0,1) - f(1,0)) (1) Where: f(0,0), f(1,0), f(0,1) and f(1,1) are the values of intensity at the four corners of the pixel Ψ. Such h(x, y) is the bivariate linear interpolation function. Let us posit: θx = [f(1,0) - f(0,0)] θy = [f(0,1) - f(0,0)] ωf = [f(1,1) + f(0,0) - f(0,1) - f(1,0)] = - θxy It follows that: h(x, y) = f(0,0) + x θx + y θy + xy ωf



(2)

∂ (h (x, y))/∂x = θx + y ωf (3) ∂ (h (x, y))/∂y = θy + x ωf







(4)

(∂2 (h(x, y))/∂x∂y ) = (∂2 (h(x, y))/∂y∂x ) = - θxy = ωf

(5)

(∂2 (h(x, y))/∂x∂y ) (0, 0) = (∂2 (h(x, y))/∂y∂x ) (0, 0) = - θxy

(6)



Primitives of h(x y) On the basis of (2), it follows that the primitives of h(x, y) with respect to y, that is Hy, and with respect to x, that is Hx, and with respect to y and x, that is Hyx = Hxy are: Hy (x, y) = f(0,0) y + x y θx + y2/2 θy + xy2/2 ωf

(7)

Hx (x, y) = f(0,0) x + x2/2 θx + y x θy + yx2/2 ωf



(8)



(9)





Hyx (x, y) = Hxy (x, y) = f(0,0) xy + yx2/2 θx + xy2/2 θy + y2x2/4 ωf Also that: ∂ (Hyx (x, y) )/∂x = Hy (x, y), and ∂ (Hyx (x, y)) /∂y = Hx (x, y).

60

The Theoretical Approach to the Improvement of the Interpolation Error

Calculation of the Intensity-Curvature Terms For a given pixel Ψ, let the intensity-curvature term at the generic intra-pixel location (x, y) be defined as: EIN = EIN (x, y) = EIN (Ψxy) + EIN (Ψyx) = x

y

∫ ∫ h (x, y) ( ∂2 (h(x, y))/∂x∂y + ∂2 (h(x, y))/∂y∂x ) dx dy = 0 0 x y ∫ ∫ h (x, y) 2 ωf dx dy = 2 ωf Hxy (x, y)



(10)

0 0

Let the intensity-curvature term at the grid point (x, y) = (0, 0) be defined as: Eo = Eo (x, y) = Eo (Ψxy) + Eo (Ψyx) = x

y

∫ ∫ f (0, 0) ( ∂2 (h(x, y))/∂x∂y + ∂2 (h(x, y))/∂y∂x ) (0, 0) dx dy =

x

y

0



0

∫ ∫ f (0, 0) 2 ωf dx dy = f (0, 0) 2 x y ωf 0

(11)

0

Definition of Intensity-Curvature Functional The Intensity-Curvature Functional is defined as the ratio between intensity-curvature terms at the grid point (0, 0) (Eo) and at the generic intra-pixel location (x, y) (EIN). Thus: ΔE (x, y) = Eo / EIN. ΔE represents the pixel intensity-curvature change determined by the interpolation function. That the novel pixel intensity resulting from re-sampling is dependent on the curvature is known because of the studies conducted earlier (Ciulla & Deek, 2005). The curvature of the interpolation function is measured in terms of second derivatives: ∂2(h (x, y))/∂x2; ∂2(h (x, y))/∂x∂y; ∂2(h (x, y))/∂y2; ∂2(h (x, y))/∂y∂x. Thus, the formulation of Eo and EIN needs to cover the cases in which the two pixels differ either in their second derivative, or their intensity, or in both. Derivatives of the interpolation function are guaranteed to be continuous within the pixel (Brock & Wiseman, 2001). In the remainder of the text the notation: (∂2(h (x, y))/∂x2) (x0) is the second order derivative of h with respect to the variable x calculated at the location (x0). Hxy (x, y) is the primitive function of h(x, y) with respect to y and x. ωf is defined as the value that expresses relationships between f(0, 0) (intensity at the pixel to re-sample) and its neighboring pixel intensity values f(1,0), f(0,1) and f(1,1). Let us posit:

61

The Theoretical Approach to the Improvement of the Interpolation Error

θx = [f(1,0) - f(0,0)] = [θx’ - f(0,0)] θy = [f(0,1) - f(0,0)] = [θy’ - f(0,0)] ωf = [f(1,1) + f(0,0) - f(0,1) - f(1,0)] = [ωf ’ + f(0,0)] [ f(1,1) + f(0,0) - θx’ - θy’ ] = ωf The formulation of the Intensity-Curvature Functional is therefore: ΔE (x, y) = Eo / EIN = - f (0, 0) x y θxy / Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]

(12)

Calculation of the First Order Partial Derivatives of the Intensity-Curvature Functional ∂ (ΔE (x, y))/∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 y2 { [θx’ - f(0,0)] + [ωf ’ + f(0,0)] y/2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2

(13)



∂ (ΔE (x, y))/∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx‘ - θy‘ ] x2/2 y2 { [θy‘ - f(0,0)] + [ωf ‘ + f(0,0)] x/2 }/ {Hxy (x, y) [ f(1,1) + f(0,0) - θx‘ - θy‘ ]}2

(14)



The math process that arrives to equations (13) and (14) is given in appendix IV.

Sub-Pix el Efficacy Region The Sub-pixel Efficacy Region exists within a pixel due to the approximate nature of the interpolation function. Geometrically, the Sub-pixel Efficacy Region can be envisioned as that set of intra-pixel points, where the local curvature of the interpolation function allows by projection the determination of novel re-sampling locations where the estimated signal is closest to the true value to be determined. Detrimental effects of the approximation are thus minimized. Let h = h (x, y) be the bivariate linear interpolation function as defined in equation (1) (Castleman, 1996). The function h is continuous within the pixel Ψ and let φ be the Sub-pixel Efficacy Region. For each pixel, the goal is to find relationships between misplacement and intensity (at the pixel to re-sample and neighbors) that are dependent from those local properties of the interpolation function and the discrete signal. Such relationships are derived from the study of the Intensity-Curvature Functional ΔE through the calculation of the first order derivatives of ΔE and their equation to zero. This is equivalent to finding the extremes of ΔE. Also, because ΔE is computed locally (i.e. for each pixel), the mentioned relationships are independent from any constant shift. It is assumed that: (i) there is one or more extremes of ΔE, and (ii) they constitute the Sub-pixel Efficacy Region of φ. Extreme points of ΔE are calculated and relationships between neighboring pixel intensities are found. The methodology is: (i) the Intensity-Curvature Functional (ΔE) is calculated,

62

The Theoretical Approach to the Improvement of the Interpolation Error

(ii) by solving the polynomial system of its first order derivatives, extreme points that locate the Subpixel Efficacy Region are found, (iii) relationships between values of intensity at neighboring pixels and misplacement are derived. These relationships are used to compute the novel re-sampling locations where the interpolation function is calculated.

Analy sis of the Intensity-Curvature Functional Definition I For the bivariate liner interpolation function, the Sub-pixel Efficacy Region is Φ = {(x, y): ∂ (ΔE(x, y))/∂x = 0 and ∂ (ΔE(x, y)) /∂y = 0}.

Approach The Intensity-Curvature Functional (ΔE) expresses a measure of the approximation produced through interpolation. Analyzing the behavior of ΔE within the pixel (i.e. increasing or decreasing), its extreme points (i.e. maximum or minimum), and its concavity (or convexity) change versus the two spatial coordinates, requires calculation of first and second order derivatives. An extreme point of ΔE can either be a maximum or a minimum, ultimately depending on the magnitude of misplacement and the nature of the relationships between neighboring pixel intensity values. From the previous paragraph the calculation of the first order derivatives of ΔE furnishes: ∂ (ΔE(x, y))/∂x = αx (x, y) / β(x, y) ∂ (ΔE(x, y))/∂y = αy (x, y) / β(x, y) αx(x, y) = -f(0,0) x2/2 y2 ωf [(f(1,0) - f(0,0)) + ωf y/2] αy(x, y) = -f (0,0) x2/2 y2 ωf [(f(0,1) - f(0,0)) + ωf x/2] β(x, y) = { Hxy (x, y) ωf }2

(15) (16) (17) (18) (19)

Posing equations (15) and (16) to zero and solving the resulting polynomial system through use of a Groebner basis (Ajwa et al., 1995; Becker & Weispfenning, 1993), four extremes are found: P1 ≡ [0, 0] P2 ≡ [0, -2 (f(1,0) - f(0,0)) / ωf ] P3 ≡ [-2 (f(0,1) - f(0,0)) / ωf, 0] P4 ≡ [-2 (f(0,1) - f(0,0)) / ωf , -2 (f(1,0) - f(0,0)) / ωf ] Points P1 through P4 determine the Sub-pixel Efficacy Region φ. Except when explicitly recalled the following vertex will be used for the experimentations: xsre = -2 (f(0,1) - f(0,0)) / ωf

(20.a)

ysre = -2 (f(1,0) - f(0,0)) / ωf

(20.b)

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The Theoretical Approach to the Improvement of the Interpolation Error

The deduction of the novel re-sampling location based on the boundary of the sub-pixel efficacy region was formerly reported (Ciulla & Deek, 2005) and it is herein recalled. The product of the interpolation function times its second order derivative is calculated at the locations (xsre - xr0, ysre - yr0) and (xsre, ysre). The ratio between the two products is equated to the actual ΔE* = EIN (xsre - x0, ysre - y0) / EIN (xsre, ysre) to obtain: { h(xsre- xr0, ysre- yr0) (∂2 (h (x,y))/∂x∂y) (xsre- xr0, ysre- yr0) }/ { h(xsre, ysre) (∂2 (h (x,y))/∂x∂y) (xsre, ysre) }= EIN (xsre-x0, ysre-y0) / EIN (xsre, ysre)



(21)

The novel re-sampling location is: xr0 = (ηxy - μx) / λx yr0 = (ηxy - μy ) / λy





(22)



(23)

Where: ηxy = [(f(0,0) + xsre θx + ysre θy + xsre ysre ωf ) ΔE* - f(0,0)]



(24)

μx = [(ysre - y0) θy + xsre (θx + (ysre - y0 )ωf )]

(25)

μy = [(xsre - x0) θx + ysre (θy + (xsre - x0 )ωf )]

(26)

λx = [(ysre - y0) ωf - θx]



(27)



(28)

λy = [(xsre - x0) ωf - θy]



What has to be acknowledged, as the MATLAB®. (c) 1984-2007 The MathWorks, Inc. code SRE2D2007 located at www.sourcecodewebsite.com shows is that the novel re-sampling location (x r0, yr0) is normalized to the range [(0, 0), (x0, y0)] where (x0, y0) is the misplacement.

CALCULATION OF THE SECOND ORDER PARTIAL DERIVATIVES OF THE INTENSITY-CURVATURE FUNCTIONAL ∂2 (ΔE (x, y))/∂x2 = f (0, 0) θxy [ f(1,1) + f(0,0) - θx‘ - θy‘ ] x y2 { [θx’ - f(0,0)] + [ωf ’ + f(0,0)] y/2 } * Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 { Hxy (x, y) - x Hy (x, y) } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 (29)

64



The Theoretical Approach to the Improvement of the Interpolation Error

∂2 (ΔE (x, y))/∂y2 = f (0, 0) θxy [ f(1,1) + f(0,0) - θx‘ - θy‘ ] x2 y { [θy‘ - f(0,0)] + [ωf ‘ + f(0,0)] x/2 } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 } * { Hxy (x, y) - y Hx (x, y) } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 (30)



∂2 (ΔE (x, y))/∂x∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx‘ - θy‘ ] x2/2 * { {2y [θx’ - f(0,0)] + 3/2 y2 [ωf ’ + f(0,0)]} * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - y { [θx’ – f(0,0)] 2y + [ωf ’ + f(0,0)] y2 } * Hx (x, y) Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 = ∂2 (ΔE (x, y))/∂x∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx‘ - θy‘ ] [ f(1,1) + f(0,0) - θx‘ - θy‘ ]2 Hxy (x, y) x2/2 * y { {2 [θx‘ - f(0,0)] + 3/2 y [ωf ’ + f(0,0)]} * Hxy (x, y) - { [θx’ - f(0,0)] 2y + [ωf ’ + f(0,0)] y2 } * Hx (x, y) } } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4



(31)

∂2 (ΔE (x, y))/∂y∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx‘ - θy‘ ] y2/2 { { 2x [θy‘ - f(0,0)] + 3/2 x2 [ωf ‘ + f(0,0)] } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - x { 2x [θy’ - f(0,0)] + [ωf ’ + f(0,0)] x2 } * Hy (x, y) Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 = ∂2 (ΔE (x, y))/∂y∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 y2/2 * x { { 2 [θy‘ - f(0,0)] + 3/2 x [ωf ‘ + f(0,0)] } * Hxy (x, y) - { 2x [θy‘ - f(0,0)] + [ωf ‘ + f(0,0)] x2 } * Hy (x, y) } }/ {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 (32)



The math process that arrives to equations (29), (30), (31) and (32) is given in appendix IV along with the assessment that the Jacobian of second order derivatives is not symmetric.

The Role of the Curvature in the Sub-pix el Efficacy Region The Sub-Pixel Efficacy Region is that set of points where the Intensity-Curvature Functional admits its extremes. In part I of the book we have seen that the Intensity-Cuvature Functional is a measure of the energy level change caused by the model interpolation function. The curvature has been seen as the key element of the notion of Chapter VI and also as the degree of match between: (i) the early definition of Sub-pixel Efficacy Region given in Chapter II and (ii) the definition derived subsequently to the study of the Intensity-Curvature Functional. To understand the role of the curvature in the essence of

65

The Theoretical Approach to the Improvement of the Interpolation Error

the Sub-pixel Efficacy Region let us undertake the following conceptualization. The task of the model interpolation function is that of estimating the signal on the basis of a sequel of discrete samples at locations where the signal is unknown. It is therefore an estimation problem that wants to determine a signal value (e.g. image pixel intensity). The degree of uncertainty in this estimation is two-folded. One issue is the signal intensity to estimate, the other issue is introduced through the interpolation function as seen as instrument for the estimation. The reason why the interpolation function introduces uncertainty is inherent to the local curvature of the model, which is itself strictly related and even descending from the values of the signal at the neighborhood of interest, but still related to the fact that the true signal is unknown. Therefore, the uncertainty remains in the signal value that is to be estimated and it is reflected into the local curvature, which can either be concave, convex or null. The task of the estimation would then be most accurate, it thus would lead to zero interpolation error, if the local curvature of the model interpolator would be exactly the same as that of the signal at the locations where the signal ought to be estimated. The task of the interpolation function is thus certainly ill posed and this is because the signal is not sampled at certain locations and thus remains unknown. Therefore, the local curvature of the signal is not known either. The Sub-Pixel Efficacy Region (SRE), as seen as that set of points where the Intensity-Curvature Functional admits its extremes, can be conceptualized as the link that wants to bridge the uncertainty in the true value of the signal with the uncertainty in the curvature of the signal. Mathematically, the SRE spatial points are the intra-voxel locations where there is maximal or minimal variation of intensity-curvature content between the two scenarios: (i) to interpolate and (ii) not to interpolate. The question to answer is then at this point: How the SRE does relate to the estimation of the unknown signals? It does relate because of its inheritance from the Intensity-Curvature Functional. The Intensity-Curvature functional (ΔE) is seen as the measure of energy change and thus is seen as an indirect measure of the interpolation error. Yet thus, it was not conceived to estimate the signal, but as it was seen in this chapter and it shall be seen in the next chapters for other interpolation functions other than the bivariate linear, it is employed to determine the novel re-sampling locations where the interpolation error is reduced. ΔE embeds the two variables (signal intensity and curvature) into one measure. The SRE descends from ΔE and it does relate to the estimation of the signal because it incorporates the two above mentioned variables, which are the determinants of the two-folded degree of uncertainty in the interpolation task. The role of the curvature in the determination of the SRE is that of the additive variable to the signal intensity. And this role is exerted within the context of the joint information content provided by signal and second order derivative of the interpolation function.

A Conceptual Pathway to Attempt Interpolation as Independent from the Resolution As it shall be investigated and widely presented in Chapter XX the novel theory and methodology to determine interpolation error improvement presented in this book is dependent on the resolution (sampling step) and this is a further confirmation of the undeniable evidence presented in a famous work (Strang & Fix, 1971). It is due to mention however the attempt that the theory makes to detach the concept of resolution from interpolation. This attempt springs from the nature of the mathematical formulation which is at the root foundations of the theory. The mathematical formulation employs the

66

The Theoretical Approach to the Improvement of the Interpolation Error

Intensity-Curvature Functional and the derived Sub-Pixel Efficacy Region to arrive through deduction to the novel re-sampling locations where the approximation properties of the model interpolation function are improved. The SRE-based interpolation function is thus the classic interpolation function re-calculated at the novel re-sampling location. This was seen in this chapter and it will be widely extended and presented through the study of several other interpolation functions other than the bivariate linear. Why then would be the SRE-based interpolation independent from the resolution? To answer this question let us see what is the implication of finding novel re-sampling loctions within the pixel. The implication is that at the novel re-sampling location the local curvature of the SRE-based interpolation function, or more generally the interpolation function, has the capability to adapt to the true and unknown local curvature of the signal to estimate. Does it happen also with varying resolutions? The answer is yes, and this shall be seen in Chapter XX where the improvement in interpolation error produced by the SRE-paradigms will reveal that the behavior of error improvement happens regardless to the step size, however, relatively to the three step sizes that shall be analized (for a better comprehension of this paragraph the reader is suggested to read provisionally paragragh: (“On the Estimation of Unknown Signals” in Chapter XX). So the next question to answer is: Why the improvement happens regardless to the sampling step? Conceptually this might be explained stating that the true local curvature is more likely to be identified and also the local unknown intensity value of the signal to estimate is more likely to be estimated when the novel re-sampling locations do vary pixel-by-pixel (e.g. node-by-node for a one-dimensional signal). The process which is the determinant of the adaptive behavior of the curvature of SRE-based interpolation is not probabilistic though. It is instead deterministic. It is a process that is driven by the geometrical projection of the sampling point into the novel resampling location (Ciulla & Deek, 2005, 2006). Finally, as explained in Chapter XX, both Intensity-Curvature Functional and Sub-pixel Efficacy Region are dependent on the original sequel of signal intensity values only for the matter of the intensity-curvature term before interpolation (Eo). The rest of their math formulation is independent from the resolution and thus being the Intensity-Curvature Functional an indirect measure of the interpolation error, the theory, to the opinion of the writer, shall be proposed as attempting to devise interpolation as independent from the resolution.

Assigning the Role of Energy to the Relationships between Neighboring Pix el Intensity Values This paragraph presents a metaphore connecting this section with the text of the preamble of the book. Such metaphore aims to give the meaning that the relationships between neighboring pixel intensity values have within the context of the determination of the novel re-sampling locations and thus within the context of interpolation error improvement. The metaphore conceptualizes the relationships between neighboring pixel intensity values and assigns to them the meaning of energy. The rationale behind this connection rests in the fact that the relationships between neighboring pixel intensity values are the main determinants of the process of locating within the voxel the location where the improved approximation properties of the model interpolation function can be at its best. At this point the reader is referred to the prelude chapter of this book where the philosophical basis underling the unifying theory were described and within this description it was envisioned that the Sub-pixel Efficacy Region is the mean

67

The Theoretical Approach to the Improvement of the Interpolation Error

that can raise the energy level of the interpolation function. As a matter of fact, the energy level of the signal as measured through the Intensity-Curvature Functional can be elevated through re-sampling which is inherent to interpolation. Consistently, the energy level of the model interpolation function, which is the capability to produce the least of the approximation, can be changed when changing the re-sampling location. The SRE is the mean to obtain the novel re-sampling location of the model interpolation function and thus to increase the potential of the interpolation function such to reduce the error and therefore to improve the approximation. It shall then be asked as to what is truly the SRE made of? Mathematically, the SRE is a polynomial equation which coefficients embed the neighboring pixel intensity values. And, what is truly the novel re-sampling location made of? Once again, the writer recalls that the novel re-sampling location is a math expression that embeds the neighboring pixel intensity values. It shall be then envisioned that the relationships between neighboring pixel intensity values are nevertheless comparable to energy quanta which are capable to raise the potentiality of the model interpolation function and to change the energy level of the re-sampled signal.

Bivariate Linear Interpolation Function: Attempting to Obtain Resilient Interpolation From equations (10) and (11) it is herein recalled that the intensity curvature functionals Eo (x, y) and EIN (x, y), before and after interpolation respectively, are: Eo = Eo (x, y) = Eo (Ψxy) + Eo (Ψyx) = x

y

∫ ∫ f (0, 0) ( ∂2 (h(x, y))/∂x∂y + ∂2 (h(x, y))/∂y∂x ) (0, 0) dx dy = 0 x



0

y

∫ ∫ f (0, 0) 2 ωf dx dy = f (0, 0) 2 x y ωf

0

0

EIN = EIN (x, y) = EIN (Ψxy) + EIN (Ψyx) = x

y

∫ ∫ h (x, y) ( ∂2 (h(x, y))/∂x∂y + ∂2 (h(x, y))/∂y∂x ) dx dy = 0 0

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The Theoretical Approach to the Improvement of the Interpolation Error

x

y

∫ ∫ h (x, y) 2 ωf dx dy = 2 ωf Hxy (x, y)





0 0

Where: Hxy (x, y) = f(0, 0) x y + y x2/2 θx + x y2/2 θy + y2 x2/4 ωf θx = [ f(1, 0) - f(0, 0) ] θy = [ f(0, 1) - f(0, 0) ] ωf = [ f(1, 1) + f(0, 0) - f(0, 1) - f(1, 0) ] = [ f(1, 1) - f(0, 1) - f(1, 0) ] + f(0, 0) It can be written that: Hxy (x, y) = f(0, 0) x y + y x2/2 [f(1, 0) - f(0, 0)] + x y2/2 [ f(0, 1) - f(0, 0) ] + y2 x2/4 { [ f(1, 1) – f(0, 1) - f(1, 0) ] + f(0, 0) } = f(0, 0) x y + y x2/2 f(1, 0) - y x2/2 f(0, 0) + x y2/2 f(0, 1) - x y2/2 f(0, 0) + y2 x2/4 [ f(1, 1) - f(0, 1) - f(1, 0) ] + y2 x2/4 f(0, 0) = f(0, 0) [x y - y x2/2 - x y2/2 + y2 x2/4 ] + y x2/2 f(1, 0) + x y2/2 f(0, 1) + y2 x2/4 [ f(1, 1) - f(0, 1) - f(1, 0) ] = f(0, 0) λ1 + λ2

(33)

Where: λ1 = [x y - y x2/2 - x y2/2 + y2 x2/4 ] λ2 = y x2/2 f(1, 0) + x y2/2 f(0, 1) + y2 x2/4 [ f(1, 1) - f(0, 1) - f(1, 0) ] Thus, equating Eo (x, y) to EIN (x, y) yields: Eo (x, y) = f (0, 0) 2 x y ωf = EIN (x, y) = 2 ωf [ f(0, 0) λ1 + λ2 ] From which it follows that: f (0, 0) x y - f(0, 0) λ1 - λ2 = 0



(34)

[ x y - λ1 ] f(0, 0) - λ2 = 0



(35)

f(0, 0) = λ2 / [ x y - λ1 ]



(36)

Equation (36) furnishes the pixel intensity value under the assumption that the interpolator has no effect on the signal. That is like saying that for the interpolation function to have no effect on the signal, the pixel intensity value should be f(0, 0) = λ 2 / [ x y - λ1 ] for given intra-pixel location (x, y). By saying so, it is like assuming that the signal f(0, 0) = λ2 / [ x y - λ1 ] is resilient to interpolation. The meaning of this novel conception herein deducted will be explained in more details in Chapter XII while studying

69

The Theoretical Approach to the Improvement of the Interpolation Error

the trivariate linear interpolation function and will lead to the understanding as to how this section of the book attempts to obtain resilient interpolation.

SUMMARY This chapter had introduced studied and solved the problem of the improvement of the bivariate linear interpolation function through the use of the Sub-pixel Efficacy Region. Two intensity-curvature terms were calculated: (i) at the grid point and (ii) at the generic intra-pixel location (x, y). These two terms determine a ratio called Intensity-Curvature Functional. The starting points of the theory are: (i) the Intensity-Curvature Functional, and (ii) the solution of the polynomial system of its first order partial derivatives. From this solution, a sub-pixel set of points in the space domain is obtained and named: Sub-pixel Efficacy Region (SRE). It is also shown that the Intensity-Curvature Functional is formulated as dependent on the node intensity and the second order derivatives of the interpolation function. At this stage, the book sets the grounds for the mathematical formulation and also explains in detailed manner how re-sampling rather than being performed at the same spatial location for each pixel, can be done at spatial locations that vary on a pixel-by-pixel basis. And these locations are called the novel re-sampling locations. This chapter has also elucidated the role that the curvature of the interpolation function plays in the determination of the SRE. Finally, the chapter undertakes a journey in formulating an attempt to obtain resilient interpolation treating the case of the bivariate linear function. The validation of such formulation is left to the willing audience of this book.

References Amidror, I. (2002). Scattered data interpolation methods for electronic imaging systems: A Survey. Journal of Electronic Imaging, 11(2), 157-176. Ajwa, I. A., Liu, Z., & Wang, P. S. (1995). Groebner bases algorithm (I.C.M. Technical Reports Series). Kent, Ohio: Kent State University. Becker, T., Weispfenning, V., & Kredel, H. (1993). Groebner bases: A computational approach to commutative algebra. London, UK: Springer-Verlag. Brock, J. S., & Wiseman, J. R. (2001). Discrete-expansions for linear interpolation functions. Computer Physics Communications, 142(1-3), 206-213. Castleman, K. R. (1996). Digital image processing. Englewood Cliffs, NJ: Prentice Hall. Ciulla C., & Deek, F. P. (2005). On the approximate nature of the bivariate linear interpolation function: A novel scheme based on intensity-curvature. ICGST - International Journal on Graphics, Vision and Image Processing, 5(7), 9-19. Ciulla, C., & Deek, F. P. (2006). Extension of the sub-pixel efficacy region to the Lagrange interpolation function. ICGST - International Journal on Graphics, Vision and Image Processing, 6(1), 1-12.

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Strang, G., & Fix, G. (1971). A Fourier analysis of the finite element variational method. In, Constructive Aspect of Functional Analysis (pp. 796-830). Rome, Italy: Edizioni Cremonese. Wang, J. (2001). Optimal design for linear interpolation of curves. Statistics in Medicine, 20(16), 24672477.

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Chapter VIII

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

INtroduction This is the first of the chapters of the book that present results obtained processing the MRI database with both classic and SRE-based interpolation paradigms. The focus of this chapter is on the bivariate linear interpolation function. An overview on the validation paradigm is given along with an explanation of the simulations that were conducted in order to validate the SRE-based bivariate liner interpolation function. Subsequently, results with real MRI are shown quantitatively through plots of the metric called RMSE Ratio which was employed to assess which one between classic and SRE-based interpolation furnishes lower root-mean-square-error (RMSE). Qualitative demonstration of the results with MRI is also given. The chapter also discusses and evaluates the case of interpolation error improvement when the novel re-sampling location within the pixel is placed on the x axis of the coordinate system (i.e. y r0 = 0) and consistently, both quantitative and qualitative results are presented through plots of the RSME Ratio and figures respectively.

Validation Paradigm Ciulla and Deek (2005) showed quantification of the advantages of the Sub-pixel Efficacy Region (SRE) and also presented the validation paradigm adopted in order to demonstrate the improvement of the bivariate liner interpolation function. For convenience to the reader the metric (RMSE Ratio) employed to assess as to which one of the two model interpolation functions delivers the smallest error is herein recalled along with the procedure employed for its calculation. The image was misplaced of (x0 y0), then motion correction and interpolation was performed in two different ways: (i) through the classic form of bivariate linear function, and (ii) through the SRE-based bivariate linear function. The latter consists of the classic bivariate interpolation function calculated at the novel re-sampling location. After processing the image with a given misplacement or rotation, the Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 1. 2D images in their original form. (a), cos (x, y), (b) ln (x, y)

ratio between the two resulting RMSE values was calculated with: (i) the classic bivariate interpolation function (RMSEbefore), and (ii) the SRE-based bivariate interpolation function (RMSEafter) respectively. The ratio was then subtracted from the numerical value of one such to obtain the: RMSE Ratio = (1 - RMSEbefore / RMSEafter) A negative value of the RMSE Ratio shows that the SRE-based function outperforms the classic in terms of interpolation error and a positive value shows the opposite. For the reminder of the book the reader is referred to above definition of RMSE Ratio. This metric is consistently employed in these works to assess the relativity of performance of the pair of interpolation functions: classic and SRE-based. Ciulla and Deek (2005) reported empirical confirmation of what was foreseen through the two conceptions that set the ground upon which the theory was developed. This was done through the employment of diverse image data among which T1-weighted MRI. The same validation paradigm is herein adopted to further illustrate the results of the application of the Sub-pixel Efficacy Region (SRE) to the bivariate linear interpolation function. This presentation starts with the results obtained on signals of known nature and continues extensively while employing T1-weighted and T2-weighted MRI along with functional MRI data (T2*).

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 2. Plot of ΔE versus misplacement after re-sampling. (a) cos (x, y), (b) ln (x, y). The white line indicates the location of the Sub-pixel efficacy Region (SRE) intra-pixel point

Simulations Ciulla and Deek (2005) presented results obtained through simulations performed employing the sin (x, y) function along with the plots of the Intensity-Curvature Functional ΔE after re-sampling and also after motion correction and interpolation through the bivariate linear interpolation function in its classic and the SRE-based forms. The results were corroborated with RMSE plots showing the improvement obtained through the SRE. Also presentation of the behavior of the x and y coordinates of the novel re-sampling locations was given for the cos(x, y) function. As shown in Figure 1, this section extends the former study and thus presents two more functions: cos (x, y) and ln (x, y). Figures 2 and 3 show the plots of the Intensity-Curvature Functional for re-sampled images (Figure 2) and for re-sampled and motion corrected images (Figure 3). Figure 4 shows the plot of x (see (a)) and y (see (b)) coordinates resulting from equations (20.a) and (20.b) of Chapter VII respectively which are used in equations (22) and (23) (Chapter VII) to compute the novel re-sampling locations. Equations (20.a) and (20.b) furnishes pure numbers that are not in any way dependent on the pixel resolution. The Sub-Pixel Efficacy Region demonstrates also how the location of the extreme of ΔE varies pixelby-pixel and therefore clarifies that the minimization of the interpolation error can be set independent of any constant shifts. Figure 5 shows the root-mean-square-error curves obtained from images that

74

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 3. Plot of ΔE versus misplacement after re-sampling, motion correction and interpolation. (a) cos (x, y), (b) ln (x, y). The white line indicates the location of the Sub-pixel efficacy Region (SRE) intra-pixel point

Figure 4. Plot of X and Y coordinates of the Sub-pixel Efficacy Region. Extreme of ΔE: point P4 ≡ [-2 (f(0,1) - f(0,0)) / ωf , -2 (f(1,0) - f(0,0)) / ωf]. The figure shows that the extreme of ΔE changes pixel-bypixel. Values were obtained from equations (20.a) and (20.b) (chapter 7). (a) cos (x, y), (b) ln (x, y)

75

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

were motion corrected with bivariate linear interpolation using: (i) the misplacement (black) and (ii) the novel re-sampling location obtained by equations (22) and (23) (white). Results shown here were obtained through equations (20.a) and (20.b) starting from the SRE point: P4 ≡ [-2 (f(0,1) - f(0,0)) / ωf , -2 (f(1,0) - f(0,0)) / ωf ]. Improvement might also be obtained starting from either of the two points: P2 ≡ [0, -2 (f(1,0) - f(0,0)) / ωf ] and P3 ≡ [-2 (f(0,1) - f(0,0)) / ωf, 0]. Since both P2 and P3 are solutions of the polynomial system of first order derivatives of ΔE, they are also extremes of ΔE. To set P2 and P3 as candidate points to obtain improvement, it is employed equation (20.b) while keeping x = 0 and this is P2, and equation (20.a) while keeping y = 0 and this is point P3.

Figure 5. Plot of RMSE versus misplacement after re-sampling, motion correction and interpolation. Through the bivariate linear function (black lines), and through the same but using the values of the novel re-sampling locations x and y obtained from equations (22) and (23) of chapter 7 (white lines). (a) cos (x, y), (b) ln (x, y). (c) For sin (x, y), the improvement obtained with P2 ≡ [0, -2 (f(1,0) - f(0,0)) / ωf], P3 ≡ [-2 (f(0,1) - f(0,0)) / ωf, 0] and P4 ≡ [-2 (f(0,1) - f(0,0)) / ωf , -2 (f(1,0) - f(0,0)) / ωf]. For all of the pictures, the white vertical line indicates the location of the Sub-pixel Efficacy Region (SRE) point

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Figure 6. Plot of the novel re-sampling locations for tan (x, y). Values were obtained from equations (22) and (23) of chapter 7, starting from (x0, y0) = (0.1, 0.1), values of x in (a), and values of y in (b)

Results obtained though equation (20.b) (using point P2) and through equation (20.a) (using point P3) are shown for sin (x, y) in figure 5c. The two lines corresponding to P2 and P3 show same behavior and indicate a larger improvement with respect to the line obtained by means of P4. For tan (x, y) figure 6 shows the behavior of the x and y coordinates of the novel re-sampling locations obtained starting from an misplacement of (0.1, 0.1).

ON THE INTERPrETATION OF THE rESULTS The presentation of the resulting RSME Ratio is consistently furnished for each slice of the MRI volumes’ database. Consequentially, the number of graphs that are displayed is large. Each of the graphs shows a behavior that is not necessarily the same. A natural question the reader may have is: how to interpret these graphs? To infer a general behavior on the basis of real data such as MRI might be attempted however at the risk of making a generalization prone to a large degree of disproof. On the other hand, it is legitimate to seek for a reasonable explanation for the variability seen in the graphs of the RSME Ratio. Before attempting such task let us consider the following reasoning. To form theories on the basis of a set of equations that explain the widest possible set of phenomena is a process that can be attempted in at least two manners. Let us call the set of phenomena: “the system”. Let us call the first manner: “the framework-based”. This process of forming theories has been in place for centuries and scientists of the calibre of Einstein and Newton are indeed in merit to be acknowledged as the most prominent providers of theories that have been developed employing such approach. Let us call the second manner: “quantum framework-based”. The difference with respect to the first manner consists in the fact that the set of equations describing the phenomena embed probabilistic properties that reflect the changing and/or adaptive behavior of the system that the theory wants to explain (Greene, 1999). For instance the probabilistic properties can generally reflect the change of the system across time.

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Let us discuss a little more into the specific while connecting above knowledge with the case study of this book which is that of forming a unifying theory for the improvement of the interpolation error. The preamble of the book has brought to the attention the following propositions. The potentiality capable to produce an approximation consists of the energy of the interpolation paradigm. The potentiality is named intensity-curvature term and the mean that can raise such potentiality is called Sub-pixel Efficacy Region. Subsequently in Chapter III, it was shown that the IntensityCurvature Functional is a measure of the energy of the signal (image). As shown through its math formulation, the Sub-pixel Efficacy region is closely related to coefficients expressing relationships between pixel intensity values that are located within the neighborhood of the pixel to re-sample. Another type of energy is introduced in Chapter VII in section: “Assigning the Role of Energy to the Relationships between Neighboring Pixel Intensity Values.” Accordingly, the relationships between neighboring pixel intensity values were conceptualized as energy quanta capable to raise the potentiality of the model interpolation function. Thus, given a model interpolation function, its energy reflects the capability of raising the goodness of the estimate of the true value of signal. Such energy can be elevated through the Sub-pixel Efficacy Region because of the energy quanta. Generalizing on the definition of the energy quanta, it can be proposed that: the energy quanta of the signal (image) are coefficients or linear/non linear combinations, expressing relationships between signal (image) intensity values. To see examples of the formulation of such energy quanta see for instance the various values of thetas of Tables I and II in Chapter XIV and Table I in Chapter XVIII. And given a signal, its energy can be measured through the Intensity curvature Functional. So the question to answer at this point is: what is the matching feature (intersection) between the energy of the signal (image) and the energy of the interpolation function? Looking at the mathematical formulation of the Intensity-Curvature Functional (ΔE) it is immediate to the writer to recognize that ΔE embeds the energy quanta of the signal which are also the features of the Sub-pixel Efficacy Region. Looking at the formulation of the model interpolation function it is immediate to see that the signal intensity is embedded. Particularly, is visible in the model interpolation functions that the signal intensity appears in coefficients that condense the values of the signal intensity at the neighborhood. For an immediate visualization of these coefficients the reader is referred to the following equations relevant to the model interpolation functions studied in this book. The equations are: (1) in Chapter VII (bivariate linear), (1) in Chapter II (trivariate linear), (3) and (4) in Chapter XIV (B-Splines), (2) and (30) in Chapter XVIII (for Lagrange and Sinc functions respectively). Generalizing on the above mentioned coefficients the writer proposes that they constitute also energy quanta of the signal. Thus, the intersection between the energy of the signal (image) and the energy of the interpolation function can be proposed to be the energy quanta of the signal (image) as per the definition given above. Furthermore, given a model interpolation function, the two determinants of the amount of the energy in the quanta are: (i) the neighborhood size and (ii) the numerical values of the intensity values within the neighborhood. This assertion allows explaining the variability of the graphs of the RSME Ratio that is seen throughout the presentation of the quantitative results for each of the interpolation functions treated in this book. When looking at the graphs of the RSME ratio in Chapter VIII (bivariate linear), Chapter XI (trivariate linear), Chapter XV (quadratic and cubic B-Splines), Chapter XIX (Lagrange and Sinc), is reasonable to

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think that the variability of the shape of behaviors seen: (i) within the same MRI volume (e.g. in figures 7 through 26 in Chapter VIII) and (ii) across model interpolation functions for same MRI volume (e.g. figures 17 though 26 in Chapter XV and figures 1 through 10 in Chapter XIX) is related to the amount of the energy in the quanta of the signal (image). This reasoning provides the writer with the intent to furnish a possible insight as to how the reader can form a reasonable interpretation of the behavior of RMSE Ratio in each figure. Naturally, each reader may have similar or different insight and thus derive a personal interpretation of the graphs. It follows a description of the observations made in each figure and the encouraging acknowledgment of the finding that in figures 96 through 99 (relevant to functional MRI) the improvement of the interpolation error obtained through the SRE-based bivariate interpolation function achieved ranges (for its best values) between 40% and 350% (across the full range of rotations seen in those graphs).

Results with Real Image Data Ciulla and Deek (2005) presented results obtained through the application of the theory and methodology to the following 2D images: Lena, anatomical (T1-weighted MRI) and functional Magnetic Resonance Imaging of the human brain. This section extends on what reported earlier reporting results obtained with T1-weighted MRI, T2- weighted MRI and functional MRI data. No thresholds were applied to the images in order to take into account all pixel intensity values. T1-weighted MRI was segmented (Zhang et al., 2001) prior processing. The terminology “processed with classic bivariate linear interpolation” refers to: (i) shifting the signal, and (ii) motion correcting and interpolating the signal using the location (x0, y0). Whereas the process of shifting of the misplacement (x0, y0), motion correcting and interpolating employing the value of the novel re-sampling location derived for (x0, y0) and which is indicated in equations (22) and (23) of Chapter VII will be referred as: “processed with SRE-based bivariate linear interpolation”. The paradigm adopted to show reduction of the interpolation error was the same as the one adopted for the simulations. Image data processed with classic and SRE-based bivariate linear interpolation employing misplacements at steps of 0.01: (i) along x or y, (ii) along x and y simultaneously, (iii) rotations at steps of 0.01 deg and 0.025 deg, and (iv) employing misplacements along x at steps of 0.0025. Consequently, the misplacements ranged within the following intervals: (i) [(0, 0), (0.99, 0)], [(0, 0), (0, 0.99)], (fractions of the pixel size), (ii) [(0, 0), (0.99, 0.99)] (fractions of the pixel size), (iii) [0 deg, 0.99 deg] [0 deg, 2.475 deg], (iv) [(0, 0), (0.2475, 0)] (fractions of the pixel size).

The Behavior of the RMSE Ratio It is herein defined the measure adopted in order to quantify the difference between the performance of the classic interpolation function versus the SRE-based. This measure is called RMSE Ratio and will be employed in this book to evaluate the performance of each of the SRE-based interpolation paradigms that were studied. In this paragraph the misplacement is either along x or y or bivariate (along x and y coordinates simultaneously). This section presents for each MRI imaging modality: T1-MRI, T2-MRI and functional MRI, the graphs depicting the behavior of the RMSE Ratio within the range of misplacement and rotation

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Figure 7. RMSE Ratio: T1-MRI-230, misplacement along x, slices 67-74. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 8. RMSE Ratio: T1-MRI-230, misplacement along x, slices 75-82. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 9. RMSE Ratio: T1-MRI-230, misplacement along x, slices 83-90. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 10. RMSE Ratio: T1-MRI-230, misplacement along x, slices 91-98. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 11. RMSE Ratio: T1-MRI-230, misplacement along x, slices 99-106. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 12. RMSE Ratio: T1-MRI-230, misplacement along x, slices 107-114. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 13. RMSE Ratio: T1-MRI-230, misplacement along x, slices 115-122. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 14. RMSE Ratio: T1-MRI-230, misplacement along x, slices 123-130. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 15. RMSE Ratio: T1-MRI-230, misplacement along x, slices 131-138. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 16. RMSE Ratio: T1-MRI-230, misplacement along x, slices 139-141. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

Figure 17. RMSE Ratio: T1-MRI-230, misplacement along y, slices 67-74. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 18. RMSE Ratio: T1-MRI-230, misplacement along y, slices 75-82. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 19. RMSE Ratio: T1-MRI-230, misplacement along y, slices 83-90. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 20. RMSE Ratio: T1-MRI-230, misplacement along y, slices 91-98. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 21. RMSE Ratio: T1-MRI-230, misplacement along y, slices 99-106. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 22. RMSE Ratio: T1-MRI-230, misplacement along y, slices 107-114. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 23. RMSE Ratio: T1-MRI-230, misplacement along y, slices 115-122. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 24. RMSE Ratio: T1-MRI-230, misplacement along y, slices 123-130. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

96

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 25. RMSE Ratio: T1-MRI-230, misplacement along y, slices 131-138. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 26. RMSE Ratio: T1-MRI-230, misplacement along y, slices 139-141. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

mentioned above. In the following figures the number placed at the upper left corner of each graph corresponds to the slice number shown in the MRI database of Chapter I. On the ordinate is the RMSE ratio (pure number) and on the abscissa is: (i) the magnitude of the misplacement (pure number, e.g. fraction of the pixel size along the direction of coordinate system: X, Y, or X and Y simultaneously), or (ii) the magnitude of the rotation (deg). In Figures 7, 10, 11, 12, 13, 14, 15 and 16 the interpolation error improvement is rapidly increasing approximately from misplacement x = 0.065 to x = 0.25 then the improvement remains quasi constant till about x = 0.63 and then starts decreasing. In Figures 8 and 9 rapid increase in interpolation error improvement is seen approximately from misplacement x = 0.065 to x = 0.38 then the improvement remains quasi constant or starts decreasing approximately from x = 0.63. Generally, the best interpolation error improvement is seen in Figures 7 through 16 in between x = 0.25 and x = 0.38. In Figure 9 the highest percentage in interpolation error improvement is approximately: 40% (graph 83), 50% (graph 84), 60% (graphs 85, 86, 87, 88, 89 and 90). In Figure 10 the highest percentage in interpolation error improvement is approximately: 60% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). In Figure 11 the highest percentage in interpolation error improvement is approximately: 70% (graphs 99, 101 and 106), 60% (graphs 100, 102, 103, 104 and 105). In Figure 12 the highest percentage in interpolation error improvement is approximately: 60% (graphs 107, 108, 112 and 114), 70% (graphs 109, 110 and 111), and 80% (graph 113). In Figure 13 the highest percentage in interpolation error improvement is approximately: 60% (graphs 115, 119, 120 and 121), 70% (graphs 116, 117, 118, 119, 120 and 122).

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 27. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 67-74. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

In Figure 14 the highest percentage in interpolation error improvement is approximately: 70% (graphs 123, 124, 125 and 127), 90% (graph 126), 80% (graphs 128, 129 and 130). In Figure 15 the highest percentage in interpolation error improvement is approximately: 90% (graphs 131, 132, 135 and 137), 70% (graphs 133, 134, 136 and 138). In Figure 16 the highest percentage in interpolation error improvement is approximately: 70% (graph 139), 90% (graphs 140 and 141). In Figures 17, 19 and 20 the interpolation error improvement is rapidly increasing approximately from misplacement y = 0.065 to y = 0.25 then the improvement starts decreasing. In Figure 17 is noticeable the bump between approximately y = 0.38 and y = 0.50. In Figures 18, 23, 24 and 26 rapid increase in interpolation error improvement is seen approximately from misplacement y = 0.065 to y = 0.25 then the improvement remains quasi constant or starts decreasing approximately from y = 0.75. In Figure18 the highest percentage in interpolation error improvement is approximately: 60% (graphs 75, 77, 78, 79 and 82), 70% (graphs 76 and 80). In Figure 19 the highest percentage in interpolation error improvement is approximately: 60% (graphs 83, 84, 85 and 87), 70% (graphs 86, 88, 89 and 90). In Figure 20 the highest percentage in interpolation error improvement is approximately: 60% (graphs 91, 92, 94 and 96), 70% (graphs 93, 97 and 98). In Figures 21, 22 and 25 rapid increase in interpolation error improvement is seen approximately from misplacement y = 0.065 to y = 0.25 then the improvement starts decreasing approximately from y = 0.5. Generally, the best interpolation error improvement is seen in Figures 17 through 26 in between y = 0.25 and y = 0.63. In Figure 22 the highest percentage in interpolation error improvement is approximately: 60% (graphs 107 and 110), 70% (graphs 108, 109, 111, 112, 113 and 114). In Figure 23 the highest percentage in interpolation error improvement is approximately: 60% (graph 119), 70% (graphs 115, 116, 117, 120, 121 and 122), 80% (graphs 118 and 120). In Figure 24 the highest percentage in interpolation error improvement is approximately: 70% (graphs 123, 124, 125, 126, 127, 128, 129 and 130). In Figure 25 the highest percentage in interpolation error improvement is approximately: 80% (graphs 131, 132, 134, 135, 136, 137 and 138), 90% (graph 133). In Figure 26 the highest percentage in interpolation error improvement is approximately: 90% (graphs 139 and 141), 80% (graph 140). In Figure 27 the increase in interpolation error improvement is manifest approximately from misplacement (x, y) = (0, 0) or (x, y) = (0.13, 0.13) then the improvement remains quasi constant or starts decreasing approximately from x = 0.38. In Figure 28 the increase in interpolation error improvement is manifest approximately from misplacement (x, y) = (0.065, 0.065) then the improvement remains quasi constant or starts decreasing approximately from (x, y) = (0.38, 0.38). In Figure 29 the increase in interpolation error improvement is manifest approximately from misplacement (x, y) = (0.065, 0.065) then the improvement remains quasi constant or starts decreasing approximately from (x, y) = (0.13, 0.13). In Figure 30 rapid increase in interpolation error improvement is manifest approximately from misplacement (x, y) = (0.065, 0.065) to (x, y) = (0.13, 0.13) then the improvement starts decreasing.

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 28. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 75-82. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

101

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 29. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 83-90. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

102

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 30. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 91-98. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 31. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 99-106. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

104

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 32. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 107-114. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 33. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 115-122. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 34. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 123-130. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

107

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 35. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 131-138. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

108

The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 36. RMSE Ratio: T1-MRI-230, misplacements along x and y, slices 139-141. The range of misplacements is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

In Figure 31 rapid increase in interpolation error improvement is manifest approximately from misplacement (x, y) = (0, 0) to (x, y) = (0.25, 0.25) then the improvement starts decreasing or remains quasi constant. In Figures 32, 33 and 34 rapid increase in interpolation error improvement is manifest approximately from misplacement (x, y) = (0, 0) to (x, y) = (0.13, 0.13) then the improvement starts decreasing or remains quasi constant. In Figure 33 the highest percentage in interpolation error improvement is approximately: 250% (graphs 115, 116, 117 and 118), 300% (graphs 119, 120, 121 and 122). In Figure 34 the highest percentage in interpolation error improvement is approximately: 100% (graph 123), 300% (graph 124), 350% (graph 125), 700% (graph 126), 500% (graph 127), 700% (graph 128), 300% (graph 129), 700% (graph 130). In Figures 35 and 36 rapid increase in interpolation error improvement is manifest approximately from misplacement (x, y) = (0, 0) to (x, y) = (0.13, 0.13) then the improvement remains quasi constant around zero or is scattered. In Figure 36 the highest percentage in interpolation error improvement is approximately: 1300% (graph 139), 800% (graph 140), 600% (graph 141). In Figure 37 the interpolation error improvement is generally manifest throughout the full range of misplacement and shows a well defined linear decrease for increasing misplacement. In graph 1 of Figure 37 is noticeable the persistent improvement increase (starting approximately from x = 0.03). In Figure 38 the interpolation error improvement is generally manifest throughout the full range of misplacement and shows a well defined linear decrease for increasing misplacement. Shown in the

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 37. RMSE Ratio: T2-MRI, misplacement along x (small), slices 1-8. The range of misplacement is [(0, 0), (0.2475, 0) at steps of 0.0025 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 38. RMSE Ratio: T2-MRI, misplacement along x (small), slices 9-16. The range of misplacement is [(0, 0), (0.2475, 0)] at steps of 0.0025 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 39. RMSE Ratio: T2-MRI, misplacement along x (small), slices 17-19. The range of misplacement is [(0, 0), (0.2475, 0)] at steps of 0.0025 along the x axis

figure is the persistent improvement increase in graph 10 (starting approximately from x = 0.12), in graph 11 (starting approximately from x = 0.09) and in graph 16 (starting approximately from x = 0.07). In graph 13 the improvement starts approximately from the misplacement x = 0.06 rapidly increases to approximately x = 0.1 then starts decreasing. In Figure 39 the interpolation error improvement is generally manifest throughout the full range of misplacement and shows a well defined linear decrease for increasing misplacement. In graph 19 is noticeable the persistent improvement increase (starting approximately from x = 0.12). In Figure 40 the interpolation error improvement is generally manifest throughout the full range of misplacement and shows a non linear decrease for increasing misplacement. In graph 1 is noticeable in the rapid increase starting approximately at x = 0.05 to approximately x = 0.23 then the improvement is quasi constant. Similar behavior is seen in Figure 41. With the exception of graphs 10, 11 and 16 where is noticeable an increase in the interpolation error improvement starting approximately at x = 0.13. In graph 13 there is a rapid increase in interpolation error improvement approximately from x = 0.05 to x = 0.1 then a rapid decrease to x = 0.25 followed by a rapid convergence to zero improvement. Similar behavior is seen in Figure 42. With the exception of graph 19 where is noticeable an increase in the interpolation error improvement starting approximately at x = 0.13 to x = 0.25 and followed by persistent decrease. In Figure 43 graphs 2, 7 and 8 is seen a persistent interpolation error improvement across the full range of misplacement with a general tendency to a non linear decrease for increasing misplacement. In graphs 1, 3, 4 and 6 is seen a rapid interpolation error improvement starting approximately from y = 0.13 to approximately y = 0.38 then the improvement starts decreasing almost linearly. In graph 5 the

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 40. RMSE Ratio: T2-MRI, misplacement along x (large), slices 1-8. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 41. RMSE Ratio: T2-MRI, misplacement along x (large), slices 9-16. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 42. RMSE Ratio: T2-MRI, misplacement along x (large), slices 17-19. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

Figure 43. RMSE Ratio: T2-MRI, misplacement along y (large), slices 1-8. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 44. RMSE Ratio: T2-MRI, misplacement along y (large), slices 9-16. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 45. RMSE Ratio: T2-MRI, misplacement along y (large), slices 17-19. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

rapid error improvement is seen starting approximately from y = 0.25 to approximately y = 0.5 then remains quasi constant. In Figure 44 graphs 9, 12, 13, 14, 15 and 16 is seen a persistent interpolation error improvement across the full range of misplacement with a general tendency to a non linear decrease for increasing misplacement (with diverse behaviors). In graph 10 is seen a rapid interpolation error improvement starting approximately at y = 0.065 to y = 0.38 then the error improvement starts decreasing quasi linearly. In graph 11 is seen a rapid interpolation error improvement starting approximately at y = 0.13 to y = 0.50 then the error improvement starts decreasing quasi linearly. In Figure 45 graphs 17, 18 and 19 is seen a rapid interpolation error improvement starting approximately at y = 0.13 to y = 0.50 then the error improvement remains quasi constant (graphs 17 and 18) or decreases quasi linearly (graph 19). In Figure 46 graph 1 is seen a rapid interpolation error improvement starting approximately at (x, y) = (0.065, 0.065) to (x, y) = (0.38, 0.38) then the error improvement starts decreasing non linearly. In graphs 2 through 8 is seen a persistent interpolation error improvement across the full range of misplacement with a general tendency to a non linear decrease for increasing misplacement (with diverse behaviors). In Figure 47 graphs 9, 10, 12, 13, 14, 15 and 16 is seen a persistent interpolation error improvement across the full range of misplacement with a general tendency to a non linear decrease (with diverse behaviors) for increasing misplacement. In graph 11 is seen a persistent interpolation error improvement starting approximately at (x, y) = (0.065, 0.065). In Figure 48 graphs 17 and 18 is seen a persistent interpolation error improvement across the full range of misplacement with a general tendency to a non linear decrease (with similar behaviors). In graph 19 is seen a rapid interpolation error improvement starting approximately at (x, y) = (0.065, 0.065) to (x, y) = (0.3, 0.3) then the error improvement starts decreasing non linearly.

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Figure 46. RMSE Ratio: T2-MRI, misplacement along x (large) and y (large), slices 1-8. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 47. RMSE Ratio: T2-MRI, misplacement along x (large) and y (large), slices 9-16. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 48. RMSE Ratio: T2-MRI, misplacement along x (large) and y (large), slices 17-19. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

In Figure 49 graphs 1, 2, 4, 5, 7 and 8 is seen a rapid interpolation error improvement starting approximately at x = 0.13 to approximately x = 0.38 then it is manifest a general tendency to a non linear decrease (with diverse behaviors). In graphs 3 and 6 the rapid interpolation error improvement starts approximately at x = 0.2 to approximately x = 0.3 then the error improvement is quasi constant (graph 3) or starts decreasing non linearly (graph 6). In Figures 50 and 51 is seen a rapid interpolation error improvement starting approximately from x = 0.13, x = 0.25 (across all graphs). Generally, the maximal interpolation error improvement is seen between x = 0.13 and x = 0.63 (across all graphs). After reaching the maximal improvement the interpolation error improvement starts decreasing non linearly with various behaviors. In Figure 51 the highest percentage in interpolation error improvement is approximately: 150% (graphs 17, 18), 300% (graph 19), 170% (graph 20). In Figure 52 graphs 1, 3, 4, 5, 6 and 8 is seen a rapid interpolation error improvement starting approximately from y = 0.25. For these graphs the maximal interpolation error improvement is seen between y = 0.25 and y = 0.5 (across all graphs) and after reaching the maximal improvement, the interpolation error improvement starts decreasing non linearly with various behaviors. In graphs 2 and 7 the rapid interpolation error improvement is seen approximately from y = 0.25, y = 0.38 and the maximal improvement is reached approximately at y = 0.75 then the improvement starts decreasing slowly. In Figure 53 is visible a rapid increase in the interpolation error improvement starting between y = 0.13 and y = 0.25 (across all graphs). The maximal interpolation error improvement is seen between y = 0.38 and y = 0.63 (across all graphs). After reaching the maximal error improvement the curves generally show a slow and non linear decrease in error improvement. In figure 54 graphs 18, 19 and 20 a rapid interpolation error improvement starts approximately in between y = 0.13 and y = 0.25, the maximal improvement is seen close to y = 0.5 (graphs 18 and 20) and

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Figure 49. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (large), slices 1-8. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 50. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (large), slices 9-16. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 51. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (large), slices 17-20. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

in between y = 0.25 and y = 0.38 (graph 19). After reaching the maximal improvement, the interpolation error improvement starts decreasing non linearly. In graph 17 the interpolation error improvement starts approximately y = 0.13 and the maximal improvement is seen approximately at y = 0.63 then decreases non linearly. In Figure 55 is visible a rapid increase in the interpolation error improvement starting between (x, y) = (0.13, 0.13) and (x, y) = (0.25, 0.25) (across all graphs). The maximal interpolation error improvement is seen between (x, y) = (0.25, 0.25) and (x, y) = (0.5, 0.5) (across all graphs). After reaching the maximal error improvement the curves generally show a slow and non linear decrease in error improvement or a quasi constant behavior (see graph 3). Some scattered data is visible. In Figure 56 is visible a rapid increase in the interpolation error improvement starting between (x, y) = (0.13, 0.13) and (x, y) = (0.25, 0.25) (across all graphs). The maximal interpolation error improvement is seen between (x, y) = (0.13, 0.13) and (x, y) = (0.75, 0.75) (across all graphs). After reaching the maximal error improvement the curves generally show a slow and non linear decrease in error improvement or a quasi constant behavior (see graph 11). The largest variability is seen in graph 12 which shows predominantly scattered data. In Figure 57 is visible a rapid increase in the interpolation error improvement starting between (x, y) = (0.1, 0.1) and (x, y) = (0.25, 0.25) (across all graphs). The maximal interpolation error improvement is seen between (x, y) = (0.25, 0.25) and (x, y) = (0.88, 0.88) (across all graphs). After reaching the maximal error improvement the curves generally show a slow and non linear decrease in error improvement. The largest variability is seen in graph 19 which shows predominantly scattered data. Figure 58 presents graphs (1 through 8) relevant to the RSME obtained processing the functional MRI data with rotations. Data is visibly more scattered in these graphs than it was in the graphs relevant to

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 52. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along y (large), slices 1-8. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 53. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along y (large), slices 9-16. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 54. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along y (large), slices 17-20. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

the RMSE ratio obtained processing the images with misplacements. Generally, the interpolation error improvement reaches its maximum approximately in between 0.5 deg and 0.63 deg and after the maximum improvement the RMSE profiles continue showing a gradual decrease of error improvement. Figure 59 presents graphs 9 through 16 for RSME Ratio profiles that are relevant to the same fMRI volume for which the RSME curves were presented in Figure 58. Generally, the interpolation error improvement reaches its maximum approximately in between 0.25 deg and 0.5 deg (graphs 9, 10, 11, 13 and 16) and approximately in between 0.63 deg and 0.88 deg (graphs 12, 14 and 15). After the maximum improvement, the RMSE profiles continue showing a gradual decrease of error improvement. Data of graph 12 are visibly scattered. Similar observations can be made on the basis of the graphs 17, 18 and 20 shown in Figure 60, with the difference that the interpolation error improvement reaches its maximum approximately in between 0.5 deg and 0.75 deg (across graphs). In graph 19 most of the RSME Ratio that is relevant to rotations greater than 0.75 deg shows almost zero error improvement.

Q ualitative Assessment of the SRE-Based Bivariate Linear Interpolation Function In this section it is presented a qualitative assessment of the difference between the classic bivariate linear interpolation function and the corresponding SRE-based interpolation function. Each error image was calculated on a pixel-by-pixel basis squaring the difference between the image processed with classic interpolation and the image processed with SRE-based interpolation.

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Figure 55. RMSE Ratio: functional MRI (fMRI-TS1), misplacements along x (large) and y (large), slices 1-8. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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Figure 56. RMSE Ratio: functional MRI (fMRI-TS1), misplacements along x (large) and y (large), slices 9-16. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 57. RMSE Ratio: functional MRI (fMRI-TS1), misplacements along x (large) and y (large), slices 17-20. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

Figure 61 shows T1-MRI images in the top row from left to right: (i) the original, (ii) the one obtained processing with classic bivariate linear interpolation, and (iii) the one obtained with SRE-based interpolation. The misplacement was (0.05, 0.05). Error images are shown in the bottom row and they were obtained processing through the two different interpolation paradigms: classic and SRE-based. It is noticeable the almost zero residual obtained through SRE-based interpolation. With the same organizational layout of Figure 61, Figures 62, 63 and 64, show for the T2-MRI, the qualitative assessment of the improvement obtainable through the SRE-based bivariate interpolation function. While in Figure 62 the image was processed with the misplacement along x alone: (0.05, 0), in Figures 63 and 64 the image was processed with concurrent misplacements along x and y: (0.05, 0.05) and (0.1, 0.1) respectively. Figures 65, 66 and 67 demonstrate the improvement obtainable through the SRE-based bivariate linear interpolation function when the images where processed with rotations. The images in these three figures are functional MRI. Figure 65 is relevant to slice 1 of the fMRI-TS1 dataset for a rotation of 0.49 deg, Figures 66 and 67 are relevant to the fMRI-snffM00587 dataset for rotations of 0.5 (slice 20) and 1 deg (slice 23) respectively. Generally, the interpolation error improvement determined through the SRE-based paradigm was clearly and consistently manifest for misplacements larger than half of the pixel for the three imaging modalities: T1-MRI, T2-MRI and functional MRI. This has been seen in the previous section where the behavior of the RMSE ratio was shown extensively. For misplacements smaller than the half of the

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 58. RMSE Ratio: functional MRI (fMRI-TS1), rotations, slices 1-8. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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Figure 59. RMSE Ratio: functional MRI (fMRI-TS1), rotations, slices 9-16. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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Figure 60. RMSE Ratio: functional MRI (fMRI-TS1), rotations, slices 17-20. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

Figure 61. Error image analysis for T1-MRI-230, misplacement along x and y. Top row, from left to right: original image, processed by classic bivariate linear interpolation, processed by SRE-based bivariate linear interpolation. Lower row: error images. From left to right: obtained processing with classic bivariate linear interpolation and SRE-based bivariate linear interpolation respectively. The misplacement was (0.05, 0.05). The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 62. Error image analysis for T2-MRI, misplacement along x. The misplacement was (0.05, 0). The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www. Dartmouth.edu/~rswenson/Atlas. The error images were calculated with the software ImageJ: http://rsb. info.nih.gov/ij/

Figure 63. Error image analysis for T2-MRI, small misplacements along x and y. The misplacement was (0.05, 0.05). The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 64. Error image analysis for T2-MRI, small misplacements along x and y. The misplacement was (0.1, 0.1). The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

Figure 65. Error image analysis for functional MRI (fMRI-TS1), rotation of 0.49 deg. The original image (the left most in the upper row) was kindly provided by INRIA Odyssee Lab (France), www-sop.inria. fr/odyssee, also courtesy of the Department of Radiology KULeuven. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 66. Error image analysis for functional MRI (fMRI-snffM00587), rotation of 0.5 deg. The original image (not shown here) was kindly provided by The Wellcome Trust Centre for Neuroimaging (University College London – UK), www.fil.ion.ucl.ac.uk. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

Figure 67. Error image analysis for functional MRI (fMRI-snffM00587), rotation of 1 deg. The original image (not shown here) was kindly provided by The Wellcome Trust Centre for Neuroimaging (University College London – UK), www.fil.ion.ucl.ac.uk. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 68. FFT Histograms for functional MRI (fMRIsnffM00587) slice 20, rotation of 0.25 deg. From top to bottom: (i) original image, (ii) image processed with classic interpolation (iii) image processed with SRE-based interpolation. The histograms presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

Figure 69. FFT Histograms for functional MRI (fMRI-snffM00587) slice 20, rotation of 0.5 deg. From top to bottom: (i) original image, (ii) image processed with classic interpolation (iii) image processed with SRE-based interpolation. The histograms presented in this figure were calculated with the software ImageJ: http://rsb. info.nih.gov/ij/

pixel the SRE-based function does not necessarily improve the interpolation error. Figures shown in this section report on cases for which the improvement is however manifest. Although Figures 61 through 67 display error images from which it is difficult to infer the frequency spectrum of the images obtained by processing with classic or SRE-based interpolation, visual inspection of Figures 68, 69 and 70 presented in the next section will show that the images obtained through the use of the SRE generally may present higher frequency components than those images obtained through classic interpolation.

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 70. FFT Histograms for functional MRI (fMRI-snffM00587) slice 23, rotation of 1 deg. From top to bottom: (i) original image, (ii) image processed with classic interpolation (iii) image processed with SRE-based interpolation. The histograms presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

Fast Fourier Transform (FFT) Analysis The effects of the application of the Sub-pixel Efficacy Region (SRE) were studied by FFT analysis. The approach undertaken is straightforward. After the images were processed with classic and SREbased bivariate linear interpolation functions, the FFT was calculated in both cases and the histogram that shows the frequency magnitude distribution across the image was displayed for either of the two conditions (classic and SRE-based) and also for the original image.

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Images used for the FFT analysis were those with the original matrix size, they were not downsampled and neither up-sampled such to avoid aliasing effects. Results are shown in Figures 68, 69 and 70 for functional MRI (fMRI-snffM00587): slices 20, 20 and 23 respectively. Figure 68 shows from top to bottom the FFT histograms of: (i) the original fMRI, (ii) the fMRI processed with classic, and (iii) the fMRI processed with SRE-based bivariate linear interpolation. Processing was performed employing a rotation of 0.25 deg. The motion correction paradigm was applied to the fMRI-snffM00587 slice 20 with rotations of 0.25 deg and 0.5 deg and this is shown in figures 68 and 69 respectively, and also to the fMRI-snffM00587 slice 23 with a rotation of 1 deg and this is shown in Figure 70. For Figures 69 and 70 the layout is consistent with that of Figure 68. The recurrent feature that is clearly visible when observing Figures 68 through 70 is that the image calculated at the novel re-sampling locations through the SRE presents the average value of the FFT histogram that is different with respect to the average value of the FFT histogram of the original image. This mismatch between average values of the FFT histograms is visibly bigger than the mismatch between average values of the FFT histograms of (i) the original image and (ii) the image re-sampled using the misplacement (x0, y0) with classic bivariate linear interpolation. This suggests that the use of the novel re-sampling locations permits incorporating some frequency components into the resulting signal that otherwise cannot be found in those images obtained through classic bivariate linear interpolation. And these frequencies tend not to match the spectrum of the original image. The above mentioned mismatch of frequency might also result to be inverted as it shall be seen in the next chapters of the book for what concerns other SRE-based interpolation paradigms that will be presented. In fact, the image processed with SRE-based interpolation function does incorporate frequencies that are matching closely with those of the original image. This degree of match is bigger than that existing between the frequency components of the image processed with classic interpolation and the frequency components of the original image. This finding will be illustrated in the subsequent chapters and it does suggest that the band-pass filtering inherent to interpolation is attenuated through the local re-sampling variable pixel-by-pixel.

On the Improvement of the Bivariate Linear Function In the preceding sections equations (22) and (23) of Chapter VII were employed to derive novel resampling locations where the bivariate interpolation assumes improved characteristics through the use of the SRE. In this section, the novel re-sampling locations can be derived using the linear system of equations (1) and (2): x0 = { [(f(0,0) + xsre θx + ysre θy + xsre ysre ωf ) ΔE* - f(0,0)] - (ysre - y0) θy - xsre [θx + ysre ωf ] + xsre y0 ωf } / [y0 ωf - (ysre ωf + θx)]



(1)

y0 = { [(f(0,0) + xsre θx + ysre θy + xsre ysre ωf ) ΔE* - f(0,0)] - ysre θy - xsre [θx + ysre ωf ] + x0 (θx + ysre

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 71. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 67-74. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 72. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 75-82. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 73. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 83-90. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 74. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 91-98. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 75. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 99-106. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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ωf ) } / [x0 ωf - (xsre ωf + θy)]



(2)

x0 = [μ1 + y0 μ2] / [y0 ωf - μ3]



(3)

y0 = [μ1 + x0 μ3] / [x0 ωf - μ2]



(4)

μ1 = [(f(0,0) + xsre θx + ysre θy + xsre ysre ωf ) ΔE* - f(0,0)] - ysre θy - xsre [θx + ysre ωf ]

(5)

μ2 = (xsre ωf + θy)



(6)

μ3 = (ysre ωf + θx)



(7)

This can be written as:

Where it is posited that:

If it is posited that x0 = yr0 and y0 = yr0 = 0 and equations (3) and (4) are solved, then the solution becomes: xr0 = - μ1 / μ3



(8.a)

yr0 = 0



(8.b)

The Behavior of the RMSE Ratio Using equations (8.a) and (8.b) results relevant to the calculation of the RMSE Ratio were obtained with MRI data. They are similar to those previously reported in this chapter (section: “Results with Real Image Data”) and they are illustrated in this section where v.2 in the plots indicates that the novel re-sampling locations were obtained through equations (8.a) and (8.b). Figures 71 through 80 show graphs depicting the behavior of the RMSE Ratio relevant to T1-MRI230 human brain slices obtained after processing the images with classic and SRE-based bivariate interpolation. The SRE-based paradigm employed equations (8.a) and (8.b) and this is indicated in each graph and figure caption through the label (v.2). Figure 71 shows a rapid interpolation error improvement starting approximately at 0.13 deg (graphs 71, 72 and 73), 0.25 deg (graphs 67, 69 and 74) 0.38 deg (graphs 68 and 70). Generally, data are scattered except for graphs 68, 70 and 73 which show well defined behavior. The maximum interpolation error improvement is reached approximately in between 0.5 deg and 0.88 deg (across graphs) then the RSME Ratio profiles continue showing a non linear decrease of interpolation error improvement. Figure 72 shows RSME Ratio profiles that are similarly behaving to those showed in Figure 71. With the exception that the rapid increase in interpolation error improvement starts approximately in between 0.13 deg and 0.25 deg (across graphs) and the maximum interpolation error improvement is reached approximately in between 0.63 deg and 0.88 deg (across graphs). Figure 73 shows RSME Ratio profiles that are less scattered than those shown in Figures 71 and 72. The rapid increase in interpolation error improvement starts approximately in between 0.13 deg and

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 76. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 107-114. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 77. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 115-122. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 78. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 123-130. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 79. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 131-138. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 80. RMSE Ratio: T1-MRI-230, rotations (v.2), slices 139-141. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

0.25 deg (across graphs) and the maximum interpolation error improvement is reached approximately in between 0.5 deg and 0.88 deg (across graphs). After reaching the maximum error improvement the RSME Ratio profiles show a relatively fast loss of error improvement for increasing rotational angles. Figure 74 shows RSME Ratio profiles that are well defined. The rapid increase in interpolation error improvement starts approximately in between 0.13 deg and 0.38 deg (across graphs) and the maximum interpolation error improvement is reached approximately in between 0.75 deg and 0.88 deg (across graphs). After reaching the maximum error improvement the RSME Ratio profiles show a relatively fast loss of error improvement for increasing rotational angles. Figure 75 shows RSME Ratio profiles that are either well defined (graphs 99, 100, 103, 105 and 106) or scattered (graphs 101, 102 and 104). The rapid increase in interpolation error improvement starts approximately in between 0.13 deg and 0.38 deg (across graphs) and the maximum interpolation error improvement is reached approximately in between 0.63 deg and 0.88 deg (across graphs). After reaching the maximum error improvement the RSME Ratio profiles show a relatively fast loss of error improvement for increasing rotational angles. Figure 76 shows RSME Ratio profiles that are either well defined (graphs 107, 108 and 109) or scattered (graphs 110 through 114). The rapid increase in interpolation error improvement starts approximately in between 0.13 deg and 0.38 deg (across graphs) and the maximum interpolation error improvement is reached approximately in between 0.5 deg and 0.88 deg (across graphs). After reaching the maximum error improvement the RSME Ratio profiles show a behavior similar to those graphs shown in figures 73 through 75. Figure 77 shows a scattered behavior in the majority of the graphs except for graph 117. Observations made as far as the start of rapid increase of interpolation error improvement are quite similar to

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 81. RMSE Ratio: T2-MRI, rotations (v.2), slices 1-8. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 82. RMSE Ratio: T2-MRI, rotations (v.2), slices 9-16. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 83. RMSE Ratio: T2-MRI, rotations (v.2), slices 17-19. The range of rotation is [0 deg, 0.99 deg] at steps of 0.01 deg

those reported for figure 76. The maximum interpolation error improvement is reached approximately in between 0.5 deg and 0.88 deg (across graphs). After approximately 0.88 deg a rapid loss of interpolation error improvement is observed in the graphs. Figure 78 shows RSME Ratio profiles that are either well defined (graphs 123, 124, 127 and 128) or scattered (graphs 125, 126, 129 and 130). The rapid increase in interpolation error improvement starts approximately in between 0.065 deg and 0.38 deg (across graphs) and the maximum interpolation error improvement is reached approximately at 0.63 deg (across graphs). After reaching the maximum error improvement the RSME Ratio profiles show a rapid loss (scattered graphs) or a slow loss (well defined graphs) of interpolation error improvement. Figure 79 shows all scattered graphs except for one (graph 136). The rapid improvement in interpolation error starts approximately at 0.13 deg (graphs 131 through 135, and 137) and approximately at 0.38 deg (graphs 136 and 138) The maximum interpolation error improvement is obtained approximately at 0.75 deg (across graphs) and is followed by a relatively fast decrease of interpolation error improvement visible in all scattered graphs. Observations that can be made about the graphs of Figure 80 are quite consistent with those made in evaluating Figure 80. Consistently, the well defined profile (graph 139) show the start of the rapid increase of interpolation error improvement approximately at 0.38 deg. Figures 81 through 83 show graphs depicting the behavior of the RMSE Ratio relevant to T2-MRI human brain slices obtained after processing with rotations the images with classic and SRE-based bivariate interpolation. The SRE-based paradigm employed equations (8.a) and (8.b) and this is indicated in each graph and in the figure caption through the label (v.2).

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 84. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (v.2), slices 1-8. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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The Results of the Sub-Pixel Efficacy Region Based Bivariate Linear Interpolation Function

Figure 85. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (v.2), slices 9-16. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

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Figure 86. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (v.2), slices 17-20. The range of misplacement is [(0, 0), (0.99, 0)] at steps of 0.01 along the x axis

Figure 81 shows two types of graphs: (i) those where the interpolation error improvement manifests itself throughout the full range of rotations (1, 2, 7 and 8) and (ii) those where the interpolation error improvement begins to increase after a certain value of degrees of rotation (3, 4, 5 and 6). For graphs 3, 4, 5 and 6 the values of rotation corresponding to the beginning of the increase in interpolation error improvement are approximately between 0.13 deg and 0.25 deg. The maximum of the interpolation improvement is approximately between 0.01 deg and 0.38 deg (graphs 1, 2 and 3) and 0.75 deg (graphs 4 through 8). Also Figure 82 shows two types of graphs. To the first type belong those graphs (13, 14, 15 and 16) for which the interpolation error improvement happens throughout the entire of range of rotations. And to the second type belong those graphs (9, 10, 11 and 12) for which a certain value of rotation sets the beginning of the interpolation error improvement. For graphs 9, 10, 11 and 12 the values of rotation corresponding to the beginning of the increase in interpolation error improvement is approximately 0.13 deg. The maximum of the interpolation improvement is approximately between 0.5 deg and 0.88 deg (graphs 9, 10, 11 and 12) and between 0.63 deg and 0.88 deg (graphs 13, 14, 15 and 16). In Figure 83, graphs 17 and 18 show a fast increase in interpolation error improvement at approximately 0.13 deg then the error improvement reaches its maximum approximately between 0.75 deg and 0.88 deg and it is followed by a fast decrease with the increase of the magnitude of the rotation. In graph 19 the interpolation error improvement happens throughout the entire of range of rotations. Figures 84 through 96 show the behavior of the RMSE Ratio obtained while processing functional MR images with misplacements along x (Figures 84 through 86), misplacements along y (Figures 87 through 89), with concurrent misplacements along x and y (Figures 90 through 92) and with rotations (Figures 93 through 99).

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Figure 87. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along y (v.2), slices 1-8. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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Figure 88. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along y (v.2), slices 9-16. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

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Figure 89. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along y (v.2), slices 17-20. The range of misplacement is [(0, 0), (0, 0.99)] at steps of 0.01 along the y axis

In Figure 84 the graphs of the RMSE Ratio are all well defined and show the increase in interpolation error improvement that begins approximately between x = 0.13 and x = 0.25 (across graphs). The maximum error improvement is approximately between x = 0.25 and x = 0.5 (across graphs) then the profiles of the RMSE Ratio show a slow loss of interpolation error improvement with the increase of the value of the misplacement. Data of Figure 85 is similar to that of Figure 84. The graphs are all well defined. The increase in interpolation error improvement begins approximately between x = 0.13 and x = 0.25 (across graphs). The maximum error improvement is however more widespread across the range of misplacement and is located approximately between x = 0.13 and x = 0.63 (across graphs) then the interpolation error improvement is reduced with the increase of the value of the misplacement. In Figure 86 the graphs demonstrate a behavior of the RSME Ratio that is consistent with data of Figures 84 and 85. The increase in interpolation error improvement begins approximately between x = 0.13 and x = 0.25 (across graphs). The maximum error improvement is located approximately between x = 0.13 and x = 0.55 (across graphs). In Figures 87, 88 and 89 the behavior of the RSME Ratio is well defined and quite consistent across the graphs. The general characteristic of these profiles is that the beginning of the increase of interpolation error improvement is located approximately between y = 0.25 and y = 0.38 (across graphs) except for graph 14 in Figure 88 and graphs 17 and 19 in Figure 89 where such increase begins approximately between y = 0.13 and y = 0.25 (across graphs). The most remarkable feature of the profiles is that after the beginning, the increase in the interpolation error improvement is persistent throughout the range of misplacement.

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Figure 90. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x and y (v.2), slices 1-8. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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Figure 91. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x and y (v.2), slices 9-16. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

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Figure 92. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x and y (v.2), slices 17-20. The range of misplacement is [(0, 0), (0.99, 0.99)] at steps of (0.01, 0.01) along the x and y axis

In Figure 88 the highest percentage in interpolation error improvement is approximately: 400% (graphs 9 and 11), 500% (graphs 10, 12, 13, 14 and 15), and 600% (graph 16). In Figure 89 the highest percentage in interpolation error improvement is approximately: 500% (graphs 17 and 18), 800% (graph 19), 600% (graph 20). Figure 90 show profiles with the following characteristics. The start of the increase of interpolation error improvement is approximately between (x, y) = (0.13, 0.13) and (x, y) = (0.25, 0.25) (across graphs). The maximal improvement is located approximately between (x, y) = (0.25, 0.25) and (x, y) = (0.63, 0.63) (across graphs). The graphs all show a reduced error improvement after reaching the maximum except for graph 3 where the improvement continues throughout the full range of misplacement. All graphs of Figure 91 show rapid increase in interpolation error improvement that starts approximately in between (x, y) = (0.13, 0.13) and (x, y) = (0.25, 0.25) (across graphs). Graph 12 show scattered data while the rest of the graphs are all well defined. The maximum improvement is widespread across the range of misplacement and located approximately between (x, y) = (0.25, 0.25) and (x, y) = (0.75, 075) (across graphs). The profiles all show a reduced interpolation error improvement after reaching the maximum. Data of Figure 92 show graphs 17 and 18 with the maximum interpolation error improvement located approximately between (x, y) = (0.38, 0.38) and (x, y) = (0.5, 0.5) (across graphs). In graph 19 the maximum is located at approximately (x, y) = (0.75, 0.75) and in graph 20 in between (x, y) = (0.25, 0.25)

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Figure 93. RMSE Ratio: functional MRI (fMRI-TS1), rotations (v.2), slices 1-8. The range of rotations is [0 deg, 0.99 deg] at steps of 0.01 deg

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Figure 94. RMSE Ratio: functional MRI (fMRI-TS1), rotations (v.2), slices 9-16. The range of rotations is [0 deg, 0.99 deg] at steps of 0.01 deg

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Figure 95. RMSE Ratio: functional MRI (fMRI-TS1), rotations (v.2), slices 17-20. The range of rotations is [0 deg, 0.99 deg] at steps of 0.01 deg

and (x, y) = (0.38, 0.38). Data of graph 19 are scattered while the rest of the profiles are well defined. All graphs show a reduced error improvement after reaching the maximum. In Figure 93 the graphs show behaviors of the RMSE Ratio that are similar to what already seen in other figures. There is an onset to the interpolation error improvement and this happens for rotations that are approximately in between 0.13 deg and 0.38 deg (across graphs) except for graph 7 where the onset is approximately in between 0.38 deg and 0.5 deg. The graphs are all well defined and they show the typical reduction in interpolation error improvement after reaching the maximum. Data of Figure 94 generally show the same features of data seen in Figure 93. Graphs 9 and 12 are scattered while the rest of the profiles of the RSME Ratio are well defined. Particularly interesting are graphs 10, 12 and 15 where the increase of the interpolation error improvement versus the value of the misplacement continues from the vale of the onset happening approximately in between 0.13 deg and 0.25 deg (across graphs) till beyond 0.88 deg (see graph 10). In Figure 95 the remarkable feature is that the value of the maximal interpolation error improvement happens at around 0.63 deg for all graphs. The remaining features of the RSME Ratio profiles shown here are common to what already seen in previous figures. In Figure 96 the graphs of the RSME Ratio show a common feature. The graphs show a maximum for the interpolation error improvement and then they continue to be generally quasi constant. Except for graphs 15, 17, and 18 all remaining profiles show improvement of the interpolation error up to 6080%. While in graph 15 the best error improvement across the full range of rotations is clearly above 80% (tending to be 100%), in graphs 17 and 18 the best error improvement is not below 40%. Data of Figure 97 is similar to that presented in Figure 96. The lowest of the best interpolation error improvements that can be seen throughout the full range of rotations is almost 60% and is seen in graph

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Figure 96. RMSE Ratio: functional MRI (fMRI-snffM00587), rotations (v.2), slices 13-20. The range of rotations is [0 deg, 2.475 deg] at steps of 0.0025 deg

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Figure 97. RMSE Ratio: functional MRI (fMRI-snffM00587), rotations (v.2), slices 21-28. The range of rotations is [0 deg, 2.475 deg] at steps of 0.0025 deg

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Figure 98. RMSE Ratio: functional MRI (fMRI-snffM00587), rotations (v.2), slices 29-36. The range of rotations is [0 deg, 2.475 deg] at steps of 0.0025 deg

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Figure 99. RMSE Ratio: functional MRI (fMRI-snffM00587), rotations (v.2), slices 37-42. The range of rotations is [0 deg, 2.475 deg] at steps of 0.0025 deg

24. While instead graphs 23 and 25 show that the improvement of the interpolation error can go up to 80% and graphs 21, 22 and 27 reach up to 100% improvement. Graphs 26 and 28 show an outstanding 150% improvement. The magnitude of the rotation is certainly a key factor in these data.

Data of Figure 98 show that the best interpolation error improvement is as high as 200% in graph 24 and as high as 250% in graphs 35 and 36. The rest of the graphs do not show an interpolation error improvement that is smaller than 100%.

Finally in Figure 99, graph 42 shows the best improvement as higher than 350%. For the remaining graphs the best interpolation error improvement is generally around 250%. Once again it is to emphasize that the relatively big rotational angles seen in the data of Figures 97, 98 and 99 play a fundamental role in elucidating the potentiality of the SRE-based bivariate linear interpolation function.

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Figure 100. Error image analysis for T1-MRI-230 (v.2). The images were processed by a rotation of 0.2 deg. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

Figure 101. Error image analysis for T2-MRI (v.2). The images were processed by a rotation of 0.2 deg. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www. Dartmouth.edu/~rswenson/Atlas. The error images were calculated with the software ImageJ: http://rsb. info.nih.gov/ij/

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Q ualitative Assessment of the SRE-Based Bivariate Linear Interpolation Function v.2 Finally this section presents the qualitative evaluation of MRI data employing equations (8.a) and (8.b) to calculate the novel re-sampling locations. Once again the remarkable feature is the net improvement of the SRE-based function (see Figures 100 and 101) over the classic function and this improvement is seen in the lower right error images: (Error Image SRE). Figures 100 and 101 confirm qualitatively that also the SRE-based Bivariate Linear Interpolation Function v.2 is effective in reducing the interpolation error when confronted with the classic form of bivariate linear function. In both figure is noticeable the qualitative remarkable improvement even for rotation angles as small as 0.2 deg.

SUMMARY This chapter has presented the results of the Sub-pixel Efficacy Region based bivariate linear interpolation function. The chapter starts with a description of the validation paradigm used to demonstrate the improvement of the bivariate linear interpolation function by showing a reduction in the interpolation error with respect to the classic form of bivariate linear interpolation. As interpolation concerns with re-sampling a signal at unknown locations, the interpolation error can be seen with substantial logic if given a-priori knowledge in the form of a reference signal (image). A motion correction paradigm is therefore used to validate the theory along with its derived mathematical formulations and this is done in the following terms. Given an image, the image is re-sampled at a fixed intra-pixel location. The image is then motion corrected and the bivariate linear interpolation is employed to re-sample the signal. The image is motion corrected and interpolated using also the novel re-sampling location calculated with the methodology that has been elucidated in Chapter VII. Quantitative and qualitative analyses are presented of the reduction of the interpolation error and of the improved approximation properties of the SRE-based interpolation function. In this chapter the reader is therefore informed as to what are the benefits of using pixel-by-pixel varying novel re-sampling locations instead of the fixed intra-pixel misplacement. The illustrations give special attention to biomedical applications such as Magnetic Resonance Imaging (MRI) of the human brain, anatomical (T1-MRI and T2-MRI) and functional (fMRI).

References Ciulla, C., & Deek, F. P. (2005). On the approximate nature of the bivariate linear interpolation function: A novel scheme based on intensity-curvature. ICGST - International Journal on Graphics, Vision and Image Processing, 5(7), 9-19. Greene, B. (1999). The elegant universe. Superstings, hidden dimensions, and the quest for the ultimate theory. New York, NY: W. W. Norton & Company. Zhang, Y., Brady, M., & Smith, S. (2001). Segmentation of brain MR images through a hidden markov random field model and the expectation maximization algorithm. IEEE Transactions on Medical Imaging, 20(1), 45-57.

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Sub-Pixel Efficacy Region of the Trivariate Linear Interpolation Function

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Interpolation Procedures

Introduction The intent of the present chapter is to expand the treatise that concerns with the presentation of the current literature in signal-image interpolation with specific focus on the classification of the procedures and the relevant applications of interpolation, in relationship to the various scientific disciplines. Also, the present chapter expands on the importance of the error bounds existing in literature for the characterization and quantification of the interpolation error. The discussion is interleaved with the characteristics features of the unifying theory that the book presents such to place the SRE-based interpolation functions within the context of the larger framework that the scientific literature has conceived so far.

CLASSIFICATION AND APPLICATIONS OF THE INTERPOLATION PROCEDURES Signal-image interpolation is normally used for several procedures, and they are hereto listed. When pixel-by-pixel correspondence has to be established for the purpose of matching images collected at different times using the same imaging modality (registration) or different imaging modalities (co-registration), interpolation finds immediate application in motion correction techniques. When signals are to be reconstructed with increased resolution (up-sampling) or decreased resolution (down-sampling). For the estimation of signals at space-time locations where the signal is unknown because of the limitation imposed by the sampling frequency of the collecting equipment. Other relevant procedures are: (i) estimation of signal parameters corresponding to particular frequencies of the spectrum through the use of linear interpolation (Borkowski, 2000), and (ii) enhancement of grayscale images and reconstruction of color images through combined use of covariance coefficients and linear interpolation (Li & Orchard, 2001). Since interpolation descends from approximation theory, it can also be grouped with extrapolation and regression as shown in an early example of a unifying methodology of linear unbiased estimations (Chow & Lin, 1971).

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Interpolation Procedures

Interpolation techniques currently in use are classified in two main categories: scene-based interpolation and object-based interpolation (Penney et al., 2004). Scene-based interpolation uses the value of the intensity at the pixels (nodes) at the aim to reconstruct the signal. In order to achieve such purpose, linear, quadratic, cubic, Lagrange and Sinc interpolation paradigms are those among the most commonly employed. Object-based interpolation uses intrinsic features captured from the signal to determine the reconstructed signal. This one category has recently extended to morphology-based methods (Lee & Wang, 2000) and feature guidance based methods (Lee & Lin, 2002). Object-based interpolation was initiated (Raya & Udupa, 1990) evaluated and extended (Herman et al., 1992), and then reinforced (Grevera & Udupa, 1996, 1998). These authors showed that linear interpolation of a surface representing the distance map of the pixels from the boundary of the object in the scene proves superior to simple linear interpolation of the pixel intensities. The aforementioned authors have also reported a statistical comparison of their shape-based method, and this was developed within the three-dimensional context against a series of other methods including the classic trivariate linear interpolation. Recent applications of interpolation procedures devote attention to a large variety of applications and paradigms for the estimation of missing data. Hereto is given a list of works that have been brought just recently to the attention of the scientific community pointing to the diversity of the approaches and the applications in science. Li and Orchard (2001) devised an improved paradigm of the bivariate linear function through the application of the adaptive covariance based approach with applications to grey scale image resolution enhancement. Varadarajan and Krolik (2003) have employed interpolation for the improvement of radar detection. Angelini et al. (2003) employed interpolation to estimate the factors of the multivariate analysis in econometric projections and in the work of these authors, the interpolation paradigm is linked indirectly to the signal through the factors of the multivariate analysis at the aim to estimate missing data in very large datasets. A Non-uniform interpolation technique based on Support Vector Machines (SVM) was reported (Rojo-Alvarez & Figuera, 2006) and employs the Sinc kernel for band-limited signal reconstruction. This work distinguishes itself for the original joint use of the Sinc kernel with the SVM and also brings to the attention that the search for improved and more efficient interpolation paradigms is nowadays still a very active research area. Within the context of image zooming (Cha & Kim, 2006, 2007), an edge forming method that eliminates the checkerboard effect seen in zooming has been developed in 2006, and also the research has been expanded in 2007 into the development of an improved paradigm of the bivariate linear interpolation function for zooming. This was achieved devising an algorithm that uses the Sobel operator to estimate the second order derivative of the interpolation function. The use of the Sobel operator for the improvement of the approximation characteristics raises substantially the accuracy of the linear function. Analogously, Goodman and Ong (2005) use first and second order derivatives of the cubic B-Spline to improve the approximation capability of the kernel through the preservation of the local convexity of the function. Chand and Kapoor (2006) make use of the derivatives of the interpolation function for the generation of a derived class of interpolators. The authors have generalized the classic Spline to a new interpolation function using the hidden variable fractal interpolation function (CHFIF) and they have calculated the derivatives of the Spline. Another application of B-Spline functions (Eldar & Dvorkind, 2006) suggests that for a more accurate signal reconstruction, preprocessing of the data with a linear transformation, which is aimed to

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minimize the squared norm error, is useful to determine an optimized approximation of the true-to-find reconstructed signal. Another type of preprocessing is that of filtering the data prior to sampling through a discrete filter prior to reconstruction. In this context, B-Spline forms furnish the representation of the impulse response of the analysis filter. A unified approach of this type has been recently presented (Eldar & Unser, 2006). The Kriging predictor has also been used recently for interpolation of large data sets in geostatistical science (Furrer et al., 2005) and in geotechnology (Largueche, 2006; Memarsadeghi & Mount, 2007). In biomedical sciences Raja et al. (2007) proposed B-Spline interpolation functions for the segmentation of images and they have compared the performance of the novel algorithm to the classic Snake image processing tools. A comparison between B-Splines (time domain techniques) and Phase domain estimators has been recently reported within the context of velocity estimation (Viola & Walker, 2006) showing the superiority of the B-Splines functions. Other applications of interpolation procedures (Woods et al., 2006; Zhang & Wu, 2006) relate to super-resolution problems which are concerned with obtaining high resolution images from aliased low-resolution images. Interpolation techniques have found applications also in psychophysics for data reconstruction (Lehky & Sejnowski, 1990). In neurophysiology, an electrical stimulus delivered to a resting muscle and to the same muscle during a voluntary contraction consists of a technique used to determine as to if neurons in the brain have been excited sufficiently to evoke all the force they can produce. When the stimulus is a single pulse this technique is called twitch interpolation (Brown, 1928; Herbert & Gandevia, 1999). Studies aimed to investigate determinants of the amplitude of the interpolated twitch have been reported (Herbert & Gandevia, 1999). Also, the twitch interpolation technique was employed to study the degree of motor unit activation during voluntary effort (Belanger & McComas, 1981), measure fatigue of the quadriceps muscle (Bulow et al., 1995) and to model muscle activation, fatigue, and recovery (Liu et al., 2002) through estimating the true maximal force from the stimulated additional force increases. Interpolation relates also to disciplines like cognitive neuroscience as recent research (Guigon, 2004) reports studies conducted on the basis of neural computation which can explain interpolation and extrapolation capacities of the nervous system in sensorimotor and cognitive tasks. Interpolation has found applications also in applied neuroscience studies concerned with the reconstruction of recorded spike waveforms (Blanche & Swindale, 2006). The novel trend of adaptive interpolation presented in this book has evolved from the previous works (Ciulla & Deek, 2005a, 2005b, 2006). Also, determination of the coefficients of the interpolation function based on local autocorrelation estimates characterized optimal cubic convolution kernels (Shi & Reichenbach, 2006). Generally, adaptive interpolation (Ciulla & Deek, 2005a, 2005b, 2006) offers more advantages over classic paradigms. The above mentioned approaches (Cha & Kim, 2006, 2007; Chand & Kapoor, 2006; Goodman & Ong, 2005) which all use in the re-formulation of the interpolation kernel the first or the second order derivatives, or both of them, or estimations of them (such as the Sobel operator) are in line with the general idea that novel concepts related to the curvature of the function needs to be taken into account to open new avenues in signal processing for what concerns signal-image interpolation. In this regard this book offers an extensive treatise as well the establishment of the

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unifying theory that bases its foundations on the novel conception for which both signal intensity and curvature are embedded in a unique and original joint formulation. This book is concerned with the determination of the non-rigid grid of misplacements as proposed by object-based methods, but in a fashion dissimilar to that proposed by current literature. In the unifying theory proposed in this book, the non-rigid grid of misplacements is determined using the Intensity-Curvature Functional and the Sub-pixel Efficacy Region (SRE). This leads to calculation of the interpolation function at the novel re-sampling locations that vary pixel-by-pixel. What distinguishes the unifying theory from above discussed methods is that the use of the SRE determines a novel class of interpolators that are mathematically dependent on the local curvature of the interpolation function. Thus, the summary of typology of interpolation methods might be revised as: (i) those classic methods based on pixel intensity, (ii) object-based methods, (iii) scene-based methods, and (iv) SRE-based methods that are introduced through the novelty presented in this book. The novelty bases its foundations on the determination of the non-rigid set of re-sampling locations through math deductions that are based on the combination of pixel intensity and curvature of the interpolation function.

Error bounds EX ISTING IN LITERATURE and the indirect measures of the interpolation error of the PROPOSED theory The relevance of this book and the unifying theory herein presented is also in the error boundary of the linear interpolation. Existent literature presents formulas that are compelling to the mathematical foundations of this book in the following manner. Waldron (1998), among all, has reported a general and explicit expression that provides an error boundary for the linear interpolation function that is dependent on the second order derivative of the same function: f(x) – Lf(x) = ½ ∑v≠Θ ∑ω≠Θ pv (x) pω (x) ∫[x, v, ω] D2 f(x)



(1)

Where Θ is the set of sampling points, the interpolation function is of type Lf = ∑v≠Θ pv f(v) such that for Θ ≠ R; Lf = f. Also, pv (x) pω (x) is a function that vanishes at all points of Θ if Lf matches the function values at Θ, and finally D2f(x) is the second order derivative of f(x). Above expression (Waldron, 1998) can be used to derive even more general error boundary which also applies to higher order interpolators such as Lagrange (Waldron, 1997) and that resembles that earlier anticipated (Strang & Fix, 1971) which asserted that the error boundary for the general interpolation function is given by: || h - hℓ ||L2 ≤ ε ℓn || h(n) ||L2



(2)

Where L2 is the square norm, ε is a given constant, and h(n) the n-th derivative of the h function as defined by hℓ (x) = Σk αk β [ (x / ℓ) – k], with ℓ being the sampling step. Waldron (1998) reports that the interpolation error can be also be obtained by the more general Kowalewski remainder (Kowalewski, 1932): f(x) – Lf(x) = - ∑v≠Θ pv (x) ∫[x, x, ω] Dv-x2 f(x)

(3)

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Equations (1) and (3) bring up to the attention, that the linear interpolation error is strictly dependent on the second order derivative of the function. This fact reconnects to the mathematical foundations of the theory presented in this book as it is true that the unifying theory herein proposed incorporates into the intensity curvature functional ΔE the second order derivative of the interpolator. And the theory uses the derivative as a measure of the curvature of the function. The two intensity-curvature terms and the resulting ΔE give an indirect measure of the interpolation error, and thus provide legitimate basis to search for the extremes of ΔE through the calculation of the set of first order derivatives with respect to the variables x, y, and z (in the specific three-dimensional case). The extremes of ΔE are then used to obtain the novel re-sampling location (xr0, yr0, zr0) where the interpolation function is calculated for given misplacement (x0, y0, z0). The extremes of ΔE, which we call (xsre, ysre, zsre), are employed to project (x0, y0, z0) into the novel re-sampling location (xr0, yr0, zr0).

SUMMARY This chapter outlines the capabilities of the theory to be extended to higher dimensionality interpolators, and also continues the treatise from Chapter I in citing relevant literature that classifies interpolation functions on the basis of the image processing task to perform and on the basis of the information content employed at the aim to reconstruct the continuous signal. The reader is informed as to what is the relevance of the interpolation error bounds characterization forms existing in literature and how they relate to the mathematical foundations of the unifying theory presented in this book. The chapter concludes discussing the meaning of the Intensity-Curvature Functional for the three dimensional case of the linear interpolation function and the use of the novel re-sampling locations obtained on the basis of the Sub-pixel Efficacy Region at the aim to improve the performance of the function.

References Belanger, A. Y., & McComas, A. J. (1981). Extent of motor unit activation during effort. Journal of Applied Physiology, 51(5), 1131-1135. Blanche, T. J., & Swindale, N. V. (2006). Nyquist interpolation improves neuron yield in multiunit recordings. Journal of Neuroscience Methods, 155(1), 81-91. Borkowski, J. (2000). LIDFT- the DFT linear interpolation method. IEEE Transactions on Instrumentation and Measurement, 49(4), 741-745. Brown, D. D. (1928). On inhibition as a reflex accompaniment of the tendon jerk and of other forms of active muscular response. Proceedings of the Royal Society of London,103, 321-336. London, UK: The Royal Society. Bulow, P. M., Nørregaard, J., Mehlsen, J., & Danneskiold-Samsøe, B. (1995). The twitch interpolation technique for study of fatigue of human quadriceps muscle. Journal of Neuroscience Methods, 62(1-2), 103-109.

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Cha, Y., & Kim, S. (2006). Edge-forming methods for image zooming. Journal of Mathematical Imaging and Vision, 25(3), 353-364. Cha, Y., & Kim, S. (2007). The error-amended sharp edge (EASE) scheme for image zooming. IEEE Transactions on Image Processing, 16(6), 1496-1505. Chand, A. K. B., & Kapoor, G. P. (2006). Spline coalescence hidden variable fractal interpolation functions. Journal of Applied Mathematics, 2006(Article ID 36829), 1-17. Chow, G. C., & Lin, A. (1971). Best linear unbiased interpolation, distribution and extrapolation of time series by related series. The Review of Economics and Statistics, 53(4), 372-375. Ciulla C., & Deek, F. P. (2005a). On the approximate nature of the bivariate linear interpolation function: A novel scheme based on intensity-curvature. ICGST - International Journal on Graphics, Vision and Image Processing, 5(7), 9-19. Ciulla, C., & Deek, F. P. (2005b). Novel schemes of trivariate linear and one-dimensional quadratic B-Spline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53. Ciulla, C., & Deek, F. P. (2006). Extension of the sub-pixel efficacy region to the Lagrange interpolation function. ICGST - International Journal on Graphics, Vision and Image Processing, 6(1), 1-12. Eldar, Y. C., & Dvorkind, T. G. (2006). A Minimum squared-error framework for generalized sampling. IEEE Transactions on Signal Processing, 54(6, Pt. 1), 2155- 2167. Eldar, Y. C., & Unser, M. (2006). Nonideal sampling and interpolation from noisy observations in shiftinvariant spaces. IEEE Transactions on Signal Processing, 54(7), 2636- 2651. Furrer, R., Genton, M. G., & Nychka, D. (2006). Covariance tapering for interpolation of large spatial datasets. Journal of Computational & Graphical Statistics, 15(3), 502-523. European Central Bank. (2003, August). Interpolation and backdating with a large information set (Working Paper 252). Frankfurt am Main, Germany: Angelini, E., Henry, J., & Marcellino, M.G. Goodman, T. N. T., & Ong, B. H. (2005). Shape-preserving interpolation by splines using vector subdivision. Advances in Computational Mathematics, 22(1), 49-77. Grevera, G. J., & Udupa, J. K. (1996). Shape-based interpolation of multidimensional grey-level images. IEEE Transactions on Medical Imaging, 15(6), 881-892. Grevera, G. J., & Udupa, J. K. (1998). An objective comparison of 3-D image interpolation methods. IEEE Transactions on Medical Imaging, 17(4), 642-652. Guigon, E. (2004). Interpolation and extrapolation in human behavior and neural networks. Journal of Cognitive Neuroscience, 16(3), 382-389. Herbert, R. D., & Gandevia, S. C. (1999). Twitch interpolation in human muscles: Mechanisms and implications for measurement of voluntary activation. Journal of Neurophysiology, 82, 2271-2283. Herman, G. T., Zheng, J., & Bucholtz, A. (1992). Shape-based interpolation. IEEE Computer Graphics and Applications, 12(3), 69-79.

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Kowalewski, G. (1932). Interpolation and genaherte quadratur. Leipzig, Germany: Teubner. Largueche, F. Z. B. (2006). Estimating soil contamination with kriging interpolation method. American Journal of Applied Sciences, 3(6), 1894-1898. Lee, T. Y., & Wang, W. H. (2000) Morphology-based three-dimensional interpolation. IEEE Transactions on Medical Imaging, 19(7), 711-721. Lee, T. Y., & Lin, C. H. (2002). Feature-guided shape-based image interpolation. IEEE Transactions on Medical Imaging, 21(12), 1479-1489. Lehky, S. R., & Sejnowski, T. J. (1990). Neural model of stereoacuity and depth interpolation based on a distributed representation of stereo disparity. The Journal of Neuroscience, 10(7) 2281-2299. Li, X., & Orchard, M. T. (2001). New edge-directed interpolation. IEEE Transactions on Image Processing, 10(10), 1521-1527. Liu, J. Z., Brown, R. W., & Yue, G. H. (2002). A dynamical model of muscle activation, fatigue, and recovery. Biophysical Journal, 82(5), 2344-2359 Memarsadeghi, N., & Mount, D. M. (2007). Efficient implementation of an optimal interpolator for large spatial data sets. In Y. Shi et al. (Eds.), Computational Science - ICCS: Vol. 4488(Pt. II). Lecture Notes in Computer Science (pp. 503-510). Berlin, Germany: Springer-Verlag. Penney, G. P., Schnabel, J. A., Rueckert, D., Viergever, M. A., & Niessen, W. J. Registration-based interpolation. (2004). IEEE Transactions on Medical Imaging, 23(7), 922-926. Raja, K. B., Madheswaran, M., & Thyagarajah, K. (2007). A general segmentation scheme for contouring kidney region in ultrasound kidney images using improved higher order spline interpolation. International Journal of Biomedical Sciences, 2(2), 81-88. Raya, S. P., & Udupa J. K. (1990). Shape-based interpolation of multidimensional objects. IEEE Transactions on Medical Imaging, 9(1), 32-42. Rojo-Alvarez, J. L., & Figuera, C. (2007). Nonuniform interpolation of noisy time series using support vector machines. IEEE Transactions on Signal Processing, 55(8), 4116-4126. Shi, J., & Reichenbach, S. E. (2006). Image interpolation by two-dimensional parametric cubic convolution. IEEE Transactions on Image Processing, 15(7), 1857-1870. Strang, G., & Fix, G. (1971). A Fourier analysis of the finite element variational method. In, Constructive Aspect of Functional Analysis (pp. 796-830). Rome, Italy: Edizioni Cremonese. Varadarajan, V., & Krolik, J. (2003). Joint space-time interpolation for bistatic STAP. Conference Record of the 37th Asilomar Conference on Signals, Systems and Computers, 1, 60- 65. Pacific Grove, CA: IEEE. Viola, F., & Walker, W. F. (2006). A comparison between spline-based and phase-domain time-delay estimators. IEEE Transaction on Ultrasonics Ferroelectrics and Frequency Control, 53(3), 515-517. Waldron, S. (1997). A multivariate form of Hardy’s inequality and Lp-error bounds for multivariate Lagrange interpolation schemes. SIAM Journal on Mathematical Analysis, 28(1), 233-258.

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Waldron, S. (1998). The error in linear interpolation at the vertices of a simplex. SIAM Journal on Numerical Analysis, 35(3), 1191-1200. Woods, N. A., Galatsanos, N. P., & Katsaggelos, A. K. (2006). Stochastic methods for joint registration, restoration, and interpolation of multiple undersampled images. IEEE Transactions on Image Processing, 15(1), 201-213. Zhang, L. & Wu, X. (2006). An edge guided image interpolation algorithm via directional filtering and data fusion. IEEE Transactions on Image Processing, 15(8), 2226-2238.

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Chapter X

The Extension of the Theory to the Trivariate Linear Interpolation Function

ORGANIZATION OF THE FORTHCOMING TEXt The organization of this chapter is similar to that of Chapter VII where it was outlined the theoretical approach to the improvement of the bivariate linear interpolation function. The methodological approach calculates the Intensity-Curvature Functional on the basis of the two intensity-curvature terms calculated before and after interpolation. The roots of the polynomial system consisting of the three first order partial derivatives of the Intensity-Curvature Functional constitute the SRE. The formula that belongs to the unifying theory and that allows the calculation of the novel re-sampling locations for the trivariate linear interpolation function is in this chapter seen in equation (1). The novel re-sampling locations (xr0, yr0, zr0) are obtained through equations (14), (15) and (16).

Scope The theory presented herein has been developed and tested in former chapters for the bivariate linear interpolation function. Hereafter the theory is extended to the three-dimensional linear interpolation (Ciulla & Deek, 2005). Because of the approximate nature of the interpolation function, it is natural to think that the Sub-pixel Efficacy Region exists within the voxel. Geometrically the Sub-pixel Efficacy Region can be thought as that set of points that allow the calculation of the novel re-sampling locations. Because of the curvature, at these locations it is minimal or maximal the discrepancy between the values estimated through the interpolation function and true values of the signal to estimate. It follows that, to achieve a more accurate approximation, the curvature of the function needs to be taken into consideration together with the intensity values. By doing so, the overall behavior of the joint parameter intensity-curvature within the neighborhood is captured. As it is true that given two intensity values that are the same, it is also true that because of

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The Extension of the Theory to the Trivariate Linear Interpolation Function

concavity or convexity in the locality of the intensity value, the function behavior may differ substantially in the neighborhood. Thus, the novel methodology proposes that within a given interpolation paradigm, the issue of resampling is addressed in the spatial domain of the Sub-pixel Efficacy Region for each voxel. Spatial location and size of such region is dependent on both the intensity values at the neighborhood of the voxel to re-sample and the curvature of the interpolation function. Furthermore, for given misplacement, consistently with the bivariate function, it is reiterated that in order to determine interpolation error improvement, re-sampling is performed at locations that are not necessarily the same as the misplacement. Instead re-sampling takes place at intra-voxel locations that change across voxels depending on the spatial extent of the Sub-pixel Efficacy Region.

Methodology To capture the variation across the voxel of both intensity and curvature of the function, the intensitycurvature term is defined as the product of the interpolation function times the sum of its second order partial derivatives: ∂2 (h (x, y, z))/δxδy, ∂2(h (x, y, z))/δxδz, and ∂2(h (x, y, z))/δyδz respectively. The intensity-curvature term is computed at the grid node and named Eo, and at the generic intra-pixel location and named EIN. The ratio Eo(x, y, z) / EIN(x, y, z) constitutes the Intensity-Curvature Functional: ΔE = ΔE(x, y, z). For ΔE, the first order partial derivatives are calculated with respect to x, y, and z. A polynomial system constituted by the first order partial derivatives of ΔE is thus obtained and its solution determines the set of points that constitute the Sub-pixel Efficacy Region. These sets of points are of the type (xsre, ysre, zsre) and they initiate the process of re-calculation of the interpolation function at each voxel. Given a misplacement (x0, y0, z0), the misplacement used for signal interpolation is calculated based on the Sub-pixel Efficacy Region in order to improve the interpolation error and it consists of the novel re-sampling location (xr0, yr0, zr0). These novel re-sampling locations are derived through equation (1), which recalls equation (21) of Chapter VII. { h(xsre-xr0, ysre-yr0, zsre-zr0) * { [(∂2 (h(x, y, z))/∂x∂y) + (∂2 (h(x, y, z))/∂x∂z) + (∂2 (h(x, y, z))/∂y∂z)] (xsre-xr0, ysre-yr0, zsre-zr0) } } / { h(xsre, ysre, zsre) * { [ (∂2 (h(x, y, z))/∂x∂y) + (∂2 (h(x, y, z))/∂x∂z) + (∂2 (h(x, y, z))/∂y∂z) ] (xsre, ysre, zsre) } } = { EIN (xsre- x0, ysre- y0, zsre- z0) / EIN (xsre, ysre, zsre)}



(1)

Definition of Intensity-Curvature Functional The definition of Intensity-Curvature Functional follows: ΔE(x, y, z) = Eo (x, y, z) / EIN(x, y, z)



(2)

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Calculation of intensity-curvature terms, at the grid node (Eo) and the generic intra-pixel location (EIN) leads to the following expression: EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) = - ωf Hxyz (x, y, z) [ θxy + θxz + θyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]

(3)

xyz

EIN (Ψxy) = ∫ ∫ ∫ h(x, y, z) (∂2 (h(x,y,z)) /∂x∂y) dx dy dz = 000

Hxyz (x, y, z) [- θxy ωf + z (1 - ζz ×) ωf ] xyz EIN (Ψzx) = ∫ ∫ ∫ h (x, y, z) (∂2(h(x,y,z)) /∂z∂x) dx dy dz =

(4)

000

Hxyz (x, y, z) [- θxz ωf + y (1 - ζy ×) ωf ]

(5)



xyz

EIN (Ψyz) = ∫ ∫ ∫ h (x, y, z) (∂2(h(x,y,z)) /∂y∂z) dx dy dz = 000

Hxyz (x, y, z) [- θzy ωf + x (1 - ζx ×) ωf ] Eo = Eo(x, y, z) = Eo (Ψxy) + Eo (Ψzx) + Eo (Ψyz) =

(6)



xyz

∫ ∫ ∫ f (0, 0, 0) (∂2 (h(x,y,z))/∂x∂y + ∂2 (h(x,y,z))/∂y∂z + ∂2 (h(x,y,z)) /∂x∂z) (0 ,0, 0) 000

dx dy dz = - x y z ωf f(0, 0, 0) [θxy + θxz + θyz]





(7)

Where, Hxyz (x, y, z) is the primitive function of h (x, y, z) with respect to x, y and z, ∂2 (h(x,y,z)) /∂x∂y = - θxy ωf, ∂2 (h(x,y,z))/∂y∂z = - θyz ωf and ∂2 (h(x,y,z))/∂x∂z = - θxz ωf. θxy, θxz, θyz and ωf express relationship between f(0, 0, 0), the intensity at the pixel to re-sample, and the pixel intensity at its neighbourhood, and they were defined in part I of the book.

Study of the Intensity-Curvature Functional Calculation of the first order derivatives of ΔE leads to the following expressions, where θxyz = θxy + θxz + θyz.

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(∂ ΔE(x, y, z) / ∂x) = { y z θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - x y z θxyz f(0, 0, 0) { Hyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζx ×) Hxyz (x, y, z) } } / { Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }2



(8)

(∂ ΔE(x, y, z) / ∂y) = { x z θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - x y z θxyz f(0, 0, 0) { Hxz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζy ×) Hxyz (x, y, z) } } / { Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }2

(9)

(∂ ΔE(x, y, z) / ∂z) = { x y θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - x y z θxyz f(0, 0, 0) { Hxy (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζz ×) Hxyz (x, y, z) } } / { Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }2

(10)

Study of the polynomial system of equations (8-10) lead to the following: z1,2 = ( -b ± Δ1/2 ) / 2 a



(11)

where: b = {[ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] y/4 ωf + (θx/2 - y/2 θxy ωf ) [- θz + y/2 θzy ωf ] θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (θxz/4 ωf - y/4 ωf ) [-f(0,0,0) - y/2 θy] } a = {(θz/2 - y/2 θzy ωf ) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (θxz/4 ωf - y/4 ωf ) (- θz + y/2 θzy ωf )} Δ = b2 - 4 a c c = { [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] - (θx/2 - y/2 θxy ωf ) [-f(0,0,0) - y/2 θy ] } y1,2 = ( -b ± Δ1/2 ) / 2 a



(12)

where:

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The Extension of the Theory to the Trivariate Linear Interpolation Function

b = { -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + (-f(0,0,0) - z θz) [(- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf )]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf )/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf ) + z θz (-θxy ωf/4 + z ωf/4) } a = { (θy/2 - z θzy ωf/2) [ ( θxy ωf /2 - z ωf/2) + (-θxy ωf/4 + z ωf/4)] } Δ = b2 - 4 a c c = { f(0,0,0) [ θx/2 - z/2 θxz ωf ] + z θz (θx/2 - z/2 θxz ωf ) } x = [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] Definition I

(13)

For the trivariate linear interpolation function, the the Sub-pixel Efficacy Region is φ = {(x, y, z): ∂ (ΔE(x, y, z)) /∂x = 0, ∂ (ΔE(x, y, z)) /∂y = 0 and ∂ (ΔE(x, y, z)) /∂z = 0}. The Sub-pixel Efficacy Region Φ is identified by equations (11-13). Based on (11-13) the SRE values (xsre(i), ysre(j), zsre(k)), i, j, k = 1, 2; can be found. The solution of (1) which is based on the SRE values does furnish the novel re-sampling locations (x r0, yr0, zr0) ≠ (xr0(i, s), yr0(j, s), zr0(k, s)), i, j, k, s = 1, 2; and also furnishes equations (14), (15) and (16) as they are given in the following paragraph. Novel Re-sampling Locations (xr0, y r0, z r0) of h(x y z) They derive from the solutions of the following second order equations respectively in (x r0 - xsre), (y - ysre), and (zr0 - zsre): r0

(xr0 - xsre)2 [θx ωf - (y0 - ysre) θxy ωf - (z0 - zsre) θxz ωf + (z0 - zsre) (y0 - ysre) ωf 2 ] + (xr0 - xsre) { f(0,0,0) ωf + θx θz + (z0 - zsre) θx ωf + θx (θx + θy) + (y0 - ysre) θx ωf - (y0 - ysre) θxy [θz + (z0 - zsre) ωf + θy + (y0 - ysre) ωf + θx] + (z0 - zsre) (y0 - ysre) ωf θyz - (z0 - zsre) θxz [θz + (z0 - zsre) ωf + θy + (y0 - ysre) ωf + θx] + (z0 - zsre) (y0 - ysre) [θz + (z0 - zsre) ωf + θy + (y0 - ysre) ωf + θx] ωf + θy (y0 – ysre) ωf + θz (z0 - zsre) ωf + θy (y0 - ysre) ωf + θz (z0 - zsre) ωf + ΔEε ε } + { f(0,0,0) (θz + θy + θx) + f(0,0,0) θz + f(0,0,0) (z0 - zsre) ωf + f(0,0,0) (y0 - ysre) ωf + (y0 - ysre) θy [θz + (z0 - zsre) ωf + θy + θx] + (y0 - ysre)2 ωf θy + (z0 - zsre) θz [θz + (y0 - ysre) ωf + θy + θx] + (z0 –

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The Extension of the Theory to the Trivariate Linear Interpolation Function

zsre)2 ωf θz - (z0 - zsre) (y0 - ysre) θyz [θz + (z0 - zsre) ωf + θy + (y0 - ysre) ωf + θx] - ΔEε ε [3 + θxy + θzx + θyz - (z0 - zsre) - (y0 - ysre)] } = 0





(14)

(yr0 - ysre)2 [θy ωf - (x0 - xsre) θxy ωf - (z0 - zsre) θyz ωf + (z0 - zsre) (x0 - xsre) ωf 2 ] + (yr0 - ysre) { f(0,0,0) ωf + θx (x0 - xsre) ωf + θy [θz + (z0 - zsre) ωf + θy + (x0 - xsre) ωf + θx] + (z0 - zsre) θz ωf - θxy (x0 - xsre)2 ωf - θxy (x0 - xsre) [θz + (z0 - zsre) ωf + θy + θx] - (z0 - zsre) θyz [θz + (z0 - zsre) ωf + θy + θx] + (z0 - zsre) (x0 - xsre) ωf θyz + (z0 - zsre) (x0 - xsre) θyz ωf - (z0 - zsre) (x0 - xsre) θxz ωf + (z0 – zsre) (x0 - xsre)2 ωf 2 + (z0 - zsre) (x0 - xsre) [θz + (z0 - zsre) ωf + θy + θx] ωf + ΔEε ε } + { f(0,0,0) (θz + θy + θx) + f(0,0,0) θz + f(0,0,0) (z0 - zsre) ωf + f(0,0,0) (x0 - xsre) ωf + (x0 - xsre) θx θz + (z0 - zsre) (x0 - xsre) θx ωf + (x0 - xsre) θx (θx + θy) + θx (x0 - xsre)2 ωf + (z0 - zsre) θz [θz + (x0 - xsre) ωf + θy + θx] + (z0 - zsre)2 ωf θz - (x0 - xsre)2 ωf (z0 - zsre) θxz - (z0 - zsre) (x0 - xsre) θxz [θz + (z0 - zsre) ωf + θy + θx] - ΔEε ε [3 + θxy + θzx + θyz - (z0 - zsre) - (x0 - xsre)] } = 0



(15)

(zr0 - zsre)2 [θz ωf - (y0 - ysre) θyz ωf - (x0 - xsre) θxz ωf + (y0 - ysre) (x0 - xsre) ωf 2 ] + (zr0 - zsre) { f(0,0,0) ωf + θx (x0 - xsre) ωf + θy (y0 - ysre) ωf + θz [θx + (y0 - ysre) ωf + θy + (x0 - xsre) ωf + θz] - (x0 - xsre) (y0 - ysre) θxy ωf - θyz (y0 - ysre) [θz + (y0 - ysre) ωf + θy + θx] + (y0 - ysre) (x0 - xsre) ωf θyz - θxz (x0 - xsre)2 ωf - θxz (x0 - xsre) [θz + (y0 - ysre) ωf + θy + θx] + (y0 - ysre) (x0 - xsre)2 ωf 2 + (y0 - ysre) (x0 - xsre) [θz + (y0 - ysre) ωf + θy + θx] ωf + ΔEε ε } + { f(0,0,0) (θz + θy + θx) + f(0,0,0) θz + f(0,0,0) (y0 - ysre) ωf + f(0,0,0) (x0 - xsre) ωf + (x0 - xsre) θx θz + (x0 - xsre) θx (θx + θy) + (y0 - ysre) (x0 - xsre) θx ωf + θx (x0 - xsre)2 ωf + (y0 - ysre) θy [θz + (x0 - xsre) ωf + θy + θx] + (y0 - ysre)2 ωf θy - (x0 - xsre)2 ωf (y0 - ysre) θxy - (y0 - ysre) (x0 - xsre) θxy [θz + (y0 - ysre) ωf + θy + θx] - ΔEε ε [3 + θxy + θzx + θyz - (y0 - ysre) - (x0 - xsre)] } = 0



(16)

Where:

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The Extension of the Theory to the Trivariate Linear Interpolation Function

ΔEε ε = h(xsre-x0, ysre-y0, zsre-z0) * { ( (∂2 (h(x, y, z))/∂x∂y) + (∂2(h(x, y, z))/∂x∂z) + (∂2(h(x, y, z))/∂y∂z) ) (xsre-x0, ysre-y0, zsre-z0) } / [ 3 + θxy + θzx + θyz - (x0 - xsre) - (y0 - ysre) - (z0 - zsre)] What has to be acknowledged, as the MATLAB®. (c) 1984-2007 The MathWorks, Inc. code SRE3D2007 located at www.sourcecodewebsite.com shows is that the novel re-sampling locations (x r0, yr0, zr0) are normalized to the range [(0, 0, 0), (x0, y0, z0)] where (x0, y0, z0) is the misplacement.

Second Order Partial Derivatives of the Intensity-Curvature Functional ΔE To determine the nature (maximum or mininum) of the extremes of ΔE the determinant Ω(3D) can be calculated on the basis of the second order partial derivatives.



(3D)

=

2

[ Eo / EIN

2 2

2

2

[ Eo / EIN

[ Eo / EIN

2

[ Eo / EIN

[ Eo / EIN

2

[ Eo / EIN

2

2

[ Eo / EIN

2

[ Eo / EIN

2

[ Eo / EIN



(17)

2

The calculation of the second order partial derivatives is reported in appendix V.

SUMMARY This chapter has outlined studied and solved the problem of the three dimensional linear interpolation improvement based on the Sub-pixel Efficacy Region. The chapter implicitly puts the emphasis on the relevance of the novel theory in terms of its capability to group together interpolators of different dimensionality. Therefore, it is revealed that even thought one dimension is added to the problem, the inclusion of all of the three second-order partial derivatives of the interpolation function into the Intensity-Curvature Functional provides the mathematical foundation to extend the theory. This is carried out with detailed description leading to a comprehensive coverage of the mathematics. The chapter stresses on the dependency of the formulation on the voxel intensity and the second order derivatives of the interpolation function and also on the consequential image re-sampling that takes place at locations that vary voxel-by-voxel. The chapter devotes most of the effort to inform the reader of the meticulous mathematics involved in the three dimensional approach. This chapter has also a clear connection with Chapter VII and can be fully understood based on the latter. It is then recalled, alike the two-dimensional case that the novel re-sampling locations depend in their spatial extent on the neighboring pixel intensity and the curvature of the interpolation function.

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Reference Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53.

187

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Chapter XI

The Results of the Sub-Pixel Efficacy Region Based Trivariate Linear Interpolation Function

Introduction This chapter presents results relevant to the evaluation of the performance of classic and SRE-based trivariate interpolation functions. The forthcoming text reports on the validation procedure employed to quantify the interpolation error of the two model functions along with the information on the resolution of the MRI database that was processed. The presentation of the results is both quantitative through the plots of the RSME Ratio and quantitative through the presentation of the spectral power evolutions of functional MRI data. As anticipated in Chapter I the spectral power evolutions were obtained processing the fMRI volume with the misplacement X = 0.49, Y = 0.49, Z = 0.49 employing the two interpolation paradigms (classic and SRE-based). Particularly, the RSME Ratio quantified the relativity of performance between the two types of trivariate linear interpolation functions while processing T1-weigthed, T2-weighted and functional MRI. The spectral power evolutions quantified for the functional MRI the differences in frequency spectral content between the two classes of interpolators. Interesting to note that the spectral power evolutions clearly show in this chapter their capability to reveal differences, between the images obtained through the two different interpolators, which otherwise would not be observed in the image space. Such differences are somehow hidden in the k-space (Fourier domain).

Validation Procedure At the aim to show the error improvement obtainable through the methodological approach of the unifying theory the same type of motion correction paradigm employed for the bivariate linear is presented in this chapter for the trivariate linear functions. An original three dimensional image is shifted. Original means prior to any processing. It is then motion corrected of the initial shift and interpolated Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Results of the Sub-Pixel Efficacy Region Based Trivariate Linear Interpolation Function

with the classic trivariate linear function. This procedure is called processing with classic interpolation as the reader may recall from Chapter VIII of the book. The root-mean-square-error (RMSE) is calculated between original and processed images. Concurrently, the original image is processed with the SRE-based trivariate linear interpolation function (that means that the image is calculated at the novel re-sampling locations). By doing so, the RMSE Ratio calculates a measure of the relativity of performance between classic and SRE-based trivariate linear interpolation functions. The performance is obviously measured on the basis of the interpolation error and so the RMSE Ratio allows elucidating which one of the two interpolation functions delivers best approximation. Real Magnetic Resonance Imaging data were employed at the aim to study the interpolation error improvement determined through the SRE-based trivariate linear interpolation function and also to study the spectral characteristics of the re-sampled images as described in the following of this chapter. The data set is the same used for the validation of the SRE-based bivariate linear interpolation function and for convenience of the reader it is hereto recalled. • • • •

T1 MRI (T1-MRI-230 and T1-MRI-450). Matrix resolution: 176 x 208 with 1.00 x 1.00 mm of pixel size, with 75 slices and an inter-slice resolution of 1.00 mm. T2 MRI. Matrix resolution: 177 x 182 with 1.00 x 1.00 mm of pixel size, with 19 slices and an inter-slice resolution of 1.00 mm. Functional MRI (fMRI-TS1). Matrix resolution: 128 x 128 with 1.72 x 1.72 mm of pixel size, with 20 slices and an inter-slice resolution of 1.72 mm. Functional MRI (fMRI-snffM00587). Matrix resolution: 53 x 63 with 3.00 x 3.00 mm of pixel size, with 30 slices and an inter-slice resolution of 3.00 mm.

Analy sis of the RMSE Ratio Obtained with MRI Data The plot of the RMSE Ratio versus the concurrent misplacement along X, Y and Z directions is shown for T1-MRI-230, T1-MRI-450, T2-MRI, fMRI-TS1 and fMRI-snffM00587 in figures 1, 2, 3, 4, and 5

Figure 1. RMSE Ratio: T1-MRI-230, concurrent misplacements along x, y and z (all volume). The range of misplacement is [(0, 0, 0), (0.99, 0.99, 0.99)] at steps of (0.01, 0.01, 0.01)

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Figure 2. RMSE Ratio: T1-MRI-450, concurrent misplacements along x, y and z (all volume). The range of misplacement is [(0, 0, 0), (0.99, 0.99, 0.99)] at steps of (0.01, 0.01, 0.01)

Figure 3. RMSE Ratio: T2-MRI, concurrent misplacements along x, y and z (all volume). The range of misplacement is [(0, 0, 0), (0.99, 0.99, 0.99)] at steps of (0.01, 0.01, 0.01)

Figure 4. RMSE Ratio: functional MRI (fMRI-TS1), concurrent misplacements along x, y and z (all volume). The range of misplacement is [(0, 0, 0), (0.99, 0.99, 0.99)] at steps of (0.01, 0.01, 0.01)

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Figure 5. RMSE Ratio: functional MRI (fMRI-snffM00587), concurrent misplacements along x, y and z (all volume). The range of misplacement is [(0, 0, 0), (0.99, 0.99, 0.99)] at steps of (0.01, 0.01, 0.01)

Figure 6. Error image analysis for functional MRI (fMRI-snffM00587), all volume. The misplacement is (0.49, 0.49, 0.49). The images were obtained processing with classic trivariate interpolation. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

respectively. Each figure plots the RMSE ratio that is relevant to the entire MRI volume. The portion of the curves below zero shows that the SRE-based trivariate interpolation function outperforms the classic trivariate interpolation function. The portion of the curves above zero shows the opposite. For the four MRI data sets the range of misplacement was: [(0, 0, 0), (0.99, 0.99, 0.99)] at steps of (0.01, 0.01, 0.01). Figure 1 shows progressive interpolation error improvement starting approximately at (0.25, 0.25, 0.25) which reaches levels that are up to approximately 150% confirming that the SRE-based trivariate interpolation function can outperform its corresponding classic paradigm. The fact that the superiority of the SRE-based paradigm is limited to a portion of the voxel is consistently noticeable also in figures 2 through 5. Figure 2 shows that the interpolation error improvement also starts approximately at (0.25, 0.25, 0.25) and reaches progressively levels that are up to approximately 200%. The similarity of the plots of the RSME Ratio shown in figures 1 and 2 consists in that the RSME profiles maintain a well defined

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Figure 7. Error image analysis for functional MRI (fMRI-snffM00587), all volume (SRE). The misplacement is (0.49, 0.49, 0.49). The images were obtained processing with SRE-based trivariate interpolation. The error images were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/

Figure 8. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 13. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -640.224

behavior till approximately (0.5, 0.5, 0.5) (the center of the voxel) and then they become scattered. This similarity can also be seen in figures 3 and 4 for T2-MRI and fMRI-TS1. In figures 3 and 4 the progression of the improvement of the interpolation error determined through the SRE-based trivariate linear interpolation function starts approximately at (0.25, 0.25, 0.25) for T2MRI and fMRI-TS1 respectively and reaches values that are approximately up 175% (see figure 3) and 700% (see figure 4). In figure 5 (fMRI-snffM00587) the behavior of the RMSE Ratio is similar to that seen in figures 1 through 4 and shows progression of the improvement of the interpolation error. The improvement starts however approximately at the voxel location of coordinates (0.38, 0.38, 0.38) and reaches values that are approximately up to 100%.

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Results with Functional Magnetic Resonance Volumes After processing the functional MRI data set (fMRI-snffM00587) with classic and SRE-based trivariate linear interpolation functions of the misplacement (0.49, 0.49, 0.49), two sets of error images were obtained through subtraction between: (i) the original image and the image obtained processing with classic interpolation and (ii) the original image and the image obtained with SRE-based interpolation. It is worth noting that the error images are calculated extracting the square root of the second power of the difference between each pixel of the images. The first set of error images are shown in figure 6 and they display the residual of the classic interpolation paradigm. The second set of error images are displayed in figure 7 and they show the residual of the SRE-based interpolation paradigm. Although it is possible to attempt visual inspection of the two sets of error images it is opinion of the writer that a more accurate evaluation of the advantages of the SRE-based trivariate interpolation function may be attainable through the spectral power evolutions. For each slice of the fMRI-snffM00587 data set figures 8 through 37 present the spectral power evolutions (Ciulla & Deek, 2005). For a thorough explanation as to how the spectral power evolutions were obtained the reader is referred to Chapter I section: “Fast Fourier Transform and Spectral Power Analysis.” In each graph the number located in the upper left corner corresponds to the slice number identifier seen in the error images in figures 6 and 7. Each of the figures quantifies the difference abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) within the discrete interval [979, 1023]. O_E is the spectral energy of the original image (prior to any processing), NOSRE_E is the spectral energy of the image processed with classic trivariate linear interpolation and SRE_E is the spectral energy of the image processed with SRE-based trivariate linear interpolation function. The evolution of the difference abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) indicates which one between classic and SRE-based interpolation (in the specific case of the trivariate linear paradigm treated in this chapter) furnishes the greatest deviation (in terms of spectral energy) from the spectral energy of the original image. As mentioned above, each functional Magnetic Resonance Image (fMRI) was processed with the misplacement (0.49, 0.49, 0.49). A negative value of the difference indicates that the spectral content

Figure 9. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 14. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1078.075

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Figure 10. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 15. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -978.442

Figure 11. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 16. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1049.307

Figure 12. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 17. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1142.59

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Figure 13. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 18. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1103.55

Figure 14. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 19. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1295.463

Figure 15. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 20. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -922.222

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Figure 16. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 21. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -810.084

Figure 17. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 22. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -604.004

Figure 18. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 23. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -615.206

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Figure 19. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 24. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -515.303

Figure 20. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 25. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -643.512

Figure 21. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 26. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -472.991

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Figure 22. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 27. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -399.643.

Figure 23. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 28. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -388.519

Figure 24. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 29. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -301.16

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Figure 25. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 30. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -63.022

Figure 26. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 31. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -53.771

Figure 27. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 32. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -21.376

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Figure 28. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 33. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -21.501

Figure 29. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 34. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -218.551

Figure 30. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 35. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -187.662

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The Results of the Sub-Pixel Efficacy Region Based Trivariate Linear Interpolation Function

Figure 31. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 36. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -125.899

Figure 32. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 37. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -233.385

Figure 33. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 38. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -187.56

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Figure 34. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 39. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -234.68

Figure 35. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 40. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -49.816

Figure 36. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 41. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -164.569

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Figure 37. Spectral power evolution, functional MRI (fMRI-snffM00587), slice 42. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is 162.105 and suggests the superiority of the classic linear interpolation function

Figure 38. Overlay in the image space, functional MRI (fMRI-snffM00587). The white pixels represent the image space location that embeds the spectral energy plotted in the graphs of figures 9 through 37. For each image, the pixels in white correspond to the interval marked by the dark dot in the spectral power evolutions. The original images are courtesy of: The Wellcome Trust Centre for Neuroimaging (University College London - UK), www.fil.ion.ucl.ac.uk.

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of the image obtained through the SRE-based trivariate linear interpolation is closer to the spectral content of the original image. The average value of abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) was calculated for each of the spectral power evolutions shown in figures 8 through 37 and suggests the superiority of the SRE-based trivariate linear interpolation function (except for figure 37). Thus, the SRE-based function is more successful than the classic function in preserving the spectral characteristics of the fMRI. On the other hand, a positive difference indicates that the classic trivariate liner function is more successful than the SRE-based. Therefore, the values of the difference abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is plotted to assess as to which one of the two interpolation functions is more conservative in terms of the spectral content of the original image. The plots shown in the spectral power evolutions seen in figures 8 through 37 are within the discrete interval [979, 1023] because such interval was found to be the one most relevant in demonstrating differences in spectral content of the three types of images: (i) original, (ii) processed with classic interpolation, (iii) processed with SRE-based interpolation. The black dots in the graphs of figures 8 through 37 points to a particular spectral content which corresponds to the sub-interval [1003, 1004] ≠ [979, 1023]. Τhe image space location corresponding to the spectral content pointed by the black dots will be shown in figure 38. For each spectral power evolution shown in figures 8 through 37 is given a quantitative description of the values of maximal and minimal difference abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) along with the average value (indicated in the figure caption). Maximum, minimum and average values were calculated in the interval [979, 1023]. In figures 8 through 14 the maximum and minimum values found in the spectral power evolutions are: 284.246, -3879.376 (figure 8), 96.259, -4457.307 (figure 9), 187.858, -4290.705 (figure 10), 381.569, -6484.609 (figure 11), 197.718, -5210.573 (figure 12), 4.416, -4047.665 (figure 13), 121.359, -6676.765 (figure 14). Also, in figures 15 through 26 the maximum and minimum values found in the spectral power evolutions are: 50.645, -2839.655 (figure 15), 257.156, -3105.717 (figure 16), 904.179, -2954.85 (figure 17), 1314.277, -4715.955 (figure 18), 642.241, -2230.093 (figure 19), 382.057, -3091.852 (figure 20), 485.269, -2455.327 (figure 21), 215.483, -1638.931 (figure 22), 395.75, -1965.087 (figure 23), 1417.443, -2059.559 (figure 24), 4281.795, -1145.743 (figure 25), 4762.889, -1813.22 (figure 26). And, in figures 27 through 37 the maximum and minimum values found in the spectral power evolutions are: 4912.802, -1638.62 (figure 27), 4726.402, -1585.586 (figure 28), 399.917, -1213.257 (figure 29), 967.7, -1003.282 (figure 30), 1909.104, -2036.66 (figure 31), 303.153, -929.21 (figure 32), 368.686, -961.983 (figure 33), 162.964, -1128.199 (figure 34), 1569.503, -884.092 (figure 35), 1414.4, -969.108 (figure 36), 5390.72, -800.62 (figure 37). Black dots in the graphs of figures 8 through 37 identifies frequency components of the images that correspond to pixels that are in white overlay in the image space in figure 38. This is visible for each of the slices of the fMRI-snffM00587 data set. Therefore, figure 38 elucidates for either the image processed with classic interpolation or the image processed with SRE-based interpolation what are the image space locations that embed frequency components closer to that of the original image. Due to acknowledge that if instead of the subinterval [1003, 1004] which shows the superiority of SRE-based interpolation (as seen in figures 8 through 36), a different interval would have been analyzed (if the black dots would have been shown at different locations in the spectral power evolutions and the corresponding image space overlay would have been produced), possibly, but not necessarily, they

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would have been found results suggesting the superiority of the classic interpolation function. Thus, naturally, the analysis presented here is demonstrative, however does not intend to assert truths that have absolute power. In conclusion, the power spectral evolutions combined with the pixels in the white overlay in the image space constitute of a relatively powerful methodology to connect k-space and image space of the MRI. This methodology was adopted also for the evaluation of the FFT characteristics of the polynomial SRE-based interpolation functions that shall be presented in forthcoming chapters of the book.

SUMMARY Chapter XI has presented the results of the Sub-pixel Efficacy Region based trivariate linear interpolation function. Also, the chapter focuses on the presentation of the spectral power evolutions and the image overlay which constitutes of a signal processing technique that is able to connect k-space and image space of functional MRI volumes processed with the two types of trivariate linear interpolation functions: classic and SRE-based.

Reference Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53.

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Chapter XII

Equating the Two Intensity-Curvature Terms, Before and After Interpolation, Attempting to Obtain Resilient Interpolation: Trivariate Linear Interpolation Function

Introduction As well accepted in literature it should be almost impossible to reconstruct the signal intensity value at time-space location where the sampling instrument has not recorded the signal. This happens because sampling is a discrete process, while instead signals (as those of biological nature among others) are continuous. Indeed this mismatch between nature and humanly invented devices gives immediate explanation as to why interpolation is a signal reconstruction technique that has quite substantial and general relevance. This chapter introduces the novel concept of resilient interpolation for the case of the trivariate linear function while continuing the treatise from Chapter VII where the concept was presented for the case of the bivariate linear function. The mathematical process is presented in full and equations are given to characterize resilient interpolation. Additionally, a discussion is provided to the reader to further clarify the methodology employed and the relevant implications. Overall, the intent of this purely theoretical chapter is that of attempting to further improve the approximation capabilities of the trivariate linear interpolation function while making use of the knowledge provided with the SRE-based methodological approach and also introducing a further conceptualization. The intent is therefore once again a more accurate signal reconstruction to be obtained through a revised form of the classic interpolation function. Details of the meaning behind the concepts herein presented indicate

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Equating the Two Intensity-Curvature Terms, Before and After Interpolation

that resilient interpolation is such that the intensity-curvature content before and after interpolation remains unchanged.

THE MATHEMATICAL PROCEDURE From equations (11) and (10) of Chapter IV it can be written that the intensity curvature terms before (Eo), and after interpolation (EIN) are: EIN = EIN(x, y, z) = - ωf Hxyz (x, y, z) [ θxy + θxz + θyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] (1) Eo = Eo(x, y, z) = - f (0, 0, 0) x y z ωf [θxy + θyz + θzx ]

(2)

Where through equations (9) through (12) of Chapter III it is posited that: θxy = θyx = - [ f(0, 0, 0) – f(1, 0, 0) – f(0, 1, 0) + f(1, 1, 0) ] / ωf

(3)

θxz = θzx = - [ f(0, 0, 0) – f(1, 0, 0) – f(0, 0, 1) + f(1 ,0, 1) ] / ωf

(4)

θyz = θzy = - [ f(0, 0, 0) – f(0, 1, 0) – f(0, 0, 1) + f(0, 1, 1) ] / ωf

(5)

ωf = [ f(1, 1, 1) - f(1, 1, 0) - f(0, 1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) - f(0, 0, 0) ] (6) Let us proceed by posing Eo(x, y, z) = EIN(x, y, z), and solving the resulting equation in the unknown f(0, 0, 0). The problem herein addressed can be formulated in the following terms. For the given re-sampling location what is the true pixel (signal) intensity value? Can the model interpolation function calculate that signal intensity value that is the one that would be sampled? It is reasonable to attempt to answer the question by posing in equality the two intensity curvature terms which is like assuming that there exists a pixel (signal) intensity value that remains the same before and after interpolation. This pixel (signal) intensity value is not the one we start from (before interpolation) nor is the value that it would be obtained through re-sampling the signal through the model interpolation function at the given placement. The question that is posed herein is: “For given placement of the re-sampling location, and given model interpolation function, what would be the signal intensity that is the same in the two conditions of non-interpolated signal and interpolated signal?” The model interpolation function that is treated in this chapter is the trivariate linear. The proposed solution to above problem virtually bypasses the effect of the interpolator onto the signal and determines a novel signal that it is assumed to be resilient to the model interpolation function, that is: it is not affected by re-sampling. In other words: “what is the model interpolation function that is such not to change the signal intensity and therefore the spectral characteristics of the signal such that no smoothing would be introduced?” The flexibility of the mathematical formulation of the two intensity-curvature terms before and after interpolation allows approaching and proposing a solution to the above posed problem. Let us thus proceed in the calculations.

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Hxyz (x, y, z) = f(0, 0, 0) x y z + x2/2 y z θx + x y2/2 z θy + x y z2/2 θz - x2/2 y2/2 z θxy ωf - x y2/2 z2/2 θzy ωf - x2/2 z2/2 y θxz ωf + x2/2 y2/2 z2/2 ωf = x y z [ f(0 ,0, 0) + x/2 θx + y/2 θy + z/2 θz - x/2 y/2 θxy ωf - y/2 z/2 θzy ωf - x/2 z/2 θxz ωf + x/2 y/2 z/2 ωf ] Hxyz (x, y, z) = x y z [ f(0, 0, 0) + x/2 θx + y/2 θy + z/2 θz - x y/4 θxy ωf - y z/4 θzy ωf - x z/4 θxz ωf + x y z/8 ωf ]



(7)

Let us rewrite equations (3) through (6) in the following form: θxy = θyx = [ - f(0, 0, 0) / ωf ] + [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0)] / ωf

(8)

θxz = θzx = [ - f(0, 0, 0) / ωf ] + [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1)] / ωf

(9)

θyz = θzy = [- f(0, 0, 0) / ωf ] + [ - f(0, 1, 0) - f(0, 0, 1) + f(0, 1, 1)] / ωf

(10)

ωf = [ f(1, 1, 1) - f(1, 1, 0) - f(0, 1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) ] - [ f(0, 0, 0) ] (11) Thus: [ θxy + θxz + θyz ] = [ - 3 f(0, 0, 0) / ωf ] + [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0)] / ωf + [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1)] / ωf + [ - f(0, 1, 0) - f(0, 0, 1) + f(0, 1, 1)] / ωf = [ - 3 f(0, 0, 0) / ωf ] + Θ



(12)

[ θxy + θxz + θyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = { [ - 3 f(0, 0, 0) / ωf ] + Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) } (13) Where it is posited that: Θ = [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0)] / ωf + [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1)] / ωf + [ - f(0, 1, 0) - f(0, 0, 1) + f(0, 1, 1)] / ωf Thus:

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(14)



Equating the Two Intensity-Curvature Terms, Before and After Interpolation

Eo = Eo(x, y, z) = - f (0, 0, 0) x y z ωf [θxy + θyz + θzx ] = - f (0, 0, 0) x y z ωf { [ - 3 f(0, 0, 0) / ωf ] + Θ } = 3 [ f(0, 0, 0) ]2 x y z - f (0, 0, 0) x y z ωf Θ (15) From definition V of Chapter II it is herein reminded that: θx = [ f(1, 0, 0) - f(0, 0, 0) ] θy = [ f(0, 1, 0) - f(0, 0, 0) ] θz = [ f(0, 0, 1) - f(0, 0, 0) ] which along with equations (8) through (11) allows writing that: Hxyz (x, y, z) = x y z [ f(0, 0, 0) + x/2 θx + y/2 θy + z/2 θz - x y/4 θxy ωf - y z/4 θzy ωf - x z/4 θxz ωf + x y z/8 ωf ] = x y z { f(0, 0, 0) + x/2 [ f(1, 0, 0) - f(0, 0, 0) ] + y/2 [ f(0, 1, 0) - f(0, 0, 0) ] + z/2 [ f(0, 0, 1) - f(0, 0, 0) ] - x y/4 [ - f(0, 0, 0) ] - x y/4 [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0) ] - y z/4 [- f(0, 0, 0) ] - y z/4 [ - f(0, 1, 0) - f(0, 0, 1) + f(0, 1, 1) ] - x z/4 [ - f(0,0,0) ] - x z/4 [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1) ] + x y z/8 [ f(1, 1, 1) - f(1, 1, 0) - f(0, 1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) ] - x y z/8 [ f(0, 0, 0) ] } = x y z { f(0, 0, 0) + x/2 [ f(1, 0, 0) ] - x/2 [ f(0, 0, 0) ] + y/2 [ f(0, 1, 0) ] - y/2 [ f(0, 0, 0) ] + z/2 [ f(0, 0, 1) ] - z/2 [ f(0, 0, 0) ] - x y/4 [ - f(0, 0, 0) ] - x y/4 [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0) ] - y z/4 [- f(0, 0, 0) ] - y z/4 [ - f(0, 1, 0) - f(0, 0, 1) + f(0, 1, 1) ] - x z/4 [ - f(0, 0, 0) ] - x z/4 [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1) ] + x y z/8 [ f(1, 1, 1) - f(1, 1, 0) - f(0, 1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) ] - x y z/8 [ f(0, 0, 0) ] } = x y z { f(0, 0, 0) [ 1 - x/2 - y/2 - z/2 + x y/4 + y z/4 + x z/4 - x y z/8 ] + x/2 [ f(1, 0, 0) ] + y/2 [ f(0, 1, 0) ] + z/2 [ f(0, 0, 1) ] - x y/4 [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0) ] - y z/4 [ - f(0, 1, 0) - f(0, 0, 1) + f(0, 1, 1) ] - x z/4 [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1)] + x y z/8 [ f(1, 1, 1) - f(1, 1, 0) - f(0,

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1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) ] } = { f(0, 0, 0) x y z [ 1 - x/2 - y/2 - z/2 + x y/4 + y z/4 + x z/4 - x y z/8] } + x y z { x/2 [ f(1, 0, 0) ] + y/2 [ f(0, 1, 0) ] + z/2 [ f(0, 0, 1) ] - x y/4 [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0) ] - y z/4 [ - f(0, 1, 0) f(0, 0, 1) + f(0, 1, 1)] - x z/4 [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1)] + x y z/8 [ f(1, 1, 1) - f(1, 1, 0) f(0, 1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) ] } Therefore: Hxyz (x, y, z) = f(0, 0, 0) Θxyz(1) + Θxyz(2)



(16)

Θxyz(1) = x y z [ 1 - x/2 - y/2 - z/2 + x y/4 + y z/4 + x z/4 - x y z/8]

(17)

Where it is posited that:

Θxyz(2) = x y z { x/2 [ f(1, 0, 0) ] + y/2 [ f(0, 1, 0) ] + z/2 [ f(0, 0, 1) ] - x y/4 [ - f(1, 0, 0) - f(0, 1, 0) + f(1, 1, 0) ] - y z/4 [ - f(0, 1, 0) - f(0, 0, 1) + f(0, 1, 1)] - x z/4 [ - f(1, 0, 0) - f(0, 0, 1) + f(1, 0, 1)] + x y z/8 [ f(1, 1, 1) - f(1, 1, 0) - f(0, 1, 1) - f(1, 0, 1) + f(0, 0, 1) + f(0, 1, 0) + f(1, 0, 0) ] } (18) Based on equations (1) and (12) it can be written that: EIN = EIN(x, y, z) = - ωf Hxyz (x, y, z) [ θxy + θxz + θyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = - ωf Hxyz (x, y, z) { [ - 3 f(0, 0, 0) / ωf ] + Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) } = - ωf { f(0, 0, 0) Θxyz(1) + Θxyz(2) } * { [ - 3 f(0, 0, 0) / ωf ] + Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) } = - ωf { f(0, 0, 0) Θxyz(1) [ - 3 f(0, 0, 0) / ωf ] + Θxyz(2) [ - 3 f(0, 0, 0) / ωf ] + f(0, 0, 0) Θxyz(1) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] + Θxyz(2) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] } = - ωf { [ f(0, 0, 0) ]2 [ - 3 Θxyz(1) / ωf ] + f(0, 0, 0) [ - 3 Θxyz(2) / ωf ] + f(0, 0, 0) Θxyz(1) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] + Θxyz(2) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] } = { [ f(0, 0, 0) ]2 [ 3 Θxyz(1) ] + f(0, 0, 0) [ 3 Θxyz(2) ] - f(0, 0, 0) ωf Θxyz(1) [ Θ - x (1 - ζx ×) -

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y (1 - ζy ×) - z (1 - ζz ×) ] - ωf Θxyz(2) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] } = [ f(0, 0, 0) ]2 [ 3 Θxyz(1) ] + f(0, 0, 0) { [ 3 Θxyz(2) ] - ωf Θxyz(1) [ Θ - x (1 - ζx ×) y (1 - ζy ×) - z (1 - ζz ×) ] } - ωf Θxyz(2) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]

(19)

Therefore: EIN = EIN(x, y, z) = [ f(0, 0, 0) ]2 Θxyz(4) + f(0, 0, 0) Θxyz(5) + Θxyz(6)

(20)

Where it is posited that: Θxyz(4) = [ 3 Θxyz(1) ] Θxyz(5) = { [ 3 Θxyz(2) ] - ωf Θxyz(1) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] } Θxyz(6) = - ωf Θxyz(2) [ Θ - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] Thus, equating (15) to (20) it is herein derived that to be Eo = Eo(x, y, z) = EIN = EIN(x, y, z) it needs to be: [ f(0, 0, 0) ]2 Θxyz(4) + f(0, 0, 0) Θxyz(5) + Θxyz(6) = 3 [ f(0, 0, 0) ]2 x y z - f (0, 0, 0) x y z ωf Θ Thus: [ f(0, 0, 0) ]2 [ Θxyz(4) - 3 x y z ] + f(0, 0, 0) [ Θxyz(5) + x y z ωf Θ ] + Θxyz(6) = 0

(21)

Equation (21) admits the following solution: f(0, 0, 0) = { - [ Θxyz(5) + x y z ωf Θ ] ± { [ Θxyz(5) + x y z ωf Θ ]2 - 4 [ Θxyz(4) - 3 x y z ] Θxyz(6) }1/2 } / { 2 [ Θxyz(4) - 3 x y z ] }



(22)

The signal intensity f(0, 0, 0) is therefore assumed the same in the two conditions of non-interpolated value and interpolated value. What is then the advantage to have found the signal intensity value that is resilient to the model interpolation function, the signal intensity that does not change its value because of re-sampling at the location (x, y, z)? To answer to this question let us undertake the following reasoning. Interpolation would be ideal if we would estimate the value of the true signal intensity at the given re-sampling location with no error. However, as pointed out in the preceding chapters, interpolation is ill posed as a problem in a two folded manner. One is that we do not know the true value, and second, the signal intensity value that the interpolator estimates is subject to an error because of the approximation properties inherent to the model interpolation function, which itself it is an estimator. To admit a signal intensity value obtained under the assumption that the signal intensity-curvature content does not change before and after interpolation is like assuming that the energy of the signal does not change after re-sampling. As to if this is the true value or not that is it impossible to determine unless we do know the signal at each location,

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such as the case of signals of known nature like the trigonometric functions defined in the domain of the real numbers. The signal intensity value that it is obtained while assuming no change in intensitycurvature content before and after interpolation is, at least theoretically, in that it waits for empirical confirmation, a possible approach to the determination of the resilient interpolator. If that is the case, equating the two intensity-curvature terms before and after interpolation, results in that signal intensity value for given re-sampling location, which makes the interpolation function a resilient estimator. That would mean that the signal reconstructed through the model interpolation function under the constraint that intensity-curvature content does not change, might be the true value of the signal that has not been sampled because of the limitations imposed by the Nyquist Theorem (Nyquist, 2002).

SUMMARY This chapter has followed the path traced by the last section of Chapter VII in attempting resilient interpolation for the case of the trivariate linear function. The problem that is addressed can be phrased through the following question: “For the given re-sampling location what is the true pixel (signal) intensity value? Can the model interpolation function calculate that signal intensity value that is the one that would be sampled?” In other words: “For given placement of the re-sampling location, and given model interpolation function, what would be the signal intensity that is the same in the two conditions of non-interpolated signal and interpolated signal?” The math deduction presented in this chapter determines the signal intensity value while assuming no change in intensity-curvature content before and after interpolation. This theoretical presentation waits for empirical confirmation. To obtain confirmation of this theoretical presentation from the audience of this book might lead to validate the conceptualization of resilient interpolation that this chapter has presented.

Reference Nyquist, H. (2002). Certain topics in telegraph transmission theory. Proceedings of the IEEE, 90(2), 280-305.

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Part IV

Improvement of One Dimensional B-Spline Functions

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Chapter XIII

On the Literature of B-Spline Interpolation Functions

THE MESSAGE OF THE CHAPTER This chapter reviews the extensive and comprehensive literature on B-Splines. In the forthcoming text emphasis is given to hierarchy and formal definition of polynomial interpolation with specific focus to the subclass of functions that are called B-Splines. Also, the literature is reviewed with emphasis on methodologies and applications of B-Splines within a wide array of scientific disciplines. The review is conducted with the intent to inform the reader and also to acknowledge the merit of the scientific community for the great effort devoted to B-Splines. The chapter concludes emphasizing on the proposition that the unifying theory presented throughout this book has for what concerns two specific cases of B-Spline functions: univariate quadratic and cubic models.

HIERARCHY AND FORMAL DEFINITION The hierarchy of the interpolation functions is organized in such a way that B-Splines descend from a larger family of functions named piecewise polynomial interpolation functions (Agarwal & Wong, 1993). If the polynomial incorporates a B-Spline function, then it is normally referred to as the polynomial B-Spline. The polynomial B-Spline is a function of nth degree, with coefficients appropriately determined to represent the signal to be approximated. For convenience to the reader it is due to report a formal definition (Chen et al., 2006) of the B-Spline function as it is due to acknowledge that the knowledge herein reported was originally provided long ago (De Boor, 1978). For any node sequence a B-Spline function of order k is defined as the sum of products between scalars αj (B-Spline coefficients) and B-Spline forms of the type: Bj,k,τ (t) : = ( τj+k - τj ) [ τj, … , τj+k ] ( - t )k-1+

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On the Literature of B-Spline Interpolation Functions

t being a real number. Bj,k,τ (t) is thus the j-th B-Spline of order k and it constitutes the k-th order divided difference built on the sequence of nodes τj with j = 1…N. It follows that [ τj, … , τj+k ] f is thus the k-th order divided difference of the function f at the nodes τj, … , τj+k. Given a signal s(t) in the space of splines of order k, it follows that the sum of products between scalars αj (B-Spline coefficients) and B-Spline forms approximates the signal s(t) such that: N

s(t) = Σj=1 αj Bj,k,τ (t) B-Splines tend to converge to the Sinc function as their degree increases. Therefore, B-Splines’ interpolation functions play a fundamental role within the context of image and signal processing applications and analysis tools.

LITERATURE: METHODOLOGIES AND APPLICATIONS Bio-signals and diagnostic images are collected routinely in time series. Within this context, the development of innovative approaches for signal interpolation assumes relevance and usefulness for the research community. Novel interpolation approaches are thus important in order to analyzing reliably the time series. Also, many image processing analysis tools make use of interpolation functions. B-Splines were demonstrated to be excellent interpolators (De Boor, 1978; Unser et al., 1993a, 1993b) and they improve significantly the interpolation error with respect to linear paradigms which are computationally less demanding. Quadratic and cubic B-Splines offer considerable approximation improvement over the linear paradigm, producing a 33.9% gain, while quintic and septic B-Splines do improve approximation over the cubic functions only slightly (3-8%) when compared to their computational demand (Mijering et al., 1999). Like other polynomial interpolators, B-Splines have the capacity to embed an approximated representation of the signal. They are continuous throughout their interval of definition (support), and their coefficients can be determined also through linear filtering (Toraichi et al., 1988). In contrast with polynomial interpolators, the characteristics of B-Splines were extensively described as determinant of a substantial degree of signal smoothness, which means that the behavior of the B-Spline do act like a band-pass filter (Unser et al., 1993a, 1993b). Extensive presentation of B-Splines was reported (De Boor, 1978; Lehmann et al., 1999; Unser et al., 1993a, 1993b). A recent and comprehensive literature review showed that the use of polynomial interpolation can be dated to the beginning of ancient astronomy (Meijering, 2002). Work, done last century gives a presentation of the foundations of B-Spline interpolation functions with special attention to how the functions’ coefficients are determined that best represent the signal (De Boor, 1978; Keys, 1981; Schoenberg, 1946a, 1946b, 1969, 1971; Unser et al., 1995). De Boor (1978) presents an outstanding and remarkable study of B-Splines. Ciulla and Deek (2005) report due mention to literature work aimed to (i) extensively characterize traditional B-Splines (Unser, 1999), (ii) conceptualize two-dimensional polynomial interpolation (Agarwal & Wong, 1993; Meijering, 1999; Unser et al., 1995) and (iii) build frameworks for the design of polynomial interpolation (Blu et al., 2001). The literature on methods for the interpolation error improvement is extensive. Specifically, relevant to the topic covered in this book is the methodology for the optimal choice of knots (Deng & Denney, 2004). Knots are space domain points where the interpolation function is defined. Along the lines of

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the more recent trend (Joyeux & Penczeck, 2002; Wang, 2001) of local optimization (e.g. pixel-bypixel optimization for a two dimensional signal) versus global optimization and within the context of the study of B-Splines, the literature reports on work done at the aim to find grid of knots (nodes) that derives from the original grid placement. Search strategies dependent on the derivative of the interpolation function were developed on the basis of error minimization, and accordingly they determine the final knots’ placement on the basis of the information content provided by the interpolation functions’ curvature (Deng & Denney, 2004). Some of the interpolation methodology and technology that is currently in use addresses re-sampling of images as a global issue (i.e. relatively to the overall signal). Also, the formulation of the interpolation error, which is a Newton-based characterisation form (Lehmann et al., 1999) is dependent on the resolution (step-size). One approach for designing optimal linear interpolation calculates the area under the interpolation curve as the sum of the square of the errors over the sampling intervals (Wang, 2001). This method is featured through the calculation of the error as sum of areas under the curve. Each area corresponds to a sampling interval and the root-mean-square of the sum is calculated (Thevenaz et al., 2000). For an excellent review of bivariate spline interpolation the reader is referred to the knowledge provided in a recent survey (Nurnberger & Zeilfelder, 2000a). These authors present knowledge for what concerns: (i) the representation of bivariate polynomials and its relation to bivariate splines, (ii) the dimension of bivariate spline, which is an issue that was brought to the attention long ago (Strang, 1973), (iii) the approximation order of the bivariate spline spaces, (iv) issues related to the relationships existing between Hermite and Lagrange interpolation methods for bivariate spline spaces. Also, Nurnberger and Zeilfelder (2000b) have contributed a general method for calculating the triangulations in the B-Spline space that are suitable for interpolation. Recent applications of B-Splines include the following approaches. B-Splines employed within the context of extracting correlation measures in gene expression data for the estimation of mutual information from a discrete data set (Daub et al., 2004). Derivatives of B-Splines functions were determined upon constraints imposed by control theory problems and such derivatives served the purpose of designing the differential equations corresponding to the particular control problem (Kano et al., 2003). The theoretical approach to the calculation of the B-Splines was based on the knowledge provided earlier (De Boor, 1978). B-Splines with equidistant knots were employed to solve Hermite interpolation through an approach aimed to the determination of the optimal parameter involved in the spacing of the knots of the Hermite function (Zeilfelder, 1996). Another research report (Davydov et al., 1998) was concerned with the use of B-Splines to determine Hermite interpolation in two dimensions and made clear distinction between Lagrange and Hermite interpolation, being the former based on the values of the function and the latter based on both the values of the function and its derivatives. B-splines were also employed to obtain trivariate Lagrange interpolation (Hecklin et al., 2008) to the extent of providing the additional points to the three dimensional interpolation grid. The additional points determined a non uniform grid which recalls the aims of the approach undertaken in this book to the extent of determining non-uniformly spaced novel re-sampling locations. Consistently with the notion provided earlier (Strang & Fix, 1971), the authors provided error bounds of their interpolation formula that are dependent on the sampling step. The data analysis method based on the Empirical Mode Decomposition (EMB) (Huang et. al., 1998, 1999) bases its foundation on the capability to extract the fluctuations of the data from the mean and has

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been found suitable for nonlinear and non stationary data such as time-frequency-energy representations. EBM has been studied further by proposing an alternative approach through a B-Spline based algorithm devised for the decomposition of the earthquake signal (Chen et al., 2006). Also, applications of B-Splines’ interpolation have just recently been added to the variety of signal processing tools employed for image segmentation (Brigger et al., 2000; Precioso et al., 2003, 2005). Particularly, it was shown that the cubic B-Spline interpolation provides the efficient combination of reducing the computational cost as well the size of the data to process (Precioso et al., 2003, 2005). B-Splines were also recently employed in elastic image registration (Ledesma-Carbayo et al., 2005) within the context of ultrasound cardiac motion estimation. Specifically, in this work, the splines determine the deformation field that leads to the motion estimation across a series of cardiac images. Another recent work that employs B-Splines for image registration (Sorzano et al., 2006) is relevant to images of gene expression and similarly proposes to determine the deformation field through splines’ models. Also, exploiting the interpolation accuracy offered through B-Splines, another recent work (Jonic et al., 2006) addresses the problem of rigid-body registration within the context of orthopaedic surgery assisted by 3D Computerized Tomography and C-arm device. Other applications of B-Splines include the solution of the Burgers’ equation (Dag et al., 2005), which are known as a mathematical model of the turbulence (Burgers, 1948). Also, within the context of computer vision applications, it was reported the squared distance minimization (SDM) method for the calculation of the B-Spline function to approximate a target surface (Wang et al., 2006). This report was followed with the formulation of the evolution of the original approach and the featuring novelty was a procedure to automatically adjust both the number and the locations of the B-Spline knots (Yang et al., 2004). A late application of B-Splines for 3D contour detection is within the context of the estimation of the shape of DNA molecules (Jacob et al., 2006). In this work the authors propose to adjust iteratively the B-Spline coefficients on the basis of the minimization of a cost function defined as energy and through the classic conjugate-gradient algorithm. B-Splines have been presented in their polyharmonic form (Van De Ville et al., 2004, 2005) and they differs from those forms presented earlier (Rabut, 1992) which have the characteristic to converge to Gaussian like forms as their order increases. The application of these types of B-Splines was to signal decomposition based on wavelets. Recent work on cardinal splines has introduced notation as well formalism for the analytical characterization of the B-Spline convolution (Unser, 2005; Unser & Blu, 2005). Also, filtering prior to sampling has been recently proposed at the aim to minimize the error that interpolation as estimator would produce when confronted with noisy data (Unser & Blu, 2006). Filtering prior to sampling results into signal deconvolution and it might furnish the advantage of removing the noise thus facilitating the task of the interpolator in the estimation of the unknown signal. Finally due to mention is the introduction of the Complex B-Splines (Forster et al., 2006) conducted as a generalization of Schoenberg’s cardinal splines (Schoenberg, 1946a, 1946b) and the demonstration that they converge to Gaussians as their degree increases.

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THE INTENT OF THE UNIFY ING THEORY IN RELATIONSHIP TO THE IMPROVEMENT OF ONE-DIMENSIONAL B-SPLINES As far as the one dimensional B-Splines are concerned, this book takes a completely different approach to the improvement of the interpolation accuracy, which is unique in literature. The approach uniqueness is determined through the combination of the following two conceptions: (i) improvement of the interpolation function error can be mathematically formulated as dependent on the joint information content of the nodes’ intensities and the curvature of the interpolation function, and (ii) re-sampling has the property of locality rather than being performed at the same location for each sampling interval. That means that when given an original grid of points and a given misplacement, re-sampling takes place at novel re-sampling locations that vary between sampling intervals and they are not necessarily the same as the misplacement. These locations are calculated based on the combined nodes’ signal intensities at the neighbourhood and also on the misplacement. They are able to determine improvement of the interpolation error. Also, by doing so, some frequencies are incorporated in the signal, which would otherwise be neglected if re-sampling would take place at the same location for each sampling interval. These concepts propose a novel theory and methodology. It is the main concern of this part of the book to extend both theoretical and methodological approaches developed for the bivariate and trivariate linear interpolation functions to other interpolation functions, regardless of their degree. It will be shown the existence of a spatial domain called Sub-pixel Efficacy Region constituted with a set of points within the sampling step of the B-Spline in its quadratic and cubic form. These points are calculated on the basis of the solution of polynomials consisting of the first order partial derivatives of the IntensityCurvature Functional (ΔE). ΔE includes the joint information content of intensity and curvature of the interpolation function. The incorporation of the curvature into the ΔE functional is driven through the intuition that to characterize the true behavior of a function at its neighbourhood, the intensity (i.e. the function value) alone is not sufficient. The second order derivative of the function, which is a measure of the local curvature, provides with supplementary information that is beneficial to the improvement of the interpolation error. Section IV of the book is organized as follows. In Chapter XIV, both theory and methodological approach are extended to the improvement of quadratic and cubic B-Spline interpolation functions. In Chapter XV, the validation paradigm based on motion correction is used to study the behavior of the root-mean-square-error (RMSE) to show the interpolation error improvement. Also, Fast Fourier Transform (FFT) and spectral power evolutions are used to study the difference between re-sampling at the given misplacement, versus re-sampling at the novel locations obtained by means of the Subpixel Efficacy Region. Finally, Chapter XVI outlines the foundations of the unifying theory within the context of the current literature.

SUMMARY This chapter informs the reader of the quite extensive literature existing for the piecewise polynomial interpolation functions with particular attention to B-Splines. The chapter emphasizes that polynomial interpolation has been object of study for quite a long time and references the excellent presentation that exists in literature for B-Spline functions and parametric piecewise polynomial functions grouped under same theoretical frameworks.

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B-Spline functions are classified in literature on the basis of their degree, order of approximation, interval of definition and differentiability (Blu et al., 2003). Also, due reference is made to methods existing for the improvement of B-Spline interpolation approximation properties. Due mention is given to a main result accepted in literature, which recognizes the superiority of B-Splines over linear interpolators, and also the lack of substantial error improvement of polynomial interpolation functions that exceed the third degree in their order. Within this context, the chapter concludes stressing on the flexibility of the novel theory proposed in the book and thus overviews on how its applicability can be extended to the one-dimensional quadratic and cubic forms of B-Spline interpolation. It is mentioned that the extension is performed with the same methodological approach employed for the linear interpolation which was illustrated in part I through part III of the book.

References Agarwal, R. P., & Wong, P. J. Y. (1993). Error inequalities in polynomial interpolation and their applications. Mathematics and its applications. Dordrecht, The Netherlands: Kluwer Academic Publishers. Blu, T., Thevenaz, P., & Unser, M. (2001). MOMS: Maximal-order interpolation of minimal support. IEEE Transactions on Image Processing, 10(7), 1069-1080. Blu, T., Thevenaz, P., & Unser, M. (2003). Complete parameterization of piecewise-polynomial interpolation kernels. IEEE Transactions on Image Processing, 12(11), 1297-1309. Brigger, P., Hoeg, J., & Unser, M. (2000). B-spline snakes: a flexible tool for parametric contour detection. IEEE Transactions on Image Processing, 9(9), 1484-1496. Burgers, J. M. (1948). A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics, 1, 171-199. Chen, Q., Huang, N., Riemenschneider, S., & Xu, Y. (2006). A B-spline approach for empirical mode decompositions. Advances in Computational Mathematics, 24, 171-195. Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53. Dag, I., Irk, D., & Sahin, A. (2005). B-spline collocation methods for numerical solutions of the Burgers’ equation. Mathematical Problems in Engineering, 5, 521-538. Daub, C. O., Steuer, R., Selbig, J., & Kloska, S. (2004, August 31). Estimating mutual information using B-spline functions - an improved similarity measure for analysing gene expression data. BMC Bioinformatics, 5(118). Retrieved October 10, 2007, from http://www.biomedcentral.com/1471-2105/5/118 Davydov, O. V., Nurnberger, G., & Zeilfelder, F. (1998). Approximation order of bivariate spline interpolation for arbitrary smoothness. Journal of computational and applied mathematics, 90(2), 117-134. De Boor, C. (1978). A practical guide to splines. Applied mathematical sciences. New York, NY: Springer-Verlag.

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Deng, X., & Denney, Jr., T. S. (2004). On optimizing knot positions for multi-dimensional B-spline models. Proceedings of SPIE, USA, 5299, 175-186. Forster, B., Blu, T., & Unser, M. (2006). Complex B-splines. Applied and Computational Harmonic Analysis, 20, 261-282. Hecklin, G., Nurnberger, G., Schumaker, L. L., & Zeilfelder, F. (2008). A local Lagrange interpolation method based on C1 cubic splines on freudenthal partitions. Mathematics of Computation, 77(262), 1017-1036. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, H. H., Zheng, Q., Yen, N. C., Tung, C. C., & Liu, H. H. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proceedings of the Royal Society. Mathematical Physical and Engineering Sciences, 454(1971), 903-995. Huang, N. E., Shen, Z., & Long, S. R. (1999). A new view of nonlinear water waves: The Hilbert spectrum. Annual Review of Fluid Mechanics, 31, 417-457. Jacob, M., Blu, T., Vaillant, C., Maddocks, J. H., & Unser, M. (2006). 3-D shape estimation of DNA molecules from stereo cryo-electron micro-graphs using a projection-steerable snake. IEEE Transactions on Image Processing, 15(1), 214- 227. Joyeux, L., & Penczeck, P. A. (2002). Efficiency of 2D alignment methods. Ultramicroscopy, 92, 3346. Jonic, S., Thevenaz, P., Zheng, G., Nolte, L. P., & Unser, M. (2006). An optimized spline-based registration of a 3D CT to a set of C-arm images. International Journal of Biomedical Imaging, 2006(Article ID 47197), 1-12. Kano, H., Egerstedt, M., Nakata, H., & Martin, C. F. (2003). B-splines and control theory. Applied Mathematics and Computation, 145(2), 263-288. Keys, R. G. (1981). Cubic convolution interpolation for digital image processing. IEEE Transactions on Acoustics, Speech, and Signal Processing, 29(6), 1153-1160. Ledesma-Carbayo, M. J., Kybic, J., Desco, M., Santos, A., Suhling, M., Hunziker, P., & Unser, M. (2005). Spatio-temporal nonrigid registration for ultrasound cardiac motion estimation. IEEE Transactions on Medical Imaging, 24(9), 1113-1126. Lehmann, T. M., Gonner, C., & Spitzer, K. (1999). Survey: Interpolation methods in medical image processing. IEEE Transactions on Medical Imaging, 18(11), 1049-1075. Meijering, E. H. W., Zuiderveld, K. J., & Viergever, M. A. (1999). Image reconstruction by convolution with symmetrical piecewise n-th order polynomial kernels. IEEE Transactions on Image Processing, 8(2), 192-201. Meijering, E. (2002). A chronology of interpolation: From ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90(3), 319-342. Nurnberger, G., & Zeilfelder, F. (2000a). Development in bivariate spline interpolation. Journal of Computational and Applied Mathematics, 121(1-2), 125-152.

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Nurnberger, G., & Zeilfelder, F. (2000b). Interpolation by spline spaces on classes of triangulations. Journal of Computational and Applied Mathematics, 119(1-2), 347-376. Precioso, F., Barlaud, M., Blu, T., & Unser M. (2003). Smoothing B-spline active contour for fast and robust image and video segmentation. Proceedings of the International Conference on Image Processing, Spain, 1, 137-140. Precioso, F., Barlaud, M., Blu, T., & Unser M. (2005). Robust real-time segmentation of images and videos using a smooth-spline snake-based algorithm. IEEE Transactions on Image Processing, 14(7), 910-924. Rabut, C. (1992). Elementary m-harmonic cardinal B-splines. Numerical Algorithms, 2(1), 39-61. Schoenberg, I. J. (1946a). Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae. Quarterly of Applied Mathematics, 4, 45-99. Schoenberg, I. J. (1946b). Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of osculatory interpolation. A second class of analytic approximation formulae. Quarterly of Applied Mathematics, 4, 112-141. Schoenberg, I. J. (1969). Cardinal interpolation and spline functions. Journal of Approximation Theory, 2, 167-206. Schoenberg, I. J. (1971). On equidistant cubic spline interpolation. Bulletin of the American Mathematical Society, 77(6), 1039-1044. Sorzano, C. O. S., Blagov, M. S., Thevenaz, P., Myasnikova, E. M., Samsonova, M. G. & Unser M. (2006). Algorithm for spline-based elastic registration in application to confocal images of gene expression. Pattern Recognition and Image Analysis, 16(1), 93-96. Strang, G., & Fix, G. (1971). A Fourier analysis of the finite element variational method. In Constructive Aspect of Functional Analysis (pp. 796-830). Rome, Italy: Edizioni Cremonese. Strang, G. (1973). Piecewise polynomials and the finite element method. Bulletin American Mathematical Society, 79(6), 1128-1137. Thevenaz, P., Blu, T., & Unser, M. (2000). Interpolation revisited. IEEE Transactions on Medical Imaging, 19(7), 739-758. Toraichi, K., Yang, S., Kamada M., & Mori R. (1988). Two-dimensional spline interpolation for image reconstruction. Pattern Recognition, 21(3), 275-284. Unser, M., Aldroubi, A., & Eden, M. (1993a). B-spline signal processing: Part I – theory. IEEE Transactions on Signal Processing, 41(2), 821-833. Unser, M., Aldroubi, A., & Eden, M. (1993b). B-spline signal processing: Part II – efficient design and applications. IEEE Transactions on Signal Processing, 41(2), 834-848. Unser, M., Thevenaz, P., & Yaroslavsky, L. (1995). Convolution-based interpolation for fast, high-quality rotation of images. IEEE Transactions on Image Processing, 4(10), 1371-1381.

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Unser, M. (1999). Splines: A perfect fit for signal and image processing. IEEE Signal Processing Magazine, 16(6), 22-38. Unser, M., & Blu, T. (2005). Cardinal exponential splines: Part I - theory and filtering algorithms. IEEE Transactions on Signal Processing, 53(4), 1425-1438. Unser, M. (2005). Cardinal exponential splines: Part II - think analog, act digital. IEEE Transactions on Signal Processing, 53(4), 1439-1449. Unser, M. & Blu, T. (2006). Generalized smoothing splines and the optimal discretization of the Wiener filter. IEEE Transactions on Signal Processing, 53(6), 2146-2159. Van De Ville, D., Blu, T., Forster, B., & Unser M. (2004). Isotropic-polyharmonic B-Splines and wavelets. Proceedings of the International Conference on Image Processing, Singapore, 1, 661-664. Van De Ville, D., Blu, T., & Unser, M. (2005). Isotropic polyharmonic B-splines: scaling functions and wavelets. IEEE Transactions on Image Processing, (14)11, 1798-1813. Wang, J. (2001). Optimal design for linear interpolation of curves. Statistics in Medicine, 20(16), 24672477. Wang, W., Pottmann, H., & Liu, Y. (2006). Fitting B-spline curves to point clouds by squared distance minimization. ACM Transactions on Graphics (TOG), 25(2), 214-238. Yang, H., Wang, W., & Sun, J. (2004). Control point adjustment for B-spline curve approximation. Computer-Aided Design, 36, 639-652. Zeilfelder, F. (1996). Interpolation und beste approximation mit periodischen splinefunktionen [Interpolation and best approximation with periodic Spline functions]. Unpublished doctoral dissertation, University of Mannheim - Germany.

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Chapter XIV

The Extension of Theory and Methodology to B-Splines

ORGANIZATION OF THE FORTHCOMING TEXt The organization of the chapter is similar to that of Chapters VII and X. The methodological approach to extend the unifying theory to the one dimensional quadratic and cubic B-Splines is herein reported along with the most relevant mathematical details. This chapter should be read along with Appendix VI where proofs are given to the assertions herein presented. In either of the two cases: quadratic and cubic B-Spline the math process starts from the calculation of the Intensity-Curvature Functional and continues with the calculation of the Sub-pixel Efficacy Region. Finally, the math process arrives to the calculation of the novel re-sampling locations through the formulas of the unifying theory seen in equations (23) and (33) for the quadratic and the cubic models respectively. The chapter concludes with a section that addresses specifically the theoretical proposition of resilient interpolation for the two classes of B-Splines. This is conducted consistently with Chapters VII and XII of the book choosing to equate the two intensity-curvature terms (before and after interpolation) as the starting point of the math deduction.

Quadratic and Cubic B-Splines Calculations were developed based on the B-Spline forms: quadratic (h3(x)), and cubic (h4(x)) as per equations (1) and (2), with 1 x 6 and 1 x 6 neighbourhoods’ nodes (see equations (3) and (4)). - 2a | x |2 + 1/2 (a+1) h3(x) = a | x |2 - (2a + 1/2) | x | + 3/4 (a+1)

0 ≤ | x | ≤ 1/2 1/2 ≤ | x | ≤ 3/2

1/2 | x |3 - | x |2 + 2/3 h4(x) =

0≤|x|≤1

(1)

(2)

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Extension of Theory and Methodology to B-Splines

-1/6 | x |3 + | x |2 - 2 | x | + 4/3

1 ≤ | x | ≤ 2

The absolute value of x assumes the meaning of representing intra-nodal misplacement within the pixel or within two nodes. It is assumed that the origin of the coordinate system of the pixel or the intra-nodal distance (sampling step) is at the corner and not at the center. Therefore the value of the misplacement is never negative. For simplicity let us consider the two functions of equations (1) and (2) written as: h3(x) = f (0) + [ f (1/2) +f (-1/2) ] * [- 2a x2 + 1/2 (a+1) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a x2 - (2a + 1/2) x + 3/4 (a+1) ]

(3)

h4(x) = f (0) + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 x3 - x2 + 2/3 ] + [f (3/2) + f (- 3/2) + f (1) +f (-1) +f (2) + f (-2)] * [ -1/6 x3 + x2 - 2 x + 4/3 ]

(4)

Calculation of the Intensity-Curvature Terms Before and After Interpolation for the Quadratic B-Spline Let us calculate the first order derivative of h3(x) with respect to the variable x: (∂ ( h3(x) ) /∂x) = { -4 a x * [ f (1/2) +f (-1/2) ] + [ 2 a x - (2 a + 1/2 ) ] * [(f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] } = A It follows that the second order derivative of h3(x) with respect to the variable x is calculated as: (∂2 ( h3(x) )/∂x2) = (∂ (A)) /∂x) = - 4 a * [ f (1/2) +f (-1/2) ] + 2 a * [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] = - 4 a * [ f (1/2) +f (-1/2) ] + 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] = - 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] Let the intensity-curvature term before interpolation be defined as: x Eo = Eo (x) = ∫ f (0) (∂2 (h3(x)) /∂x2) (0) dx =



(5)



0

x ∫ {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } * f (0) dx = 0

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The Extension of Theory and Methodology to B-Splines

x ∫ θ[a,0] dx = x θ[a,0]



(6)

0

where it is posited that: θ[a,0] = {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } * f (0)



(7)

Let the intensity-curvature term after interpolation be defined as: x EIN = EIN (x) = ∫ h3 (x) (∂2 (h3(x)) /∂x2) dx = 0

x ∫ { f (0) + [ f (1/2) +f (-1/2) ] * [- 2a x2 + 1/2 (a+1) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) 0 + f (-3/2) ] * [a x 2 - (2a + 1/2) x + 3/4 (a+1) ] } * {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) +

f (1) + f (3/2) + f (-3/2) ] } dx = x EIN (x) = ∫ θ[a,0] + θ[a,1/2] [ -2a x2 + 1/2 ( a + 1 ) ] + 0

θ[a,3/2] [ a x2 - ( 2a + 1/2 ) x + 3/4 ( a + 1) ] dx = x θ[a,0] + [ -2/3 a x3 + 1/2 ( a +1 ) x ] θ[a,1/2] + [ a/3 x3 - ( 2a + 1/2 ) x2 /2 + 3/4 ( a +1 ) x ] θ[a,3/2] = { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] ( 2a + 1/2 )/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1 )/2 + θ[a,3/2] 3( a +1 )/4 ] }

(8)



Table 1. Theta values for equations (6) through (8)

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The Extension of Theory and Methodology to B-Splines

Where the positions of Table 1 hold true.

CALcULATioN of THe INTeNsiTY-CURVATURe FUNcTioNAL ΔE of THe Quadratic B-Spline From the previous paragraph it follows that the Intensity-Curvature Functional ∆E(x) = Eo(x) / EIN(x) is computed for the quadratic B-Spline h3(x) as: ∆E(x) = θ[a,0] x / { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] (2a + 1/2)/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] }



(9)

STUdY of THe INTeNsiTY-CURVATURe FUNcTioNAL ΔE of THe Quadratic B-Spline (∂ ( ∆E(x) ) /∂x) = 0 furnishes: x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x [ θ[a,3/2] (2a + 1/2)/2 ] + [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] - 3 x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] + 2 x [ θ[a,3/2] (2a + 1/2)/2 ] - [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] = 0



(10)

The proof to above assertion is given in appendix VI.

The Sub-pix el Efficacy Region of the Quadratic B-Spline From equation (10) it follows that the Sub-pixel Efficacy Region of the quadratic B-Spline h3(x) is: - 2 x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] + x [ θ[a,3/2] (2a + 1/2)/2 ] = 0 x(1) = 0

(11.a)

x(2) = [ θ[a,3/2] (2a + 1/2)/2 ] / { 2 * [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] }

226

(11)

(11.b)

The Extension of Theory and Methodology to B-Splines

Solution of a 4th Degree Poly nomial (2008, Carlo Ciulla, IGI Global) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the quartic equation is: http://en.wikipedia.org/wiki/Quartic_equation Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. Let the polynomial: ψ 4 φ4 + ψ 3 φ3 + ψ 2 φ2 + ψ φ1 + φ0 = 0

(12)

be written as: ψ 4 + ψ 3 ( φ3 / φ4) + ψ 2 ( φ2 / φ4) + ψ ( φ1 / φ4) + ( φ0 / φ4) = 0

(13)

The quartic equation: ψ 4 + ψ 3 ( φ3 / φ4) + ψ 2 ( φ2 / φ4) + ψ ( φ1 / φ4) + ( φ0 / φ4) = 0 is solved by: ψ = ± { { { ( α + 2y )1/2 } + { ( α + 2y ) - 4 * { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } }1/2 } / 2 } - ( φ3 / 4 φ4))



(14)

α = [6 ( φ3 / 4 φ4)2 - 3 ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ]

(15)

β = [ -4 ( φ3 / 4 φ4)3 + 3 ( φ3 / φ4) ( φ3 / 4 φ4)2 - 2 ( φ3 / 4 φ4) ( φ2 / φ4) + ( φ1 / φ4) ]

(16)

γ = [ ( φ3 / 4 φ4)4 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) ]

(17)

p = [ (5/6) (5/3) α2 + ( (5/6) α )2 - (5/3) α2 (5/2) + (2 α2 - γ ) ]

(18)

q = - ( (5/6) α )3 + ( (5/6) α )2 (5/2) α - (2 α2 - γ ) (5/6) α + [ (α3 - α γ - β2 /4 ) / 2 ]

(19)

u = { ( q /2 ) ± { ( q /2 )2 + ( p3 / 27) }1/2 }1/3

(20)

v = [ ( p / 3 u ) - u ]

(21)

y = v - (5/6) α

(22)





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The Extension of Theory and Methodology to B-Splines

Table 2. Theta values for equations (25) and (26)

The mathematics that precedes the solution of the quartic equation is reported in appendix VI. The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve the quartic equation. Also, the author wishes to acknowledge that the main author of the solution of the quartic equation is the Italian mathematician Lodovico Ferrari, also the main reference to the solution of the quartic equation as well as to the Galois Theory (Stewart, 2004) as reported at: http://en.wikipedia.org/wiki/Quartic_equation.

The Derivation of the Novel Re-sampling Locations of the Quad ratic B-Spline The entire math process is illustrated in appendix VI. The novel re-sampling locations for the B-Spline h3 (x) are derived from the following equation: { h((xsre - xr0 )/ xsre) * (∂2 (h(x)) /∂x2) (xsre - xr0) } / { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { EIN (xsre-x0) / EIN (xsre) }





(23)

And they are: xr0(k, i) = (xsre(i) - ρ(k) xsre(i)) with i, k = 1, 2. (24) Where: ρ(k) = { f 2 (2a + 1/2) ± [ Δq ]1/2 } / [ 2 * (- 2 f 1 + f 2) a ] With:

228



The Extension of Theory and Methodology to B-Splines

f 1 = [ f (1/2) + f (-1/2) ]; f 2 = [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ]; Δq = { [ f 2 (2a + 1/2) ]2 + 4 * (- 2 f 1 + f 2) a * { [ EIN (xsre-x0) / EIN (xsre) ] * [ h(xsre) ] - f (0) - f 1 1/2 ( a +1 ) - f 2 3/4 ( a + 1 ) } } What has to be acknowledged, as the MATLAB®. (c) 1984-2007 The MathWorks, Inc. code h3SRE2007 located at www.sourcecodewebsite.com shows is that the solutions ρ(k) are normalized to the range [0, x0] (where x0 is the misplacement). This happens just before equation (24) is employed to determine the novel re-sampling locations x r0(k, i).

Calculation of the Intensity-Curvature Terms Before and After Interpolation for the Cubic B-Spline The intensity-curvature term before interpolation is: x Eo = Eo (x) = ∫ f (0) (∂2 (h4(x))/∂x2) (0) dx =



0

x ∫ {- 2 * [ f (1/2) +f (-1/2) + f (1) + f (-1) ] + 0

2 * [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] } * f (0) dx =

x ∫ θ[a,10] dx = x θ[a,10]



(25)

0

where it is posited that: θ[a,10] = (∂2 (h4(x))/∂x2) (0, 0) * f (0) = {- 2 * [ f (1/2) + f (-1/2) + f (1) + f (-1) ] + 2 * [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] } * f (0) The intensity-curvature term after interpolation is: x EIN = EIN (x) = ∫ h4 (x) (∂2 (h4(x))/∂x2) dx =



0

θ[a,1] * [ 3/2 x2 - 2 x ] + θ[a,2] * [ - 1/2 x2 + 2 x ] + θ[a,11] * [ 3/10 x5 - x4 + 2/3 x3 + x2 - 4/3 x ] + θ[a,12] * [ - 1/5 x5 + 13/12 x4 - 10/3 x3 + 22/6 x2 - 4/3 x ] + θ[a,22] * [ 1/30 x5 - 4/12 x4 + 4/3 x3 -

229

The Extension of Theory and Methodology to B-Splines

16/6 x2 + 8/3 x ]



(26)

The proofs to these two assertions are reported in appendix VI. The positions of Table 2 hold true:

CALcULATioN of THe INTeNsiTY-CURVATURe FUNcTioNAL ΔE of THe Cubic B-Spline From the previous paragraph it follows that the Intensity-Curvature Functional ∆E(x) = Eo(x) / EIN(x) is computed for the cubic B-Spline h4(x) as: ∆E(x) = θ[a,10] x / { θ[a,1] * [ 3/2 x2 - 2 x ] + θ[a,2] * [ - 1/2 x2 + 2 x ] + θ[a,11] * [ 3/10 x5 - x4 + 2/3 x3 + x2 - 4/3 x ] + θ[a,12] * [ - 1/5 x5 + 13/12 x4 - 10/3 x3 + 22/6 x2 - 4/3 x ] + θ[a,22] * [ 1/30 x5 - 4/12 x4 + 4/3 x3 - 16/6 x2 + 8/3 x ] } = θ[a,10] / { θ[a,1] * [ 3/2 x - 2 ] + θ[a,2] * [ - 1/2 x + 2 ] + θ[a,11] * [ 3/10 x4 - x3 + 2/3 x2 + x - 4/3 ] + θ[a,12] * [ - 1/5 x4 + 13/12 x3 - 10/3 x2 + 22/6 x - 4/3 ] + θ[a,22] * [ 1/30 x4 - 4/12 x3 + 4/3 x2 - 16/6 x + 8/3 ] }



(27)

STUdY of THe INTeNsiTY-CURVATURe FUNcTioNAL ΔE of THe CUBic B-Spline The calculation of the first order derivative of the Intensity-Curvature Functional of the cubic B-Spline h4(x) with respect to the variable x yields: x3 { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } + x2 {- 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22]} + x { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } + { 3/2 θ[a,1] - 1/2 θ[a,2] + θ[a,11] + 11/3 θ[a,12] - 8/3 θ[a,22] } = 0 The full math process is reported in appendix VI.

230

(28)



The Extension of Theory and Methodology to B-Splines

The Sub-pix el Efficacy Region of the Cubic B-Spline To find the Sub-pixel Efficacy Region of the the cubic B-Spline h4(x), equation (28) is solved (2008, Carlo Ciulla, IGI Global) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the cubic equation is: http://it.wikipedia.org/wiki/Equazione_di_terzo_grado Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. Let us posit: φ0 = { 3/2 θ[a,1] - 1/2 θ[a,2] + θ[a,11] + 11/3 θ[a,12] - 8/3 θ[a,22] } φ1 = { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } φ2 = {- 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22]} φ3 = { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } Equation (28) becomes: x3 φ3 + x2 φ2 + x φ1 + φ0 = 0 x3 + x2 (φ2 / φ3) + x (φ1 / φ3) + ( φ0 / φ3) = 0 and it is rewritten in the form: y3 + py + q = 0 with x = [ y - (φ2 / 3 φ3) ] Such to obtain: zi = { ( q / 2 ) ± { ( q / 2 )2 + ( p3 / 27 ) }1/2 }1/3 i = 1, 2, 3

(29)

yi = zi - ( p / 3zi ) with:

i = 1, 2, 3

(30)

xsre(i) = [ yi - (φ2 / 3 φ3) ]

i = 1, 2, 3

(31)

where: q = [ (2/27) (φ2 / φ3)2 - (φ2 / 3 φ3) (φ1 / φ3) + ( φ0 / φ3) ] = { (2/27) [ { - 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ]2 -

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[ { - 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22] } / 3 { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ] * [ { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] }] + [ { 3/2 θ[a,1] - 1/2 θ[a,2] + θ[a,11] + 11/3 θ[a,12] - 8/3 θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ] }

(32.a)

p = [ - (φ2 / 9 φ3)2 + (φ1 / φ3) ] = [ - [{ - 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22] } / 9 { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ]2 + [ { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ] ]

(32.b)

The full math process is reported in appendix VI. The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve the cubic equation. Also, the author wishes to acknowledge the main authors of the solution of the cubic equation, the following Italian mathematicians: Scipione del Ferro (1465 - 1526), Niccolò Fontana (Tartaglia) and Gerolamo Cardano, also the main reference (Boyer, 1968) to the solution of the cubic equation as reported at: http://it.wikipedia. org/wiki/Equazione_di_terzo_grado.

The Derivation of the Novel Re-sampling Locations of the Cubic B-Spline On the basis of xsre(i), i = 1, 2, 3; of equation (31), to obtain the novel re-sampling locations that improve the interpolation error for the B-Spline h4 (x), the equation employed is: { h(xr0) * (∂2 (h(x)) /∂x2) (xr0) } / { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { EIN (xsre-x0) / EIN (xsre) } (33) And through the math deduction reported in appendix VI, its solution is: ψ (k, i)) = ± { { { ( α + 2y )1/2 } + { ( α + 2y ) - 4 * { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } }1/2 } / 2 } - ( φ3 / 4 φ4))

(34)

k = 1, 2, 3, 4; and where i = 1, 2…3 as per the degree of the SRE polynomial system as per equation (31). The reader is referred to appendix VI for a full description of the coefficients seen in equations (34) through (42).

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α = [6 ( φ3 / 4 φ4)2 - 3 ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ]

(35)

β = [ -4 ( φ3 / 4 φ4)3 + 3 ( φ3 / φ4) ( φ3 / 4 φ4)2 - 2 ( φ3 / 4 φ4) ( φ2 / φ4) + ( φ1 / φ4) ]

(36)

γ = [ ( φ3 / 4 φ4)4 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) ]



(37)

p = [ (5/6) (5/3) α2 + ( (5/6) α )2 - (5/3) α2 (5/2) + (2 α2 - γ ) ]

(38)

q = - ( (5/6) α )3 + ( (5/6) α )2 (5/2) α - (2 α2 - γ ) (5/6) α + [ (α3 - α γ - β2 /4 ) / 2 ]

(39)

u = { ( q /2 ) ± { ( q /2 )2 + ( p3 / 27) }1/2 }1/3

(40)

v = [ ( p / 3 u ) - u ]



(41)

y = v - (5/6) α



(42)

The value of xr0 is the novel re-sampling location. It is posited that: (xr0) = ψ The total number of novel re-sampling locations is thus twelve for the cubic B-Spline h4(x) obtained from: xr0(i, k) = ψ (k, i)) i = 1, 2, 3; k = 1, 2, 3, 4.



(43)

What has to be acknowledged, as the MATLAB®. (c) 1984-2007 The MathWorks, Inc. code h4SRE2007 located at www.sourcecodewebsite.com shows is that the novel re-sampling location x r0(i, k) = ψ (k, i)) is normalized to the range [0, x0] where x0 is the misplacement.

Eq uating the Intensity-Curvature Terms for Quadratic and Cubic B-Splines: The Deduction Q uadratic B-Spline From equation (6), the intensity-curvature term before interpolation is: x Eo = Eo (x) = ∫ f (0) (∂2 (h3(x)) /∂x2) (0) dx = x θ[a,0] = x θ[a,0]’ * f (0) 0

where it is posited that:

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θ[a,0] = {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } * f (0) = θ[a,0]’ * f (0)





(44)

θ[a,0]’ = {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } From equation (8) the intensity-curvature term after interpolation is: x EIN = EIN (x) = ∫ h3 (x) (∂2 (h3(x)) /∂x2) dx = 0

{ x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] ( 2a + 1/2 )/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1 )/2 + θ[a,3/2] 3( a +1 )/4 ] }

=



{ x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] ( 2a + 1/2 )/2 ] + x [ ( θ[a,0]’ * f (0) ) + θ[a,1/2] ( a + 1 )/2 + θ[a,3/2] 3( a +1 )/4 ] } It follows that equating Eo (x) to EIN (x) yields to: Eo (x) = x θ[a,0]’ * f (0) = EIN (x) = { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] ( 2a + 1/2 )/2 ] + x [ ( θ[a,0]’ * f (0) ) + θ[a,1/2] ( a + 1 )/2 + θ[a,3/2] 3( a +1 )/4 ] } x θ[a,0]’ * f (0) = { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] ( 2a + 1/2 )/2 ] + x [ ( θ[a,0]’ * f (0) ) + θ[a,1/2] ( a + 1 )/2 + θ[a,3/2] 3( a +1 )/4 ] } { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] ( 2a + 1/2 )/2 ] + x [ θ[a,1/2] ( a + 1 )/2 + θ[a,3/2] 3( a +1 )/4 ] } = 0



(45)

{ x3 [ θ[a,3/2] a/3 ] - x3 [ θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] ( 2a + 1/2 )/2 ] + x [ θ[a,1/2] ( a + 1 )/2 ] + x [ θ[a,3/2] 3( a +1 )/4 ] } = 0

(46)

θ[a,3/2] { x3 a/3 - x2 ( 2a + 1/2 )/2 + x 3( a +1 )/4 } + θ[a,1/2] { - x3 2a/3 + x ( a + 1 )/2 } = 0 (47)

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Considering that: θ[a,1/2] = [ f (1/2) + f (-1/2) ] * {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } θ[a,3/2] = { [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] } * {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } And dividing equation (47) by the term: {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } the same equation (47) can be written as: { [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] } * { x3 a/3 - x2 ( 2a + 1/2 )/2 + x 3( a +1 )/4 } + [ f (1/2) + f (-1/2) ] * { - x3 2a/3 + x ( a + 1 )/2 } = 0



(48)

Which implies that: f (1/2) * { x3 a/3 - x2 ( 2a + 1/2 )/2 + x 3( a +1 )/4 } + [ f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * { x3 a/3 - x2 ( 2a + 1/2 )/2 + x 3( a +1 )/4 } + f (1/2) * { - x3 2a/3 + x ( a + 1 )/2 } + f (-1/2) * { - x3 2a/3 + x ( a + 1 )/2 } = 0



(49)

Let us posit: Θ10 = { x3 a/3 - x2 ( 2a + 1/2 )/2 + x 3( a +1 )/4 } Θ1 = [ f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * Θ10 Θ20 ={ - x3 2a/3 + x ( a + 1 )/2 } Θ2 = f (-1/2) * Θ20 Equation (49) is thus written as:

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f (1/2) * Θ10 + Θ1 + f (1/2) * Θ20 + Θ2 = 0 f (1/2) * [ Θ10 + Θ20 ] = - [ Θ1 + Θ2] f (1/2) = - [ Θ1 + Θ2] / [ Θ10 + Θ20 ]



(50)

Equation (50) furnishes the value of the pixel intensity at the location nearest to the pixel to re-sample (the pixel to re-sample has signal intensity f(0)) and that has been deducted equating the two intensitycurvature terms before and after interpolation. Therefore, equation (50) answers the question as what it would be the pixel intensity value f(1/2) when equating the two intensity-curvature terms. The answer is therefore given under the following assumption:, if the two intensity-curvature terms are the same before and after interpolation, then the interpolation function has no effect on the signal for the location x where re-sampling is to happen.

Cubic B-Spline Under the same assumption stated above let us undertake the calculations for the cubic B-Spline. The intensity-curvature term before interpolation is: x Eo = Eo (x) = ∫ f (0) (∂2 (h4(x))/∂x2) (0) dx = x θ[a,10] = x θ[a,10]’ * f (0) 0

where it is posited that: θ[a,10] = θ[a,10]’ * f (0) θ[a,10]’ = {- 2 * [ f (1/2) + f (-1/2) + f (1) + f (-1) ] + 2 * [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] } The intensity-curvature term after interpolation is: x EIN = EIN (x) = ∫ h4 (x) (∂2 (h4(x))/∂x2) dx =



0

θ[a,1] * [ 3/2 x2 - 2 x ] + θ[a,2] * [ - 1/2 x2 + 2 x ] + θ[a,11] * [ 3/10 x5 - x4 + 2/3 x3 + x2 - 4/3 x ] + θ[a,12] * [ - 1/5 x5 + 13/12 x4 - 10/3 x3 + 22/6 x2 - 4/3 x ] + θ[a,22] * [ 1/30 x5 - 4/12 x4 + 4/3 x3 – 16/6 x2 + 8/3 x ] Where the following positions hold true:

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The Extension of Theory and Methodology to B-Splines

θ[a,1] = f (0) [ f (1/2) +f (-1/2) + f (1) + f (-1) ] = f (0) * θ[a,1]’ θ[a,1]’ = [ f (1/2) +f (-1/2) + f (1) + f (-1) ] θ[a,2] = f (0) [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] = f (0) * θ[a,2]’ θ[a,2]’ = [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] θ[a,123] = θ[a,11] * [ 3/10 x5 - x4 + 2/3 x3 + x2 - 4/3 x ] + θ[a,12] * [ - 1/5 x5 + 13/12 x4 - 10/3 x3 + 22/6 x2 - 4/3 x ] + θ[a,22] * [ 1/30 x5 - 4/12 x4 + 4/3 x3 - 16/6 x2 + 8/3 x ] Therefore: EIN (x) = θ[a,1] * [ 3/2 x2 - 2 x ] + θ[a,2] * [ - 1/2 x2 + 2 x ] + θ[a,123] = f (0) * θ[a,1]’ * [ 3/2 x2 - 2 x ] + f (0) * θ[a,2]’ * [ - 1/2 x2 + 2 x ] + θ[a,123]



(51)

By equating the two terms Eo (x) and EIN (x), it follows that: Eo (x) = x θ[a,10]’ * f (0) = EIN (x) = f (0) * θ[a,1]’ * [ 3/2 x2 - 2 x ] + f (0) * θ[a,2]’ * [ - 1/2 x2 + 2 x ] + θ[a,123] x θ[a,10]’ * f (0) = f (0) * θ[a,1]’ * [ 3/2 x2 - 2 x ] + f (0) * θ[a,2]’ * [ - 1/2 x2 + 2 x ] + θ[a,123] f (0) { x θ[a,10]’ - θ[a,1]’ * [ 3/2 x2 - 2 x ] - θ[a,2]’ * [ - 1/2 x2 + 2 x ] } = θ[a,123]

(52)

f (0) = θ[a,123] / { x θ[a,10]’ - θ[a,1]’ * [ 3/2 x2 - 2 x ] - θ[a,2]’ * [ - 1/2 x2 + 2 x ] }

(53)

Under the assumption: if the two intensity-curvature terms are the same before and after interpolation, then the interpolation function has no effect on the signal for the location x where re-sampling is to happen, equation (53) answers the question as what it would be the pixel intensity value f (0) given by the cubic B-Spline interpolation function.

SUMMARY Chapter XIV details the methodological approach that descends from the novel theory which is undertaken for the quadratic and cubic B-Spline interpolation functions as per extension of the unifying approach that has characterized SRE-based bivariate and trivariate linear interpolation in the previous chapters. The chapter starts with a preliminary outline of the two fundamental equations of the method and emphasizes that for given value at location x the function h(x) behavior in the neighborhood can be substantially different because of local concavity or convexity.

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Then, the chapter reiterates that the second order derivative of the function incorporates the curvature information content and thus that the function times its second order derivative ( h(x) * ∂2(h(x)) /∂x ) describes the behavior of the intensity-curvature content at the neighborhood of the node to re-sample. The information content is calculated overall the sub-pixel distribution through ∫ h(x) * ∂2(h(x)) /∂x such to determine the intensity-curvature terms: at the grid point (Eo) and at the generic intra-nodal location (EIN). The ratio between Eo and EIN is computed and named the Intensity-Curvature Functional (ΔE). The Intensity-Curvature Functional comprises of the intensity-curvature joint information content within the pixel and varies across pixels because of its dependency on signal intensity and second order derivative. The mathematics detailed in the chapter furnish the solution of the polynomial first order derivative of ΔE and define such solution as the Sub-pixel Efficacy Region: Φ = {x: ∂ (ΔE(x)) /∂x = 0} (Ciulla & Deek, 2005). By means of the SRE, the chapter explains how to compute the novel re-sampling locations. Finally, the chapter ends the treatise presenting the math process that explains the deduction of the signals that are herein defined and proposed as resilient to the two B-Splines’ model interpolation functions.

REFERENCES Boyer, C. B. (1968). A History of Mathematics. New York, NY: John Wiley & Sons. Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53. Stewart, I. (2004). Galois theory. Boca Raton, Florida: Chapman & Hall/CRC.

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Chapter XV

The Results of the Sub-Pixel Efficacy Region-Based B-Spline Interpolation Functions

Introduction The results obtained processing the MRI database with classic and SRE-based one dimensional quadratic and cubic B-Splines are presented in this chapter. The chapter opens up with information relevant to the image resolution of the MRI database employed for validation. The assessment of the performance of the two classes of interpolators (classic and SRE-based) is conducted both quantitatively and qualitatively. The RSME Ratio is plotted to ascertain which ones of the classic or the SRE-based models deliver the smaller interpolation error. Also, the analysis of error images obtained after processing with either of the two model interpolators and the display of the maps of novel re-sampling locations along with spectral power evolutions corroborates the presentation of the characteristic features of the performances of the interpolation functions treated in this chapter.

Data Ciulla and Deek (2005) presented results obtained applying theory and methodology for the improvement of the quadratic B-Spline employing the following 2D images: Lena, T1 MRI and functional MRI. This section extends on what reported earlier presenting results obtained with two-dimensional T1 MRI, T2 MRI and functional MRI data. MRI images’ resolution and matrix size were: • • •

T1 MRI (T1-MRI-230 and T1-MRI-450). Matrix resolution: 176 x 208 with 1.00 x 1.00 mm of pixel size, with 75 slices and an inter-slice resolution of 1.00 mm. T2 MRI: matrix resolution was 177 x 182 with 1.00 x 1.00 mm of pixel size, with 19 slices and an inter-slice resolution of 1.00 mm. Functional MRI (fMRI-TS1). Matrix resolution was 128 x 128 with 1.72 x 1.72 mm of pixel size, with 20 slices and an inter-slice resolution of 1.72 mm.

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Results of the Sub-Pixel Efficacy Region-Based B-Spline Interpolation Functions

The motion correction paradigm that was applied is described hereto. A misplacement x0 was applied to the signal, then motion correction and interpolation was performed in two different ways: (i) with the classic form of B-Spline and (ii) with SRE-based B-Spline. The latter is the classic B-Spline calculated at the novel re-sampling location as per equation (24) for h3 (x), and equation (43) for h4 (x) respectively (see Chapter XIV for the two equations). For the purpose of simplification, the remainder of this discussion will use the term processed to indicate that the signal or the image was motion corrected and interpolated. At the misplacement x0 the signal or image is termed as processed with classic forms of B-Spline as per equations (3) and (4) of Chapter XIV, whereas at the novel re-sampling location xr0 obtained from equations (24) and (43), the signal or image is termed as processed with SRE-based B-Spline interpolation.

Results with the Quadratic B-Spline This section expands what reported earlier (Ciulla & Deek, 2005) presenting additional results that were collected employing T1 MRI, T2 MRI and functional MRI data.

The Behavior of the RMSE Ratio for Varying Values of the “a” Constant As it can be seen in equation (3) of Chapter XIV, the quadratic B-Spline h3(x) is parametric, which means that it is dependent on the value of the parameter “a”. An investigation was conducted to determine the behavior of the interpolation error improvement with the change of the value of “a”. The signal was processed with classic quadratic B-Spline and SRE-based quadratic B-Spline. Figures 1 through 16 show the behavior of the RMSE Ratio with varying values of the “a” constant. This presentation is given for T1-MRI (T1-MRI-230) in figures 1 through 10, T2-MRI, in figures 11 through 13 and functional MRI (fMRI-TS1), in figures 14 through 16. The definition of the RMSE Ratio was given in Chapter VIII of the book. A negative value of the RMSE Ratio shows that the Sub-pixel Efficacy Region based quadratic interpolation function outperforms the classic in terms of interpolation error, while a positive value shows the opposite. In figure 1 as in all of the figures that relevant to the processing of T1-MRI it can be clearly seen that the plot of the RMSE Ratio versus the value of the “a” constant of the Quadratic B-Spline is well defined. The profiles show a slow but persistent increase in the improvement of the interpolation error for values of the “a’ constant that goes from 2.34 to -2.34. This behavior indicates that the RMSE Ratio (measure of the interpolation error improvement or worsening) converge to zero (no improvement in the interpolation error) from a percentage increase of the interpolation error improvement that is relatively high and that is seen approximately for a = -2.5 to a = -1.9. In graphs 67, 73 and 74, after reaching the maximum increase, the improvement is subject to a decrease and such decrease starts approximately in between a = -2.5 to a = -1.9. This feature is visible also in other graphs of figures 1 through 10. Across the graphs of figure 1 the highest interpolation error improvement achieved through the SRE-based quadratic B-Spline is approximately 350% (see graph 67). In figure 2 the highest percentage in interpolation error improvement is approximately: 175% (graph 75), 45% (graph 76), 250% (graphs 77 and 78), 35% (graph 79), 225% (graph 80), 100% (graph 81), 30% (graph 82).

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Figure 1. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 6774. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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The Results of the Sub-Pixel Efficacy Region-Based B-Spline Interpolation Functions

Figure 2. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 7582. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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The Results of the Sub-Pixel Efficacy Region-Based B-Spline Interpolation Functions

Figure 3. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 8390. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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Figure 4. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 9198. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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In figure 3 the highest percentage in interpolation error improvement is approximately: 100% (graph 83), 200% (graph 84), 50% (graph 85), 80% (graphs 86, 87 and 88), 250% (graph 89), 30% (graph 90). In figure 4 the highest percentage in interpolation error improvement is approximately: 250% (graphs 91 and 92), 80% (graph 93), 250% (graph 94), 180% (graph 95), 60% (graph 96), 200% (graph 97), 300% (graph 98). In figure 5 the highest percentage in interpolation error improvement is approximately: 10% (graph 99), 40% (graph 100), 80% (graph 101), 200% (graphs 102 and 103), 180% (graphs 104 and 105), 250% (graph 106). In graph 99 it is observed a behavior that is not consistent with the profiles of the RMSE shown for the T1-MRI in figures 1 through 10. In figure 6 the highest percentage in interpolation error improvement is approximately: 250% (graph 107), 160% (graph 108), 35% (graph 109), 6% (graph 110), 125% (graph 111), 20% (graph 112), 200% (graph 113), 70% (graph 114). In figure 7 the highest percentage in interpolation error improvement is approximately: 200% (graph 115), 250% (graphs 116, 117, 118, 119 and 120), 75% (graph 121), 250% (graph 122). In figure 8 the highest percentage in interpolation error improvement is approximately: 250% (graph 123), 160% (graph 124), 120% (graph 125), 250% (graphs 126, 127 and 128), 20% (graph 129), 50% (graph 130). In figure 9 the highest percentage in interpolation error improvement is approximately: 250% (graphs 131, 132, 133 and 134), 16% (graph 135), 250% (graph 136 and 137), 100% (graph 138). In figure 10 the highest percentage in interpolation error improvement is approximately: 30% (graph 139), 80% (graph 140), 250% (graph 141). Figures 11 through 13 are relevant to T2-MRI and the general behavior seen for T1-MRI is not clearly visible in these graphs. Generally, data is scattered and in some graphs many points are located along the zero improvement line. It is possible however in each of these graphs to find several values of the “a” constant indicating interpolation error improvement. Hereto is reported a more detailed description of the highest improvement in interpolation error achievable through the SRE-based quadratic B-Spline. In figure 11 the highest percentage in interpolation error improvement is approximately: 12.5% (graphs 1 and 2), 300% (graph 3), 15% (graph 4), 20% (graph 5), 4.5% (graph 6), 25% (graph 7), 400% (graph 8). In figure 12 the highest percentage in interpolation error improvement is approximately: 400% (graph 9), 350% (graph 10), 7% (graph 11), 15% (graph 12), 400% (graph 13), 0.75% (graph 14), 500% (graph 15), 8% (graph 16). In figure 13 the highest percentage in interpolation error improvement is approximately: 25% (graphs 17 and 18), 300% (graph 19). Figures 14, 15 and 16 are relevant to functional MRI and present a behavior of the RMSE Ratio that is quite similar to what seen for T1-MRI as far as the shape of the profiles is concerned. In figure 14 the highest percentage in interpolation error improvement is approximately: 125% (graph 1), 100% (graph 2), 125% (graph 3), 400% (graph 4), 45% (graph 5), 175% (graph 6), 125% (graphs 7 and 8). In figure 15 the highest percentage in interpolation error improvement is approximately: 75% (graph 9), 100% (graph 10), 125% (graphs 11 and 12), 200% (graph 13), 60% (graph 14), 130% (graph 15), 200% (graph 16). In figure 16 the highest percentage in interpolation error improvement is approximately: 150% (graph 17), 125% (graph 18 and 19), 75% (graph 20).

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Figure 5. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 99106. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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Figure 6. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 107114. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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The Results of the Sub-Pixel Efficacy Region-Based B-Spline Interpolation Functions

Figure 7. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 115122. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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Figure 8. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 123130. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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The Results of the Sub-Pixel Efficacy Region-Based B-Spline Interpolation Functions

Figure 9. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 131138. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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Figure 10. Behavior of the RMSE Ratio for varying values of the “a” constant, T1-MRI-230, slices 139-141. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

The Behavior of the RMSE Ratio for Varying Values of the Misplacement Like wise for the study conducted to investigate the behavior of the RMSE Ratio for varying values of the “a” constant, the paradigm adopted to show reduction in the interpolation error was that described in the section: “Data” of this chapter. The image was processed with classic and SRE-based quadratic B-Splines with misplacements along the x coordinate at steps of: (i) 0.01, (ii) 0.0025, and (iii) 0.00001. Therefore the misplacements ranged within the following intervals: (i) [0, 0.99], (ii) [0, 0.2475], (iii) [0, 0.00099], all fractions of the pixel size. This section presents the results of obtained employing T1-MRI (T1-MRI-230 and T1-MRI-450), T2-MRI and functional MRI (fMRI-TS1) data sets. The results show that the SRE-based quadratic B-Spline interpolation function achieves an improvement of the error when compared to the classic form of quadratic B-Spline. The majority of the graphs seen in figures 17 through 26 show a behavior of the RMSE Ratio that indicate a constant and persistent improvement of the interpolation error across the full range of misplacement. The graphs are relevant to T1-MRI and the value of the “a” constant is kept the same for all values of the misplacement. For some of them the interpolation error improvement shows a decrease after the maximum improvement is reached (see for example graphs 67 and 73 in figure 17) while for some others the improvement remains quasi constant after the maximum (see graph 69 in figure 17). In figure 17 the highest percentage in interpolation error improvement is approximately: 350% (graph 67), 300% (graphs 68 and 69), 140% (graph 70), 250% (graph 71), 300% (graph 72), 250% (graph 73), 300% (graph 74). In figure 18 the highest percentage in interpolation error improvement is approximately: 160% (graph 75), 40% (graph 76), 250% (graphs 77 and 78), 35% (graph 79), 250% (graph 80), 100% (graph 81), 30% (graph 82).

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Figure 11. Behavior of the RMSE Ratio for varying values of the “a” constant, T2-MRI, slices 1-8. The “a” constant range is [-4.34, 0.54]. The images were processed with a misplacement of 0.05 along the x direction.

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Figure 12. Behavior of the RMSE Ratio for varying values of the “a” constant, T2-MRI, slices 9-16. The “a” constant range is [-4.34, 0.54]. The images were processed with a misplacement of 0.05 along the x direction.

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Figure 13. Behavior of the RMSE Ratio for varying values of the “a” constant, T2-MRI, slices 17-19. The “a” constant range is [-4.34, 0.54]. The images were processed with a misplacement of 0.05 along the x direction.

In figure 19 the highest percentage in interpolation error improvement is approximately: 100% (graph 83), 200% (graph 84), 50% (graph 85), 80% (graph 86), 70% (graphs 87 and 88), 245% (graph 89), 30% (graph 90). In figure 20 the highest percentage in interpolation error improvement is approximately: 250% (graph 91), 200% (graph 92), 80% (graph 93), 250% (graph 94), 150% (graph 95), 60% (graph 96), 200% (graph 97), 250% (graph 98). In figure 21 the highest percentage in interpolation error improvement is approximately: 0.0005% (graph 99), 250% (graph 100), 70% (graph 101), 200% (graph 102), 200% (graph 103), 150% (graphs 104 and 105), 240% (graph 106). In figure 22 the highest percentage in interpolation error improvement is approximately: 250% (graph 107), 150% (graph 108), 40% (graph 109), 7% (graph 110), 125% (graph 111), 18% (graph 112), 200% (graph 113), 80% (graph 114). In figure 23 the highest percentage in interpolation error improvement is approximately: 200% (graph 115), 280% (graph 116), 300% (graphs 117 and 118), 250% (graph 119), 250% (graph 120), 80% (graph 121), 240% (graph 122). In figure 24 the highest percentage in interpolation error improvement is approximately: 280% (graph 123), 160% (graph 124), 125% (graph 125), 300% (graph 126), 250% (graph 127), 240% (graph 128), 20% (graph 129), 50% (graph 130). In figure 25 the highest percentage in interpolation error improvement is approximately: 300% (graph 131), 250% (graph 132), 250% (graph 133), 250% (graph 134), 15% (graph 135), 50% (graphs 136 and 137), 100% (graph 138).

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Figure 14. Behavior of the RMSE Ratio for varying values of the “a” constant, functional MRI (fMRITS1), slices 1-8. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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Figure 15. Behavior of the RMSE Ratio for varying values of the “a” constant, functional MRI (fMRITS1), slices 9-16. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

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Figure 16. Behavior of the RMSE Ratio for varying values of the “a” constant, functional MRI (fMRITS1), slices 17-20. The “a” constant range is [-2.54, 2.34]. The images were processed with a misplacement of 0.001 along the x direction.

In figure 26 the highest percentage in interpolation error improvement is approximately: 30% (graph 139), 70% (graph 140), 230% (graph 141). In figures 27 through 36 the graphs are relevant to T1-MRI. Some of the graphs show a behavior that is dissimilar to any other graph seen so far (this can be seen in figure 27 for example). Generally though, the graphs of figures 27 through 36 display a behavior of the RSME that is such to furnish a constant increase of the interpolation error improvement till when a maximum improvement is reached (local maximum given that in one single graph there could be up to two) and this is generally followed by a decrease of error improvement. A more detailed analysis is herein reported with focus on the highest error improvement seen in each graph. In figure 27 the highest percentage in interpolation error improvement is approximately: 300% (graph 67), 250% (graph 68), 300% (graph 69), 250% (graph 70), 125% (graph 71), 150% (graph 72), 250% (graph 73), 350% (graph 74). In figure 28 the highest percentage in interpolation error improvement is approximately: 250% (graph 75), 300% (graph 76), 250% (graph 77), 200% (graph 78), 45% (graph 79), 300% (graph 80), 230% (graphs 81 and 82). In figure 29 the highest percentage in interpolation error improvement is approximately: 100% (graph 83), 175% (graph 84), 150% (graph 85), 200% (graphs 86 and 87), 100% (graph 88), 125% (graph 89), 100% (graph 90).

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Figure 17. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 67-74. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 18. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 75-82. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 19. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 83-90. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 20. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 91-98. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 21. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 99-106. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 22. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 107-114. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 23. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 115-122. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 24. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 123-130. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 25. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 131-138. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 26. RMSE Ratio: T1-MRI-230, misplacement along x, quadratic B-Spline, slices 139-141. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

In figure 30 the highest percentage in interpolation error improvement is approximately: 225% (graphs 91 and 92), 87.5% (graph 93), 250% (graph 94), 200% (graph 95), 230% (graph 96), 150% (graph 97), 200% (graph 98). In figure 31 the highest percentage in interpolation error improvement is approximately: 225% (graphs 99, 100, 101 and 102), 250% (graphs 103 and 104), 200% (graph 105), 250% (graph 106). In figure 32 the highest percentage in interpolation error improvement is approximately: 250% (graphs 107, 108, 109 and 110), 200% (graph 111), 250% (graph 112), 275% (graph 113), 220% (graph 114). In figure 33 the highest percentage in interpolation error improvement is approximately: 250% (graph 115), 150% (graphs 116 and 117), 175% (graph 118), 225% (graph 119), 225% (graph 120), 225% (graph 121), 250% (graph 122). In figure 34 the highest percentage in interpolation error improvement is approximately: 250% (graphs 123, 124 and 125), 300% (graph 126), 200% (graph 127 and 128), 250% (graph 129), 270% (graph 130). In figure 35 the highest percentage in interpolation error improvement is approximately: 250% (graph 131), 300% (graph 132), 250% (graph 133), 250% (graph 134), 250% (graph 135), 150% (graphs 136 and 137), 150% (graph 138). In figure 36 the highest percentage in interpolation error improvement is approximately: 225% (graph 139), 200% (graph 140), 175% (graph 141).

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Figure 27. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 67-74. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 28. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 75-82. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 29. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 83-90. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 30. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 91-98. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 31. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 99-106. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 32. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 107-114. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 33. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 115-122. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 34. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 123-130. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 35. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 131-138. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 36. RMSE Ratio: T1-MRI-450, misplacement along x, quadratic B-Spline, slices 139-141. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

The graphs of figures 37 through 39 are relevant to T2-MRI and they present a behavior of the interpolation error improvement that is constantly increasing, unless a maximum improvement is reached, in which case the profile continues with a decrease the improvement. In figure 37 the highest percentage in interpolation error improvement is approximately: 100% (graph 1), 10% (graph 2), 600% (graph 3), 16% (graph 4), 180% (graph 5), 8.75% (graph 6), 87.5% (graph 7), 350% (graph 8). In figure 38 the highest percentage in interpolation error improvement is approximately: 400% (graph 9), 350% (graph 10), 35% (graph 11), 70% (graph 12), 350% (graph 13), 17.5% (graph 14), 500% (graph 15), 35% (graph 16). In figure 39 the highest percentage in interpolation error improvement is approximately: 200% (graph 17), 65% (graph 18), 250% (graph 19). Figures 40 through 42 (T2-MRI) generally indicate a constant increase of the interpolation error improvement. In some graphs however, a maximum improvement is reached and this is followed with a decrease of error improvement. The maximum is local though since in one single graph there could be even up to three (for example see graph 10 in figure 41). In figure 40 the highest percentage in interpolation error improvement is approximately: 10% (graph 1), 22.5% (graph 2), 30% (graph 3), 14% (graph 4), 6% (graph 5), 11% (graph 6), 30% (graph 7), 450% (graph 8). In figure 41 the highest percentage in interpolation error improvement is approximately: 250% (graph 9), 300% (graph 10), 9% (graph 11), 11% (graph 12), 600% (graph 13), 11% (graph 14), 600% (graph 15), 9% (graph 16).

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Figure 37. RMSE Ratio: T2-MRI, misplacement along x, quadratic B-Spline, slices 1-8. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 38. RMSE Ratio: T2-MRI, misplacement along x, quadratic B-Spline, slices 9-16. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 39. RMSE Ratio: T2-MRI, misplacement along x, quadratic B-Spline, slices 17-19. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.2475] at steps of 0.0025.

In figure 42 the highest percentage in interpolation error improvement is approximately: 2.5% (graph 17), 0.75% (graph 18), 250% (graph 19). Figures 43 through 45 (functional MRI) generally indicate only a constant increase of the interpolation error improvement. In figure 43 the highest percentage in interpolation error improvement is approximately: 120% (graph 1), 200% (graph 2), 120% (graph 3), 270% (graph 4), 90% (graph 5), 300% (graph 6), 180% (graph 7), 120% (graph 8). In figure 44 the highest percentage in interpolation error improvement is approximately: 100% (graph 9), 180% (graph 10), 100% (graph 11), 120% (graph 12), 200% (graph 13), 70% (graph 14), 120% (graph 15), 160% (graph 16). In figure 45 the highest percentage in interpolation error improvement is approximately: 140% (graph 17), 100% (graph 18), 70% (graphs 19 and 20).

Q ualitative Assessment of the SRE-based Q uadratic B-Spline Interpolation Function Figure 46 shows for slice 83 of the T1-MRI-230 data set the qualitative assessment of the performance of the SRE-based B-Spline interpolation function. In the top row, from left to right, they are shown: (i) the original image, (ii) the image processed with classic quadratic B-Spline interpolation and (iii) the image processed with SRE-based quadratic B-Spline. The misplacement along the x direction was 0.003 and the value of the “a” constant was -2.36.

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Figure 40. RMSE Ratio: T2-MRI, misplacement along x, quadratic B-Spline, slices 1-8. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 41. RMSE Ratio: T2-MRI, misplacement along x, quadratic B-Spline, slices 9-16. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 42. RMSE Ratio: T2-MRI, misplacement along x, quadratic B-Spline, slices 17-19. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.99] at steps of 0.01.

The three FFT maps are shown in the second row from the top. They do not reveal significant differences. In the third row from the top, from left to right are shown: (i) the map of the novel re-sampling locations, (ii) the error image obtained from the original image and the image processed with classic quadratic B-Spline, and (iii) the error image obtained from the original image and the image processed with SRE-based quadratic B-Spline. Dark corresponds to low error, whereas white corresponds to high error. It is clear the advantage of the SRE-based quadratic B-Spline over the classic form. This is further confirmed in the two histograms of the error images shown in the fourth row from the top. Figure 47 shows the spectral power evolution that elucidates the differences in spectral characteristics between processing with the classic form of quadratic B-Spline and the SRE-based quadratic B-Spline. In this case, the evolution demonstrates that the SRE-based quadratic B-Spline interpolation is capable to preserve the spectral content of the original image more than the classic form of the quadratic BSpline is able to do. The reader is referred to chapter 1 section “Fast Fourier Transform and Spectral Power Analysis” for the description of the methodology used to build the spectral power evolutions shown in figure 47 and in other figures of this chapter. For the first three rows of figure 48 (slice 87 of the T1-MRI-230 data set) and figure 49 (slice 82 of the T1-MRI-450 data set) the organizational layout of the picture is same as that of figure 46. The misplacement employed to process the images was 0.001 along the X direction and a = -2.54 (figure 48) and 0.23 along the X direction and a = -0.254 (figure 49). In the fourth row from the top it is shown the spectral power evolution. Observations that can be drawn from figures 48 and 49 are quite similar to those already outlined by figure 46. The power spectral evolution and the error images shown in figure 49 are interesting in that

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Figure 43. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, quadratic B-Spline, slices 1-8. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 44. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, quadratic B-Spline, slices 9-16. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 45. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, quadratic B-Spline, slices 17-20. The “a” constant value is indicated in each graph. The range of misplacements is [0, 0.00099] at steps of 0.00001.

reveal areas of the k-space where the SRE-based quadratic B-Spline does not outperform the classic quadratic B-Spline. This fact is visible in both the error images and the power spectral evolutions. Also, figure 50 (slice 78 of the T1-MRI-230 data set), for which the misplacement along the X direction was 0.001 with a = -2.54, suggests similar observations made for figure 46. Due to mention is the fact that the map of the novel re-sampling locations for this particular case was not as detailed as that seen in figures 46, 48 and 49. This means that there was not much visible difference on a pixel-by-pixel basis for the values of the novel re-sampling locations and therefore the map resulted flat and not as such to closely match the features of the brain image. The presentation continues with the T2-MRI data set for slices 8, 13, 12, 10 and 9 for which the results are shown in figures 51, 52, 53, 54 and 55 respectively. The misplacement along the x direction was 0.1375, 0.1725, 0.23, 0.001 and 0.23 respectively and the values of the “a” constant were: -0.66, -0.36, -0.012, -2.44, and -0.012 respectively. Due the following comments about these five figures. Figure 51 (slice 8) shows the superiority of the SRE-based quadratic B-Spline interpolation in terms of residual and this is shown in the error images in the third row from the top. Also, the superiority of the SRE-based quadratic B-Spline is manifest through the capability to preserve the spectral content of the original image as shown in the fourth row from the top. Finally, the map of novel re-sampling locations is detailed as it resembles the features of the brain image. Figure 52 (slice 13) shows the superiority of the SRE-based quadratic B-Spline in terms of interpolation error. For what pertains to the preservation of the spectral characteristics of the original image,

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Figure 46. T1-MRI-230 data qualitative assessment, slice 83. The misplacement along the x direction is 0.003 (fraction of pixel size) and the value of the “a” constant is -2.36. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The histograms, the error images and the FFT maps presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 47. T1-MRI-230 data qualitative assessment, quadratic B-Spline, slice 83, spectral power evolution. The evolution is plotted for the intervals 979-1023.

the SRE-based quadratic B-Spline performed better although the map of novel re-sampling locations resulted flat. Figure 53 (slice 12) is interesting because it shows that even though there was not clear and remarkable difference in the error images, the images obtained processing with classic and SRE-based forms of the quadratic B-Spline have visible difference in their spectral content with respect to the original image. This difference is manifest in the oscillations of the power spectral evolution. Also, the map of the novel re-sampling locations (shown in the third row from the top) is detailed and resembles the brain image. In figure 54 (slice 10) although clear and remarkable difference in the error images was not visible, and the map of novel re-sampling locations was flat, there is a manifest superiority of the SRE-based quadratic B-Spline over the corresponding classic form in the capability to preserve the spectral content of the original image. Figure 55 (slice 9) shows the superiority of the SRE-based quadratic B-Spline in terms of residual error accompanied only partially with the capability of preserving the spectral characteristics of the original image. And this is visible in the oscillations seen in the spectral power evolution. Furthermore, one manifest common denominator seen in figures 51 through 55 is that the FFT maps are not quite effective in revealing differences in the spectral content of the three images: (i) original, (ii) processed with classic quadratic B-Spline and (iii) processed with SRE-based quadratic B-Spline. This fact plays in favour of the necessity of the spectral power evolutions to reveal image characteristics in the k-space (Fourier domain) that would otherwise remain hidden. This presentation concludes showing in figure 56 a typical example of the functional MRI data set (fMRI-TS1). Figure 56 is relevant to slice 1 (misplacement along x of 0.001 and a = -2.44). Visual inspection did not reveal differences in the error images neither in the FFT maps. The map of the novel re-sampling locations was flat. However the power spectral evolution reveals relatively moderate superiority of the SRE-based quadratic B-Spline interpolation function.

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Figure 48. T1-MRI-230 data qualitative assessment, quadratic B-Spline, slice 87. The misplacement along the x direction is 0.001 (fraction of pixel size) and the value of the “a” constant is -2.54. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb. info.nih.gov/ij/.

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Figure 49. T1-MRI-450 data qualitative assessment, quadratic B-Spline, slice 82. The misplacement along the x direction is 0.23 (fraction of pixel size) and the value of the “a” constant is -0.254. The evolution is plotted for the intervals 979-1018. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb. info.nih.gov/ij/.

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Figure 50. T1-MRI-230 data qualitative assessment, quadratic B-Spline, slice 78. The misplacement along the x direction is 0.001 (fraction of pixel size) and the value of the “a” constant is -2.54. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb. info.nih.gov/ij/.

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Figure 51. T2-MRI data qualitative assessment, quadratic B-Spline, slice 8. The misplacement along the x direction is 0.1375 (fraction of pixel size) and the value of the “a” constant is -0.66. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 52. T2-MRI data qualitative assessment, quadratic B-Spline, slice 13. The misplacement along the x direction is 0.1725 (fraction of pixel size) and the value of the “a” constant is -0.36. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 53. T2-MRI data qualitative assessment, quadratic B-Spline, slice 12. The misplacement along the x direction is 0.23 (fraction of pixel size) and the value of the “a” constant is -0.012. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 54. T2-MRI data qualitative assessment, quadratic B-Spline, slice 10. The misplacement along the x direction is 0.001 (fraction of pixel size) and the value of the “a” constant is -2.44. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

Results with the Cubic B-Spline The Behavior of the RMSE Ratio for Varying Values of the Misplacement Classic and SRE-based cubic interpolation functions’ performances were compared through the RMSE Ratio. As the reader shall see the RMSE Ratio is consistently presented in this book as the measure

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Figure 55. T2-MRI data qualitative assessment, quadratic B-Spline, slice 9. The misplacement along the x direction is 0.23 (fraction of pixel size) and the value of the “a” constant is -0.012. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 56. Functional MRI (fMRI-TS1) data qualitative assessment, quadratic B-Spline, slice 1. The misplacement along the x direction is 0.001 (fraction of pixel size) and the value of the “a” constant is -2.44. The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee, also courtesy of the Department of Radiology KULeuven. The FFT maps presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

chosen to elucidate the differences between classic and SRE-based interpolation functions. Like wise the motion correction paradigm is the same throughout the book. Evidently the RMSE Ratio formulation descends from the manifest goal to compare the interpolation error determined through the two classes of interpolators studied: classic and SRE-based. Consistently with what was presented for the quadratic B-Spline, this paragraph reports on the results that were obtained employing the classic and SRE-based cubic B-Spline interpolation functions while applying the motion correction paradigm to the MRI data set. Image data was processed with classic and Sub-pixel Efficacy Region based cubic interpolation functions with misplacements along the x coordinate at steps of: (i) 0.01, (ii) 0.0025, (iii) 0.00001, and with misplacements (all fractions of the pixel size) ranging within the following intervals: (i) [0, 0.99], (ii) [0, 0.2475], (iii) [0, 0.00099]. Like wise done for the quadratic B-Spline, this section presents the

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Figure 57. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 67-74. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 58. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 75-82. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 59. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 83-90. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 60. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 91-98. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 61. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 99-106. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 62. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 107-114. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 63. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 115-122. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 64. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 123-130. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 65. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 131-138. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 66. RMSE Ratio: T1-MRI-230, misplacement along x, cubic B-Spline, slices 139-141. The range of misplacements is [0, 0.00099] at steps of 0.00001.

results obtained with T1-MRI (T1-MRI-230 and T1-MRI-450), T2-MRI and functional MRI (fMRITS1) data sets. Figures 57 through 66 are relevant to RSME profiles obtained while processing T1-MRI through the cubic B-Splines. The behavior is quite consistent across all the graphs showing a constant linear increase of the interpolation error improvement versus the misplacement. The magnitude of the improvement seen in these figures is generally much smaller than the improvement that was obtained while processing the same data with the quadratic B-Splines. Hereto is reported a detailed description of the highest improvement seen in each graph. In figure 57 the highest percentage in interpolation error improvement is approximately: 6% (graphs 67, 72, 73 and 74), 7% (graphs 68, 69, 70, and 71). In figure 58 the highest percentage in interpolation error improvement is approximately: 7% (graphs 75, 76, 77, 78, 79and 80), 8% (graphs 81 and 82). In figure 59 the highest percentage in interpolation error improvement is approximately: 8% (graph 83), 7% (graphs 84 and 85), 6% (graphs 86, 87, 88, 89 and 90). In figure 60 the highest percentage in interpolation error improvement is approximately: 6% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). The same happens for figure 61 where the highest percentage in interpolation error improvement is approximately: 6% (graphs 99, 100, 101, 102, 103, 104, 105 and 106). And for figure 62 where the highest percentage in interpolation error improvement is approximately: 6% (graphs 107, 108, 109, 110, 111, 112, 113 and 114). In figure 63 the highest percentage in interpolation error improvement is approximately: 6% (graphs 116, 117, 118, 120, 121 and 122), 7% (graphs 115 and 119).

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Figure 67. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 67-74. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 68. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 75-82. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 69. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 83-90. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 70. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 91-98. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 71. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 99-106. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 72. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 107-114. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 73. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 115-122. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 74. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 123-130. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 75. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 131-138. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 76. RMSE Ratio: T1-MRI-450, misplacement along x, cubic B-Spline, slices 139-141. The range of misplacements is [0, 0.99] at steps of 0.01.

In figure 64 the highest percentage in interpolation error improvement is approximately: 6% (graphs 123, 126, 128, 129 and 130), 7% (graphs 124, 125 and 127). In figure 65 the highest percentage in interpolation error improvement is approximately: 6% (graph 132), 6.5% (graphs 131, 133, 134 and 135), 7% (graphs 136, 137 and 138). In figure 66 the highest percentage in interpolation error improvement is approximately: 6.5% (graph 140), 7% (graph 139), 7.5% (graph 141). In figures 67 through 76 are reported graphs relevant to T1-MRI processing with cubic B-Splines. The common characteristic in all the graphs is a rapid increase of interpolation error improvement that reaches a single maximum and is followed by a decrease. In the following descriptions are reported the values of the maximum error improvement reached for each graph seen in the pictures. Such improvement is attributable to the SRE-based cubic B-Spline. The magnitude of the improvement seen in these figures is generally way much bigger than the improvement seen for figures 57 though 66 and the determinant of this fact might be the magnitude of the misplacement. In figure 67 the highest percentage in interpolation error improvement is approximately: 700% (graph 67), 1500% (graph 68), 1300% (graph 69), 1200% (graph 70), 1500% (graph 71), 750% (graph 72), 1100% (graph 73), 1800% (graph 74). In figure 68 the highest percentage in interpolation error improvement is approximately: 4200% (graph 75), 5400% (graph 76), 4000% (graph 77), 3500% (graphs 78, 79), 2500% (graphs 80 and 81), 1700% (graph 82). In figure 69 the highest percentage in interpolation error improvement is approximately: 1600% (graphs 83 and 84), 2200% (graph 85), 1500% (graphs 86 and 87), 1400% (graph 88), 2300% (graph 89), 1500% (graph 90).

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Figure 77. RMSE Ratio: T2-MRI, misplacement along x, cubic B-Spline, slices 1-8. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 78. RMSE Ratio: T2-MRI, misplacement along x, cubic B-Spline, slices 9-16. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 79. RMSE Ratio: T2-MRI, misplacement along x, cubic B-Spline, slices 17-19. The range of misplacements is [0, 0.2475] at steps of 0.0025.

In figure 70 the highest percentage in interpolation error improvement is approximately: 1200% (graph 91), 750% (graph 92), 1300% (graph 93), 1500% (graph 94), 2200% (graph 95), 1700% (graph 96), 1700% (graph 97), 2100% (graph 98). In figure 71 the highest percentage in interpolation error improvement is approximately: 1400% (graph 99), 1100% (graphs 100 and 102), 1000% (graph 101), 900% (graph 103), 1400% (graph 104), 1200% (graph 105), 800% (graph 106). In figure 72 the highest percentage in interpolation error improvement is approximately: 1100% (graph 107), 1500% (graphs 108 and 109), 950% (graph 110), 1200% (graph 111), 900% (graph 112), 1500% (graph 113), 900% (graph 114). In figure 73 the highest percentage in interpolation error improvement is approximately: 1400% (graph 115), 1200% (graph 116), 1500% (graph 117), 1600% (graph 118), 1300% (graphs 119 and 120), 800% (graph 121), 1600% (graph 122). In figure 74 the highest percentage in interpolation error improvement is approximately: 1400% (graph 123), 1700% (graph 124), 1500% (graph 125), 600% (graph 126), 1300% (graph 127), 1700% (graph 128), 2500% (graph 129), 1700% (graph 130). In figure 75 the highest percentage in interpolation error improvement is approximately: 2300% (graph 131), 1000% (graph 132), 2200% (graph 133), 2600% (graph 134), 2200% (graphs 135, 136), 3000% (graph 137), 2600% (graph 138). In figure 76 the highest percentage in interpolation error improvement is approximately: 1800% (graph 139), 2800% (graphs 140 and 141). Figures 77 through 82 show RMSE profiles relevant to T2-MRI, hereto follow is detailed information about the interpolation error improvement achieved through the SRE-based cubic B-Spline.

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Figure 80. RMSE Ratio: T2-MRI, misplacement along x, cubic B-Spline, slices 1-8. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 81. RMSE Ratio: T2-MRI, misplacement along x, cubic B-Spline, slices 9-16. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 82. RMSE Ratio: T2-MRI, misplacement along x, cubic B-Spline, slices 17-19. The range of misplacements is [0, 0.99] at steps of 0.01.

In figure 77 it is observable a constant increase of the interpolation error improvement. Particularly the profiles show the following highest improvement percentages: 0.6% (graph 1), 0.18% (graph 2), 3% (graph 3), 0.25% (graph 4), 0.125% (graph 5), 0.175% (graph 6), 0.075% (graph 7), 80% (graph 8). In figure 78 the highest percentage in interpolation error improvement is approximately: 15% (graph 9), 40% (graph 10), 0.25% (graph 11), 0.0002% (graph 12), 40% (graph 13), 0.4% (graph 14), 17% (graph 15), 0.4% (graph 16). In figure 79 the highest percentage in interpolation error improvement is approximately: 0.6% (graph 17), 0.4% (graph 18), 4% (graph 19). In figure 80 the highest percentage in interpolation error improvement is approximately: 1.125% (graph 1), 0.4% (graph 2), 4.5% (graph 3), 0.6% (graph 4), 0.225% (graph 5), 0.4% (graph 6), 0.15% (graph 7), 125% (graph 8). In figure 81 the highest percentage in interpolation error improvement is approximately: 35% (graph 9), 60% (graph 10), 0.6% (graph 11), 0.0006% (graph 12), 50% (graph 13), 0.8% (graph 14), 35% (graph 15), 0.075% (graph 16). In figure 82 the highest percentage in interpolation error improvement is approximately: 1.25% (graph 17), 0.75% (graph 18), 7% (graph 19). In figure 83 through 85 data is relevant to functional MRI and the graphs show a constant increase of interpolation error improvement achieved through the SRE-based cubic B-Spline. In figure 83 the highest percentage in interpolation error improvement is approximately: 80% (graph 1), 100% (graph 2), 50% (graph 3), 60% (graphs 4, 5), 180% (graph 6), 70% (graph 7), 80% (graph 8). In figure 84 the highest percentage in interpolation error improvement is approximately: 80% (graph 9), 90% (graph 10), 60% (graphs 11, 12), 100% (graph 13), 80% (graphs 14, 15), 225% (graph 16).

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Figure 83. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, cubic B-Spline, slices 1-8. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 84. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, cubic B-Spline, slices 9-16. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Figure 85. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, cubic B-Spline, slices 1720. The range of misplacements is [0, 0.00099] at steps of 0.00001.

In figure 85 the highest percentage in interpolation error improvement is approximately: 100% (graphs 17 and 18), 80% (graphs 19), 90% (graph 20). Finally figures 86, 87 and 88 show profiles of RSME that demonstrate the existence of a maximum of the interpolation error improvement which is followed by a rapid loss of error improvement with the increase of the magnitude of the misplacement. The RSME profiles are relevant to functional MRI. In figure 86 the highest percentage in interpolation error improvement is approximately: 500% (graph 1), 1200% (graph 2), 1100% (graph 3), 2200% (graph 4), 1200% (graph 5), 1800% (graph 6), 2000% (graph 7), 800% (graph 8). In figure 87 the highest percentage in interpolation error improvement is approximately: 1300% (graph 9), 650% (graphs 10 and 11), 1400% (graph 12), 1000% (graph 13), 650% (graph 14), 1000% (graph 15), 1400% (graph 16). In all graphs of figure 88 the highest percentage in interpolation error improvement is approximately 1000%.

Q ualitative Assessment of the SRE-Based Cubic B-Spline Interpolation Function This section reports on the qualitative assessment of the performance of the SRE-based cubic interpolation function. The figures herein shown were obtained while processing the data with both classic and SRE-based cubic interpolation functions and all misplacements employed in the experiments were along the x direction.

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Figure 86. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, cubic B-Spline, slices 1-8. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 87. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, cubic B-Spline, slices 9-16. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 88. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, cubic B-Spline, slices 1720. The range of misplacements is [0, 0.2475] at steps of 0.0025.

Figures 89, 90 and 91 are relevant to the T1-MRI data set. Their organizational layout is as follow. In the top row, from left to right, it is shown; (i) the original MRI, (ii) the MRI processed with classic cubic B-Spline interpolation, and (iii) the MRI processed with cubic SRE-based B-Spline interpolation. In the second row from the top, the FFT maps are shown for each of the images. In the third row from the top, from left to right are shown: (i) the map of the novel re-sampling locations, (ii) the error image obtained through the classic interpolation paradigm, and (iii) the error image obtained through the SRE-based interpolation paradigm. In the fourth row is displayed the spectral power evolution. Figure 89 (T1-MRI-230) was obtained processing slice 70 with the misplacement of 0.1, figure 90 (T1-MRI-230) was obtained processing slice 77 with the misplacement of 0.001 and figure 91 (T1-MRI450) was obtained processing slice 90 with the misplacement of 0.23. Both in figures 89 and 91 the interpolation error visible in the error images is considerably smaller for the SRE-based function but the preservation of the frequency components of the original image determined by the SRE-based function is manifest in figure 91 but not figure 89. Visual inspection of the error images in figure 90 does not allow distinguishing which paradigm between classic and SRE-based interpolation delivers better approximation properties, and the preservation of the spectral components is marked in favour of the SRE-based paradigm. In all of the figures 89, 90, and 91 the map of the novel re-sampling locations nicely resembles the brain image features and this is in line with the adaptive behavior of the pixel-by-pixel re-sampling approach embedded in the SRE-based interpolation functions. Also, the FFT maps are quite similar across the images and the spectral power evolutions are beneficial aids to them in order to reveal hidden differences in the frequency components of the images.

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Figure 89. T1-MRI-230 data qualitative assessment, cubic B-Spline, slice 70. The misplacement along the x direction is 0.1 (fraction of pixel size). The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 90. T1-MRI-230 data qualitative assessment, cubic B-Spline, slice 77. The misplacement along the x direction is 0.001 (fraction of pixel size). The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 91. T1-MRI-450 data qualitative assessment, cubic B-Spline, slice 90. The misplacement along the x direction is 0.23 (fraction of pixel size). The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 92. T2-MRI data qualitative assessment, cubic B-Spline, slice 11. The misplacement along the x direction is 0.23 (fraction of pixel size). The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth. edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 93. T2-MRI data qualitative assessment, cubic B-Spline, slice 8. The misplacement along the x direction is 0.023 (fraction of pixel size). The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth. edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 94. Functional MRI (fMRI-TS1) data qualitative assessment, cubic B-Spline, slice 1. The misplacement along the x direction is 0.001 (fraction of pixel size). The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee, also courtesy of the Department of Radiology KULeuven. The FFT maps presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

This was seen for the qualitative evaluation of the quadratic B-Splines and is a recurrent feature also for the cubic B-Splines treated in this section. Figure 92 (T2-MRI) was obtained processing slice 11 with the misplacement of 0.23 and figure 93 (T2-MRI) was obtained processing slice 8 with the misplacement of 0.023. In figure 92 the error images are quite similar suggesting similar approximation capabilities for both classic and SRE-based cubic interpolation functions. Also, preservation of the spectral components of the original image visible in the spectral power evolution is similarly delivered by classic and SRE-based cubic B-Splines. In figure 93 instead, a substantial difference in the error images is in favour to the cubic SRE-based interpolation function and is accompanied by larger but variable capability of preserving the spectral

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Figure 95. Functional MRI (fMRI-TS1) data qualitative assessment, cubic B-Spline, slice 1. The misplacement along the x direction is 0.025 (fraction of pixel size). The evolution is plotted for the intervals 979-1023. The original image (the left most in the upper row) was kindly provided by INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee, also courtesy of the Department of Radiology KULeuven. The FFT maps presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

components of the original image, which is manifest for both classic and SRE-based cubic interpolation functions. For the T2-MRI like wise the case of T1-MRI, as shown in figures 92 and 93, the maps of novel re-sampling locations resemble the features of the brain image. Figures 94 and 95 are relevant to functional MRI (fMRI-TS1) data. The error images were similar both for classic and SRE-based cubic interpolation functions suggesting similar approximation capabilities. The interesting feature shown by the two figures is the manifest increase of capability of the SRE-based cubic interpolation function to preserve the spectral characteristics of the original image with the increase of misplacement as seen in the spectral evolution of the two figures. Fair to note that also the classic form of cubic B-Spline had an increase of the capability of preservation of the spectral content (this is seen in figure 95). While in figure 94 the images were processed with the misplacement of 0.001, in figure 95 the images were processed with the misplacement of 0.025, suggesting that the magnitude of the misplacement plays a role in changing the band-pass characteristics of the images processed with cubic B-Spline interpolation.

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SUMMARY Chapter XV presents results of the experimentation based on the motion correction validation paradigm used to validate the SRE-based quadratic and cubic B-Spline interpolation functions. The main difference in the presentation of the results of the two B-Spline forms descends from the nature of the functions. While the quadratic is parametric, the cubic is not. Therefore, for the former a preliminary investigation reveals the behavior of the interpolation error improvement obtained using the SRE versus the variation of the B-Spline parameter. Also, a subsequent investigation reveals the behavior of the error versus the sub-pixel image shift (misplacement). For the cubic B-Splines, due to the lack of the parameter, the behavior of the error is studied versus the misplacement only. Results are presented for high resolution Magnetic Resonance Images (T1-MRI and T2-MRI) and also low resolution functional MRI. Investigation is reported through FFT analysis for the two conditions: (i) classic interpolation and (ii) SRE-based interpolation. The SRE-based paradigm is novel and is obtained through the Sub-pixel Efficacy Region and the subsequently derived novel re-sampling locations. This method reveals differences determined through the interpolation method in the spectral energy content of the resulting images. It is shown that the methodological approach herein proposed is capable to reduce the filtering effect inherent to interpolation. This is consistent with results obtained for the linear interpolation that were presented in parts II and III of the book.

Reference Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53.

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Chapter XVI

On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region

Introduction Concepts of efficiency, approximation, and efficacy are discussed in this chapter while referencing the existing literature. Frameworks grouping classes of interpolation functions are acknowledged. Also, properties of the unifying theory are addressed along with the characteristics of the methodology employed to derive the SRE-based interpolation functions. Properties of the unifying theory are also discussed and the reasoning pertaining to the methodology is provided. As it can be seen in this book, both properties and characteristics featuring the methodological development of the SRE-based interpolation function are quite consistent across interpolators and it is true that the empirical observations provided through this works have set the grounds for the legitimate assertion that there exist a unifying theory for the improvement of the interpolation error. Therefore, such consistency across the spectrum of interpolation functions embraced through the unifying theory shows that the concepts embedded in both properties and characteristics of the methodology employed to derive the SRE-based interpolation functions remain the same across the diversity of model interpolators. It is also due to acknowledge that Section V of the book is devoted to Lagrange and Sinc functions and will further expand on the consistency just mentioned. The last section of this chapter addresses the Fourier properties of the Sub-pixel Efficacy Region.

Efficiency, Approx imation and Efficacy When the efficiency of algorithms for alignment of brain images was studied, a comparative evaluation of the interpolation function computational demand was reported (Joyeux & Penczeck, 2002; Meijer-

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On the Properties of the Unifying Theory and the Derived Sub-Pixel Efficacy Region

ing et al., 2001; Ostuni et al., 1997). Several interpolation methods have been analyzed at the aim of finding the trade off between accuracy and computational cost, which is defined as the function having time and number of multiplications and additions as dependent and independent variables respectively (Thevenaz et al., 2000). Extensive evaluation of accuracy and efficiency of interpolation paradigms was reported (Lehmann et al., 1999) through the analysis of: (i) the Fourier properties, (ii) the visual quality, (iii) the quantitative error estimation, and (iv) the computational demand. An approach was designed to equate the effects of an ideal interpolator for the linear paradigm (Wang, 2001), while focusing on error estimations and minimizations over intervals rather than over the entire interpolation curve. This book introduces the concept of interpolation function efficacy as the capability to generate minimal approximation. For the interpolation functions seen in the preceding chapters (bivariate and trivariate linear, quadratic and cubic B-Spline) it was shown that the capability of the interpolation function to determine minimal approximation is dependent on: (i) the magnitude of the misplacement, (ii) the relationships between intensity values at the target pixel to re-sample and the intensity values at the neighboring pixels, and (iii) the local curvature of the function. This constitutes of a confirmation and also of an extension of the findings reported while studying the Newton-based description of the interpolation error given within the context of mathematical analysis (Agarwal & Wong, 1993; Thevenaz et al., 2000). According to these studies the interpolation error depends: (i) the on step size (resolution), (ii) the location of the re-sampled point relative to the initial coordinate system (misplacement) and (iii) the value of the interpolation function at the node points. It was also reported a specific expression of the Hermite polynomial interpolation error formulated as dependent on optimized constants (Agarwal & Wong, 1993). These constants minimize the approximation of the kernel and are dependent on both resolution and misplacement. Analogously, the concept of Sub-pixel Efficacy Region introduced in part I of the book is aimed to improve the approximation properties of the interpolation functions. The difference between this exploration and previous works on interpolation (Agarwal & Wong, 1993; De Boor, 1992) consists in that in order to improve the approximation; relationships between the misplacement and the pixel intensity are formulated through the Sub-pixel Efficacy Region, instead of using optimized constants. The Subpixel Efficacy Region is calculated and scaled to the actual misplacement in order to compute novel re-sampling locations where the interpolation function is improved. The extremes of the Intensity-Curvature Functional ΔE could be either a maximum or a minimum and this is ultimately depending upon the basis of the relationships between the value of the pixel intensity to re-sample and the neighboring pixel intensities and also on the magnitude of the misplacement. The novel re-sampling locations are computed on the basis of the SRE spatial points which assume the role of references. Minimal approximation of the interpolation function can thus be achieved regardless of the nature of the extreme of ΔE (e.g. maximum or minimum). Thus, the intent of the book is to achieve minimization of interpolation error through the employment of a novel methodology that explores the behavior of the interpolation functions through non uniform re-sampling. More recent research (Blu et al., 2003) reported a general formulation of parameterized piecewise polynomial kernels which provides an interpolation function with a classification for given degree, support (interval of definition), regularity (i.e. differentiability) and order of approximation. It was shown that in most of the cases the kernel coefficients are determined from above features. As far as the placement of the concept of Sub-pixel Efficacy Region within the current literature is concerned, it is relevant to note that the SRE allows studying the approximation property of the interpolation function

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through a novel formulation which is dependent on the node intensity and the second order derivative of the interpolation function. The theory presented in this book makes use of the Intensity-Curvature Functional ΔE and this functional is dependent on the curvature of the interpolation function. One implication of the approach presented here is that it permits finding one or more SRE locations within the pixel that are useful to compute the novel re-sampling locations, which determine the improvement of the interpolation error. Also, the interpolation error can be analyzed locally (i.e. pixel-by-pixel), on the basis of the nature of ΔE (which is calculated as sub-pixel function). This type of analysis is different with respect to the more general least square expressions reported earlier (Thevenaz et al., 2000). An alternative approach to the improvement of the interpolation approximation was reported (Blu et al., 2002, 2004) and considers a constant shift. It calculates the coefficients of the interpolator at each node on the basis on that constant shift. Even though this approach made the improvement local to the node, the resulting improvement was dependent on the shift calculated equally at each node. Along this line of thought and outgrowing the conceptualization of the research presented earlier (Blu et al., 2002, 2004) the methodology presented in this book determines the Sub-pixel Efficacy Region and on the basis of the SRE calculates locally novel re-sampling locations that constitute of shifts varying on a pixel-by-pixel basis. The behavior of the map of the novel re-sampling locations is such to be adaptive to the local curvature of the model interpolation function. Results herein presented confirm that the interpolation approximation can be improved on the basis of the pixel intensity at the location to re-sample and neighbors and on the basis of the curvature of the interpolation function. Thus, the pixel intensity along with the second order derivative of the interpolation function becomes a unified parameter that features the interpolation error and the potentiality of the improvement of the approximation. The foremost implication of the Intensity-Curvature Functional ΔE consists in the fact that the novel re-sampling locations vary across pixels. Also as a consequence of the mathematical formulation of the unifying theory, examining re-sampling as local issue (rather than a global issue) improves the approximation properties of the interpolation function. Practical implications of theory and methodology were shown though the improvement of MRI motion correction. Extension of the study of practical implications that the unifying theory determines for B-Spline models up to the third order are presented in Section IV of this book. Cubic B-Spline models allow considerable improvement over linear interpolation (Meijering et al., 1999) and they were also optimized through approaches aimed to recalculate the node position (De Boor, 1978; Deng & Denney, 2004). The research herein reported calculates the novel re-sampling locations on the basis of the Sub-pixel Efficacy Region and in a manner that is novel: dependent on the joint information content of pixel intensity and curvature of the interpolation function. Finally, due to emphasize is that re-sampling at locations that are determined through the SRE produce a substantial difference in the frequency distribution of the resulting signal contrary to what happens when the signal is re-sampled at the same constant intra-pixel location. Results shown in the previous chapter suggest that there exists another practical implication that derives from the Sub-pixel Efficacy Region, which is that of the preservation of the higher frequencies and a general reduction of the smoothing properties of the interpolation function.

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Frameworks Ex isting in Literature The foundations of B-Splines, wherein frameworks group together different classes of interpolators were examined (Agarwal & Wong, 1993; De Boor, 1978). Comprehensive mathematical characterization of B-Splines was reported (Unser et al., 1993a) along with extensive description of their application in signal processing (Unser et al., 1993b; Unser, 1999). A unifying formulation of parametric piecewise polynomial interpolation was recently reported (Blu et al., 2003) and the characterization of B-Spline digital filters was also studied (Samadi et al., 2004). Above works portrait recent studies on image interpolation within frameworks that fulfill the role to unify the knowledge that is found in the literature. This book derives a framework also. It is of relevance to place the unifying theory presented in this book within the context of the existing literature. To the best of the author’s knowledge, at the aim to improve the interpolation error, the combined use of pixel intensity and curvature of the interpolation function is the distinguishing feature that outstands while comparing the SRE approach presented herein to the various forms of interpolation schemes that are reported in literature since the inception of interpolation (Newton & Huygens, trans. 1934) and its subsequent applications (Blu & Unser, 1999; De Boor, 1994; Strang & Fix, 1971). What determines novelty and originality of the SRE approach is in first instance the act of embedding signal-image intensity values and also curvature of the model interpolation function into a combined formulation called the Intensity-Curvature Functional. In second instance, the calculation of the novel re-sampling locations reveals that the intra-pixel behavior of the model interpolation function shows improved approximation as long as the local curvature of the function at the novel re-sampling location is as close as possible to the true curvature of the unknown signal. The calculation of the novel re-sampling locations, as it can be intuitively understood, is an ill posed problem and this problem is quite similar in its nature to the approximation task demanded to the interpolator. The encouraging truth is certainly that through the employment of the formulas presented in this book the interpolation error show a consistent improvement across a diverse spectrum of interpolators thus allowing the statement that the theoretical foundations of these formulas constitute a unifying approach that permits the improvement of the approximation characteristics of the model interpolators. The theory presented in this book has been thus demonstrated applicable to various interpolation functions regardless of their degree and dimensionality. Therefore, the possibility to achieving error improvement rests on the mathematical framework that consists of the formulas of the unifying theory. These formulas determine the Intensity-Curvature Functional, and the novel re-sampling locations where the interpolation function is calculated. The implication of this framework is that the conceptualization of re-sampling is local: varying pixel-by-pixel.

Summary of the Properties of the SRE-based InteRpolation FunctionS For the SRE-based interpolation functions, relationships were found between neighboring nodes’ intensities and magnitude of misplacement and they allow calculation of novel re-sampling locations within the pixel (node). Such relationships are dependent on pixel intensity of the neighboring nodes and on

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the curvature of the interpolation function and they vary from node-to-node thus depicting re-sampling as a problem that is local and curvature-adaptive. These relationships are independent from any constant shift and do allow improvement of the approximation properties, thus reducing the interpolation error. The novelty of the present approach is two-fold as it consists of: (i) formulating interpolation error improvement as dependent on the node intensity and the curvature of the interpolator and (ii) re-sampling performed at novel locations that can be found based on the knowledge provided through the intensity-curvature behavior at the neighborhood of the node to re-sample.

Characteristics of the Methodology FOR THE IMPROVEMENT OF THE INTERPOLATION ERROR The unifying theory presented in this book makes use of the Intensity-Curvature Functional expressed as ratio (ΔE) between intensity-curvature terms calculated at the grid point and at the generic intranode location. These two intensity-curvature terms are dependent on the interpolation function and its second order partial derivative, which is the measure the local curvature. ΔE is calculated within the nodes and furnishes relationships between neighboring intensity values and misplacement. One or more extremes of ∆E within the sampling step are found and constitute the intra-pixel space domain of points called Sub-pixel Efficacy Region. Equations defining the extreme points of ∆E are calculated on the basis of the polynomial resulting from its first order partial derivatives and are used to determine relationships between neighbouring pixel intensity and magnitude of misplacement. Thus, because of its definition, an extreme of ∆E could be a maximum or a minimum ultimately on the basis of the relationships between (i) the node intensity at the location to re-sample and its neighbours and (ii) the magnitude of the misplacement. Whether the extreme of ∆E is a maximum or a minimum is not an issue of particular relevance because the extreme assumes a role of reference employed in the projection to calculate the novel re-sampling locations. This approach is applied on a node-by-node basis and so is the mathematical tool called Intensity-Curvature Functional. Interpolation approximation is thus approached in an innovative manner that although previously studied (Deng & Denney, 2004; Joyeux & Penczeck, 2002; Wang, 2001), has not yet been fully explored. Solution of the polynomial derived from the first order partial derivatives of ΔE furnishes a set of points that are named Sub-pixel Efficacy Region. These set of points are then candidates to be used to calculate the location where the interpolation function yields improved approximation. One consequence of this approach is that it fosters the discovery of one or more locations within the pixel where is possible to calculate the interpolation function in order to determine interpolation error improvement. As a consequence of the definition of ΔE, which is an intra-node function, re-sampling locations are therefore computed locally thus varying pixel-by-pixel on the basis of the Sub-pixel Efficacy Region. This constitutes a new approach with respect to least square expressions reported earlier (Agarwal & Wong, 1993) which allow motion correction based on a fixed misplacement that is the same for all the nodes. Whereas in the present approach the misplacement and the Sub-pixel Efficacy Region points are used to calculate locally the novel re-sampling locations of the interpolation function and this yields improved approximation.

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It is demonstrated that the location of the set of points that constitute the Sub-pixel Efficacy Region is formulated as dependent on the values of intensity at the neighboring points and the curvature of the interpolator. The view expressed in this book is consistent with what was earlier seen (Agarwal & Wong, 1993) to the extent that interpolation accuracy is dependent on the location of the point to resample relative to the original coordinate system (e.g. magnitude of misplacement) and also dependent on the value of intensity at the neighboring pixels. It then follows that the Sub-pixel Efficacy Region can be different across nodes because of different neighbors’ intensity values. Thus, within an image and for given motion estimate it is extensively demonstrated that re-sampling is an issue that assumes local relevance on the basis of both: (i) magnitude of misplacement and (ii) relationships between neighboring node intensities. It is shown that such relationships can be found through the calculation of the extremes of the Intensity-Curvature Functional and it is also shown that such extremes allow calculation of the novel re-sampling location in order to obtain an improvement of the interpolation error. Limitations of this approach might include the fairly high calculation demand of derivatives, primitives, integrals and the solution of the polynomials derived in order to compute the locations where the interpolation function offers improved approximation. But nevertheless, the calculation demand can be seen as an opportunity for the interested reader and the interested student to practice skills in algebra and calculus, and as an opportunity to break free from customary interpolation schemes, to go into researching in signal (image) interpolation with the intent to discover novel solutions to interpolation and more generally to signal estimation. With this in mind it shall be manifest to the reader that this book offers sections that are explored only theoretically and thus empirical confirmation and expansion of the knowledge is within reach.

Fourier Properties of the Sub-pix el Efficacy Region Ciulla and Deek (2005) reports mention to studies conducted to elucidate: (i) the Fourier properties of the B-Spline functions (Lehmann et al., 1999, 2001) with focus on the tight and undeniable connection between the k-space (frequency) and the image space domain of the model interpolation functions, (ii) filtering capabilities of the B-Splines (Unser et al., 1993a, 1993b; Unser, 1999), (iii) the determination of the B-Spline coefficients (Samadi et al., 2004). Comments are due as far as the Fourier properties of the Sub-pixel Efficacy Region (SRE) based interpolation improvement approach and they shall be addressed to the extent of admitting that the SRE provides the interpolator with the capability of changing the band-pass filtering characteristics. Such property is readily manifest through the data presented in the studies undertaken with quadratic and cubic B-Splines and it was already apparent for the trivariate linear interpolation. And the property derives from the fact that the SRE-based interpolation functions adopt a curvature-adaptive and thus pixel-by-pixel (node-by-node) changing re-sampling approach effectuated through the use of the so called novel re-sampling locations. These are calculated through deduction with the mathematical methodology that is at the foundations of the unifying theory.

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SUMMARY Chapter XVI concludes Section IV of the book. The chapter gives attention to the most relevant literature. Given a re-sampling location, to achieve interpolation error improvement the signal is re-sampled at a novel location obtained on the basis of the Sub-pixel Efficacy Region (SRE). This chapter discusses the two conceptions that were object of experimentation in Chapter XV. They are: (i) interpolation error improvement can be formulated as dependent on the joint information content of the node intensity and the second order derivative of the interpolation function, and (ii) re-sampling is an issue of local relevance which depends on the neighboring signal intensity values and the local curvature of the model interpolation function. The discussion is extended to concepts of efficiency, accuracy of approximation and efficacy, and it is pointed out how the novel theory differs from previous work. The literature is referenced to inform the reader of the works developed by others at the aim to characterize interpolation error and efficiency in terms of computational demand, and also a comparison is made with the novel approach proposed through the unifying theory. Limitations are also addressed and grounds are set for the extension of the theory to other interpolation paradigms, regardless of their degree (the biggest exponent of the model interpolation function) and the dimensionality (the number of spatial dimensions covered with the model function). Characteristics of the methodology are referenced to the current literature pointing out to the flexibility of the novel unifying theory and its capability to group interpolators under same methodological approach regardless of their dimensionality. Finally this chapter discusses the other relevant practical implications of the Sub-pixel Efficacy Region: (i) the preservation of spectral frequencies into the images, and (ii) the reduction of the smoothing effect of the interpolation functions.

References Agarwal, R. P., & Wong, P. J. Y. (1993). Error inequalities in polynomial interpolation and their applications. Mathematics and its applications. Dordrecht, The Netherlands: Kluwer Academic Publishers. Blu, T., & Unser, M. (1999). Quantitative Fourier analysis of approximation techniques: Part I – interpolators and projectors. IEEE Transactions on Signal Processing, 47(10), 2783-2795. Blu, T., Thevenaz, P., & Unser, M. (2002). How a simple shift can significantly improve the performance of linear interpolation. Proceedings of the International Conference on Image Processing, USA, 3, 377-380. Blu, T., Thevenaz, P., & Unser, M. (2003). Complete parameterization of piecewise-polynomial interpolation kernels. IEEE Transactions on Image Processing, 12(11), 1297-1309. Blu, T., Thevenaz, P., & Unser, M. (2004). Linear interpolation revitalized. IEEE Transactions on Image Processing, 13(5), 710-719. Ciulla, C., & Deek, F. P. (2005). Novel schemes of trivariate linear and one-dimensional quadratic BSpline interpolation functions based on the sub-pixel efficacy region. ICGST - International Journal on Graphics, Vision and Image Processing, 5(8), 43-53.

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De Boor, C. (1978). A practical guide to splines. Applied mathematical sciences. New York, NY: Springer-Verlag. De Boor, C. (1992). On the error in multivariate polynomial interpolation. Applied Numerical Mathematics, 10(3-4), 297-305. De Boor, C., Devore, R. A., & Ron, A. (1994). Approximation from shift invariant subspaces of L2 (Rd). Transactions of the American Mathematical Society, 341(2), 787-806. Deng, X., & Denney, Jr., T. S. (2004). On optimizing knot positions for multi-dimensional B-spline models. Proceedings of SPIE, USA, 5299, 175-186. Joyeux, L., & Penczeck, P. A. (2002). Efficiency of 2D alignment methods. Ultramicroscopy, 92, 3346. Lehmann, T. M., Gonner, C., & Spitzer, K. (1999). Survey: Interpolation methods in medical image processing. IEEE Transactions on Medical Imaging, 18(11), 1049-1075. Lehmann, T. M., Gonner, C., & Spitzer, K. (2001). Addendum: B-spline interpolation in medical image processing. IEEE Transactions on Medical Imaging, 20(7), 660-665. Meijering, E. H. W., Zuiderveld, K. J., & Viergever, M. A. (1999). Image reconstruction by convolution with symmetrical piecewise n-th order polynomial kernels. IEEE Transactions on Image Processing, 8(2), 192-201. Meijering, E. H. W., Niessen, W. J., & Viergever, M. A. (2001). Quantitative evaluation of convolutionbased methods for medical image interpolation. Medical Image Analysis, 5(2), 111-126. Newton I., & Huygens, C. (1934). The motion of the moon’s nodes. In R. M. Aynard Hutchins (Ed.), Mathematical Principles of Natural Philosophy: Optics, Treatise on light (A. Motte, Trans.). (pp. 338339). William Benton. Ostuni, J. L., Santha, A. K. S., Mattay, V. S., Weinberger, D. R., Levin, R. L., & Frank, J. A. (1997). Analysis of interpolation effects in the reslicing of functional MR Images. Journal of Computer Assisted Tomography, 21(5), 803-810. Samadi, S., Ahmad, M. O., & Swamy, M. N. S. (2004). Characterization of B-spline digital filters. IEEE Transactions on Circuits and Systems, 51(4), 808-816. Strang, G., & Fix, G. (1971). A Fourier analysis of the finite element variational method. In Constructive Aspect of Functional Analysis (pp. 796-830). Rome, Italy: Edizioni Cremonese. Thevenaz, P., Blu, T., & Unser, M. (2000). Interpolation revisited. IEEE Transactions on Medical Imaging, 19(7), 739-758. Unser, M., Aldroubi, A., & Eden, M. (1993a). B-spline signal processing: Part I – theory. IEEE Transactions on Signal Processing, 41(2), 821-833. Unser, M., Aldroubi, A., & Eden, M. (1993b). B-spline signal processing: Part II – efficient design and applications. IEEE Transactions on Signal Processing, 41(2), 834-848.

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Unser, M. (1999). Splines: A perfect fit for signal and image processing. IEEE Signal Processing Magazine, 16(6), 22-38. Wang, J. (2001). Optimal design for linear interpolation of curves. Statistics in Medicine, 20(16), 24672477.

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Section V

Extension to Lagrange and Sinc Interpolation Functions

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Chapter XVII

The Main Innovation Determined By the Sub-Pixel Efficacy Region

Introduction This chapter introduces the reader to Section V of the book. The chapter opens up with a discussion on the undeniable evidence reported in literature that the magnitude of the interpolation error is strictly related to the magnitude of the sampling resolution. While reference to the literature on the Lagrange interpolation function is reported elsewhere (Ciulla & Deek, 2006), the chapter devotes attention to the literature and the applications related to the Sinc function. The core of the chapter reports a section that condenses the message to the reader of this book about the main innovation determined through the Sub-pixel Efficacy Region. It is delivered to the reader the realization that combining signal intensity with the curvature of the interpolation function, the approximation properties of the model function can be improved. This message is linked to the bridging concept between classic and SRE-based interpolation which is that of the curvature of the interpolation function.

LITERATURE For a discussion on Lagrange approximation, literature and more generally on interpolation approximation error bounds the reader is referred to Ciulla and Deek (2006). For what concerns the present chapter it is necessary to recall that the literature extensively admits and reports the evidence that the goodness of approximation of any interpolation function is inversely proportional to the sampling step (resolution). Conceptually this is straightforward since the degree of uncertainty of the estimate of the model interpolation increases proportionally to the length of the curve connecting two adjacent nodes. That is to say that the degree of uncertainty about the true local curvature between two adjacent nodes increases with the increase of the sampling step. Such increase is detrimental because of two fundamental reasons. Both reasons descend from the nature of the ill posed problem that the model interpolator

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The Main Innovation Determined By the Sub-Pixel Efficacy Region

attempt to solve. The first and main reason is that the true signal is unknown and from this concept it descends that the true local curvature is consequentially unknown. The second reason is consequential to the first one and it reveals itself by the nature of the task the interpolator has to perform which is that of approximation. To approximate within a large sampling step is thus far more difficult thus prone to a larger error because the length of the curve that the model interpolator has to reconstruct is extended. This concept applies in either image space or k-space (frequency domain). Nevertheless, as demonstrated by early experimental research (Whittaker, 1915, 1935), the ideal approach to interpolation remains the Sinc interpolation function. The reason why Sinc interpolation proves to be superior stands in its theoretical background as it considers the signal to be defined from -∞ to +∞ through infinite convolution of the sample node intensities. The typical Sinc function is: ∞

∞

x (t) = Σ -∞ x (nT) sinc( π(t-nT)/T) = Σ -∞ x (nT) sin[ (π(t-nT)/T) ] / (π(t-nT)/T)

(1)

where T is the sampling period. Two main limitations arise from equation (1) and they are: (i) the number of intensity values (nT) are finite, and (ii) values of the sampling point close or equal to t = nT furnish an undetermined expression of the type 0/0. Methods of approximating the Sinc interpolation function given by equation (1) through a finite summation of samples have been explored (Schanze, 1995). It was also demonstrated that the approximated form of the Sinc function can also be represented by the Lagrange function (Whittaker, 1915, 1935). Schanze (1995) reported the following expressions of the finite sum approximation of the Sinc function:

M –1



M –1

x (t) = sin[ πt / N ] * Σn = - L xn (-1n) cot [π (t - n )/ N], x (t) = sin[ πt / N ] * Σn = - L xn (-1n) csc [π (t - n )/ N],

N even

(2)

N odd

(3)

Alternative formulations were reported (Candocia & Principe, 1998; Dooley & Nandi, 2000). The formulation of Dooley and Nandi (2000) was chosen to extend the unifying theory by means of the Sub-pixel Efficacy Region (SRE) and this shall be seen in the next chapter. It is important to recall common applications of Sinc interpolation for signal-image reconstruction. One of the most important is up-sampling which consists of the estimation of the signal at time points that could not be sampled because of the limitations due to the Nyquist’s theorem. In other words, even though the signal is sampled at frequencies that are as close as possible to the Nyquist rate, values of intensity may be needed at interleaved time locations and so up-sampling of the signal is required. The most straightforward method is that of calculating the FFT of the signal, zero filling the Fourier Series and subsequently to calculate the inverse FFT such to obtain a spatially expanded signal. Various methods for the improvement of the efficiency of this approach have been reported (Markel, 1971; Screenivas & Rao, 1979; Skinner, 1976; Smit et al., 1990; Smith & Nichols, 1988; Yaroslavsky, 1996). Other applications of Sinc interpolation use the powered data to be convoluted by the Sinc function to obtain a more accurate estimate (Lavalle, 1992). Other approaches provide with estimates of the Sinc interpolation error as dependent on the curvature (slope) of the signal (Jenq, 1994) and also provide with improved signal reconstruction in Magnetic Resonance Imaging (MRI) (Hajnal et al., 1995; Thacker et al., 1999). Because of the convolution functions that are used to approximate the signal, interpola-

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tion determines smoothing through cutting the higher frequency range of the original signal and this is detrimental because determines loss of resolution.

MESSAGE TO THE READER The SRE approach applied to the Sinc interpolation function exerts one of its main advantages in reducing the smoothing effects of the convolution. Within this context the literature is missing of a mathematical formulation that (i) captures the effects of the interpolator on the signal, and to reduce its detrimental effects (ii) merges the pixel (node) intensity value together with the curvature. This is the main innovation determined through the SRE and also the main message to the reader: with the combined use of signal and curvature, the SRE interpolation functions can be more powerful than the classic functions. The significance of the use of the curvature (second order derivative) is intuitively connected to the task of approximating the true behavior of the signal at nodes that have not been sampled. But most importantly, as well straightforward in its truth, the curvature is conceptually linked to the approximation order of the interpolator. To determine a neat improvement of the interpolation approximation characteristics, ideally the re-sampling location needs to be adaptive to the curvature of the true signal. But practically, since the true signal remains unknown, the re-sampling location is tight to be adaptive to the curvature of the model interpolation function. The task of calculating the re-sampling location as adaptive to the curvature of the model interpolation function is achieved through the SRE approach with the re-sampling strategy that takes place at novel locations varying node-by-node on the basis of the local curvature of the model interpolation function. Development of the mathematical formulation with above features and the derivation through deduction of a unifying theory for the improvement of the interpolation error was achieved for several interpolation paradigms. These include bivariate linear (Part II of the book), trivariate linear, (Part III) and B-Splines (Part IV). Intuitions made of the Intensity-Curvature Functional (ΔE) and the Sub-pixel Efficacy Region (SRE), the former being a mathematical formulation embedding the effect of the interpolator before and after interpolation, and the latter a space domain set of points obtained from the study of ΔE have the advantage: (i) to conceptualize interpolation error improvement as the formulation dependent on the node intensity and the curvature of the interpolation function and (ii) to perform re-sampling at locations that vary node-by-node. These intuitions have the keynote in the practical use of the concept of curvature. They have permitted to derive the novel theory and also to bridge the gap with the theoretical basis of the classic forms of interpolation functions. This part of the book extends the theoretical framework to Lagrange and Sinc functions, formulates ΔE for them, and also shows that by the study of ΔE it is possible to determine an intra-pixel set of points in the space domain designated as the Sub-pixel Efficacy Region. The SRE provides the interpolation function with the basis that lead to improved approximation. The following chapter will outline the theory relevant to the Lagrange and Sinc functions and will extend the theoretical framework for the improvement of the interpolation error. It will also recall the demonstration given earlier (Ciulla & Deek, 2006) for which both interpolation error and interpolation error improvement are bounded by two constants. In chapter 19, spectral power evolutions and RMSE

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analyses of the improved function are presented. In Chapter XX it is discussed the SRE approach to the improvement of the interpolation error placing the emphasis on the benefits and the limitations of the unifying theory.

SUMMARY Chapter XVII introduces to the reader the intent of Section V of the book which is that of treating two interpolators that have been shown through Fourier analysis to present excellent band-pass characteristics, they are: one-dimensional Lagrange and Sinc interpolation functions. Literature on these two functions is selective and the chapter references the work done on this topic with emphasis on the Fourier properties of these interpolators. Especially Sinc interpolation has been shown in literature to limit very well the detrimental effects that are inherent to interpolation and that are manifest through smoothing of the signal. This chapter finally recalls to the reader the main innovation introduced with the SRE-based interpolation functions developed under the umbrella of the unifying theory.

References Candocia, F., & Principe, J. C. (1998). Comments on “Sinc interpolation of discrete periodic signals”. IEEE Transactions on Signal Processing, 46(7), 2044-2047. Ciulla, C., & Deek, F. P. (2006). Extension of the sub-pixel efficacy region to the Lagrange interpolation function. ICGST - International Journal on Graphics, Vision and Image Processing, 6(1), 1-12. Dooley, S. R., & Nandi, A. K. (2000). Notes on the interpolation of discrete periodic signals using sinc function related approaches. IEEE Transactions on Signal Processing, 48(4), 1201-1203. Hajnal, J. V., Saeed, N., Soar, E. J., Oatridge, A., Young, I. R., & Bydder, G. M. (1995). A registration and interpolation procedure for subvoxel matching of serially acquired MR images. Journal of Computer Assisted Tomography, 19(2), 289-296. Jenq, Y. C. (1994). Sinc interpolation errors in finite data record length. 10th Conference on Instrumentation and Measurement Technology, 2, 704-707. Hamamatsu, Japan: IEEE. Lavalle, C. (1992). The sinc interpolation in the impulse response function analysis. International Geoscience and Remote Sensing Symposium, 2, 842-844. Huston, TX: IEEE. Markel, J. D. (1971). FFT pruning. IEEE Transactions on Audio and Electroacoustics, 19(4), 305-311. Schanze, T. (1995). Sinc interpolation of discrete periodic signals. IEEE Transactions on Signal Processing, 43(6), 1502-1503. Screenivas, T. V., & Rao, P. V. S. (1979). FFT algorithm for both input and output pruning. IEEE Transactions on Acoustics, Speech, and Signal Processing, 27(3), 291-292. Skinner, D. P. (1976). Pruning the decimation-in-time FFT algorithm. IEEE Transactions on Acoustics, Speech, and Signal Processing, 24(2), 193-194.

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Smit, T., Smith, M. R., & Nichols, S. T. (1990). Efficient sinc function interpolation technique for center padded data. IEEE Transactions on Acoustics, Speech, and Signal Processing, 38(9), 1512-1517. Smith M. R. & Nichols, S. T. (1988). Efficient algorithms for generating interpolated (zoomed) MR images. Magnetic Resonance in Medicine, 7(2), 156-171. Thacker, N. A., Jackson, A., Moriarty, D., & Vokurka, E. (1999). Improved quality of re-sliced MR images using re-normalized sinc interpolation. Journal of Magnetic Resonance Imaging, 10(4), 582-588. Whittaker, E. T. (1915). On the functions which are represented by the expansions of interpolation theory. Proceedings of the Royal Society Edinburgh, UK, Sec. A(35), 181-194. Whittaker, J. M. (1935). Interpolatory function theory (Cambridge Tracts in Mathematics and Mathematical Physics No. 33). New York, NY: Cambridge University Press. Yaroslavsky, L. P. (1996). Signal sinc-interpolation: A fast computer algorithm. Bioimaging, 4(4), 225231.

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

ORGANIZATION OF THE FORTHCOMING TEXt This chapter is devoted to the mathematics of the Lagrange and Sinc SRE-based interpolation functions. The organization of the text of this chapter is consistent with that of chapters VII, X, and XIV. The basic aim of this chapter is to employ the methodology outlined in the book such to develop a mathematical formulation that allows interpolation error improvement also for Lagrange and Sinc interpolation functions. This is achieved through two instruments that bridge classic interpolation with the present innovative theory. The instruments are the Intensity-Curvature Functional (ΔE) and the Sub-pixel Efficacy Region (SRE). Math processes are thus presented that start from the calculation of the intensitycurvature terms and the corresponding Intensity-Curvature Functional, determine the SRE and employ the formula of the unifying theory (see equations [16] and [38] for Lagrange and Sinc respectively) to calculate the novel re-sampling locations for the two model interpolation functions. A section of this chapter is delegated to recall to the reader the characterization of upper and lower bounds of interpolation error improvement and interpolation error respectively. Details of this section are reported elsewhere (Ciulla & Deek, 2006). Finally the theoretical presentation of resilient interpolation is extended also to Lagrange and Sinc as it was already presented for the two linear functions and the two B-Splines that were object of treatise in Parts II, III and IV of this book. Although the logic behind the math of resilient interpolation is explained and characterized, resilient interpolation remains in this book a theoretical conceptualization which looks forward to empirical confirmation.

Lagrange Interpolation Function The one dimensional Lagrange interpolation function studied here was reported earlier (Lehmann et al., 1999) as follow:

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

(1/2) | x |3 - | x |2 - (1/2) | x | + 1 0≤|x|≤1 LGR3(x) = 1≤|x|≤2 -(1/6) | x |3 + | x |2 - (11/6) | x | + 1

(1)

Herein it assumes a 1 x 4 pixels’ neighborhood: f (-1), f (1), f (-2), f (2) centered at f (0) (see equation (2)). For simplicity let us consider the function: LGR3(x) = f (0) + [ f (1) + f (-1) ] * [ (1/2) x3 - x2 - 1/2 x + 1 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ -(1/6) x3 + x2 - (11/6) x + 1 ]

(2)

Let us calculate the first and second order derivatives of LGR3(x) with respect to the variable x: (∂ ( LGR3(x) ) /∂x) = [ f (1) + f (-1) ] * [ (3/2) x2 - 2 x - 1/2 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ -(1/2) x2 + 2 x - (11/6) ] (∂2 ( LGR3(x) ) /∂x2) = [ f (1) + f (-1) ] * [ 3 x - 2 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ - x + 2 ]

(3)

Let us posit: θ(1) = [ f (1) + f (-1) ]; θ(2) = [ f (1) + f (2) + f (-1) + f (-2) ]; Thus: (∂2 ( LGR3(x) ) /∂x2) (x) = θ(1) * [ 3 x - 2 ] + θ(2) * [ - x + 2 ]

(4)

and (∂2 ( LGR3(x) ) /∂x2) (0) = -2 θ(1) + 2 θ(2) = θ

(5)

Calculation of the Lagrange Intensity-Curvature Terms Before and After Interpolation Let Eo and EIN be the intensity-curvature terms before and after interpolation. While the full math process employed for their calculation is reported in appendix VII, this paragraph illustrates the two resulting formulations:

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

x Eo = Eo (x) = ∫ f (0) (∂2 ( LGR3(x) ) /∂x2) (0) dx =



0

x ∫ f(0) θ dx = f(0) θ x



(6)

0

x EIN = EIN (x) = ∫ LGR3(x) (∂2 ( LGR3(x) ) /∂x2) dx =





0

x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1

(7)

Where it is posited that: θ5 = { (3/10) θ(1) θ(1) -(1/5) θ(1) θ(2) + (1/30) θ(2) θ(2) } θ4 = { - θ(1) θ(1) + (1/2) θ(1) θ(2) + (5/6) θ(2) θ(1) - (1/3) θ(2) θ(2) } θ3 = { (1/6) θ(1) θ(1) - (1/2) θ(1) θ(2) - (15/6) θ(2) θ(1) + (23/18) θ(2) θ(2) } θ2 = { (3/2) f (0) θ(1) - (1/2) f (0) θ(2) + 2 θ(1) θ(1) - θ(1) θ(2) - (20/9) θ(2) θ(1) - (7/3) θ(2) θ(2) } θ1 = { - 2 f (0) θ(1) + 2 f (0) θ(2) - 2 θ(1) θ(1) + 2 θ(2) θ(2) }

Calculation of the Lagrange Intensity-Curvature FUNcTioNAL ΔE ΔE(x) = Eo (x) / EIN (x) = { f(0) θ x } / { x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1 }



(8)

Table 1. Theta values for equations (4) through (8).

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

STUdY of THe LAGRANGe INTeNsiTY-CURVATURe FUNcTioNAL ΔE To determine the SRE of the Lagrange function let us calculate the first order derivative of the IntensityCurvature Functional ΔE and pose it equal to zero. In equating the first order derivative of ΔE to zero, the study of the Intensity-Curvature Functional of Lagrange interpolation is performed as follows: ∂ (ΔE(x)) / ∂x = [ Eo(x) * ∂ (EIN (x)) /∂x + EIN (x) * ∂ (Eo(x)) /∂x ] / [ EIN(x) ]2 = 0 And based on equation (8) it can be written that: ∂ { { f(0) θ x } / { x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1 } } / ∂x = 0 ∂ { { f(0) θ x } / { x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1 } } / ∂x = { { f(0) θ x } * { 5 x4 * θ5 + 4 x3 * θ4 + 3 x2 * θ3 + 2 x * θ2 + θ1 } - { f(0) θ } * { x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1 } } / { x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1 } = 0 { { f(0) θ x } * { 5 x4 * θ5 + 4 x3 * θ4 + 3 x2 * θ3 + 2 x * θ2 + θ1 } - { f(0) θ } * { x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1 } } = 0 f(0) θ * { 5 x5 * θ5 + 4 x4 * θ4 + 3 x3 * θ3 + 2 x2 * θ2 + θ1 * x - x5 * θ5 - x4 * θ4 - x3 * θ3 - x2 * θ2 - x * θ1 } = f(0) θ * { 4 x5 * θ5 + 3 x4 * θ4 + 2 x3 * θ3 + x2 * θ2 } = 0



(9)

The Sub-pix el Efficacy Region of the Lagrange Interpolation Function Given the theta values as per Table I, from equation (9) it follows that the SRE of the Lagrange interpolation function is: { 4 x5 * θ5 + 3 x4 * θ4 + 2 x3 * θ3 + x2 * θ2 } = x2 { 4 x3 * θ5 + 3 x2 * θ4 + 2 x * θ3 + θ2 } = 0 (10) There are five SRE intra-nodal points and they result from the solution of above equation as given as follows. xsre(4) = xsre(5) = 0



The remaining three are given by the solution of:

356

(11)

The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

{ 4 x3 * θ5 + 3 x2 * θ4 + 2 x * θ3 + θ2 } = 0 (12) (2008, Carlo Ciulla, IGI Global)



This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the cubic equation is: http://it.wikipedia.org/wiki/Equazione_di_terzo_grado Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. { x3 + x2 * (3 θ4 / 4 θ5 ) + x * ( 2 θ3 / 4 θ5 ) + ( θ2 / 4 θ5) } = 0 Let us posit: φ3 = 4 θ5 φ2 = 3 θ4 φ1 = 2 θ3 φ0 = θ2 Thus: { x3 + x2 * (3 θ4 / 4 θ5 ) + x * ( 2 θ3 / 4 θ5 ) + ( θ2 / 4 θ5) } = { x3 + x2 (φ2 / φ3) + x (φ1 / φ3) + ( φ0 / φ3) } = 0 It follows that: zi = { ( q / 2 ) ± { ( q / 2 )2 + ( p3 / 27 ) }1/2 }1/3 i = 1, 2, 3 yi = zi - ( p / 3zi ) with: xsre(i) = [ yi - (φ2 / 3 φ3) ]

(13)

i = 1, 2, 3

(14)

i = 1, 2, 3

(15)

where: q = [ (2/27) ( 3 θ4 / 4 θ5 )2 - ( 3 θ4 / 12 θ5 ) ( 2 θ3 / 4 θ5 ) + ( θ2 / 4 θ5 ) ] p = [ - ( 3 θ4 / 36 θ5 )2 + ( 2 θ3 / 4 θ5 ) ] The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve the cubic equation. Also, the author wishes to acknowledge the main authors of the solution of the cubic equation, the following Italian mathematicians: Scipione del Ferro (1465 - 1526), Niccolò

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

Fontana (Tartaglia) and Gerolamo Cardano, also the main reference (Boyer, 1968) to the solution of the cubic equation as reported at: http://it.wikipedia.org/wiki/Equazione_di_terzo_grado.

The Derivation of the Novel Re-sampling Locations of the Lagrange Interpolation Function The SRE points furnish novel locations where the Lagrange interpolation has improved approximation properties. Given a misplacement x0, any novel re-sampling locations xr0 ≠ (xr0(1,i), xr(2,i), xr(3,i), xr(4,i)) are found through the solution of equation (16) while considering xsre = xsre(i), i =1, 2, 3, 4, 5. { LGR3(xr0) * { (∂2 (LGR3 (x)) /∂x2) (xr0) } } / { LGR3(xsre) * { (∂2(LGR3 (x)) /∂x2) (xsre) } } = { EIN (xsre-x0) / EIN (xsre) }





(16)

While the full math process is reported in appendix VII, hereto is presented the equation that provides with the novel re-sampling locations: ψ 4 φ4 + ψ 3 φ3 + ψ 2 φ2 + ψ φ1 + φ0 = 0



(17)

For a full characterization of the coefficients in equation (17) the reader is referred to appendix VII (2008, Carlo Ciulla, IGI Global) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the quartic equation is: http://en.wikipedia.org/wiki/Quartic_equation Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”.

Table 2. Positions for equation (17)

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

ψ 4 + ψ 3 ( φ3 / φ4) + ψ 2 ( φ2 / φ4) + ψ ( φ1 / φ4) + ( φ0 / φ4) = 0 where ψ = (xr0) and it admits the following four solutions (k) for each (i): ψ (k, i)) = ± { { { ( α + 2y )1/2 } + { ( α + 2y ) - 4 * { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } }1/2 } / 2 } - ( φ3 / 4 φ4)) (18)



k = 1, 2, 3, 4; and i = 1, 2…5 as per the degree of the SRE polynomial system of equation (10). α = [6 ( φ3 / 4 φ4)2 - 3 ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ]

(19)

β = [ -4 ( φ3 / 4 φ4)3 + 3 ( φ3 / φ4) ( φ3 / 4 φ4)2 - 2 ( φ3 / 4 φ4) ( φ2 / φ4) + ( φ1 / φ4) ]

(20)

γ = [ ( φ3 / 4 φ4)4 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) ]



(21)

p = [ (5/6) (5/3) α2 + ( (5/6) α )2 - (5/3) α2 (5/2) + (2 α2 - γ ) ]

(22)

q = - ( (5/6) α )3 + ( (5/6) α )2 (5/2) α - (2 α2 - γ ) (5/6) α + [ (α3 - α γ - β2 /4 ) / 2 ]

(23)

u = { ( q /2 ) ± { ( q /2 )2 + ( p3 / 27) }1/2 }1/3

(24)

v = [ ( p / 3 u ) - u ] y = v - (5/6) α



(25)

The total number of novel re-sampling locations is thus twenty for the Lagrange interpolation function obtained from: xr0(i, k) = ψ (k, i)) i = 1, 2, 3, 4, 5; k = 1, 2, 3, 4.



(26)

The mathematics that is relevant to the solution of the quartic equation is reported in appendix VII. The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve the quartic equation. Also, the author wishes to acknowledge that the main author of the solution of the quartic equation is the Italian mathematician Lodovico Ferrari, also the main reference to the solution of the quartic equation as well as to the Galois Theory (Stewart, 2004) as reported at: http://en.wikipedia.org/wiki/Quartic_equation. What has to be acknowledged, as the MATLAB®. (c) 1984-2007 The MathWorks, Inc. code LGRSRE2007 located at www.sourcecodewebsite.com shows is that the novel re-sampling location xr0(i, k) = ψ (k, i)) is normalized to the range [0, x0] where x0 is the misplacement.

359

The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

Characterization of the Interpolation Error Improvement and Interpolation Error Bounds Let f : R n → R n the n-th dimensional function to approximate, and let x : R n → R n its domain. Let the interpolation function be the n-th dimensional h : R n → R n. By virtue of the unifying theory it results that there exists: For h, a spatial set of points called the Sub-pixel Efficacy Region: xsre(i)≠ {x : ∂ (ΔE(xp)) / ∂xp = 0, p = 1,2…n}, i =1, 2…s. where s is the degree of the polynomial ∂ (ΔE(xp)) / ∂xp = 0. For the sampling location x = x0, a novel re-sampling location: xr0≠ {xr(k,i), k =1, 2…t, i =1, 2…s}. The following relationships exits within the sampling step (resolution) ℓ: || x0 - xr0 ||L2 ≤ ℓ || x0 - xsre(i) ||L2 ≤ ℓ || xr0 - xsre(i) ||L2 ≤ ℓ, i =1, 2…s. Let IE(x0) = || h(x0) - h(xr0) ||L2 be the interpolation error improvement at x0, and C(1) and C(2) two arbitrary constants. Let Iε(x0) = || f(x0) - h(xr0) ||L2 be the interpolation error at x0 and, C(3) and C(4) two arbitrary constants. Ciulla and Deek (2006) (paragraph 2.6) demonstrate that: C(1) ≤ || h(x0) - h(xr0) ||L2 ≤ C(2)



(27)

C(3) ≤ || f(x0) - h(xr0) ||L2 ≤ C(4)



(28)

The Intensity-Curvature Functional of Sinc Interpolation Sinc Interpolation Function The formulation of the Sinc interpolation function was selected from recent work (Dooley & Nandi, 2000), with the neighborhood being defined within [–L, M]; with L + M = N. For N odd number of neighboring nodes and with the sampling frequency f 0 = (1/NT), the function assumes the following characterization:

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

M –1

(N – 1) / 2

hO(x) = Σn = - L f (nT) (1/N) [ 1 + 2 Σm = 1 cos (2 π m f 0 x) ]

(29)

With T = 1 we obtain the function: M –1

(N – 1) / 2

hO(x) = Σn = - L f (n) (1/N) [ 1 + 2 Σm = 1 cos (2 π m x / N) ] In the following the mathematics are relevant to equation (30).

(30)

The Intensity Curvature Terms x Eo = Eo (x) = ∫ f (0) ( ∂2 hO(x) / ∂x2 ) (0) dx = 0



x x M –1 (N – 1) / 2 2 = ∫ f (0) Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N) dx = ∫ f (0) θ0 dx = f (0) θ0 x



(31)

( ∂2 hO(x) / ∂x2 ) (0) = Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 = θ0

(32)

0

0

Where it is true that: M –1



(N – 1) / 2

Let us posit the following: (2 π m / N) = θ3m; (2 π m / N)2 = θ4m; cos( θ3m x ) = λ2m; sin( θ3m x ) = λ1m; (2 π m x / N) = θ6;

M –1

- { Σn = - L f (n) (1/N) } = θ5;

it follows that: ∂ λ1m /∂x = θ3m cos( θ3m x ) = θ3m λ2m ∂ λ2m /∂x = - θ3m sin( θ3m x ) = - θ3m λ1m ∂ θ6 /∂x = θ3m The intensity-curvature term EIN (x) can be written as: x EIN = EIN (x) = ∫ hO(x) ( ∂2 hO(x) / ∂x2 ) dx = 0 (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } -

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

{ [ θ2 ]2 } Σm = 1 { Σm = 1 - λ2m } * { θ3m λ1m } (N – 1) / 2

(N – 1) / 2

- { [ θ2 ]2 } (1/2) Σm = 1 { Σm = 1 λ1m * λ2m + θ6 } * θ3m



(33)

The Intensity Curvature Functional ΔE(x) = Eo (x) / EIN (x) = f (0) θ0 x / { (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } { Σm = 1 { Σm = 1 λ1m } * { { - θ3m λ2m } + { θ3m λ2m } } (N – 1) / 2

- (1/2) { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } } }

=

f (0) θ0 x / { (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } - { [ θ2 ]2 /2} Σm = 1 { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } }



(34)

The details subsequent to the math deductions above asserted are reported in appendix VII.

The Sub-pix el Efficacy Region of Sinc Interpolation ( ∂ ΔE(x) / ∂x ) = 0 implies that: (N – 1) / 2

{ f (0) θ0 * { θ5 * θ2 { Σm = 1 θ4m λ1m - x θ4m θ3m λ2m } ( N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 /2 } Σm = 1 { Σm = 1 λ1m * θ3m λ2m - θ6 θ3m + x θ3m θ3m [ (λ2m)2 - λ1m + 1] } } } = 0 (35) In the following, equation (35) is solved for the case of N = 3 using Newton’s iterative method. Let be true that: f ( x ) = θ5 * θ2 { θ4m λ1m - x θ4m θ3m λ2m } { [ θ2 ]2 /2 } * { λ1m * θ3m λ2m - θ6 θ3m + x θ3m θ3m [ (λ2m)2 - λ1m + 1] }

362

(36)

The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

Solution of Equation (35) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the equation (35) is found at: http://en.wikipedia.org/wiki/Newton%27s_method Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. The value of the xsre location is given by the following formulation: xsre = xsre (n+1) = xsre (n) - { f ( xsre (n) ) / [ ∂ f ( xsre (n) ) /∂x ] } = xsre (n) - { θ5 * θ2 { θ4m λ1m( xsre (n) ) - ( xsre (n) ) θ4m θ3m λ2m ( xsre (n) ) } { [ θ2 ]2 /2 } * { λ1m( xsre (n) ) * θ3m λ2m( xsre (n) ) - θ6( xsre (n) ) θ3m + ( xsre (n) ) θ3m θ3m [ λ2m( xsre (n) ) ]2 - λ1m( xsre (n) ) + 1] } } / { θ5 * θ2 θ3m θ3m { [ θ4m λ2m ( xsre (n) ) / θ3m ] - [ θ4m λ2m ( xsre (n) ) / θ3m ] + [ x λ1m( xsre (n) ) θ4m ] } - { [ θ2 ]2 /2 } θ3m θ3m * { [ λ2m( xsre (n) ) ]2 - [λ1m( xsre (n) )]2 - 1 + [ λ2m( xsre (n) ) ]2 - [ λ1m( xsre (n) ) ] + 1 + x [ -2 λ2m( xsre (n) ) θ3m λ1m( xsre (n) ) - θ3m λ2m( xsre (n) ) ] }} (37) Where: λ1m( xsre (n) ) = sin( θ3m xsre (n) ) λ2m ( xsre (n) ) = cos( θ3m xsre (n) ) θ6( xsre (n) ) = (2 π m xsre (n) / N) The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve equation (35). Also, the author wishes to acknowledge the main author of the iterative method herein employed: ISAAC NEWTON, also the reference (Tjalling J. Ypma, 1995) to the Newton’s Iterative method as reported at: http://en.wikipedia.org/wiki/Newton%27s_method. The details subsequent to the math deductions above asserted are reported in appendix VII.

The Novel Re-sampling Locations of Sinc Interpolation The starting formula is the one of the unifying theory:

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

{ hO(xsre - xr0) * { (∂2 (hO(x)) /∂x2) (xsre - xr0) } } / { hO(xsre) * { (∂2(hO(x)) /∂x2) (xsre) } } = { EIN (xsre-x0) / EIN (xsre) }





(38)

The following two equations are derived: cos (2 π (xsre - xr0) / N) = { - [ θ2 ]2 / 2 * (2 π / N)2 + { ( [ θ2 ]4 / 4) * (2 π / N)4 - 4 * [ θ2 ]2 * (2 π / N)2 * Λ(xsre -x0) }1/2 } / { 2 * [ θ2 ]2 * (2 π / N)2 } = Γ+



(39)

cos (2 π (xsre - xr0) / N) = { - [ θ2 ]2 / 2 * (2 π / N)2 - { ( [ θ2 ]4 / 4) * (2 π / N)4 - 4 * [ θ2 ]2 * (2 π / N)2 * Λ(xsre -x0) }1/2 } / { 2 * [ θ2 ]2 * (2 π / N)2 } = Γ-



(40)

(2 π (xsre - xr0) / N) = acos ( Γ+ )



(41)

(2 π (xsre - xr0) / N) = acos ( Γ- )



(42)

The novel re-sampling locations are the following two: xr1 = xsre - [ 3 * acos ( Γ+ ) / 2 π ] xr2 = xsre - [ 3 * acos ( Γ- ) / 2 π ]





(43)



(44)

What has to be acknowledged, as the MATLAB®. (c) 1984-2007 The MathWorks, Inc. code SINCOSRE2007 located at www.sourcecodewebsite.com shows is that the novel re-sampling locations x r1 and xr2 are normalized to the range [0, x0] where x0 is the misplacement. Where from equation (37): xsre = xsre (n+1) = xsre (n) - { f ( xsre (n) ) / [ ∂ f ( xsre (n) ) /∂x ] } = xsre (n) - { θ5 * θ2 { θ4m λ1m( xsre (n) ) - ( xsre (n) ) θ4m θ3m λ2m ( xsre (n) ) } { [ θ2 ]2 /2 } * { λ1m( xsre (n) ) * θ3m λ2m( xsre (n) ) - θ6( xsre (n) ) θ3m + ( xsre (n) ) θ3m θ3m [ λ2m( xsre (n) ) ]2 - λ1m( xsre (n) ) + 1] } } / { θ5 * θ2 θ3m θ3m { [ θ4m λ2m ( xsre (n) ) / θ3m ] - [ θ4m λ2m ( xsre (n) ) / θ3m ] + [ x λ1m( xsre (n) ) θ4m ] } –

364

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{ [ θ2 ]2 /2 } θ3m θ3m * { [ λ2m( xsre (n) ) ]2 - [λ1m( xsre (n) )]2 - 1 + [ λ2m( xsre (n) ) ]2 - [ λ1m( xsre (n) ) ] + 1 + x [ -2 λ2m( xsre (n) ) θ3m λ1m( xsre (n) ) - θ3m λ2m( xsre (n) ) ] }} m = 1 All the mathematical deductions employed to reach above results are reported in appendix VII. Through the study of the Intensity Curvature Functional and the derived Sub-pixel Efficacy Region, novel re-sampling locations were derived for Lagrange and Sinc interpolation functions. The unifying theory has thus been here extended and in the next chapter results will be presented to validate the improvement of the approximation characteristics of these two novel SRE-based functions.

Eq uating the Intensity-Curvature Terms for Lagrange and Sinc Interpolation Functions: The Deduction Lagrange From equation (6) it herein recalled that: x Eo = Eo (x) = ∫ f (0) (∂2 ( LGR3(x) ) /∂x2) (0) dx = f(0) θ x



0

Where it is posited that: θ = (∂2 ( LGR3(x) ) /∂x2) (0) = -2 θ(1) + 2 θ(2) θ(1) = [ f (1) + f (-1) ]; θ(2) = [ f (1) + f (2) + f (-1) + f (-2) ]; From equation (7) it is herein recalled that: EIN (x) = x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1 Where the following positions hold true: θ5 = { (3/10) θ(1) θ(1) -(1/5) θ(1) θ(2) + (1/30) θ(2) θ(2) } θ4 = { - θ(1) θ(1) + (1/2) θ(1) θ(2) + (5/6) θ(2) θ(1) - (1/3) θ(2) θ(2) } θ3 = { (1/6) θ(1) θ(1) - (1/2) θ(1) θ(2) - (15/6) θ(2) θ(1) + (23/18) θ(2) θ(2) } θ2 = { (3/2) f (0) θ(1) - (1/2) f (0) θ(2) + 2 θ(1) θ(1) - θ(1) θ(2) - (20/9) θ(2) θ(1) - (7/3) θ(2) θ(2) } θ2 = f (0) { (3/2) θ(1) - (1/2) θ(2) } + θ2’ θ2’ = { 2 θ(1) θ(1) - θ(1) θ(2) - (20/9) θ(2) θ(1) - (7/3) θ(2) θ(2) } θ1 = { - 2 f (0) θ(1) + 2 f (0) θ(2) - 2 θ(1) θ(1) + 2 θ(2) θ(2) } θ1 = f (0) * (- 2 θ(1) + 2 θ(2) ) + θ1’ θ1’ = { - 2 θ(1) θ(1) + 2 θ(2) θ(2) }

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Therefore: EIN (x) = x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * [ f (0) { (3/2) θ(1) - (1/2) θ(2) } + θ2’ ] + x * [ f (0) * (- 2 θ(1) + 2 θ(2) ) + θ1’ ] = x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * f (0) [ (3/2) θ(1) - (1/2) θ(2) ] + x2 * θ2’ + x * [ f (0) * (- 2 θ(1) + 2 θ(2) ) ] + x * θ1’ EIN (x) = x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2’ + x * θ1’ + f (0) { x2 * [ (3/2) θ(1) - (1/2) θ(2) ] + x * [ - 2 θ(1) + 2 θ(2) ] }



(45)

f (0) = - θ6 / - ( θ7 - θ8 ) = [ θ6 / ( θ7 - θ8 ) ]

(46)

Equating EIN (x) as per equation (45) to Eo (x), it is derived that: EIN (x) = x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2’ + x * θ1’ + f (0) { x2 * [ (3/2) θ(1) - (1/2) θ(2) ] + x * [ - 2 θ(1) + 2 θ(2) ] } = Eo (x) = f(0) θ x Which implies that: x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2’ + x * θ1’ + f (0) { x2 * [ (3/2) θ(1) - (1/2) θ(2) ] + x * [ - 2 θ(1) + 2 θ(2) ] } - f(0) θ x = 0 And by the positions: θ6 = x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2‘ + x * θ1’ θ7 = { x2 * [ (3/2) θ(1) - (1/2) θ(2) ] + x * [ - 2 θ(1) + 2 θ(2) ] } θ8 = θ x It is derived that:

Sinc From equations (31) and (32) we recall that: x

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Eo = Eo (x) = ∫ f (0) ( ∂2 hO(x) / ∂x2 ) (0) dx = f (0) θ0 x 0



M –1



(N – 1) / 2

( ∂2 hO(x) / ∂x2 ) (0) = Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 = θ0 and being m = 1, L + M = N, N=3: θ0 = [ f (-1) + f(0) + f(1) ] * [ - (2/3) (2 π / 3)2 ] = [ f (-1) + f(1) ] * [ - (2/3) (2 π / 3)2 ] + f(0) * [ - (2/3) (2 π / 3)2 ]

(47)



Recalling equation (33), the intensity-curvature term EIN (x) is: x EIN = EIN (x) = ∫ hO(x) ( ∂2 hO(x) / ∂x2 ) dx = 0 (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } { [ θ2 ]2 } Σm = 1 { Σm = 1 - λ2m } * { θ3m λ1m } (N – 1) / 2

(N – 1) / 2

- { [ θ2 ]2 } (1/2) Σm = 1 { Σm = 1 λ1m * λ2m + θ6 } * θ3m and since m = 1 and N=3, it follows that: EIN (x) = θ5 * { θ2 θ41 λ11 } + { [ θ2 ]2 } { λ11 } * { - θ31 λ21 } - { [ θ2 ]2 } { - λ21 } * { θ31 λ11 } - { [ θ2 ]2 } (1/2) { λ11 * λ21 + θ6 } * θ31 = θ5 * { θ2 θ41 λ11 } - { [ θ2 ]2 λ11 θ31 λ21 } + { [ θ2 ]2 } { λ21 θ31 λ11 } - { [ θ2 ]2 } (1/2) { λ11 * λ21 + θ6 } * θ31

(48)

And being: M –1

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(2 π m / N) = θ3m;

(2 π m / N)2 = θ4m;

sin( θ3m x ) = λ1m;

cos( θ3m x ) = λ2m;

- { Σn = - L f (n) (1/N) } = θ5;

(2 π m x / N) = θ6; M –1

[ Σn = - L f (n) (2/N) ] = θ2 It descends that when m = 1 and N= 3, the following is true: (2 π / 3) = θ31; (2 π / 3)2 = θ41; cos( θ31 x ) = λ21; sin( θ31 x ) = λ11; (2 π x / 3) = θ6; θ2 = (2/3) [ f (-1) + f(0) + f(1) ] θ5 = - (1/3) * [ f (-1) + f(0) + f(1) ] θ2 ’ = (2/3) [ f (-1) + f(1) ] θ5 ’ = - (1/3) [ f (-1) + f(1) ] θ2 = (2/3) f(0) + θ2 ’



(49)

θ5 = - (1/3) f(0) + θ5 ’



(50)

Equating the two intensity-curvature terms it can be written that: Eo (x) = f (0) θ0 x = EIN (x) = θ5 * { θ2 θ41 λ11 } - { [ θ2 ]2 λ11 θ31 λ21 } + { [ θ2 ]2 } { λ21 θ31 λ11 } - { [ θ2 ]2 } (1/2) { λ11 * λ21 + θ6 } * θ31 Thus: f (0) θ0 x = θ5 * { θ2 θ41 λ11 } - { [ θ2 ]2 λ11 θ31 λ21 } + { [ θ2 ]2 } { λ21 θ31 λ11 } - { [ θ2 ]2 } (1/2) { λ11 * λ21 + θ6 } * θ31 { f (0) θ0 x / [ θ2 ]2 } = { θ5 * { θ41 λ11 } / θ2 } { λ11 θ31 λ21 } + { λ21 θ31 λ11 } - (1/2) { λ11 * λ21 + θ6 } * θ31 Let us posit: Θ0 = - { λ11 θ31 λ21 } + { λ21 θ31 λ11 } - (1/2) { λ11 * λ21 + θ6 } * θ31

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

Therefore the equation of the two intensity-curvature terms yields: { f (0) θ0 x / [ θ2 ]2 } - { θ5 * { θ41 λ11 } / θ2 } = Θ0



(51)

Rewritng equation (47) as: θ0 = Θ1 + f(0) * [ - (2/3) (2 π / 3)2 ] Θ1 = [ f (-1) + f(1) ] * [ - (2/3) (2 π / 3)2 ] And placing equations (49), (50) into equation (51): { f (0) θ0 x / [ (2/3) f(0) + θ2 ’ ]2 } - { [ - (1/3) f(0) + θ5 ’ ] * [ θ41 λ11 ] / [ (2/3) f(0) + θ2 ’ ] } = Θ0 { f (0) θ0 x } - { [ - (1/3) f(0) + θ5 ’ ] * [ (2/3) f(0) + θ2 ’ ] * [ θ41 λ11 ] } = Θ0 [ (2/3) f(0) + θ2 ’ ]2 { f (0) θ0 x } - { [ - (2/9) f(0)2 - (1/3) f(0) θ2 ’ + (2/3) f(0) θ5 ’ + θ5 ’ θ2 ’ ] * [ θ41 λ11 ] } = Θ0 [ (4/9) f(0)2 + (4/3) f(0) θ2 ’ + (θ2 ’)2 ] { f (0) θ0 x } + { [ (2/9) f(0)2 + (1/3) f(0) θ2 ’ - (2/3) f(0) θ5 ’ - θ5 ’ θ2 ’ ] * [ θ41 λ11 ] } = Θ0 [ (4/9) f(0)2 + (4/3) f(0) θ2 ’ + (θ2 ’)2 ] { f (0) θ0 x } + { [ (2/9) f(0)2 + (1/3) f(0) θ2 ’ - (2/3) f(0) θ5 ’ - θ5 ’ θ2 ’ ] * [ θ41 λ11 ] } Θ0 [ (4/9) f(0)2 + (4/3) f(0) θ2 ’ + (θ2 ’)2 ] = 0 f (0) { θ0 x + { (1/3) θ2 ’ - (2/3) θ5 ’ }* [ θ41 λ11 ] - (4/3) θ2 ’ Θ0 } + { - θ5 ’ θ2 ’ * [ θ41 λ11 ] } + f(0)2 [ -(4/9) Θ0 ] - [ Θ0 (θ2 ’)2 ] = 0



(52)

Let us posit: Θ1 = { θ0 x + { (1/3) θ2 ’ - (2/3) θ5 ’ }* [ θ41 λ11 ] - (4/3) θ2 ’ Θ0 } Θ3 ={ - θ5 ’ θ2 ’ * [ θ41 λ11 ] } - [ Θ0 (θ2 ’)2 ] Θ2 = [ -(4/9) Θ0 ] Therefore equation (52) can be written as: f(0)2 Θ2 + f (0) Θ1 + Θ3 = 0



(53)

f(0) = { - Θ1 ± { [ Θ1 ]2 - 4 Θ2 Θ3 }1/2 } / { 2 Θ2 }

(54)

And it admits the following solution:

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The Unifying Theory Embraces Lagrange and Sinc Interpolation Functions

Equations (46), (54) furnish for the Lagrange and Sinc interpolation functions respectively the value of signal intensity obtained under the assumption that if the two intensity-curvature terms are the same, interpolation has no effect on the signal.

SUMMARY Chapter XVIII addresses studies and solves the problem of the calculation of the Sub-pixel Efficacy Region of the one-dimensional third degree Lagrange interpolation function and the one-dimensional Sinc interpolation function. For each of the two interpolation function models, through a detailed mathematical description the two intensity-curvature terms (defined as the integral of the function multiplied its second order derivative) are calculated and the characterization of the Intensity-Curvature Functional ΔE is provided. The study of the polynomial resulting from the first order derivative of ΔE furnishes the Sub-pixel Efficacy Region (SRE) and thus determines the basis for computing the novel re-sampling locations where the signal (image) is calculated. The methodological approach descends from the unifying methodology that the book proposes for each interpolation function regardless of degree or dimensionality. The reader is informed within the context of the SRE-based Lagrange and the SRE-based Sinc interpolation functions as to what is the methodology to compute the SRE, what is the resulting SRE, and at what is the novel re-sampling location that allow interpolation error improvement. The organization of the chapter follows the style already used for the writings of Chapters VII, X and XIV. Finally, the chapter presents the deduction of the formulas that characterize resilient interpolation for the two model functions that were object of study.

References Boyer, C. B. (1968). A History of Mathematics. New York, NY: John Wiley & Sons. Ciulla, C., & Deek, F. P. (2006). Extension of the sub-pixel efficacy region to the Lagrange interpolation function. ICGST - International Journal on Graphics, Vision and Image Processing, 6(1), 1-12. Dooley, S. R., & Nandi, A. K. (2000). Notes on the interpolation of discrete periodic signals using sinc function related approaches. IEEE Transactions on Signal Processing, 48(4), 1201-1203. Lehmann, T. M., Gonner, C., & Spitzer, K. (1999). Survey: Interpolation methods in medical image processing. IEEE Transactions on Medical Imaging, 18(11), 1049-1075. Stewart, I. (2004). Galois theory. Boca Raton, Florida: Chapman & Hall/CRC. Tjalling J. Ypma. (1995). Historical development of the Newton-Raphson method. SIAM Review, 37(4), 531–551.

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Chapter XIX

The Results of the Sub-Pixel Efficacy Region-Based Lagrance and Sinc Interpolation Functions

Introduction This chapter presents results obtained processing the MRI dataset with classic and SRE-based models of one dimensional Lagrange and Sinc interpolation functions. The presentation is consistent with chapters VIII, XI, and XV where bivariate linear, trivariate linear and B-Splines’ results were reported and consists of a compilation that is both quantitative and qualitative. The RSME Ratio is once again employed as the metric to assess the relativity of performance in terms of interpolation error between classic and SRE-based functions. Qualitative results show MRI images obtained through the validation paradigm that are corroborated with the information provided with error images. In several cases the information from the image space is not enough to reveal hidden differences between classic and SRE-based interpolation and this happens because the error images of the two classes of interpolation functions are quite similar with each other. Hence, spectral power evolutions supply the mean to evaluate the spectral content of each image resulting after processing, in relationship to the original image. The spectral power evolutions demonstrate that even if the error images are almost the same, differences exist in the Fourier domain and the determinant of such differences is the non equal filtering behavior of classic and SRE-based interpolation functions.

VALIDATION METHODOLOGY AND MRI DATASET The validation methodology adopted to validate both SRE-based Lagrange SRE-based Sinc interpolation functions is same as that reported earlier (Ciulla & Deek, 2006) (section 3.1) and same as that seen previously in Section IV of this book for one dimensional quadratic and cubic SRE-based B-Splines. As previously clarified, the term processed will be used to indicate that the image was motion corrected and Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

interpolated. When using the misplacement x0 the image is termed as processed with classic Lagrange and Sinc interpolation functions as per equations (2) and (30) (chapter 18) respectively. When using the novel re-sampling location xr0 obtained from equation (26) and equations (43), (44) of chapter 18 the image is termed as processed with Lagrange and Sinc SRE-based interpolation functions respectively. The misplacements were all along the x coordinate at steps of: (i) 0.01, (ii) 0.0025, (iii) 0.00001 and their range were: (i) [0, 0.99], (ii) [0, 0.2475], (iii) [0, 0.00099], all fractions of the pixel size. The image database (T1 MRI, T2 MRI and functional MRI) used for the validation of the SRE-based interpolation functions is the same as that used in Part III of the book for the study of the SRE-based B-Splines and for convenience to the reader the MRI images’ resolution and matrix size are herein recalled: • • •

T1 MRI (T1-MRI-230 and T1-MRI-450). Matrix resolution: 176 x 208 with 1.00 x 1.00 mm of pixel size, with 75 slices and an inter-slice resolution of 1.00 mm. T2 MRI: matrix resolution was 177 x 182 with 1.00 x 1.00 mm of pixel size, with 19 slices and an inter-slice resolution of 1.00 mm. Functional MRI (fMRI-TS1). Matrix resolution was 128 x 128 with 1.72 x 1.72 mm of pixel size, with 20 slices and an inter-slice resolution of 1.72 mm.

Also, this chapter constitutes of an extension of the investigation reported earlier (Ciulla & Deek, 2006) where a presentation of the capability of the SRE-based Lagrange interpolation function was reported on a variety of imaging modalities including cellular imaging. This section of the book focuses on the presentation of results obtained employing T1-MRI (T1-MRI-230 and T1-MRI-450), T2-MRI and functional MRI (fMRI-TS1) data sets.

Lagrange Interpolation: Results with T1, T2 and functional MRI Data The Behavior of the RMSE Ratio for Varying Values of the Misplacement The presentation of the results of the investigation conducted to compile this chapter is consistent with that given for all of the classic and SRE-based interpolation functions studied in the preceding chapters. The RMSE Ratio is the measure employed to reveal the difference in approximation properties of the two classes of interpolation functions (classic and SRE-based). Consistently, also the motion correction paradigm is the same. This implies that interpolation is performed twice on the images. The first time, to the image the misplacement is applied and so re-sampling is performed through the classic paradigm. The second time re-sampling is performed either through the classic Lagrange formula (2) or the SRE-based formula (26) (see chapter 18). Re-sampling with the SRE-based formula consists of calculating the image at the novel re-sampling locations of equation (26). In evaluating the results it is due to acknowledge that the spectral power evolutions presented in this chapter as well as those presented in chapters 11 (Trivariate Linear) and 15 (B-Splines) reflect a process that applies interpolation twice: forward and backwards to correct for the motion. Figures 1 through 10 are relevant to the T1-MRI-230 data for misplacements of 0.00001 progressing from 0 to 0.00099. Figures 11 through 20 are relevant to the T1-MRI-450 data for misplacements of 0.01 progressing from 0 to 0.99. Figures 21 through 23 are relevant to T2-MRI data for misplacements of

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

0.0025 progressing from 0 to 0.2475. Figures 24 through 26 are relevant to the same T2-MRI data set and for misplacements of 0.01 progressing from 0 to 0.99. Figures 27 through 29 are relevant to functional MRI (fMRI-TS1) for misplacements of 0.00001 progressing from 0 to 0.00099. Figures 30 through 32 are relevant to the same functional MRI data set for misplacements of 0.0025 progressing from 0 to 0.2475. A negative value of the RMSE Ratio indicates the superiority of the Lagrange SRE-based interpolation function over the classic in terms of interpolation error, while a positive value shows the opposite. The presentation of the graphs displaying the behavior of the RMSE Ratio is corroborated through the report of the highest values of percentage interpolation error improvement that were achieved through the SRE-based Lagrange function within the full range of misplacement seen in each graph. In figures 1 through 10 (T1-MRI-230) the common behavior of the RMSE Ratio is that of a persistent linear error improvement of the SRE-based Lagrange function versus the classic Lagrange function. The percentage improvement increases with the increase of the misplacement and reaches its highest value at the location of the largest misplacement. In figure 1, the highest percentage improvement ranges approximately between: 2% and 2.5% (graphs 67, 68, 69, 70, 71, 72, 73 and 74). In figure 2, the highest percentage improvement ranges approximately between: 2% and 3% (graphs 75, 76, 77, 78, 79, 80, 81 and 82). Also in figure 3, the highest percentage improvement ranges approximately between: 2% and 3% (graphs 83, 84, 85, 86, 87, 88, 89 and 90). In figure 4, the highest percentage improvement is approximately 2% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). In figure 5, the highest percentage improvement ranges approximately between: 2% and 2.5% (graphs 99, 100, 101, 102, 103, 104,105 and 106). In figure 6, the highest percentage improvement is approximately 2% (graphs 107, 108, 109, 110, 112, 113 and 114) and ranges approximately between 1.5% and 2% in graph 111. In figure 7, the highest percentage improvement ranges approximately between: 2% and 2.5% (graphs 115, 116, 118, 119, 120, 121 and 122) and is approximately 1.5% in graph 117. In figure 8, the highest percentage improvement ranges approximately between: 1.5% and 2% (graphs 123, 124, 125, 126, 127, 128, 129 and 130). In figure 9, the highest percentage improvement ranges approximately between: 1.5% and 2.5% (graphs 131, 132, 133, 134, 135, 136, 137 and 138). And in figure 10, the highest percentage improvement ranges approximately between: 2% and 2.5% (graphs 139, 140 and 141). In figures 11 through 20 (T1-MRI-450) the common behavior of the RMSE Ratio is that of revealing an interpolation error improvement profile characterized through a maximum improvement followed by a decrease in percentage improvement. This behavior is predominant in showing superiority of the SRE-based Lagrange function versus the classic Lagrange function. In some cases the maximum improvement is local (being two the maximums seen within the full range of misplacement as shown for example in graph 88 of figure13). Nevertheless, also the second maximum is followed by decrease of percentage improvement. In some graphs however there is also a minimum of the improvement of the interpolation error which is reached after the maximum (see graph 90 in figure13). In figure 11, the highest percentage improvement ranges approximately between: 37.5% and 170% (graphs 67, 68, 69, 70, 71, 72, 73 and 74).

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 1. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 67-74. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 2. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 75-82. The range of misplacements is [0, 0.00099] at steps of 0.00001.

375

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 3. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 83-90. The range of misplacements is [0, 0.00099] at steps of 0.00001.

376

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 4. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 91-98. The range of misplacements is [0, 0.00099] at steps of 0.00001.

377

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 5. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 99-106. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 6. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 107-114. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 7. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 115-122. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 8. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 123-130. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 9. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 131-138. The range of misplacements is [0, 0.00099] at steps of 0.00001.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 10. RMSE Ratio: T1-MRI-230, misplacement along x, Lagrange, slices 139-141. The range of misplacements is [0, 0.00099] at steps of 0.00001.

In figure 12, the highest percentage improvement ranges approximately between: 10% and 140% (graphs 75, 76, 77, 78, 79, 80, 81 and 82). In figure 13, the highest percentage improvement ranges approximately between: 20% and 160% (graphs 83, 84, 85, 86, 87, 88, 89 and 90). In figure 14, the highest percentage improvement ranges approximately between: 15% and 150% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). In figure 15, the highest percentage improvement ranges approximately between: 35% and 175% (graphs 99, 100, 101, 102, 103, 104,105 and 106). In figure 16, the highest percentage improvement is ranges approximately between: 45% and 160% (graphs 107, 108, 109, 110, 11, 112, 113 and 114). In figure 17, the highest percentage improvement ranges approximately between: 150% and 175% (graphs 115, 116, 117, 118, 119, 120, 121 and 122). In figure 18, the highest percentage improvement ranges approximately between: 70% and 180% (graphs 123, 124, 125, 126, 127, 128, 129 and 130). In figure 19, the highest percentage improvement ranges approximately between: 30% and 200% (graphs 131, 132, 133, 134, 135, 136, 137 and 138). And in figure 20, the highest percentage improvement ranges approximately between: 160% and 180% (graphs 139, 140 and 141). In figures 1 through 10, the range of misplacement is [0, 0.00099] and in figures 11 through 20 is [0, 0.99]. This difference is worth noting as it suggests that the interpolation error improvement of the SRE-based Lagrange function versus the classic function is affected from the magnitude of the mis-

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 11. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 67-74. The range of misplacements is [0, 0.99] at steps of 0.01.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 12. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 75-82. The range of misplacements is [0, 0.99] at steps of 0.01.

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 13. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 83-90. The range of misplacements is [0, 0.99] at steps of 0.01

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Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 14. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 91-98. The range of misplacements is [0, 0.99] at steps of 0.01.

387

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 15. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 99-106. The range of misplacements is [0, 0.99] at steps of 0.01.

388

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 16. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 107-114. The range of misplacements is [0, 0.99] at steps of 0.01.

389

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 17. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 115-122. The range of misplacements is [0, 0.99] at steps of 0.01.

390

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 18. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 123-130. The range of misplacements is [0, 0.99] at steps of 0.01.

391

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 19. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 131-138. The range of misplacements is [0, 0.99] at steps of 0.01.

392

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 20. RMSE Ratio: T1-MRI-450, misplacement along x, Lagrange, slices 139-141. The range of misplacements is [0, 0.99] at steps of 0.01.

placement. A similar behavior will be observed in figures 21 through 26 (for T2-MRI) and 27 through 32 (for functional MRI) but for different ranges of misplacement. In figures 21 through 26 (T2-MRI) the common behavior of the RMSE Ratio is that of revealing an interpolation error improvement profile characterized generally through persistent percentage increase. In some cases however, the graphs show a maximum improvement followed by rapid decrease in percentage improvement which will tends to zero (see for example graphs 10, 13 and 15 in figure 22). Overall though, graphs of figures 21 through 26 confirm the superiority of the SRE-based Lagrange function versus the classic Lagrange function. In figure 21, the highest percentage improvement ranges approximately between: 40% and 150% (graphs 1, 2, 3, 4, 5, 6 and 7). A peak is seen at approximately 3500% in graph 8. In figure 22, the highest percentage improvement ranges approximately between: 40% and 150% (graphs 9, 11, 12, 14 and 16). A peak is seen at approximately 3500% in graphs 10 and 13 and another one at approximately 2800% is seen in graph 15. In figure 23, the highest percentage improvement is approximately 40% (graphs 17 and 18). The peak is seen in graph 19 with a percentage improvement of approximately 2400%. In figure 24, the highest percentage improvement ranges approximately between: 160% and 500% (graphs 1, 2, 3, 4, 5, 6, and 7). A peak is seen in graph 8 at approximately 1000% improvement. In figure 25, the highest percentage improvement ranges approximately between: 150% and 800% (graphs 9, 10, 11, 12, 13, 14, 15 and 16). In figure 26, the highest percentage improvement is approximately 175% (graphs 17 and 18). A peak is seen in graph 19 with a percentage improvement of approximately 500%.

393

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 21. RMSE Ratio: T2-MRI, misplacement along x, Lagrange, slices 1-8. The range of misplacements is [0, 0.2475] at steps of 0.0025.

394

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 22. RMSE Ratio: T2-MRI, misplacement along x, Lagrange, slices 9-16. The range of misplacements is [0, 0.2475] at steps of 0.0025.

395

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 23. RMSE Ratio: T2-MRI, misplacement along x, Lagrange, slices 17-19. The range of misplacements is [0, 0.2475] at steps of 0.0025.

In figures 27 through 29 (fMRI-TS1) the interpolation error improvement profiles are characterized through a persistent increasing percentage improvement across the full range of misplacement. In figures 30 through 32 the graphs show a maximum improvement followed by rapid decrease in percentage improvement which tends to zero. Also the graphs of figures 27 through 32 generally confirm the superiority of the SRE-based Lagrange function versus the classic Lagrange function. In figure 27, the highest percentage improvement ranges approximately between: 2% and 2.5% (graphs 1, 2, 3, 4, 5, 6, 7 and 8). In figure 28, the highest percentage improvement ranges approximately between: 2% and 3.5% (graphs 9, 10, 11, 12, 13, 14, 15 and 16). In figure 29, the highest percentage improvement ranges approximately between: 2% and 3% (graphs 17, 18, 19 and 20). In figure 30, the highest percentage improvement ranges approximately between: 350% and 2400% (graphs 1, 2, 3, 4, 5, 6, 7 and 8). In figure 31, the highest percentage improvement ranges approximately between: 250% and 1600% (graphs 9, 10, 11, 12, 13, 14, 15 and 16). In figure 32, the highest percentage improvement ranges approximately between: 150% and 1200% (graphs 17, 18, 19 and 20).

Q ualitative Assessment of the SRE-Based Lagrange Interpolation Function The presentation of the qualitative assessment of the performances of the SRE-based Lagrange interpolation function starts with figures 33 (T1-MRI-230) and 34 (T1-MRI-450).

396

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 24. RMSE Ratio: T2-MRI, misplacement along x, Lagrange, slices 1-8. The range of misplacements is [0, 0.99] at steps of 0.01.

397

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 25. RMSE Ratio: T2-MRI, misplacement along x, Lagrange, slices 9-16. The range of misplacements is [0, 0.99] at steps of 0.01.

398

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 26. RMSE Ratio: T2-MRI, misplacement along x, Lagrange, slices 17-19. The range of misplacements is [0, 0.99] at steps of 0.01.

The organizational layout is as follow. In the top row, from left to right, is presented: (i) the original image, (ii) the image processed with the classic Lagrange interpolation function and (iii) the image processed with the SRE-based interpolation function. In the second row from the top, from left to right, the FFT maps are shown for each image. In the third row from the top, from left to right, are shown: (i) the novel re-sampling locations, (ii) the error image obtained with the classic Lagrange formula, and (iii) the error image obtained with the SRE-based Lagrange formula. The misplacement was 0.045 (figure 33) and 0.023 (figure 34) along the x direction and in either case was a fraction of the pixel size. Consistently with Part IV of the book, the spectral power evolutions seen in the fourth row from the top in both figures 33 and 34 were built on 2000 equally spaced FFT resulting magnitudes, and the analysis was focused on the highest energy range. Thus, intervals from 970 through 1023 were analyzed. The value abs(O_E - SRE_E) - abs(O_E - NOSRE_E) was calculated at each interval and plotted. Its evolution elucidates the capability of both classic and Lagrange interpolation functions to preserve the spectral components of the original image. The spectral power evolutions will be shown also in figures 35 and 36 for the T2-MRI and the fMRI-TS1 datasets respectively. The following observations can be made based on visual inspection the two pictures. The error images were quite similar in either case of classic or SRE-based interpolation functions’ processing. Differences between the two interpolation paradigms became clearly manifest when the spectral power evolutions were taken into account and were such to reveal that the SRE-based Lagrange function embeds the capability to preserve the spectral components of the original image more than the classic Lagrange function is capable to do. The spectral power evolutions are additive to the FFT maps in order to reveal differences in the spectra of the images.

399

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 27. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, Lagrange, slices 1-8. The range of misplacements is [0, 0.00099] at steps of 0.00001.

400

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 28. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, Lagrange, slices 9-16. The range of misplacements is [0, 0.00099] at steps of 0.00001.

401

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 29. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, Lagrange, slices 17-20. The range of misplacements is [0, 0.00099] at steps of 0.00001.

Finally, the maps of the novel re-sampling locations are well defined and match closely the brain image features in either figure 33 or 34. The dark dots in the spectral evolutions are relevant to frequency components corresponding to image space locations that are shown for the four datasets: T2-MRI (in figure 35), fMRI-TS1 (in figure 36), T1-MRI-230 (in figure 37) and T1-MRI-450 (in figure 38). The image space locations seen in figures 35 and 36 correspond to the intervals 992-993 (T2-MRI, see the left most image in the third row from the top in figure 35) and 994-996 (fMRI-TS1, see the right most image in the second row from the top in figure 36). The organizational layout of figures 35 and 36 is similar to that of figures 33 and 34 and the observations that can be made are consistent with those made for the T1-MRI data. Worth noting that the interpolation error for both T2-MRI and fMRI-TS1 was lower when processing the brain image with the SRE-based Lagrange interpolation function, than it was when processing with the classic Lagrange formula. This is manifest in the error images in figure 35 (T2-MRI) and the histograms of the error images in figure 36 (fMRI-TS1). The misplacements were 0.045 and 0.023 (fraction of the pixel size) for the T2-MRI in figure 35 and for the fMRI-TS1 in figure 36 respectively. As far as the maps of the novel re-sampling locations are concerned they resulted flat in both of the experiments presented in figures 35 and 36.

Analysis of the Spectral Power Evolutions This section consists of an extension of the investigation reported formerly (Ciulla & Deek, 2006). The investigation was conducted with the same signal processing tool used for the quadratic and cubic SRE-

402

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 30. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, Lagrange, slices 1-8. The range of misplacements is [0, 0.2475] at steps of 0.0025.

403

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 31. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, Lagrange, slices 9-16. The range of misplacements is [0, 0.2475] at steps of 0.0025.

404

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 32. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x, Lagrange, slices 17-20. The range of misplacements is [0, 0.2475] at steps of 0.0025.

based B-Splines as shown in Section IV of the book. The purpose is to reveal frequency components that are embedded into the images processed with SRE-based interpolation and that are different from those frequency components contained in the images processed with classic interpolation. The spectral power evolutions that were presented show the behavior of the difference between absolute values of two differences between spectral energies. The absolute values of the two differences are calculated as: (i) the spectral energy of the original image minus the spectral energy of the image obtained with SRE-based interpolation, and (ii) the spectral energy of the original image minus the spectral energy of the image obtained with classic interpolation. It is therefore possible to study each image-space location of the signal resulting from interpolation processing (either the classic or the SRE-based paradigm) and to discover as to if the image-space location embeds frequency components that are as close as possible to those of the original image. The purpose is inherent to the approximate nature of interpolation, which determines a change in the bandpass characteristics of the signal it processes. To achieve the above mentioned purpose, an additional analysis of the spectral power evolutions was done by overlaying in white onto the image-space locations. The white overlay corresponds to frequency components. The energy of the frequency components is plotted in the spectral power evolutions versus the intervals and each interval corresponds to a range of resulting magnitudes. Therefore through the white overlay, the image-space is connected to the frequency domain and each interval (each value seen in the abscissa of the spectral power evolutions) corresponds to a range of frequency components. The white overlay is thus the complement to the spectral power evolution. The spectral power evolution is capable to reveal which of the frequency components of the two images: (i) processed with classic interpolation or (ii) processed with SRE-based interpolation, are

405

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 33. T1-MRI-230 data qualitative assessment, Lagrange, slice 90. The misplacement along the x direction is 0.045. The evolution is plotted for the intervals 979-1023. The dark dots correspond to the intervals 1001-1002. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

406

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 34. T1-MRI-450 data qualitative assessment, Lagrange, slice 82. The misplacement along the x direction is 0.023. The evolution is plotted for the intervals 979-1023. The dark dots correspond to the intervals 1001-1002. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

407

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 35. T2-MRI data qualitative assessment, Lagrange, slice 9. The misplacement along the x direction is 0.045. The evolution is plotted for the intervals 979-1023. The dark dots correspond to the intervals 992-993. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

408

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 36. Functional MRI (fMRI-TS1) data qualitative assessment, Lagrange, slice 2. The misplacement along the x direction is 0.023. The evolution is plotted for the intervals 979-1023. The dark dots correspond to the intervals 994-996. The original image (the left most in the upper row) was kindly provided by INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee, also courtesy of the Department of Radiology KULeuven. The histograms and the FFT maps presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

409

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 37. Difference error image and image-space overlay, T1-MRI-230 data, Lagrange, slice 90. The difference error image presented in this figure was calculated with the software ImageJ: http://rsb.info. nih.gov/ij/.

Figure 38. Difference error image and image-space overlay, T1-MRI-450 data, Lagrange, slice 82. The difference error image presented in this figure was calculated with the software ImageJ: http://rsb.info. nih.gov/ij/.

410

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 39. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 67-74. The range of misplacements is [0, 0.2475] at steps of 0.0025.

411

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 40. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 75-82. The range of misplacements is [0, 0.2475] at steps of 0.0025.

412

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 41. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 83-90. The range of misplacements is [0, 0.2475] at steps of 0.0025.

413

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 42. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 91-98. The range of misplacements is [0, 0.2475] at steps of 0.0025.

414

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 43. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 99-106. The range of misplacements is [0, 0.2475] at steps of 0.0025.

415

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 44. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 107-114. The range of misplacements is [0, 0.2475] at steps of 0.0025.

416

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 45. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 115-122. The range of misplacements is [0, 0.2475] at steps of 0.0025.

417

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 46. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 123-130. The range of misplacements is [0, 0.2475] at steps of 0.0025.

418

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 47. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 131-138. The range of misplacements is [0, 0.2475] at steps of 0.0025.

419

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 48. RMSE Ratio: T1-MRI-230, misplacement along x (small), Sinc, slices 139-141. The range of misplacements is [0, 0.2475] at steps of 0.0025.

closer to the original image. Overlay images presented in the figures 37 and 38 (to the right) show the image-space location corresponding to the frequency components identified through the dark dots of the power spectral power evolutions of figures 33 and 34 for T1-MRI-230 (slice 90) and T1-MRI-450 (slice 82) respectively. The error images are calculated as: the original MRI image minus the MRI image obtained processing with classic or SRE-based interpolation functions respectively. Finally, the difference error images seen in figure 37 and 38 (to the left) were obtained subtracting the error image resulting through processing with SRE-based Lagrange interpolation from the error image resulting through processing with the classic form of Lagrange interpolation. The value of such difference error images is in that they reveal that the SRE-based Lagrange interpolation paradigm and the classic Lagrange interpolation paradigm do offer approximation properties that are not the same with each other.

Sinc Interpolation: Results with T1, T2 and functional MRI Data The analysis of the results is presented as follow. RMSE Ratio, analysis of the spectral power evolutions to complement and support the information provided by the FFT maps were performed employing T1MRI, T2-MRI and functional MRI volumes. It is worth noting that the Sinc interpolation formula as per equation (30) determines a smoothing effect that increases with the increase of neighborhood size

420

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 49. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 67-74. The range of misplacements is [0, 0.99] at steps of 0.01.

421

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 50. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 75-82. The range of misplacements is [0, 0.99] at steps of 0.01.

422

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 51. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 83-90. The range of misplacements is [0, 0.99] at steps of 0.01.

423

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 52. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 91-98. The range of misplacements is [0, 0.99] at steps of 0.01.

424

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 53. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 99-106. The range of misplacements is [0, 0.99] at steps of 0.01.

425

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 54. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 107-114. The range of misplacements is [0, 0.99] at steps of 0.01.

426

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 55. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 115-122. The range of misplacements is [0, 0.99] at steps of 0.01.

427

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 56. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 123-130. The range of misplacements is [0, 0.99] at steps of 0.01.

428

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 57. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 131-138. The range of misplacements is [0, 0.99] at steps of 0.01.

429

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 58. RMSE Ratio: T1-MRI-230, misplacement along x (large), Sinc, slices 139-141. The range of misplacements is [0, 0.99] at steps of 0.01.

and also produces an increase of root-mean-square-error clearly measurable after the motion correction paradigm. To lower the smoothing effect, two techniques were employed. The first was that of simply keeping the neighborhood as small as possible and this was done setting N = 3. Also, a convolution was applied to the pixel intensity series, which consisted of a Gaussian with very small Full Width at Half Magnitude (FWHM) and this is like to multiply a set of coefficients to the original pixel value before any processing. The FWHM used varied across experimental sessions within the range [0.0087, 1] determining βn for n = 1…3 and on the basis of this pre-processing step, the value of θ2 given in the formulations of the Intensity-Curvature Functional in Chapter XVIII changed accordingly from: M –1

[ Σn = - L f (n) (2/N) ] to:

M –1

θ2 = [ Σn = - L βn * f (n) (2/N) ]

The Behavior of the RMSE Ratio for Varying Values of the Misplacement Figures 39 through 48 are relevant to the T1-MRI-230 data for misplacements of 0.0025 in the range [0, 0.2475]. Figures 49 through 58 are relevant to the T1-MRI-230 data for misplacements of 0.01 in the range [0, 0.99]. Figures 59 through 68 are relevant to the T1-MRI-450 data for misplacements of 0.0025 in the range [0, 0.2475]. Figures 69 through 78 are relevant to the T1-MRI-450 data for misplacements of 0.01 in the range [0, 0.99]. Figures 79 through 81 are relevant to T2-MRI data for misplacements

430

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 59. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 67-74. The range of misplacements is [0, 0.2475] at steps of 0.0025.

431

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 60. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 75-82. The range of misplacements is [0, 0.2475] at steps of 0.0025.

432

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 61. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 83-90. The range of misplacements is [0, 0.2475] at steps of 0.0025.

433

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 62. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 91-98. The range of misplacements is [0, 0.2475] at steps of 0.0025.

434

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 63. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 99-106. The range of misplacements is [0, 0.2475] at steps of 0.0025.

435

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 64. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 107-114. The range of misplacements is [0, 0.2475] at steps of 0.0025.

436

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 65. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 115-122. The range of misplacements is [0, 0.2475] at steps of 0.0025.

437

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 66. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 123-130. The range of misplacements is [0, 0.2475] at steps of 0.0025.

438

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 67. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 131-138. The range of misplacements is [0, 0.2475] at steps of 0.0025.

439

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 68. RMSE Ratio: T1-MRI-450, misplacement along x (small), Sinc, slices 139-141. The range of misplacements is [0, 0.2475] at steps of 0.0025.

of 0.025 in the range [0, 0.2475]. Figures 82 through 84 are relevant to the same T2-MRI data set and for misplacements of 0.01 in the range [0, 0.99]. Figures 85 through 87 are relevant to functional MRI (fMRI-TS1) for misplacements of 0.0025 in the range [0, 0.2475]. Figures 88 through 90 are relevant to the same functional MRI data set for misplacements of 0.01 in the range [0, 0.99]. In figures 39 through 48 (Sinc interpolation, T1-MRI-230) the behavior of the improvement of the interpolation error is non linear and the percentage improvement keeps increasing within the full range of misplacement. The range of misplacement is [0, 0.2475]. A similar behavior is seen in figures 49 through 58 (Sinc interpolation, T1-MRI-230) for misplacements that range in between [0, 0.99] but with a percentage improvement of the interpolation error that is approximately 10 times bigger (value across graphs) than that seen in figures 39 through 48. These results suggests that the magnitude of the misplacement plays an effect on the interpolation error improvement and this effect is likely to be such to magnify the improvement for increasing magnitudes of the misplacement. Also, these results extend further to the Sinc interpolation function the confirmation of the superiority of the SRE-based paradigm over the classic one, in terms of approximation capabilities. In figure 39, the highest percentage improvement ranges approximately between: 0.32% and 3.2% (graphs 67, 68, 69, 70, 71, 72, 73 and 74). In figure 40, the highest percentage improvement ranges approximately between: 0.16% and 1.2% (graphs 75, 76, 77, 78, 79, 80, 81 and 82). In figure 41, the highest percentage improvement ranges approximately between: 0.09% and 0.32% (graphs 83, 84, 85, 86, 87, 88, 89 and 90).

440

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 69. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 67-74. The range of misplacements is [0, 0.99] at steps of 0.01.

441

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 70. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 75-82. The range of misplacements is [0, 0.99] at steps of 0.01

442

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 71. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 83-90. The range of misplacements is [0, 0.99] at steps of 0.01.

443

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 72. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 91-98. The range of misplacements is [0, 0.99] at steps of 0.01.

444

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 73. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 99-106. The range of misplacements is [0, 0.99] at steps of 0.01.

445

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 74. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 107-114. The range of misplacements is [0, 0.99] at steps of 0.01.

446

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 75. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 115-122. The range of misplacements is [0, 0.99] at steps of 0.01.

447

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 76. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 123-130. The range of misplacements is [0, 0.99] at steps of 0.01.

448

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 77. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 131-138. The range of misplacements is [0, 0.99] at steps of 0.01.

449

Sub-Pixel Efficacy Region-Based Lagrange and Sinc Interpolation Functions

Figure 78. RMSE Ratio: T1-MRI-450, misplacement along x (large), Sinc, slices 139-141. The range of misplacements is [0, 0.99] at steps of 0.01.

In figure 42, the highest percentage improvement ranges approximately between: 0.2% and 0.4% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). In figure 43, the highest percentage improvement ranges approximately between: 0.14% and 0.36% (graphs 99, 100, 101, 102, 103, 104,105 and 106). In figure 44, the highest percentage improvement is ranges approximately between: 0.24% and 0.4% (graphs 107, 108, 109, 110, 11, 112, 113 and 114). In figure 45, the highest percentage improvement ranges approximately between: 0.2% and 0.4% (graphs 115, 116, 117, 118, 119, 120, 121 and 122). In figure 46, the highest percentage improvement ranges approximately between: 0.3% and 0.5% (graphs 123, 124, 125, 126, 127, 128, 129 and 130). In figure 47, the highest percentage improvement ranges approximately between: 0.2% and 0.4% (graphs 131, 132, 133, 134, 135, 136, 137 and 138). And in figure 48, the highest percentage improvement ranges approximately between: 0.12% and 0.25% (graphs 139, 140 and 141). In figure 49, the highest percentage improvement ranges approximately between: 0.9% and 6% (graphs 67, 68, 69, 70, 71, 72, 73 and 74). In figure 50, the highest percentage improvement ranges approximately between: 1.8% and 5.6% (graphs 75, 76, 77, 78, 79, 80, 81 and 82). In figure 51, the highest percentage improvement ranges approximately between: 0.9% and 4% (graphs 83, 84, 85, 86, 87, 88, 89 and 90).

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Figure 79. RMSE Ratio: T2-MRI, misplacement along x (small), Sinc, slices 1-8. The range of misplacements is [0, 0.2475] at steps of 0.025.

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Figure 80. RMSE Ratio: T2-MRI, misplacement along x (small), Sinc, slices 9-16. The range of misplacements is [0, 0.2475] at steps of 0.025.

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Figure 81. RMSE Ratio: T2-MRI, misplacement along x (small), Sinc, slices 17-19. The range of misplacements is [0, 0.2475] at steps of 0.025.

In figure 52, the highest percentage improvement ranges approximately between: 2.5% and 4% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). In figure 53, the highest percentage improvement ranges approximately between: 2% and 5% (graphs 99, 100, 101, 102, 103, 104,105 and 106). In figure 54, the highest percentage improvement is ranges approximately between: 3.5% and 5% (graphs 107, 108, 109, 110, 11, 112, 113 and 114). In figure 55, the highest percentage improvement ranges approximately between: 3% and 5% (graphs 115, 116, 117, 118, 119, 120, 121 and 122). In figure 56, the highest percentage improvement ranges approximately between: 3.6% and 6% (graphs 123, 124, 125, 126, 127, 128, 129 and 130). In figure 57, the highest percentage improvement ranges approximately between: 2.5% and 5% (graphs 131, 132, 133, 134, 135, 136, 137 and 138). And in figure 58, the highest percentage improvement ranges approximately between: 1.5% and 3.4% (graphs 139, 140 and 141). In figures 59 through 68 (Sinc interpolation, T1-MRI-450) the behavior of the improvement of the interpolation error is quite similar to that one seen for T2-MRI-230 (non linear and constantly increasing within the full range of misplacement). The range of misplacement is [0, 0.2475]. However the improvement reached for T1-MRI-450 through the SRE-based Sinc function is generally 10 times lower than that one reached for T2-MRI-230. To have an insight as to how to interpret this substantial difference the reader is suggested to read section: “ON THE INTERPrETATION OF THE rESULTS” in Chapter VIII. In figure 59, the highest percentage improvement ranges approximately between: 0.00175% and 0.015% (graphs 67, 68, 69, 70, 71, 72, 73 and 74). 453

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Figure 82. RMSE Ratio: T2-MRI, misplacement along x (large), Sinc, slices 1-8. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 83. RMSE Ratio: T2-MRI, misplacement along x (large), Sinc, slices 9-16. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 84. RMSE Ratio: T2-MRI, misplacement along x (large), Sinc, slices 17-19. The range of misplacements is [0, 0.99] at steps of 0.01.

In figure 60, the highest percentage improvement ranges approximately between: 0.006% and 0.04% (graphs 75, 76, 77, 78, 79, 80, 81 and 82). In figure 61, the highest percentage improvement ranges approximately between: 0.0035% and 0.03% (graphs 83, 84, 85, 86, 87, 88, 89 and 90). In figure 62, the highest percentage improvement ranges approximately between: 0.02% and 0.035% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). In figure 63, the highest percentage improvement ranges approximately between: 0.025% and 0.04% (graphs 99, 100, 101, 102, 103, 104,105 and 106). In figure 64, the highest percentage improvement is ranges approximately between: 0.0096% and 0.04% (graphs 107, 108, 109, 110, 11, 112, 113 and 114). In figure 65, the highest percentage improvement ranges approximately between: 0.01% and 0.04% (graphs 115, 116, 117, 118, 119, 120, 121 and 122). In figure 66, the highest percentage improvement ranges approximately between: 0.015% and 0.038% (graphs 123, 124, 125, 126, 127, 128, 129 and 130). In figure 67, the highest percentage improvement ranges approximately between: 0.028% and 0.032% (graphs 131, 132, 133, 134, 135, 136, 137 and 138). And in figure 68, the highest percentage improvement ranges approximately between: 0.013% and 0.032% (graphs 139, 140 and 141). In figures 69 through 78 (Sinc interpolation, T1-MRI-450) the behavior of the improvement of the interpolation error is quite similar to that one seen in figures 59 through 68 (where the range of mis-

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Figure 85. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (small), Sinc, slices 1-8. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 86. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (small), Sinc, slices 9-16. The range of misplacements is [0, 0.2475] at steps of 0.0025.

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Figure 87. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (small), Sinc, slices 17-20. The range of misplacements is [0, 0.2475] at steps of 0.0025.

placements is [0, 0.2475]) however the improvement achieved through the SRE-based Sinc is higher. The range of misplacement is [0, 0.99]. In figure 69, the highest percentage improvement ranges approximately between: 0.02% and 0.5% (graphs 67, 68, 69, 70, 71, 72, 73 and 74). In figure 70, the highest percentage improvement ranges approximately between: 0.06% and 0.45% (graphs 75, 76, 77, 78, 79, 80, 81 and 82). In figure 71, the highest percentage improvement ranges approximately between: 0.04% and 0.35% (graphs 83, 84, 85, 86, 87, 88, 89 and 90). In figure 72, the highest percentage improvement ranges approximately between: 0.25% and 0.4% (graphs 91, 92, 93, 94, 95, 96, 97 and 98). In figure 73, the highest percentage improvement ranges approximately between: 0.28% and 0.45% (graphs 99, 100, 101, 102, 103, 104,105 and 106). In figure 74, the highest percentage improvement is ranges approximately between: 0.1% and 0.45% (graphs 107, 108, 109, 110, 11, 112, 113 and 114). In figure 75, the highest percentage improvement ranges approximately between: 0.11% and 0.5% (graphs 115, 116, 117, 118, 119, 120, 121 and 122). In figure 76, the highest percentage improvement ranges approximately between: 0.16% and 0.5% (graphs 123, 124, 125, 126, 127, 128, 129 and 130). In figure 77, the highest percentage improvement ranges approximately between: 0.3% and 0.4% (graphs 131, 132, 133, 134, 135, 136, 137 and 138).

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Figure 88. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (large), Sinc, slices 1-8. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 89. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (large), Sinc, slices 9-16. The range of misplacements is [0, 0.99] at steps of 0.01.

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Figure 90. RMSE Ratio: functional MRI (fMRI-TS1), misplacement along x (large), Sinc, slices 17-20. The range of misplacements is [0, 0.99] at steps of 0.01.

And in figure 78, the highest percentage improvement ranges approximately between: 0.15% and 0.35% (graphs 139, 140 and 141). The behavior of the SRE-based Sinc interpolation error improvement for T2-MRI, as far as the shape of the RSME profiles is concerned, is quite similar to that seen for T1-MRI. Hereto follow is the report of the highest values of error improvement discernible across the graphs of each picture. In figure 79, the highest percentage improvement ranges approximately between: 0.03% and 0.038% (graphs 1, 2, 3, 4, 5, 6, 7 and 8). In figure 80, the highest percentage improvement ranges approximately between: 0.03% and 0.035% (graphs 9, 10, 11, 12, 13, 14, 15 and 16). In figure 81, the highest percentage improvement ranges approximately between: 0.03% and 0.035% (graphs 17, 18, 19 and 20). In figures 82 through 84 (T2-MRI) the behavior of the RMSE Ratio does not change except for graphs 1 through 4 in figure 82 where a maximum improvement is seen and it is followed by rapid decrease of improvement of the interpolation error. In figure 82, the highest percentage improvement ranges approximately between: 0.008% and 0.05% (graphs 1, 2, 3, 4, 5, 6, 7 and 8). In figure 83, the highest percentage improvement ranges approximately between: 0.0088% and 0.013% (graphs 9, 10, 11, 12, 13, 14, 15 and 16).

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Figure 91. T1-MRI-230 data qualitative assessment, Sinc, slice 72. The misplacement along the x direction is 0.045 (fraction of pixel size), the FWHM is 0.587. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 92. T1-MRI-450 data qualitative assessment, Sinc, slice 70. The misplacement along the x direction is 0.445 (fraction of pixel size), the FWHM is 0.587. The original image (the left most in the upper row) was kindly provided by the Open Access Structural Imaging Series (OASIS), www.oasis-brains.org. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 93. T2-MRI data qualitative assessment, Sinc, slice 8. The misplacement along the x direction is 0.023 (fraction of pixel size), the FWHM is 0.887. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih. gov/ij/.

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Figure 94. T2-MRI data qualitative assessment, Sinc, slice 8. The misplacement along the x direction is 0.34 (fraction of pixel size), the FWHM is 0.887. The original image (the left most in the upper row) was provided by courtesy of: R. S. Swenson, www.Dartmouth.edu/~rswenson/Atlas. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih. gov/ij/.

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Figure 95. Functional MRI data qualitative assessment, Sinc, slice 5. The misplacement along the x direction is 0.034 (fraction of pixel size), the FWHM is 0.487. The original image (the left most in the upper row) was kindly provided by INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee, also courtesy of the Department of Radiology KULeuven. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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Figure 96. Functional MRI data qualitative assessment, Sinc, slice 5. The misplacement along the x direction is 0.45 (fraction of pixel size), the FWHM is 0.487. The original image (the left most in the upper row) was kindly provided by INRIA Odyssee Lab (France), www-sop.inria.fr/odyssee, also courtesy of the Department of Radiology KULeuven. The FFT maps and the error images presented in this figure were calculated with the software ImageJ: http://rsb.info.nih.gov/ij/.

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And in figure 84, the highest percentage improvement ranges approximately between: 0.008% and 0.01% (graphs 17, 18, 19 and 20). Finally figures 85 through 90 present the behavior of the RMSE Ratio for the functional MRI (fMRITS1). Particularly the interpolation error improvement is displayed versus the range of misplacement [0, 0.2475] (figures 85 through 87) and versus the range of misplacement [0, 0.99] (figures 88 through 90). In figure 85, the highest percentage improvement ranges approximately between: 0.2% and 0.35% (graphs 1, 2, 3, 4, 5, 6, 7 and 8). In figure 86, the highest percentage improvement ranges approximately between: 0.18% and 0.5% (graphs 9, 10, 11, 12, 13, 14, 15 and 16). In figure 87, the highest percentage improvement ranges approximately between: 0.3% and 0.4% (graphs 17, 18, 19 and 20). There is a visible increase in the percentage improvement of the interpolation error of the SRE-based Sinc function for the results relevant to the range of misplacement [0, 0.99]. This is visible in figures 88 through 90. The improvement is approximately 10 times bigger than the improvement seen for the same functional MRI data (fMRI-TS1) in figures 85 through 87 (range of misplacement [0, 0.2475]). In figure 88, the highest percentage improvement ranges approximately between: 2% and 4.8% (graphs 1, 2, 3, 4, 5, 6, 7 and 8). In figure 89, the highest percentage improvement ranges approximately between: 2.5% and 6% (graphs 9, 10, 11, 12, 13, 14, 15 and 16). In figure 90, the highest percentage improvement ranges approximately between: 3.6% and 5% (graphs 17, 18, 19 and 20).

Q ualitative Assessment of the SRE-Based Sinc Interpolation Function The organizational layout of figures 91 through 96 is the following. In the top row, from left to right are shown: (i) the original image, (ii) the image processed with classic Sinc interpolation and (iii) the image processed with SRE-based Sinc interpolation. In the second row from the top, the FFT maps are shown for each of the three images of the first row. In the third row, from left to right are shown: (i) the map of the novel re-sampling locations, (ii) the error image obtained processing with the classic Sinc function, and (iii) the error image obtained processing with the SRE-based Sinc function. In the fourth row is shown the spectral power evolution. The presentation of the qualitative evaluation of the T1-MRI data is given in figures 91 (T1-MRI-230) and 92 (T1-MRI-450) for slices 72 and 70 respectively. The images were processed with the misplacement along the x direction of 0.045 (slice 72) and 0.445 (slice 70). The qualitative evaluation of the T2-MRI data is given in figures 93 and 94. Slice 8 of the T2-MRI data was processed with the misplacement along the x direction of 0.023 (figure 93) and 0.34 (figure 94). The functional MRI (fMRI-TS1) data is evaluated in figures 95 and 96. Slice 5 was processed with a misplacement of 0.034 (figure 95) and 0.45 (figure 96). The FWHM of the Gaussian was 0.587 for the T1-MRI data (T1-MRI-230 and T1-MRI-450), 0.887 for the T2-MRI data and 0.487 for the functional MRI (fMRI-TS1) data. The observations that can be made are hereto reported. The error images do not show considerable difference neither for the classic nor for the SRE-based Sinc interpolation functions except in the case

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shown in figure 96 (fMRI-TS1). Also, the FFT maps do not show much difference across the three types of images: original, processed with classic Sinc and processed with SRE-based Sinc. The information about the capability to retain the frequency of the original image is well seen in the spectral power evolutions where it is shown that generally the SRE-based Sinc interpolation function determines frequencies that are matching those of the original image more than the frequencies that are determined through the classic Sinc function. And this is manifest in the negative values of the spectral power evolutions. The opposite is true also as seen in figures 92 (T1-MRI-450) and 93 (T2-MRI). Finally, the maps of the novel re-sampling locations are well defined in all three MRI imaging modalities as they match closely the brain features. In figure 96 (fMRI-TS1) is it visible the effect of the background noise on the calculation of the novel re-sampling locations and this is seen in the map shown in the left most image in the third row.

SUMMARY Chapter XIX presents results of the application of the Sub-pixel Efficacy Region to the cubic Lagrange and the Sinc interpolation functions in their one dimensional form. Results are obtained consistently with the same motion correction validation paradigm outlined in the preceding chapters. The aim of this chapter is to inform the reader of the possible improvement of the interpolation error. In summary, on the basis of the data herein presented, the capability to evolve from the classic interpolation function to the SRE-based interpolation function, while producing an improvement of the interpolation error in favor of the SRE-based function, is much higher for the Lagrange polynomial herein studied than it is for the Sinc function that was object of analysis. This result is significant because allows forming an idea on how large is the capability of the unifying theory to determine improvement of the interpolation error in relationship to the interpolator that is being studied. Direct implications of these results are within the domain of signal processing applications in MRI such as image registration and co-registration. Also, specific focus is given to the possible improved band-pass filtering effect added through the SRE. Since the work presented in the book performs re-sampling locally (varying pixel-by-pixel in its location) filtering effects of interpolation are generally changed. This feature was earlier shown in previous chapters for the SRE-based trivariate linear and the SRE-based B-Splines. The tool used to reveal how the Sub-pixel Efficacy Region influences the smoothing effect of the Lagrange interpolator is the analysis of the spectral power evolutions of the interpolated images. Such analysis is extended to the images resulting after processing with and without the application of the SRE. Magnetic Resonance Imaging data: T1 and T2 and functional MRI were used for experimentation.

Reference Ciulla, C., & Deek, F. P. (2006). Extension of the sub-pixel efficacy region to the Lagrange interpolation function. ICGST - International Journal on Graphics, Vision and Image Processing, 6(1), 1-12.

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Introduction The last chapter of the book reports on concluding remarks recalling to the reader the message given through these works and also recalling the proposed novelty. The novelty is discussed within the context of the current literature with specific attention to other works devoted to the improvement of the interpolation error. The reader is acknowledged that the methodological approach outlined through the theory can be seen as a viable pathway to follow in order to conceptualize interpolation in an innovative and alternative manner. This descends from the adoption of the mathematical formulation that is dependent on the joint information content of node intensity and curvature of the interpolation function and has brought to the determination of a viable option to adopt when re-sampling the signal (image). Re-sampling inherent to interpolation can be performed at sub-pixel locations that are not necessarily the same as the given misplacement neither are they necessarily the same pixel-by-pixel. This is because of the variability of pixel intensity and curvature of the interpolation function at the neighborhood and between neighborhoods and also because such variability corresponds to various signal shape characteristics. The reader is informed that within the context of a paradigm to be used for the improvement of the interpolation error it is of relevance to include the curvature in the methodology that is chosen to improve the approximation capability of a given interpolation function. The focus is also towards the evidence that local re-sampling is capable of changing the band-pass filtering property of the interpolation functions. Also, a study is undertaken to determine how beneficial is the application of the Sub-pixel Efficacy Region in the estimation of signals at unknown time-space locations and this is done for the one-dimensional interpolation functions presented in the book. It is also shown that the SRE-based interpolation functions are capable to determine error improvement and to be more accurate with respect to the classic interpolation functions in the estimation of signals at Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

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locations that are not captured by the sampling frequency because of the Nyquist’s theorem constraint. Also, based on the same data, the effect of the sampling resolution is studied on the interpolation error and the interpolation error improvement. Consequentially, it is outlined the licit conclusion that under the umbrella of the unifying framework proposed within the theory, the sampling resolution influences both interpolation error and interpolation error improvement obtained from the SRE-based functions. Finally the chapter reports on the investigation of the influence of the SCALE parameter and on the performance comparison across classic and SRE-based interpolation functions. The SCALE parameter is employed to scale the convolution of the pixel intensities determined through the polynomials forms: quadratic and cubic B-Splines, Lagrange, and also to scale the numerical values of the sums of cosine and sine functions of the Sinc interpolation function.

NOVEL CONCEPTIONS AND LIMITATIONS OF THE UNIFY ING THEORY Ciulla and Deek (2006) report a review of Lagrange polynomials for image processing applications (Francesconi et al., 1993; Fuchs & Delyon, 2002; Lehmann et al., 1999; Xie & Zhou, 2001; Ye, 2003). Similarities between Lagrange and Sinc interpolation functions were investigated in literature (Lehmann et al., 1999; Wittaker, 1915) along with an interesting study that presents work on the improvement of the stop-band characteristics of the Lagrange function (Dumpster & Murphy, 1999). This book presents an improved version of the Lagrange interpolation and this is done through the extension of the theoretical framework herein outlined. As far as the more general description of benefits and limitations of the application of the SRE to the interpolation functions, the knowledge reported in the book shall be connected to the knowledge reported earlier (Ciulla & Deek, 2006). The above mentioned work point out however that the non uniform re-sampling inherent to the nature of the Sub-pixel Efficacy Region is not novel in literature (Amidror, 2002; De Boor, 1978). The formulation of the SRE is strictly related to the originality of the methodology which merges the pixel intensity values together with the curvature of the interpolation function. This is a novelty that differentiates the SRE-based interpolation functions with respect to previously reported approaches (Blu et al., 2001; Blu & Unser, 1999; Thevenaz et al., 2000). The unifying theory presented in this book descends from novel conceptions, which fairly admitting, present potentialities that are confronted with undeniable limitations. These limitations consist in the non absolute power of the theory, as shown through the experimental results relevant to the improvement of the interpolation error and the not absolute successfulness in improving the band-pass characteristics of the interpolation functions. In these two contexts: (i) results that show the boundaries of the potentiality of the theory have been presented through graphs showing the analysis of the RMSE Ratio along with (ii) an objective study of the spectral components of the images obtained with the SRE-based interpolation functions. And by the studies conducted through the spectral power evolutions, which were appropriately chosen as signal processing tools fulfilling the role of revealing the spectral characteristics hidden in the interpolated images, demonstrations have been given that certain frequency components are preserved in the images processed with the SRE-based interpolation functions. To reinforce on the significance of the present work however, this book has presented results relevant to the improvement of various interpolation functions with diverse degree and diverse dimensionality. Thus, this chapter reminds us that a new class of interpolators has been derived: the SRE-based

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interpolation functions. It is important to emphasize on the relevance of the bridging concept between the classic interpolation paradigms and the SRE-based interpolation: the curvature of the interpolation function. The curvature is embedded into the Intensity-Curvature Functional, which study reveals novel re-sampling sub-pixel locations allowing non-uniform re-sampling along with improved approximation properties of the interpolation function.

On the Estimation of Unknown Signals It is necessary to comment on the paradigm adopted to validate the resulting unifying theory for the improvement of the interpolation error. In first instance this validation paradigm shifts an image (or a signal) of the misplacement x0. Subsequently, based on the SRE-based calculated novel re-sampling location xr0, the paradigm motion corrects the signal using both the misplacement x0 and the novel location xr0. At this point the paradigm computes the root-mean-square-error obtained through x0, which is called RMSEbefore and the root-mean-square-error obtained through x r0, which is called RMSEafter. While RMSEbefore is thus obtained through the classic interpolation function, RMSEafter is obtained through the SRE-based function. The motivation behind the adoption of the above mentioned validation paradigm is justified by the necessity to make available a term of comparison, and at this aim, one viable option is that of applying interpolation twice. The first time using x0, and the second time using the novel re-sampling location xr0. It follows that the basic difference between the classic interpolation scheme and the SRE-based scheme is that of using a novel re-sampling location x r0 ≠ x0. One would expect that the root mean square-error would vanish because it is computed after motion correction and interpolation. This would happen if interpolation was perfect. On the other hand, the fact that the validation paradigm demonstrates an improvement in interpolation error revealed through the RMSE Ratio: R = (1 - RMSEbefore / RMSEafter) reinforces on the capability of the theory to determine the novel re-sampling location xr0 thereby effecting an improvement in the approximation characteristics of the interpolation function. This improvement expresses itself through the SRE-based interpolation paradigm which consists of the classic interpolation formula calculated at xr0. Having highlighted and demonstrated the potentials and capabilities of the theory through the motion correction paradigm, further remarks are warranted. One point is that the novel re-sampling location xr0 is useful for re-sampling techniques in signal processing applications, specifically but not limited to the context of image registration. Another point is that it would be informative to see how this theory might prove useful in the estimation of signals of unknown nature, for which there are no actual sets of samples at given intra-node locations in time because of the sampling frequency of the equipment used for recording. To determine as to whether or not the novel re-sampling location xr0 can insure, or at least suggest, an improved estimation with respect to the classic interpolation paradigm, one should devise a set of experiments in the following manner. Let be given a one dimensional signal of known nature such as the sinusoidal function sin (x), and let be given an interpolation model to determine the value of the signal at locations for which the true value is known. If we are able to demonstrate that the true value is estimated through the SRE-based interpolation function with higher accuracy than it is estimated through the classic interpolation function and given that we do actually know the true value of the signal, we could then infer that the SRE-based

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interpolation function is of usefulness in the estimation of signals of unknown nature and also that its approximation characteristics are more accurate than the classic schemes. With this goal in mind, the following sets of experiments have been conducted. Three functions: cos (x), sin (x), log (x) were object of experimentation choosing a given interpolation scheme (say the quadratic B-Spline). The interpolation scheme was used to estimate the value of the function at x0. Further, using the SRE-based version of the same interpolation scheme the function (signal) was estimated at xr0. Therefore, given that the function (sin (x) for instance) is known at any location, it is possible to ascertain the accuracy of the two interpolation models through the use of the RMSE Ratio which is able to measure the magnitude of the improvement. Figure 1. RMSE Ratio: estimation of known signals, quadratic B-Spline, step size 0.0001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively. The value of the “a” constant is -0.00045.

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Figure 2. RMSE Ratio: estimation of known signals, quadratic B-Spline, step size 0.001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively. The value of the “a” constant is -0.00045.

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As a note to the reader, it is due to report that in this chapter the notation log (x) indicates the natural logarithm and not the logarithm base 10. This experiment was conducted for a set of misplacements ranging from 0 to 0.0099, 0.099 and 0.04455, with x0 increasing 100 times (100 steps) from 0 at steps sizes of 0.0001, 0.001 and 0.00045 respectively. The interpolation functions that were evaluated were: B-Splines in quadratic and cubic forms, Lagrange and Sinc functions. For each of the experiments, the value of the RMSE Ratio was computed and it was plotted versus the misplacement. Results are presented hereto.

Figure 3. RMSE Ratio: estimation of known signals, quadratic B-Spline, step size 0.00045. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively. The value of the “a” constant is -0.00045.

Figure 4. RMSE Ratio: estimation of known signals, cubic B-Spline, step size 0.0001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

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Figure 5. RMSE Ratio: estimation of known signals, cubic B-Spline, step size 0.001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

Figure 6. RMSE Ratio: estimation of known signals, cubic B-Spline, step size 0.00045. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

For the quadratic B-Spline, figure 1 shows the plot of the RMSE Ratio versus the misplacement furnished through classic and SRE-based functions for cos (x) in (A), sin (x) in (B) and log (x) in (C), for the value of step size of 0.0001. Also, values of the RMSE Ratio for the step size of 0.001 are shown in figure 2, in (A) for cos (x), in (B) for sin (x) and in (C) for log (x) respectively. The corresponding values of the RMSE Ratio for the step size of 0.00045 are shown in figure 3: in (A) for cos (x), in (B) for sin (x) and in (C) for log (x). In figures 1 through 3 (quadratic B-Splines) is observed that the interpolation error improvement increases versus misplacement with a clear and well defined linear behavior. At the largest misplacement

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Figure 7. RMSE Ratio: estimation of known signals, Lagrange, step size 0.0001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

Figure 8. RMSE Ratio: estimation of known signals, Lagrange, step size 0.001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

corresponds the largest improvement (across the three functions cos (x), sin (x) and log (x)) approximately in between 3% and 4% (see figure 2, step size 0.001). When the step size is 0.0001, as shown in figure 1, the largest interpolation error improvement is approximately 0.5%, and when the step size is 0.00045 (figure 3) the largest improvement is between 1.5% and 2% (values across cos (x), sin (x) and log (x)). Thus the percentage improvement increases with increasing step size. With the same organizational layout of figures 1, 2 and 3, results for the cubic B-Spline are presented in figures 4, 5 and 6 for the step sizes of 0.0001, 0.001 and 0.00045 respectively. Also, figures 7, 8 and

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Figure 9. RMSE Ratio: estimation of known signals, Lagrange, step size 0.00045. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

Figure 10. RMSE Ratio: estimation of known signals, Sinc, step size 0.0001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

9 present the results for the Lagrange interpolation functions for the step sizes of 0.0001, 0.001 and 0.00045 respectively. And consistently, figures 10, 11 and 12 present the results for the Sinc interpolation functions for the step sizes of 0.0001, 0.001 and 0.00045 respectively. The linear behavior that features the increase of interpolation error improvement with increasing misplacement is also clearly distinguishable in figures 4 through 6 (cubic B-Splines) except for figure 4(A) (cos (x), step size 0.0001) where the percentage improvement is constant (approximately 0.13%). In the rest of the graphs of figure 4 (step size 0.0001) it can be seen that the highest improvement is approximately: 1.3% (sin (x)) and 0.32% (log (x)). In figure 5 (step size 0.001) the highest improvement is approximately: 0.28% (cos (x)), 14% (sin (x)) and 2% (log (x)). In figure 6 (step size 0.00045) the highest

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Figure 11. RMSE Ratio: estimation of known signals, Sinc, step size 0.001. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

Figure 12. RMSE Ratio: estimation of known signals, Sinc, step size 0.00045. Estimation at location xr0 in place of x0: (A), (B), (C) for cos (x), sin (x) and log (x) respectively.

improvement is approximately: 0.15% (cos (x)), 6% (sin (x)) and 1% (log (x)). Values are given across the three functions and confirm that the percentage improvement generally increases with increasing step size. Figures 7 through 9 (Lagrange) show the highest interpolation error improvement to be approximately: 0.12%, 1.6%, 1.6% (see figure 7, step size 0.0001); 0.16%, 14%, 18% (see figure 8, step size 0.001); 0.2%, 7%, 7% (see figure 9, step size 0.00045). Figures 10 through 12 (Sinc) show the highest interpolation error improvement to be approximately: 0.0024%, 0.002%, 0.0024% (see figure 10, step size 0.0001); 7%, 7%, 8% (see figure 11, step size 0.001); 1.5%, 1.5%, 1.8% (see figure 12, step size 0.00045). Data of figure 10 is visibly scattered and together

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with figures 11 and 12 show that the behavior of the interpolation error improvement is non linear, nevertheless still increasing with increasing values of the misplacement. Like wise the quadratic and cubic B-Splines the values confirm that the percentage improvement increases with increasing step size. Regardless of the intra-node resolution chosen for experimentation it is remarkable to see the consistent improvement in signal estimation which is manifest for all interpolation schemes and this is an effect of the SRE. This evidence shows that the unifying theory can reliably estimate unknown signals. It further suggests that the use of the Sub-pixel Efficacy Region (SRE) for the calculation of the novel re-sampling location improves the estimation capabilities of the interpolation function. As far as the incapability of the theory seen in some cases for misplacements that are generally but not necessarily less then half of the pixel size (specifically for the case of bivariate and trivariate linear interpolation functions), what can be commented is that the theory has found a domain where is falsified and thus faces a limitation. On the other hand it is fair to emphasize that the theory was empirically confirmed for the rest of the interplation schemes that were evaluated. What can be learned from the experiments presented in this section is that for given misplacement x0, the application of the SRE conception yields a different re-sampling location (xr0). This is conceptually equivalent to the re-organization of the independent variable within the domain of existence of the interpolation function and in such manner that the re-sampling location x0 is replaced with the novel re-sampling location xr0. This descends from the mathematical formulation of the Sub-pixel Efficacy Region and an illustration of this re-organization process was given earlier (Ciulla & Deek, 2006) (figure 9, section 4.3). It has been extensively seen through empirical validation that re-sampling at the novel location x r0 improves the interpolation error. By pointing out to the improvement of signal value estimations at unknown location as illustrated in the simulations presented in this chapter, it is demonstrated that is licit to re-sample at the novel re-sampling location xr0 which is not the same as x0. Also, it is provided a further justification as to why motion correcting and interpolating the signal, as it was done through the validation paradigm employed in the testing sessions, the novel re-sampling location xr0 does not have to be necessarily the misplacement x0. Instead, xr0 can be used as obtained from the SRE and this yields improved approximation characteristics of the interpolation function.

On the Interpolation Error Representation Ciulla and Deek (2006) discuss the work reported in literature that concerns with the interpolation error representation and that advances the general proposition that the interpolation error tends to be nullified for sampling steps tending to zero (Bojanov et al., 1993; De Boor, 1978; Strang & Fix, 1971; Unser & Daubechies, 1997). This error bound representation has served the role to instruct the scientific community and thus a quite extensive literature was reported in the years to follow (Blu et al., 1999, 2001; De Boor et al., 1994; Deng & Denney, 2004; Meijering, 2002; Meijering & Unser, 2003; Thevenaz et al., 2000; Unser, 2000). Also, the error bound representation has lead to: (i) the study of the Gaussian interpolation function (Wendland, 2001), (ii) the parameterization of piecewise-polynomial interpolation kernels (Blu et al., 2003), and (iii) a novel and improved form of linear interpolation (Blu et al., 2004). Early results (Strang & Fix, 1971) were further elaborated within the context of B-Spline knot placement (De Boor, 1978). Ciulla and Deek (2006) give due mention to the quite extensive effort conducted in literature at the aim to minimize the interpolation error through the optimization of the choice of the

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locations of the knots along the grid and also through the determination of the optimal sampling step of the interpolation function (De Boor, 1978). As far as the existence of a spatial support of the error bound of the bivariate linear interpolation function, Ciulla and Deek (2006) also mention the discovery that elucidates that the spatial support of such error bound consists of a rectangular region (Waldron, 1998). Later, the same author reported error bound representations which expressions were derived as dependent on the functions’ nodes at the neighborhood. This study was done within the context of polynomial interpolation. It is worth noting the similarity of the rectangular region consisting of the spatial support of the error bound (Waldron, 1998) with the Sub-Pixel Efficacy Region reported in this book and discovered for the bivariate linear interpolation function. Such similarity was earlier discussed (Ciulla & Deek, 2006). As pointed out in earlier work (Ciulla & Deek, 2006) other error bounds are reported in literature for the Lagrange interpolation function and they are dependent on the sampling step (Radzyner & Bason, 1972), and the Hilbert transform (Kubayi & Lubinsky, 2004). Furthermore the Chebyshev polynomial is used (Xie & Zhou, 2001) within the context of an extension of the work earlier undertaken (Gopengauz, 1967; Vertesi, 1979) in order to optimize the choice of the nodes and to formulate the error bound representation.

The Concept Bridging Classic Interpolation with the Unify ing Theory Ciulla and Deek (2006) discuss an overview of the evolution of the theoretical basis employed to design interpolation functions with specific mention to: (i) the inception of interpolation (Newton & Huygens, trans. 1934) and (ii) the followers whom established a school of thought which designed interpolation functions based on the pixel signal intensity (Agarwal & Wong, 1993; Blu et al., 2001, 2004; Blu & Unser, 1999; De Boor, 1978; Schoenberg, 1946a, 1946b, 1969; Thevenaz et al., 2000; Unser et al., 1993a, 1993b), and (iii) the scene-based interpolation paradigms (Grevera & Udupa, 1996; Herman et al., 1992; 1998; Raya & Udupa, 1990). Whereas the new class of interpolators introduced in this book which are called the SRE-based functions, are different from above mentioned works as far as their capability to improve the approximation properties. The SRE-based functions yields a reduced interpolation error through the novel and original concept that sets the ground for the unifying theory presented in this book. The novel concept consists of merging together pixel intensity and curvature of the function such to adapt locally to the behavior of the true and unknown signal to estimate. It is also apparent, that the interpolation error improvement as expressed by equation (27) in Chapter XVIII is bounded by two constants. And on the basis of the Sub-pixel Efficacy Region, a unifying interpolation error characterization form determines the two constants as dependent on values of the function to approximate and the SRE-based interpolation function. The unifying interpolation error characterization form should be emphasized since the biomedical and image processing scientific community has accepted (Meijering, 2002) that the interpolation error is dependent on the sampling step (resolution). There is also necessity of further clarification. The claim made by the unifying theory is that the mathematical formulation and the approach used for the improvement attempts to detach itself from the concept of sampling resolution and this conceptualiza-

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tion is expressed through the Intensity-Curvature Functional (ΔE) and its study which leads to the set of spatial points namely the Sub-pixel Efficacy Region (SRE). On the other hand, although the interpolation error characterization form across the functions embraced by the theory is bounded by two constants that are not dependent on the resolution, the interpolation error itself remains dependent on the resolution as it will be seen in the following sections of this chapter. This concept can be intuitively understood considering that the local curvature built through any interpolator assumes degree of uncertainty as big as is the resolution simply because the shape and thus the curvature of the function are dependent on the resolution. Therefore, it is beneficial to search for the re-sampling location through the measure that incorporates the curvature, and this measure is ΔE. This concept reinforces on the significance of the mathematical formulation that has sets the grounds of the unifying theory presented in this book. This book has thus presented a novel methodology and a novel framework that converge to a unifying theory that asserts that the classic interpolation functions and the novel SRE-based interpolation schemes are linked through the curvature of the interpolator as expressed through the second order derivative (or derivatives). Although the theory is derived through deduction and as such, extended to six interpolation paradigms, it has been empirically and extensively demonstrated that the use of the curvature as part of the information content of the Intensity-Curvature Functional produces an improvement in the interpolation error and also consistent improvement of the estimation of signals at unknown locations. Therefore, the concept that bridges classic interpolation to SRE-based interpolation is the curvature of the interpolator.

The Basic Formulas of the Unify ing Theory For the calculation of the novel re-sampling locations of the SRE-based bivariate linear function EIN is calculated at the SRE location (xsre, ysre) and the location (xsre- x0, ysre- y0), and the second order derivative (∂2 (h (x,y))/∂x∂y) is calculated at the SRE location (xsre, ysre) and at the location (xsre- xr0, ysre- yr0). The following equation is employed: { h(xsre- xr0, ysre- yr0) (∂2 (h (x,y))/∂x∂y) (xsre- xr0, ysre- yr0) }/ { h(xsre, ysre) (∂2 (h (x,y))/∂x∂y) (xsre, ysre) }= EIN (xsre-x0, ysre-y0) / EIN (xsre, ysre) For the calculation of the novel re-sampling locations of the SRE-based trivariate linear function EIN is calculated at the SRE location (xsre, ysre, zsre) and the location (xsre- x0, ysre- y0, zsre- z0), and the second order derivatives (∂2 (h(x, y, z))/∂x∂y), (∂2 (h(x, y, z))/∂x∂z), (∂2 (h(x, y, z))/∂y∂z) are calculated at the SRE location (xsre, ysre, zsre) and at the location (xsre- xr0, ysre- yr0, zsre- zr0). The following equation is employed: { h(xsre-xr0, ysre-yr0, zsre-zr0) * { [(∂2 (h(x, y, z))/∂x∂y) + (∂2 (h(x, y, z))/∂x∂z) + (∂2 (h(x, y, z))/∂y∂z)] (xsre-xr0, ysre-yr0, zsre-zr0) } } / { h(xsre, ysre, zsre) * { [ (∂2 (h(x, y, z))/∂x∂y) +

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(∂2 (h(x, y, z))/∂x∂z) + (∂2 (h(x, y, z))/∂y∂z) ] (xsre, ysre, zsre) } } = { EIN (xsre- x0, ysre- y0, zsre- z0) / EIN (xsre, ysre, zsre)} For either of the SRE-based bivariate or the trivariate linear interpolation functions, the numerical value of the novel re-sampling locations ((x r0, yr0) and (xr0, yr0, zr0) respectively) is deducted as dependent on the numerical value of the SRE location ((xsre, ysre) and (xsre, ysre, zsre) respectively). For the calculation of the novel re-sampling locations of the SRE-based quadratic B-Spline function EIN is calculated at the SRE location xsre and the location (xsre- x0), and the second order derivative (∂2 (h(x)) /∂x2) is calculated at the SRE location xsre and at the location ( (xsre - xr0 ) / xsre). The following equation is employed: { h((xsre - xr0 ) / xsre) * (∂2 (h(x)) /∂x2) (xsre - xr0) } / { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { EIN (xsre-x0) / EIN (xsre) } Where the numerical value of ((xsre - xr0 ) / xsre) is the percentage decrease of the numerical value of the SRE location xsre. And the novel re-sampling locations (as reported in Chapter XIV) are given by: xr0(k, i) = (xsre(i) - ρ(k) xsre(i)) with i, k = 1, 2 Where ρ(k) are normalized to the range [0, x0] and are greater than zero. This formula thus admits that the numerical value of the novel re-sampling location x r0 is smaller than the numerical value of the SRE location xsre. For the calculation of the novel re-sampling locations of the SRE-based cubic B-Spline function EIN is calculated at the SRE location xsre and the location (xsre- x0), and the second order derivative (∂2 (h(x)) /∂x2) is calculated at the SRE location xsre. The following equation is employed: { h(xr0) * (∂2 (h(x)) /∂x2) (xr0) } / { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { EIN (xsre-x0) / EIN (xsre) } For the calculation of the novel re-sampling locations of the SRE-based Lagrange function EIN is calculated at the SRE location xsre and the location (xsre- x0), and the second order derivative (∂2 (h(x)) /∂x2) is calculated at the SRE location xsre. The following equation is employed: { LGR3(xr0) * { (∂2 (LGR3 (x)) /∂x2) (xr0) } } / { LGR3(xsre) * { (∂2(LGR3 (x)) /∂x2) (xsre) } } = { EIN (xsre-x0) / EIN (xsre) } In the preceding two cases of SRE-based cubic B-Spline and SRE-based Lagrange interpolation functions the numerical value calculated for x r0 is not calculated as a percentage of the numerical value of the SRE location xsre however its value descends from xsre.

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For the calculation of the novel re-sampling locations of the SRE-based Sinc function EIN is calculated at the SRE location xsre and the location (xsre- x0), and the second order derivative (∂2 (h(x)) /∂x2) is calculated at the SRE location xsre and at the location (xsre - xr0). The following equation is employed: { hO(xsre - xr0) * { (∂2 (hO(x)) /∂x2) (xsre - xr0) } } / { hO(xsre) * { (∂2(hO(x)) /∂x2) (xsre) } } = { EIN (xsre-x0) / EIN (xsre) } For the case of the SRE-based Sinc interpolation function the numerical value of the novel resampling location xr0 becomes a fraction of the SRE location xsre as seen by the equations reported in Chapter XVIII: xr1 = xsre - [ 3 * acos ( Γ+ ) / 2 π ] xr2 = xsre - [ 3 * acos ( Γ- ) / 2 π ] Hence, the novel re-sampling locations always descends from the SRE locations. The SRE locations descend from the Intensity-Curvature Functional which is a measure of the intensity-curvature content of the signal in the two cases of interpolation and non-interpolation. Particularly, the SRE locations are the extremes of the Intensity-Curvature Functional. Also, they serve the purpose to project re-sampling onto a novel location where the approximation properties of the interpolation function are improved. This assertion is considerably supported by the results presented in this book.

Influence of the Resolution on Interpolation Error and its Improvement It is now necessary to address the following research questions that are compelling to the present theory and relate directly to the existing literature: (i) how to define an approach for the improvement of the interpolation error that is independent or dependent on the sampling resolution, (ii) how to define an interpolation paradigm that is independent or dependent on the sampling resolution, and (iii) how to differentiate between these. To start the investigation it is necessary to recall the undisputed evidence that the literature provides (Strang & Fix, 1971). These authors asserted and demonstrated that the interpolation error is inversely proportional to the sampling density (sampling step) at the nth power, where n is the order of the interpolation function. The second and consequential supporting evidence is that the interpolation accuracy is inversely proportional to the number of sampling points (Fraser, 1989) in either cases of windowed or un-windowed signal. Earlier work (Schoenberg, 1946a, 1946b, 1969) provided the basis for further development (Strang & Fix, 1971) and thus a school of interpolation functions and approaches to interpolation has been developed throughout the years (Agarwal & Wong, 1993; Blu et al., 2001, 2004; Blu & Unser, 1999; De Boor, 1978; Deng & Denney, 2004; Thevenaz et al., 2000; Unser et al., 1993a, 1993b). Also, due to

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report is the demonstration formulated on the basis of the study of polynomial Splines (De Boor & Fix, 1973), which similarly asserted that interpolation accuracy is inversely proportional to the sampling step. De Boor and Fix (1973) formulated the dependence of the interpolation error on the sampling step (resolution) in the following terms: || f(r) – (Fπ f)(r) ||∞ ≤ K r ω(f(k+1), | π |) | π |k-1-r , for r < k and k ≥ 1 Where | π | = max 0
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Another method was validated with the linear interpolation function (Keith & Prata, 1996). Starting from the original set of discrete sample points, the method determines through an iterative search the optimal number of node points which preserves a prescribed level of interpolation error. An application of cubic B-Spline for the determination of snakes for contour detection was also presented (Brigger et al., 2000), which features flexible spacing between knots through an iterative minimization and also determines the optimal knot placement for contour detection starting the search from the initial grid placement. As far as the error bound characterization form is concerned, such technique (Brigger et al., 2000) maintains the approach consistent to the findings earlier reported (Strang & Fix, 1971). Other approaches all started from the original grid of knots and determined the optimized knot placement based on the original grid. Such approaches are based: on least square solutions (Prahasto & Bedi, 2000), on a neural network (Yiu et al., 2001), and on the attempt to merge B-Spline curves (Tai et al., 2003). These approaches showed consistency in calculating the B-Spline functions in a fashion that is quite similar to the one previously outlined (De Boor, 1978) that is to calculating the B-Spline functions recurrently as a function of the knot’s position. Consistent with these approaches, Deng and Denney (2004) determine a cost function on the basis of the condition earlier reported (Stang & Fix, 1971). Through an iterative search that is based on the calculation of derivatives at the original grid placement, Deng and Denney (2004) determine a nonuniform knot placement with the aim of improved interpolation error for multidimensional B-Splines. All of above mentioned optimization strategies constitute of methods for the improvement of the interpolation error. They are either dependent on the original knot placement or dependent on the calculation of derivatives at the original grid points. Consequently, they can be classified as methods that are dependent on the sampling resolution. Therefore, in order to be independent from the sampling resolution, an ideal requirement can be demanded to an interpolation paradigm is that of demonstrating, regardless of the sampling resolution, the same interpolation error. And the ideal requirement that can be demanded to a method for the improvement of the interpolation error is that the interpolation error improvement remains the same regardless of the resolution. Thus, the key to understand what differentiates the theory proposed by this book from the typology of approaches that are above mentioned is found in the mathematical description of the Intensity-Curvature Functional (ΔE). ΔE is a ratio of terms: the intensity-curvature terms Eo (at the grid point) and EIN (at the generic intra-pixel location). Eo is dependent on the derivatives calculated at the original grid points and EIN is independent from the grid points and from the resolution (defined as the intra-node distance between samples of the discrete series). In principle, it is the formulation what differentiates the theoretical framework presented in this book from the work of others. Formulas herein reported attempts to characterize the interpolation error and the interpolation error improvement as independent from the resolution. This can be seen from the definition of ΔE and particularly from the definition of the EIN intensitycurvature term through which the SRE-based interpolation functions are detached from the concept of resolution. However, the SRE-based functions still show some dependency from the original grid of points and the values of second order derivatives that are computed at such locations. This stems from the mode of calculation of the intensity-curvature term Eo.

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Figure 13. RIFE: estimation of known signals, SRE-based quadratic B-Spline. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively; the value of the “a” constant is -0.00045.

Figure 14. RIFE: estimation of known signals, SRE-based cubic B-Spline. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively.

To place the theory and its mathematical formulation in the appropriate context within the body of research literature it is necessary to address the question: “How dependent on the resolution are the results obtained through the unifying theory herein proposed?” Through the exploration of the theorems that express the condition necessary and sufficient in order to bound the interpolation error improvement and the interpolation error characterization forms (Ciulla & Deek, 2006), it is portrayed theoretically and demonstrated consequentially an assertion that depicts the interpolation error as bounded by two constants. The assumption that Fπ f – f = O(| π |k) (De Boor & Fix, 1973) was empirically validated in literature (extensively and consistently) and this fact is recalled in this chapter through the mention and discussion of the research reports existing in literature. 487

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Figure 15. RIFE: estimation of known signals, SRE-based Lagrange. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively.

Figure 16. RIFE: estimation of known signals, SRE-based Sinc. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively.

Under the present theory, error bounds of interpolation error and its improvement are determined as shown in equations (27), (28) of Chapter XVIII wherein the novel re-sampling location is calculated as dependent on the EIN and the second order derivative (∂2 (h(x)) /∂x2). While, EIN is independent from the resolution, Eo is calculated based on pixel intensities and derivatives at the original grid placement locations. It is also true that the Sub-pixel Efficacy Region (SRE) point location xsre is determined from ΔE, As is the novel re-sampling location x r0. Therefore, both SRE and xr0 are also dependent on the Eo intensity-curvature term and thus on the original grid placement locations.

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The fact that no assumption was made on the nature of the relationship between interpolation error and sampling step induces to think that a viable option in order to ascertain the magnitude of dependency on the resolution of interpolation error improvement and the interpolation error is the following. It is possible to employ data obtained through the estimation of signals at unknown locations at the specific purpose to clarify for different values of sampling resolution the behavior of variation of the interpolation error and its improvement obtained on the basis of the SRE. We then look back to the data previously reported in this chapter where it was seen that estimating the signals at unknown locations demonstrates that the interpolation error improvement is achieved regardless of the intra-node resolution. The next question to answer is then related to how the interpolation error and its improvement behave with varying with intra-node resolution (Ciulla & Deek, 2006). This presentation was anticipated (Ciulla & Deek, 2006) and is herein extended in comprehensive manner presenting the analysis that is relevant to all of the one dimensional interpolation functions covered by the book. The influence of the resolution on the interpolation error (RIFE) was calculated summing three values of absolute difference between interpolation errors. Each of the interpolation errors is obtained through the SRE-based function while estimating the signal at a given intra-node resolution. The intranode resolutions are: 0.0001, 0.001 and 0.00045 (they correspond to the three range of misplacements [0, 0.0099], [0, 0.099] and [0, 0.04455] respectively) and the differences of interpolation errors (IE) are: (i) IE obtained with resolution 0.0001 minus IE obtained with 0.001; (ii) IE obtained with 0.0001 minus IE obtained with 0.00045; (iii) IE obtained with 0.001 minus IE obtained with 0.00045. The sum of these three values of absolute differences of IE should ideally be zero, were there no effect caused by the resolution on the interpolation error, and it should be ideally zero were there no effect of the resolution on the improvement of the interpolation error obtained through the SRE-based function versus the corresponding classic method (Ciulla & Deek, 2006). RIFE is clearly a measure of the effect of the intra-node resolution on the interpolation error. Figure 13 shows for cos (x) in (A), for sin (x) in (B), and log (x) in (C) the influence of the resolution on the interpolation error for the SRE-based quadratic B-Spline as indicated by the Resolution Influence Factor on Error (RIFE). Figures 14, 15 and 16 display the RIFE calculated on the three functions cos (x) in (A), sin (x) in (B), and for log (x) in (C), for: (i) the SRE-based cubic B-Spline (figure 14), (ii) the SRE-based Lagrange (figure 15), and (iii) the SRE-based Sinc (figure 16). The steps sizes for which the interpolation error was computed were 0.0001, 0.001 and 0.00045 corresponding to the three range of misplacements [0, 0.0099], [0, 0.099] and [0, 0.04455] respectively. The display of the RIFE is versus the normalized misplacement along X (see Norm. Mispl. Along X in the figures) where each point on the abscissa corresponds to m * { [ 1 / (0.0001 * 0.00045) ] + [ 1 / (0.001 * 0.00045) ] + [ 1 / (0.0001 * 0.00045) ] }, with m = 1…99. The SCALE factor was -0.00045. In figures 13 through 16 the effect of the resolution on the interpolation error of the SRE-based functions is visibly non linear except for figures 14(A) and 15(A) (SRE-based cubic B-Spline and SREbased Lagrange respectively, for cos (x)). The general behavior is that of an increase of the effect of the resolution on the interpolation error with increasing values of the misplacement and the effect is that of magnifying the error. The highest values of the percentage increase of the interpolation error are approximately: 0.0008%, 0.001%, 0.03% (figure 13); 0.018%, 0.0016%, 0.007% (figure 14); 0.024%, 0.0014%, 0.028% (figure 15); 1%, 1%, 1.2% (figure 16). The triplets of values for each figure correspond to cos (x), sin (x) and log (x)

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Figure 17. RIFI: estimation of known signals, quadratic B-Spline. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively.

Figure 18. RIFI: estimation of known signals, cubic B-Spline. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively.

respectively. Figures 13 through 16 thus demonstrate that the resolution has an effect on the interpolation error. The influence of the resolution on the interpolation error improvement (RIFI) was calculated summing three values of absolute difference between RMSE Ratios. Each of the RMSE Ratio is obtained though the classic and the SRE-based functions while estimating the signal at a given intra-node resolution. Consistently with what presented for the RIFE, the intra-node resolutions are: 0.0001, 0.001 and 0.00045 (they correspond to the three range of misplacements [0, 0.0099], [0, 0.099] and [0, 0.04455] respectively) and the differences of RMSE Ratio were calculated as: (i) RMSE Ratio obtained with resolution 0.0001 minus RMSE Ratio obtained with 0.001; (ii) RMSE Ratio obtained with 0.0001

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Figure 19. RIFI: estimation of known signals, Lagrange. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively.

Figure 20. RIFI: estimation of known signals, Sinc. (A), (B), and (C) for cos (x), sin (x) and log (x) respectively.

minus RMSE Ratio obtained with 0.00045; (iii) RMSE Ratio obtained with 0.001 minus RMSE Ratio obtained with 0.00045. The sum of these three values of absolute differences of RMSE Ratio should ideally be zero, were there no effect caused by the resolution on the RMSE Ratio. Since the RMSE Ratio measures how big is the improvement (or worsening) determined by the SRE-based interpolation function over the classic function, RIFI becomes an investigative tool to assess the influence of the resolution on the performance of the two classes of interpolation functions (classic and SRE-based) relative to one another. As previously shown in figures 1 through 12 the estimation of the unknown signal showed that the SRE-based interpolation functions perform better than the classic. Therefore the RIFI shown in figures

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17 through 20 measures the influence of the intra-node resolution on the interpolation error improvement. Also the display of the RIFI factor is versus the normalized misplacement along X consistently with the presentation given for the RIFE. Figure 17 shows the Resolution Influence Factor on Improvement (RIFI) for the quadratic B-Spline. Figures 18, 19 and 20 show the RIFI for: (i) cubic B-Spline, (ii) Lagrange, and (iii) Sinc. The signals of known nature were the same employed for the presentation of the RIFE: cos (x) in (A), for sin (x) in (B), and log (x) in (C). In figures 17 through 20 the effect of the resolution on the interpolation error improvement of the SRE-based functions over the classic functions is visibly linear except for figure 20 (Sinc). The general behavior is that of an increase of the effect of the resolution on the interpolation error improvement with increasing values of the misplacement and the effect is that of magnifying the error improvement. The highest values of the percentage increase of the interpolation error improvement are approximately: 6%, 6%, 6% (figure 17); 0.32%, 23%, 3.5% (figure 18); 0.12%, 25%, 30% (figure 19); 15%, 15%, 15% (figure 20). The triplets of values for each figure correspond the RSME Ratios calculated with cos (x), sin (x) and log (x) respectively. Even though the ideal value of zero for the RIFE was not achieved, the influence of the sampling resolution on the interpolation error was below 0.015 (1.5%) overall cases presented in figures 13 through 16. The consistent feature shown through the RIFE is that it increases for increasing lengths of the sampling location (misplacement). The sampling location, which is certainly located within the range of the misplacement, is seen on the abscissa of figures 13 through 16. That the RIFE is directly proportional to the length of the sampling location means that the effect that the intra-node resolution has on the interpolation error magnifies itself with increasing values of the sampling location. Therefore, increasing the length of the sampling location within the range of the misplacement makes the SRE-based interpolation function subject to a magnified effect of the intra-node resolution resulting in an increased interpolation error. This is seen visually in all figures 13 through 16 where the RIFE increases with increasing values of the sampling location. Overall, the RIFI was below 0.35 (35%) as seen in figures 17 through 20. The consistent behavior seen in the figures is that of an increase of RIFI for increasing length of the sampling location (misplacement). The RIFI is a measure of the interpolation error improvement (or worsening) of the SRE-based function versus the classic. RIFI increases with misplacements of increasing magnitude. Therefore, it can be inferred that the magnitude of the interpolation error improvement produced through the SREbased functions is directly proportional to the magnitude of the misplacement. And given that the RIFI, as it was calculated, comprises of three different intra-node resolutions, there exists an effect of the intra-node resolution on the interpolation error improvement. Thus it is manifest that there exist an effect that resolution exerts on both interpolation error and the interpolation error improvement obtained through the SRE-based functions. This result admits the conclusion that even though the mathematical formulation proposed by the theory has been determined to detach itself from the concept of sampling resolution, and also without any a-priori assumption on the nature of the relationships existing between interpolation error and resolution, it is possible to attain interpolation error improvement regardless of the resolution. But as the results demonstrate both the interpolation error and the interpolation error improvement are dependent on the intra-node resolution.

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Table I. Quantitative evaluation of the influence of the SCALE parameter, quadratic B-Spline. The value of the “a” constant is -2.36. The images were processed with a misplacement of 0.0032 along the x direction.

Table II. Quantitative evaluation of the influence of the SCALE parameter, cubic B-Spline. The images were processed with a misplacement of 0.0032 along the x direction.

An Investigation on the Influence of the SCALE Parameter and A Performance Comparison Across Classic and SRE-based Interpolation Functions The MRI data set employed for the experimentations presented in this section consisted of three images: (i) T1-MRI (T1-MRI-230, slice 93), (ii) T2-MRI (slice 7), (iii) functional MRI (fMRI-TS1, slice 5). The validation paradigm was the same as that employed in all of the chapters of the book and consisted of: (i) shifting the image of a given misplacement, (ii) re-sampling and (iii) shifting back the image to the original image space location. At this point the root-mean-square-error (RMSE) was measured on the basis of the original image. This process was done twice: the first time through the classic function, and the second time through the SRE-based function. Consistently with the description given throughout the book this means that each of the three images was processed with classic and SRE-based interpolation functions of the same misplacement along the x direction. In these sets of experiments the misplacement was 0.0032 (fraction of the pixel size). Each processing was repeated for four values of the SCALE parameter: 0.1, 0.01, 0.001 and 0.0001 respectively.

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Table III. Quantitative evaluation of the influence of the SCALE parameter, Lagrange. The images were processed with a misplacement of 0.0032 along the x direction.

Table IV. Quantitative evaluation of the influence of the SCALE parameter, Sinc. The images were processed with a misplacement of 0.0032 along the x direction. The value of the FWHM of the Gaussian is 0.487.

Investigation on the Influence of the SCALE Parameter This section presents results pertaining to quantitative and qualitative evaluation of the influence of the SCALE parameter. The SCALE parameter is employed to scale the convolution of the pixel intensities determined through the polynomials forms: quadratic and cubic B-Splines, Lagrange, and to scale the numerical values of the sums of cosine and sine functions of the Sinc interpolation function. Table I (quadratic B-Spline), table II (cubic B-Spline), table III (Lagrange) and table IV (Sinc) show from left to right columns: (i) the MRI data, (ii) SCALE, (iii) RMSE NO SRE, and (iv) RMSE SRE. In each table RMSE NO SRE is the root-mean-square-error (RMSE) obtained employing the classic interpolation function and RMSE SRE is the RMSE obtained employing the SRE-based interpolation function. The value of the “a” constant was set to -2.36 for the quadratic B-Splines (both classic and SREbased). The value of the FWHM of the Gaussian was set to 0.487 for the Sinc interpolation function. The influence of the SCALE parameter on the interpolation error while processing the images

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Figure 21. Qualitative evaluation of the influence of the SCALE parameter, T1-MRI-230, slice 93, maps of novel re-sampling locations. The images were processed with a misplacement of 0.0032 along the x direction. For the Sinc function the value of the FWHM of the Gaussian is 0.487.

through classic and SRE-based interpolation functions is clearly distinguished in tables I, II, III and IV. The influence is that of direct proportionality of the SCALE parameter to the magnitude of the interpolation error. Values in bold in the tables point to the few cases for which the classic interpolation function achieved a smaller interpolation error with respect to that achieved by the corresponding SRE-based function.

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Figure 22. Qualitative evaluation of the influence of the SCALE parameter, T2-MRI, slice 7, maps of novel re-sampling locations. The images were processed with a misplacement of 0.0032 along the x direction. For the Sinc function the value of the FWHM of the Gaussian is 0.487.

Figures 21, 22 and 23 show a qualitative evaluation of the influence of the SCALE parameter on the map of novel re-sampling locations. The organizational layout of figure 21 (T1-MRI-230, slice 93), figure 22 (T2-MRI (slice 7)) and figure 23 (fMRI-TS1, slice 5) is the same in each figure. Each image consists of a map of novel re-sampling locations. Each row of the four shown in each of the three figures is relevant to one of the four SRE-based interpolation functions studied: (i) quadratic B-Spline, (ii) cubic B-Spline, (iii) Lagrange and (iv) Sinc. In each row, from left to right, the maps of novel re-sampling locations were obtained with the SCALE parameter value of 0.1, 0.01, 0.001 and 0.0001 respectively. The following observations can be made. Figure 21 show that the SCALE parameter does not affect the map of the novel re-sampling locations in the case of the SRE-based polynomial interpolation functions: quadratic B-Spline, cubic B-Spline, Lagrange respectively. This means that changing the SCALE parameter does not change the magnitude of the novel re-sampling location. This happens because the SCALE parameter scales the convolution of the MRI signal determined through the polynomial interpolation functions, and the convolution of the MRI signal is not involved in the calculation of the novel

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Figure 23. Qualitative evaluation of the influence of the SCALE parameter, fMRI-TS1, slice 5, maps of novel re-sampling locations. The images were processed with a misplacement of 0.0032 along the x direction. For the Sinc function the value of the FWHM of the Gaussian is 0.487.

re-sampling locations. This statement is relevant to the forms of polynomial interpolation functions that were studied in this book. Conversely, for the case of the SRE-based Sinc function (see maps in the fourth row of figure 21), changing the SCALE parameter does change the magnitude of the novel re-sampling locations. This happens because for the SRE-based Sinc interpolation function the SCALE parameter influences the numerical values of the sums of cosine and sine functions and these are employed to calculate the numerical value of the novel re-sampling locations. Similar behavior is observable in figure 22 for T2-MRI (slice 7), with the difference that both SREbased quadratic B-Spline and SRE-based Lagrange presents for this particular set of experiments a map of novel re-sampling locations that is flat. The map is visibly not matching the brain images’ features as shown by the images in figure 21 (T1-MRI-230, slice 93) and the images in the second and fourth row of figure 22 (SRE-based cubic B-Spline and SRE-based Sinc). Figure 23 is relevant to the maps of novel re-sampling locations for the functional MRI data (fMRITS1, slice 5). The maps are notably flat in all cases except for the case of the SRE-based Sinc function 497

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(SCALE = 0.1). It is natural to infer that the resolution plays an influence on the variability of the magnitude of the novel re-sampling locations. Low resolutions images like functional MRI have had flat maps for the novel re-sampling locations in the majority of the cases treated by this book. The resolution however might not be the only determinant. In fact, as extensively shown in this book, the curvature of the interpolation function has a direct influence on the calculation of the novel re-sampling locations. And the curvature of the interpolation function is strictly related to the variability of the pixel intensity in the MRI images. Since the functional MRI data generally presents less variability in signal intensity than other MRI modalities like T1-MRI or T2-MRI, the curvature of the interpolation function is reduced and this might be the determinant of the lack of variability in the values of the magnitude of the novel re-sampling locations.

Comparison Across Classic and SRE-Based Interpolation Functions This section presents results collected specifically to determine a performance comparison across SREbased interpolation functions. The interpolation functions object of comparison were: (i) Quadratic BSpline (Table V), (ii) Cubic B-Spline (Table VI), (iii) Lagrange (Table VII) and (iv) Sinc (Table VIII). Tables V, VI, VII and VIII condense the results presented in Tables I, II, III and IV, presenting the sum of the RMSE across the four values (0.1, 0.01, 0.001 and 0.0001) of the SCALE parameter. The following observations can be made. Generally, the approximation capabilities of classic and SRE-based functions are similar. They were measured in terms of RMSE. In fact as it is visible in Tables V, VI, VII and VIII, the two classes of interpolation functions determine an error of the same order of magnitude. As shown in the tables, except for the Sinc function, the SRE-based interpolation functions determine a lower RMSE than the RMSE determined by the classic interpolation functions. Thus, the capability to improve the interpolation error (the RMSE was chosen as its measure) is generally greater for the SRE-based interpolation functions. This capability was seen extensively both qualitatively and quantitatively in the chapters of the book that present the results obtained for the two classes of interpolation functions: classic and SRE-based. This section provides with further evidence of such capability through the presentation of results obtained with varying values of the SCALE parameter. Given the limited amount of data herein presented, to infer a general behavior is prone to mistake. Nevertheless, what can be observed is: (i) the SRE-based Lagrange interpolation function provided best approximation properties (across interpolation paradigms) for the functional MRI (fMRI-TS1 slice 5) and (ii) the SRE-based Sinc interpolation function provided generally better approximation capability (across interpolation paradigms) for both T1-MRI (T1-MRI-230, slice 93) and T2-MRI (slice 7). It is worth noting the following phenomenon. Overall, in Tables V, VI, VII and VIII, the smallest magnitude of RMSE is seen for the functional MRI data. In the tables, this is true for all the polynomial interpolation functions: quadratic and cubic B-Splines, Lagrange, but not for the Sinc. The fact that the functional MRI data has smallest magnitude of RMSE may be obviously a contradiction because the fMRI-TS1 data set has the coarser resolution. In fact, the T1-MRI-230 data set has in plane matrix resolution of 176 x 208 with 1.00 x 1.00 mm of pixel size (i.e. Field of View – FOV of 176 x 208 mm2), the T2-MRI data set, after down-sampling has in plane matrix resolution of 177 x 182 with 1.00 x 1.00 mm of pixel size (i.e. Field of View – FOV of 176 x 182 mm2), the fMRI-TS1 data set has in plane matrix resolution of 128 x 128 with 1.72 x 1.72 mm of pixel size (i.e. Field of View – FOV of 220.16 x 220.16 mm2).

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Table V. Comparison across classic and SRE-based interpolation functions, quadratic B-Spline. Sums of the RMSE across the four values (0.1, 0.01, 0.001, 0.0001) of the SCALE parameter. The value of the “a” constant is -2.36. The images were processed with a misplacement of 0.0032 along the x direction.

Table VI. Comparison across classic and SRE-based interpolation functions, cubic B-Spline. Sums of the RMSE across the four values (0.1, 0.01, 0.001, 0.0001) of the SCALE parameter. The images were processed with a misplacement of 0.0032 along the x direction.

Table VII. Comparison across classic and SRE-based interpolation functions, Lagrange. Sums of the RMSE across the four values (0.1, 0.01, 0.001, 0.0001) of the SCALE parameter. The images were processed with a misplacement of 0.0032 along the x direction.

Table VIII. Comparison across classic and SRE-based interpolation functions, Sinc. Sums of the RMSE across the four values (0.1, 0.01, 0.001, 0.0001) of the SCALE parameter. The images were processed with a misplacement of 0.0032 along the x direction. The value of the FWHM of the Gaussian is 0.487.

Also, the magnitude of the pixel intensity was scaled to the same range for all the three imaging modalities: T1-MRI, T2-MRI, functional MRI and this fact excludes the influence of the magnitude of the pixel intensity on the value of the RMSE. The magnitudes of the SCALE parameter (0.1, 0.01, 0.001 and 0.0001) were the same for all the interpolation functions and this fact excludes influences of the SCALE parameter on the values of the sum of RMSE seen in Tables V, VI, VII and VIII. Nevertheless, the neighborhood of the Sinc function was set to N = 3 and the neighborhoods of the quadratic B-Spline, cubic B-Spline and Lagrange polynomials were set to: 1 x 6, 1 x 6 and 1 x 4 respec-

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tively. It is therefore natural to think that the size of the neighborhood employed to convolve the signal through the interpolation function has an influence on the magnitude of the interpolation error. This thought is supported by the fact that the smallest neighborhood of the four interpolation functions was that of the Sinc (N = 3, i.e. 1 x 3) and it was accompanied by the smallest RMSE across interpolation functions for T1-MRI-230 and T2-MRI (see Tables V, VI, VII and VIII). Due also to recall that the Sinc interpolation function usually admits best approximation properties and this is widely accepted in literature. This does not explain however why the smallest RMSE is seen for the polynomial interpolation functions for the functional MRI data, which has the coarser resolution of the three data sets. It might be possible, as supported by the data herein presented, that there exists a combined influence on the interpolation error (RMSE in this investigation) exerted through the resolution and the size of the neighborhood of the polynomial interpolation function. This combined influence seems to be such that the effect of the resolution on the interpolation error is counter-effected by the size of the neighborhood of the polynomial interpolation function. Intuitively, the combined influence is such that the interpolation error is inversely proportional to the resolution and directly proportional to the size of the neighborhood. Data presented here therefore raises the following two research questions: (i) Does the size of the neighborhood of the polynomial interpolation function has an effect on the interpolation error? And (ii) what is the combined effect of resolution and size of the neighborhood of the polynomial interpolation function on the interpolation error? To finalize the comparison across SRE-based interpolation functions, figures 24 through 35 present the spectral power evolutions of experiments conducted for the writing of this section of the book. The experiments were conducted processing the brain images while employing the four different SRE-based interpolation functions: quadratic B-Spline, cubic B-Spline, Lagrange and Sinc. The misplacement as earlier mentioned was 0.0032 (fraction of the pixel size) along the x direction and the SCALE parameter was set to 0.1 for all experiments. The intervals shown in figures 24 through 35 are those between 979 and 1023. Particularly, figures 24, 28 and 32 were obtained employing the quadratic B-Splines (classic and SRE-based), figures 25, 29 and 33 were obtained employing the cubic B-Splines, figures 26, 30 and 34 were obtained employing the two classes (classic and SRE-based) of Lagrange function, and finally figures 27, 31 and 35 are relevant to the two classes of Sinc functions. Figures 24, 25, 26 and 27 are relevant to the T1-MRI-230 (slice 93). Figures 28, 29, 30 and 31 are relevant to T2-MRI (slice 7). Figures 32, 33, 34 and 35 are relevant to functional MRI (fMRI-TS1, slice 5). The average of abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) was calculated for each of the spectral power evolutions shown in figures 24 through 35 and its numerical value is reported in the figures’ legend. The average was a negative value (except for figure 35) suggesting the superiority of the SREbased interpolation functions. Also, for each figure is given a quantitative description of the values of maximal and minimal difference abs(O_E - SRE_E) minus abs(O_E - NOSRE_E). In figures 24 through 27 (T1-MRI-230) the maximum and minimum values seen in the spectral power evolutions are: 2.834, -19.479 (figure 24), 2.976, -23.28 (figure 25), 7.716, -20.757 (figure 26), 4 E-05, -0.00011 (figure 27). In figures 28 through 31 (T2-MRI) the maximum and minimum values seen in the spectral power evolutions are: 119.316, -343.065 (figure 28), 94.843, -120.792 (figure 29), 58.009, -110.87 (figure 30), 14.523, -14.402 (figure 31).

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Figure 24. Spectral power evolution, quadratic B-Spline, T1-MRI-230, slice 93. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1.029.

Figure 25. Spectral power evolution, cubic B-Spline, T1-MRI-230, slice 93. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1.445.

In figures 32 through 35 (fMRI-TS1) the maximum and minimum values seen in the spectral power evolutions are: 56.078, -101.397 (figure 32), 93.210, -176.25 (figure 33), 43.209, -73.063 (figure 34), 0.00074, -0.0007 (figure 35). Preservation of the spectral components of the original brain image after processing with either classic or SRE-based interpolation paradigms is manifest in all of the figures. It is fair as well as quite beneficial to conclude that the figures strongly suggest that combining classic and SRE-based interpolation on a pixel-by-pixel basis would determine a strong preservation of spectral characteristics of the MRI signal. This means that combining the pixel-by-pixel adaptive re-sampling typical of SRE-based interpolation with the uniform re-sampling of the classic interpolation function might be more beneficial than using only one of the two re-sampling techniques: either the uniform, associated with classic

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Figure 26. Spectral power evolution, Lagrange, T1-MRI-230, slice 93. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -1.952.

Figure 27. Spectral power evolution, Sinc, T1-MRI-230, slice 93. The images were processed with a misplacement of 0.0032 along the x direction. The value of the FWHM of the Gaussian is 0.487. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -6.666 E-07.

Figure 28. Spectral power evolution, quadratic B-Spline, T2-MRI, slice 7. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -33.89.

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interpolation, or the SRE-based adaptive approach. This is feasible because, as shown in earlier sections of the book, the methodology developed in this work allows identifying in the image space the exact location (pixels) that corresponds to the frequency components that are relevant to each of the intervals of the power spectral evolutions.

Summary This paragraph gives a brief summary of the research approach, results and implications of the theory presented in this book. The theory involves the determination of a mathematical formulation that sets the foundation for a unifying theory for the improvement of the interpolation error so as to embrace,

Figure 29. Spectral power evolution, cubic B-Spline, T2-MRI, slice 7. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -10.96.

Figure 30. Spectral power evolution, Lagrange, T2-MRI, slice 7. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -12.454.

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Figure 31. Spectral power evolution, Sinc, T2-MRI, slice 7. The images were processed with a misplacement of 0.0032 along the x direction. The value of the FWHM of the Gaussian is 0.487. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -0.772.

Figure 32. Spectral power evolution, quadratic B-Spline, fMRI-TS1, slice 5. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -7.751.

Figure 33. Spectral power evolution, cubic B-Spline, fMRI-TS1, slice 5. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is -12.371.

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Figure 34. Spectral power evolution, Lagrange, fMRI-TS1, slice 5. The images were processed with a misplacement of 0.0032 along the x direction. The average abs(O_E - SRE_E) minus abs(O_E NOSRE_E) is -5.867.

Figure 35. Spectral power evolution, Sinc, fMRI-TS1, slice 5. The images were processed with a misplacement of 0.0032 along the x direction. The value of the FWHM of the Gaussian is 0.487. The average abs(O_E - SRE_E) minus abs(O_E - NOSRE_E) is 1.533 E-05.

under the same methodology, different interpolators regardless of their degree and dimensionality. The book proposes what is missing in literature as starting point. This is a methodology that consequential to its developmental approach formulates interpolation error improvement as dependent from the joint information content of node intensity and curvature of the interpolation function, and from which it descends that re-sampling is an issue of local relevance. Re-sampling is therefore determined at locations that vary pixel-by-pixel (node-by-node) depending on the pixel intensity distribution across the neighborhood and the local curvature of the interpolator. To achieve this goal, two main conceptions were formulated and then developed in mathematics. These two primary conceptions are the named: (i) the Intensity-Curvature Functional, and (ii) the Subpixel Efficacy Region (SRE). They are capable of: (i) formulating interpolation error improvement as dependent from node intensity and curvature, (ii) capturing the effect of the interpolator on the data at the grid node (that means, as if not interpolated), and anywhere at the generic intra-pixel location (that

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means, as if interpolated), (iii) to act as indirect measure of the interpolation error, and given its flexibility (iv) to be adaptive to interpolation functions of diverse degree and dimensionality. The Intensity-Curvature Functional (ΔE) is a measure of the effect of the interpolation function on the signal. ΔE comprises of signal intensity and curvature of the model function (the second order derivative). The Intensity-Curvature Functional is derived from the ratio of the two Intensity-Curvature terms, which distinguish re-sampling (EIN) versus the absence of re-sampling (E0). The study of the Intensity-Curvature Functional that is subsequent to the solution of the polynomial of its first order partial derivatives demonstrates that it is possible to determine the Sub-pixel Efficacy Region (SRE). The SRE is a set of sub-pixel points located at the extremes of the Intensity-Curvature Functional. The SRE allows through projection the determination of a novel location within the pixel where the same interpolation function can be calculated with improved approximation characteristics. This yields to lower interpolation error. Because of the pixel-by-pixel variability of such novel re-sampling locations, the band-pass characteristics of the interpolator are altered. Beneficially, in some cases, as there is an attenuation of the smoothing characteristics that are inherent to interpolation. Finally, frequency components that otherwise would be neglected are incorporated in the re-sampled signal. The novelty is therefore in determining new theory and methodology for the improvement of the interpolation error. The formulation that emanates from the unifying theory is dependent on the joint information content given by node intensity and curvature of the interpolation function. Because of the flexibility of the mathematics, the theory leads to a unified methodology in an effective framework for the following interpolation paradigms: bivariate and trivariate linear interpolation, B-Spline in its quadratic and cubic forms, Lagrange and Sinc (with odd number of nodes). The theorem proposed shows that while considering the curvature of the interpolation function, the interpolation error improvement can be demonstrated as bounded by two constants. This was seen earlier (Ciulla & Deek, 2006) by presenting the definition of interpolation error improvement. Three paradigms were used to validate the theory. One is a motion correction paradigm that measures quantitatively the improvement attained through the SRE-based interpolation on the basis of a ratio between the root-mean-square-error (RMSE) of the classic paradigm and the RMSE obtained with the SRE-based interpolation function. A validation paradigm parallel to this study was developed. This focused on the spectral power evolutions obtained from the spectral energy of the original image and the images resulting after processing with classic and SRE-based interpolation. This validation allowed the comparisons of the evolution of the spectral energy to show which frequency components are affected by the interpolation functions. It is possible to see how the resulting spectral power evolution reflects the preservation of the spectral components of the original image despite processing with interpolation (either classic or SRE-based). Finally, to prove the benefits of the theory in the estimation of signals of unknown nature, for instance when it is necessary to determine the signal a time-space locations that cannot be recorded because of the limitations imposed by the Nyquist theorem, the SRE-based interpolation paradigms were tested to ascertain accuracy in estimating signals at unknown spatial locations. Results indicate that for the large majority of data employed for experimentation, and consistently for the large majority of the interpolation functions so far tested: (i) The RMSE analysis reveals an improvement of the interpolation error that is fairly consistent across the range of misplacements.

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(ii) The frequency spectrum of the SRE-based resulting signal incorporates frequencies that match with the spectrum of the original image more closely than the spectrum resulting from classic interpolation. This statement is not absolutely true however. The frequency spectrum of the SREbased resulting signal was presented to elucidate that there exist the possibility that a particular pixel of an image has the feature of a reduced band-pass filtering inherent to interpolation. (iii) Evidence shows that the SRE-based interpolation functions derived though the unifying theory are more accurate than the classic functions in estimating signals at unknown spatial locations. This last result suggests that for practical applications, the SRE-based interpolation might be preferred to the classic when is necessary to increase the resolution of a signal that otherwise would be restricted by the constraints imposed by the sampling instrument. Most of the current experimentation has been conducted within the context of Magnetic Resonance Imaging (MRI), however reported elsewhere (Ciulla & Deek, 2006) also within the context of other types of signals not of biological nature. With the aim of finalizing the details of the theory, both the formulation of the interpolation error improvement boundaries and the interpolation error boundary characterization form have been presented (Ciulla & Deek 2006). It was seen that both interpolation error and its improvement as determined by the SRE-based functions are limited by two constants that are dependent on the value of the signal to be estimated, and the value of the interpolator at the SRE locations (these determined by the study of the Intensity-Curvature Functional). It is true that because of the mathematical formulation of the theory, the claim that can be made is that the improvement of the interpolation function is formulated as dependent on node intensity and curvature of the interpolation function, and this is an outgrowth of the conceptualization of the Intensity-Curvature Functional and the Sub-pixel Efficacy Region. Whereas it is confirmed that the interpolation error itself clearly diminishes to zero as the resolution tends to zero, as it is largely demonstrated and accepted in literature. This fact is consistent with the concept of curvature which makes the task of estimation of the interpolation function more uncertain with increasing resolution. This is because the local curvature of the interpolator incorporates higher degree of variability for increasing resolution. Interpolation schemes were originally based on finite differences, and later developed their approximation properties based on the pixel-signal intensity and diversified also into object-based interpolation paradigms based on pre-processing that extracts significant information from the object in the image. In this book, the novel class of interpolators so called SRE-based interpolation functions distinguish themselves by virtue of their development which is based on the intensity-curvature content. The intensity is extracted from the image and the curvature is that of the interpolator. Through the application of the concept of curvature, the novel theory bridges the gap between classic interpolation and SREbased interpolation. This reveals the new approach of using the curvature to build a unifying theory capable to improve the approximation characteristics of the interpolation functions. Improvement of the interpolation error is achieved through the novel re-sampling locations where the function is calculated and such intra-node locations change pixel-by-pixel adapting the process of re-sampling to the local curvature of the interpolator.

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Ye, Z. (2003). Linear phase Lagrange interpolation filter using odd number of basepoints. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, China, 6, 237- 240. Yiu, K. F. C., Wang, S., Teo, K. L., & Tsoi, A. C. (2001). Nonlinear system modeling via knot-optimizing B-spline networks. IEEE Transactions on Neural Networks, 12(5), 1013-1022.

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Appendix Section

APPENDIX I. PROOFS TO THE MATHEMATICAL ASSERTIONS OF CHAPTER III While reading the following proofs the reader is reminded that in the one dimensional case, that is: for an integral with respect to the sole variable x it is true the following assertion. Let f(x) = g(x) = x, the following integral in the variable x is solved as: ∫ x · x = ∫ f(x) g(x) = F(x) g(x) - ∫ F(x) (∂ (g(x)) / ∂x) = (x2 / 2) x - ∫ (x2 / 2) · 1 = (x3 / 2) - ∫ (x2 / 2) = (x3 / 2) - (x3 / 6) = x3 [ (1 / 2) - (1 / 6) ] = x3 [ (3 - 1) / 6) ] = x3 [ (2 / 6) ] = x3 (1 / 3) where F(x) is the primitive of f(x), and (∂ (g(x)) / ∂x) is the first order derivative of g(x) with respect to the variable x. Proof: (∂ x3 (1 / 3) ) = 3 x2 (1 / 3) = x2 = x · x The Intensity-Curvature Term EIN (Ψxy) Let the demonstration start from the term EIN (Ψxy). From equation (2) of Chapter III it can be written that:

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Appendix Section



x/2 y/2 z/2

EIN (Ψxy) = ∫ ∫ ∫ h(x, y, z) (∂2 (h(x, y, z)) /∂x∂y) dx dy dz -x/2 -y/2 -z/2



z/2

y/2

x/2

z/2

=

y/2

= ∫ dz ∫ dy ∫ h(x, y, z) (∂ (h(x, y, z)) /∂x∂y) dx = ∫ dz ∫ [ Hx (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) ] x/2 -z/2 -y/2 -z/2 -y/2 -x/2 2

[ Hx (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) ] -x/2 dy

(1)

Above is possible because (∂2 (h(x, y, z)) /∂x∂y) = - θxy ωf + z ωf is a variable of the sole z, and thus being a constant when integrating with respect to the variable x or the variable y. Thus equation (1) can be written as: z/2

∫ dz [ [ Hxy (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) ] x/2 - [ Hxy (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) ] -x/2 ] y/2

-z/2

z/2

∫ dz [ [ Hxy (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) ] x/2 - [ Hxy (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) ] -x/2 ] -y/2 -z/2

(2)

Keeping in mind that: (∂2 (h(x, y, z)) /∂x∂y) = - θxy ωf + z ωf it can be stated that (∂ (∂2 (h(x, y, z)) /∂x∂y) / ∂z) = ωf and also equation (2) can be written as: { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] y/2 - [Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] y/2 ] z/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] y/2 - [Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] y/2 ] -z/2 z/2

∫ dz [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] y/2

-z/2

- [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] y/2 } {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] -y/2 - [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] -y/2 ]z/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] -y/2 - [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] -y/2 ] -z/2 z/2

∫ dz [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2] -y/2

-z/2

- [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2] -y/2 }

(3)

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Appendix Section

Since (∂ (∂2 (h(x, y, z)) /∂x∂y) / ∂z) = (∂3 (h(x, y, z)) /∂x∂y∂z) = ωf is a constant and using the results of Lemma III (Chapter III), equation (3) is solved as: A - B - C - D

(4)

where: A = { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] y/2 ] z/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] y/2 ] -z/2 } B = {[ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] y/2 [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] y/2 ] z/2 [ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] y/2 [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] y/2 ] -z/2 } C = {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] -y/2 ] z/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] -y/2 ] -z/2 } D = {[ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -y/2 [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -y/2 ] z/2 [ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -y/2 [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -y/2 ] -z/2 } equation (4) can be solved keeping in mind that: (∂2 (h(x, y, z)) /∂x∂y) = (∂2 (h(x, y, z)) /∂x∂y) | y/2 = (∂2 (h(x, y, z)) /∂x∂y) | -y/2 = - θxy ωf + z ωf (∂2 (h(x, y, z)) /∂x∂y) = (∂2 (h(x, y, z)) /∂x∂y) | x/2 = (∂2 (h(x, y, z)) /∂x∂y) | -x/2 = - θxy ωf + z ωf

514

Appendix Section

(∂2 (h(x, y, z)) /∂x∂y) = (∂2 (h(x, y, z)) /∂x∂y) | z/2 = - θxy ωf + z/2 ωf (∂2 (h(x, y, z)) /∂x∂y) = (∂2 (h(x, y, z)) /∂x∂y) | -z/2 = - θxy ωf - z/2 ωf (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) | y/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) | -y/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 = ωf Therefore: A = { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] y/2 ] z/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] y/2 ] -z/2 } = [ (∂ (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | x/2 y/2 - (∂2 (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | -x/2 y/2 ] z/2 2

[ (∂2 (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | x/2 y/2 - (∂2 (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | -x/2 y/2 ] -z/2 = (∂2 (h(x, y, z)) /∂x∂y) * { [ Hxyz (x, y, z) | x/2 y/2 - Hxyz (x, y, z) | -x/2 y/2 ] z/2

[Hxyz (x, y, z) | x/2 y/2 - Hxyz (x, y, z) | -x/2 y/2 ] -z/2 } =

(∂2 (h(x, y, z)) /∂x∂y) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 }

(5)

B = {[ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] y/2 [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] y/2 ] z/2 [ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] y/2 [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] y/2 ] -z/2 } =

515

Appendix Section

[(∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) | x/2 y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) | -x/2 y/2 ] z/2 [(∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) | x/2 y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) | -x/2 y/2 ] -z/2 =

(∂3 (h(x, y, z)) /∂x∂y∂z) * { [Þz (Hxyz (x, y, z)) | x/2 y/2 - Þz (Hxyz (x, y, z)) | -x/2 y/2 ] z/2

[Þz (Hxyz (x, y, z)) | x/2 y/2 - Þz (Hxyz (x, y, z)) | -x/2 y/2 ] -z/2 } =

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þz (Hxyz (x, y, z)) | x/2 y/2 z/2 - Þz (Hxyz (x, y, z)) | -x/2 y/2 z/2

Þz (Hxyz (x, y, z)) | x/2 y/2 -z/2 + Þz (Hxyz (x, y, z)) | -x/2 y/2 -z/2 }

(6)

C = {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] -y/2 ] z/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | x/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂x∂y) | -x/2 ] -y/2 ] -z/2 } =

[ (∂2 (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | x/2 -y/2 - (∂2 (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | -x/2 -y/2 ] z/2 [ (∂2 (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | x/2 -y/2 - (∂2 (h(x, y, z)) /∂x∂y) * Hxyz (x, y, z) | -x/2 -y/2 ] -z/2 = (∂2 (h(x, y, z)) /∂x∂y) * { [ Hxyz (x, y, z) | x/2 -y/2 - Hxyz (x, y, z) | -x/2 -y/2 ] z/2

[ Hxyz (x, y, z) | x/2 -y/2 - Hxyz (x, y, z) | -x/2 -y/2 ] -z/2 } =

(∂2 (h(x, y, z)) /∂x∂y) * { Hxyz (x, y, z) | x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 z/2

516

Hxyz (x, y, z) | x/2 -y/2 -z/2 + Hxyz (x, y, z) | -x/2 -y/2 -z/2 }

(7)

Appendix Section

D = {[ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -y/2 [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -y/2 ] z/2 [ [Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -y/2

[Þz (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -y/2 ] -z/2 } =

[(∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) |x/2 -y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) |-x/2 -y/2 ] z/2 [(∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) |x/2 -y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þz (Hxyz (x, y, z)) |-x/2 -y/2 ] -z/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) * { [ Þz (Hxyz (x, y, z)) | x/2 -y/2 - Þz (Hxyz (x, y, z)) | -x/2 -y/2 ] z/2

[ Þz (Hxyz (x, y, z)) | x/2 -y/2 - Þz (Hxyz (x, y, z)) | -x/2 -y/2 ] -z/2 } =

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þz (Hxyz (x, y, z)) | x/2 -y/2 z/2 - Þz (Hxyz (x, y, z)) | -x/2 -y/2 z/2 Þz (Hxyz (x, y, z)) | x/2 -y/2 -z/2 + Þz (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } (8) Therefore equation (4) can be written as: A-B-C-D= (∂2 (h(x, y, z)) /∂x∂y) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 } –

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þz (Hxyz (x, y, z)) | x/2 y/2 z/2 - Þz (Hxyz (x, y, z)) | -x/2 y/2 z/2

Þz (Hxyz (x, y, z)) | x/2 y/2 -z/2 + Þz (Hxyz (x, y, z)) | -x/2 y/2 -z/2 } -

(∂2 (h(x, y, z)) /∂x∂y) * { Hxyz (x, y, z) | x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 z/2

Hxyz (x, y, z) | x/2 -y/2 -z/2 + Hxyz (x, y, z) | -x/2 -y/2 -z/2 } – 517

Appendix Section

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þz (Hxyz (x, y, z)) | x/2 -y/2 z/2 - Þz (Hxyz (x, y, z)) | -x/2 -y/2 z/2



Þz (Hxyz (x, y, z)) | x/2 -y/2 -z/2 + Þz (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } =

(∂2 (h(x, y, z)) /∂x∂y) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 – Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 +



Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { - Þz (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þz (Hxyz (x, y, z)) | -x/2 y/2 z/2 +

Þz (Hxyz (x, y, z)) | x/2 y/2 -z/2 - Þz (Hxyz (x, y, z)) | -x/2 y/2 -z/2 Þz (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þz (Hxyz (x, y, z)) | -x/2 -y/2 z/2 +

Þz (Hxyz (x, y, z)) | x/2 -y/2 -z/2 - Þz (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } (9) Equation (9) consists in an intermediate step in quantifying the effect of the model interpolation function in terms of the change of energy level as measured by the intensity-curvature term EIN (Ψxy).

The Intensity-Curvature Term EIN (Ψzx) Now let us proceed to derive an equation similar to (9) to measure the change of energy level determined by the intensity-curvature term EIN (Ψzx). From equation (3) of Chapter III it can be written that:

x/2 y/2 z/2

EIN (Ψzx) = ∫ ∫ ∫ h(x, y, z) (∂2 (h(x, y, z)) /∂z∂x) dx dy dz = -x/2 -y/2 -z/2

z/2

y/2

∫ dz ∫ [ Hx (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) ] x/2 - [ Hx (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) ] -x/2 dy (10) -z/2 -y/2 Above is possible because (∂2 (h(x, y, z)) /∂z∂x) = - θxz ωf + y ωf is a variable of the sole y, and thus being a constant when integrating with respect to the variable x or the variable z. Thus equation (10) can be written as: y/2

∫ dy [ [ Hxz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) ] x/2 - [ Hxz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) ] -x/2 ] z/2 -y/2

518

Appendix Section

y/2

∫ dy [ [ Hxz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) ] x/2 - [ Hxz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) ] -x/2 ] -z/2 -y/2

(11)

Keeping in mind that: (∂2 (h(x, y, z)) /∂z∂x) = - θxz ωf + y ωf it can be stated that (∂ (∂2 (h(x, y, z)) /∂z∂x) / ∂y) = ωf and also equation (11) can be written as: { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] z/2 - [Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] z/2 ] y/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] z/2 - [Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] z/2 ] -y/2 y/2

∫ dy [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] z/2

-y/2

- [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ]z/2 } {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] -z/2 - [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] -z/2 ]y/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] -z/2 - [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] -z/2 ] -y/2 y/2

∫ dy [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2] -z/2 -y/2

- [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2] -z/2}

(12)

Since (∂ (∂2 (h(x, y, z)) /∂z∂x) / ∂y) = (∂3 (h(x, y, z)) /∂x∂y∂z) = ωf is a constant and using the results of Lemma III (Chapter III), equation (12) is solved as: A - B - C - D where:

(13)



A = { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] z/2 ] y/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] z/2 ] -y/2 }

519

Appendix Section

B = {[ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] z/2 ] y/2 [ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] z/2 ] -y/2 } C = {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] -z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] -z/2 ] y/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] -z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] -z/2 ] -y/2 } D = {[ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -z/2 ] y/2 [ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -z/2 ] -y/2 } equation (13) can be solved keeping in mind that: (∂2 (h(x, y, z)) /∂z∂x) = (∂2 (h(x, y, z)) /∂z∂x) | z/2 = (∂2 (h(x, y, z)) /∂z∂x) | -z/2 = - θxz ωf + y ωf (∂2 (h(x, y, z)) /∂z∂x) = (∂2 (h(x, y, z)) /∂z∂x) | x/2 = (∂2 (h(x, y, z)) /∂z∂x) | -x/2 = - θxz ωf + y ωf (∂2 (h(x, y, z)) /∂z∂x) = (∂2 (h(x, y, z)) /∂z∂x) | y/2 = - θxz ωf + y/2 ωf (∂2 (h(x, y, z)) /∂z∂x) = (∂2 (h(x, y, z)) /∂z∂x) | -y/2 = - θxz ωf - y/2 ωf Therefore: A = { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] z/2 ] y/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] z/2 ] -y/2 } =

520

Appendix Section

[ (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | x/2 z/2 - (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | -x/2 z/2 ] y/2 [ (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | x/2 z/2 - (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | -x/2 z/2 ] -y/2 = (∂2 (h(x, y, z)) /∂z∂x) * { [ Hxyz (x, y, z) | x/2 z/2 - Hxyz (x, y, z) | -x/2 z/2 ] y/2

[Hxyz (x, y, z) | x/2 z/2 - Hxyz (x, y, z) | -x/2 z/2 ] -y/2 } =

(∂2 (h(x, y, z)) /∂z∂x) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 }

(14)

B = {[ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] z/2 ] y/2 [ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] z/2 ] -y/2 } = [(∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) | x/2 z/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) | -x/2 z/2 ] y/2 [(∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) | x/2 z/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) | -x/2 z/2 ] -y/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) * { [Þy (Hxyz (x, y, z)) | x/2 z/2 - Þy (Hxyz (x, y, z)) | -x/2 z/2 ] y/2

[Þy (Hxyz (x, y, z)) | x/2 z/2 - Þy (Hxyz (x, y, z)) | -x/2 z/2 ] -y/2 } =

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þy (Hxyz (x, y, z)) | x/2 y/2 z/2 - Þy (Hxyz (x, y, z)) | -x/2 y/2 z/2 Þy (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þy (Hxyz (x, y, z)) | -x/2 -y/2 z/2 }

(15)

C = {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] -z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] -z/2 ] y/2 -

521

Appendix Section

[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | x/2 ] -z/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂z∂x) | -x/2 ] -z/2 ] -y/2 } = [ (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | x/2 -z/2 - (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | -x/2 -z/2 ] y/2 [ (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | x/2 -z/2 - (∂2 (h(x, y, z)) /∂z∂x) * Hxyz (x, y, z) | -x/2 -z/2 ] -y/2 = (∂2 (h(x, y, z)) /∂z∂x) * { [ Hxyz (x, y, z) | x/2 -z/2 - Hxyz (x, y, z) | -x/2 -z/2 ] y/2

[ Hxyz (x, y, z) | x/2 -z/2 - Hxyz (x, y, z) | -x/2 -z/2 ] -y/2 } =

(∂2 (h(x, y, z)) /∂z∂x) * { Hxyz (x, y, z) | x/2 y/2 -z/2 - Hxyz (x, y, z) | -x/2 y/2 -z/2

Hxyz (x, y, z) | x/2 -y/2 -z/2 + Hxyz (x, y, z) | -x/2 -y/2 -z/2 }

D = {[ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -z/2 ] y/2 [ [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | x/2 ] -z/2 [ Þy (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -x/2 ] -z/2 ] -y/2 } = [(∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) |x/2 -z/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) |-x/2 -z/2 ] y/2 [(∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) |x/2 -z/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þy (Hxyz (x, y, z)) |-x/2 -z/2 ] -y/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) * { [Þy (Hxyz (x, y, z)) | x/2 -z/2 - Þy (Hxyz (x, y, z)) | -x/2 -z/2 ] y/2

522

[Þy (Hxyz (x, y, z)) | x/2 -z/2 - Þy (Hxyz (x, y, z)) | -x/2 -z/2 ] -y/2 } =

(16)

Appendix Section

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þy (Hxyz (x, y, z)) | x/2 y/2 -z/2 - Þy (Hxyz (x, y, z)) | -x/2 y/2 -z/2 Þy (Hxyz (x, y, z)) | x/2 -y/2 -z/2 + Þy (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 }

(17)

Therefore equation (13) can be written as: A-B-C-D= (∂2 (h(x, y, z)) /∂z∂x) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 } -

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þy (Hxyz (x, y, z)) | x/2 y/2 z/2 - Þy (Hxyz (x, y, z)) | -x/2 y/2 z/2 Þy (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þy (Hxyz (x, y, z)) | -x/2 -y/2 z/2 } - (∂2 (h(x, y, z)) /∂z∂x) * { Hxyz (x, y, z) | x/2 y/2 -z/2 - Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 -z/2 + Hxyz (x, y, z) | -x/2 -y/2 -z/2 } (∂3 (h(x, y, z)) /∂x∂y∂z) * { Þy (Hxyz (x, y, z)) | x/2 y/2 -z/2 - Þy (Hxyz (x, y, z)) | -x/2 y/2 -z/2

Þy (Hxyz (x, y, z)) | x/2 -y/2 -z/2 + Þy (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } =

(∂2 (h(x, y, z)) /∂z∂x) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 +



Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { - Þy (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þy (Hxyz (x, y, z)) | -x/2 y/2 z/2 +

Þy (Hxyz (x, y, z)) | x/2 -y/2 z/2 - Þy (Hxyz (x, y, z)) | -x/2 -y/2 z/2 Þy (Hxyz (x, y, z)) | x/2 y/2 -z/2 + Þy (Hxyz (x, y, z)) | -x/2 y/2 -z/2 +

Þy (Hxyz (x, y, z)) | x/2 -y/2 -z/2 - Þy (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 }

(18)

523

Appendix Section

The Intensity-Curvature Term EIN (Ψyz) The third intensity-curvature term EIN (Ψyz) will allow as shown in the following to derive equations similar to (9) and (18) such to measure the change of energy level. From equation (4) of Chapter III it can be written that:

x/2 y/2 z/2

EIN (Ψyz) = ∫ ∫ ∫ h(x, y, z) (∂2 (h(x, y, z)) /∂y∂z) dx dy dz -x/2 -y/2 -z/2

z/2

y/2

x/2

x/2

y/2

= ∫ dz ∫ dy ∫ h(x, y, z) (∂ (h(x, y, z)) /∂y∂z) dx = ∫ dx ∫ [ Hz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) ] z/2 -x/2 -y/2 -z/2 -y/2 -x/2 2

[ Hz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) ] -z/2 dy (19) Above is possible because (∂2 (h(x, y, z)) /∂y∂z) = - θzy ωf + x ωf is a variable of the sole x, and thus being a constant when integrating with respect to the variable y or the variable z. Thus equation (19) can be written as: x/2

∫ dx [ [ Hzy (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) ] z/2 - [ Hzy (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) ] -z/2 ] y/2

-x/2

x/2

∫ dx [ [ Hzy (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) ] z/2 - [ Hzy (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) ] -z/2 ] -y/2 -x/2

(20)

Keeping in mind that: (∂2 (h(x, y, z)) /∂y∂z) = - θzy ωf + x ωf it can be stated that (∂ (∂2 (h(x, y, z)) /∂y∂z) / ∂x) = ωf and also equation (20) can be written as: { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] y/2 - [Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] y/2 ] x/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] y/2 - [Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] y/2 ] -x/2 x/2

∫ dx [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] y/2

-x/2

- [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ]y/2} {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] -y/2 - [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] -y/2 ]x/2 -

524

Appendix Section

[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] -y/2 - [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] -y/2 ] -x/2 x/2

∫ dx [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2] -y/2

-x/2

- [Hxyz (x, y, z) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2] -y/2}

as:

(21)

Since (∂ (∂2 (h(x, y, z)) /∂y∂z) / ∂x) = (∂3 (h(x, y, z)) /∂x∂y∂z) = ωf is a constant, equation (21) is solved

A - B - C - D

(22)

where: A = { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] y/2 ] x/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] y/2 ] -x/2 } B = {[ [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] y/2 [Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] y/2 ] x/2 [ [Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] y/2 [Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] y/2 ] -x/2 } C = {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] -y/2 ] x/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] -y/2 ] -x/2 } D = {[ [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] -y/2 -

525

Appendix Section

[ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] -y/2 ] x/2 [ [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] -y/2 [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] -y/2 ] -x/2 } Equation (22) can be solved keeping in mind that: (∂2 (h(x, y, z)) /∂y∂z) = (∂2 (h(x, y, z)) /∂y∂z) | y/2 = (∂2 (h(x, y, z)) /∂y∂z) | -y/2 = - θzy ωf + x ωf (∂2 (h(x, y, z)) /∂y∂z) = (∂2 (h(x, y, z)) /∂y∂z) | z/2 = (∂2 (h(x, y, z)) /∂y∂z) | -z/2 = - θzy ωf + x ωf (∂2 (h(x, y, z)) /∂y∂z) = (∂2 (h(x, y, z)) /∂y∂z) | x/2 = - θzy ωf + x/2 ωf (∂2 (h(x, y, z)) /∂y∂z) = (∂2 (h(x, y, z)) /∂y∂z) | -x/2 = - θzy ωf - x/2 ωf Therefore: A = { [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] y/2 ] x/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] y/2 ] -x/2 } = [ (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | z/2 y/2 - (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | -z/2 y/2 ] x/2 [ (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | z/2 y/2 - (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | -z/2 y/2 ] -x/2 = (∂2 (h(x, y, z)) /∂y∂z) * { [ Hxyz (x, y, z) | z/2 y/2 - Hxyz (x, y, z) | -z/2 y/2 ] x/2

[Hxyz (x, y, z) | z/2 y/2 - Hxyz (x, y, z) | -z/2 y/2 ] -x/2 } =

(∂2 (h(x, y, z)) /∂y∂z) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | x/2 y/2 -z/2

Hxyz (x, y, z) | -x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 }

B = {[ [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] y/2 [Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] y/2 ] x/2 [ [Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] y/2 -

526

(23)

Appendix Section

[Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] y/2 ] -x/2 } = [(∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) | z/2 y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) | -z/2 y/2 ] x/2 [(∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) | z/2 y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) | -z/2 y/2 ] -x/2 =

(∂3 (h(x, y, z)) /∂x∂y∂z) * { [Þx (Hxyz (x, y, z)) | z/2 y/2 - Þx (Hxyz (x, y, z)) | -z/2 y/2 ] x/2

[Þx (Hxyz (x, y, z)) | z/2 y/2 - Þx (Hxyz (x, y, z)) | -z/2 y/2 ] -x/2 } =

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þx (Hxyz (x, y, z)) | x/2 y/2 z/2 - Þx (Hxyz (x, y, z)) | x/2 y/2 -z/2 Þx (Hxyz (x, y, z)) | -x/2 y/2 z/2 + Þx (Hxyz (x, y, z)) | -x/2 y/2 -z/2 }

(24)

C = {[ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] -y/2 ] x/2 [ [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | z/2 ] -y/2 [ Hxyz (x, y, z) (∂2 (h(x, y, z)) /∂y∂z) | -z/2 ] -y/2 ] -x/2 } = [ (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | z/2 -y/2 - (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | -z/2 -y/2 ] x/2 [ (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | z/2 -y/2 - (∂2 (h(x, y, z)) /∂y∂z) * Hxyz (x, y, z) | -z/2 -y/2 ] -x/2 =

(∂2 (h(x, y, z)) /∂y∂z) * { [ Hxyz (x, y, z) | z/2 -y/2 - Hxyz (x, y, z) | -z/2 -y/2 ] x/2

[ Hxyz (x, y, z) | z/2 -y/2 - Hxyz (x, y, z) | -z/2 -y/2 ] -x/2 } =

527

Appendix Section

(∂2 (h(x, y, z)) /∂y∂z) * { Hxyz (x, y, z) | x/2 -y/2 z/2 - Hxyz (x, y, z) | x/2 -y/2 -z/2

Hxyz (x, y, z) | -x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 -z/2 }

(25)

D = {[ [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] -y/2 [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] -y/2 ] x/2 – [ [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | z/2 ] -y/2 [ Þx (Hxyz (x, y, z)) (∂3 (h(x, y, z)) /∂x∂y∂z) | -z/2 ] -y/2 ] -x/2 } =

[(∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) |z/2 -y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) |-z/2 -y/2 ] x/2 [(∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) |z/2 -y/2 (∂3(h(x, y, z)) /∂x∂y∂z) * Þx (Hxyz (x, y, z)) |-z/2 -y/2 ] -x/2 = (∂3 (h(x, y, z)) /∂x∂y∂z) * { [Þx (Hxyz (x, y, z)) | z/2 -y/2 - Þx (Hxyz (x, y, z)) | -z/2 -y/2 ] x/2

[Þx (Hxyz (x, y, z)) | z/2 -y/2 - Þx (Hxyz (x, y, z)) | -z/2 -y/2 ] -x/2 } =

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þx (Hxyz (x, y, z)) | x/2 -y/2 z/2 - Þx (Hxyz (x, y, z)) | x/2 -y/2 -z/2 Þx (Hxyz (x, y, z)) | -x/2 -y/2 z/2 + Þx (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } Therefore equation (22) can be written as: A-B-C-D= (∂2 (h(x, y, z)) /∂y∂z) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | x/2 y/2 -z/2

Hxyz (x, y, z) | -x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 } -

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þx (Hxyz (x, y, z)) | x/2 y/2 z/2 - Þx (Hxyz (x, y, z)) | x/2 y/2 -z/2

528

Þx (Hxyz (x, y, z)) | -x/2 y/2 z/2 + Þx (Hxyz (x, y, z)) | -x/2 y/2 -z/2 } -

(26)

Appendix Section

(∂2 (h(x, y, z)) /∂y∂z) * { Hxyz (x, y, z) | x/2 -y/2 z/2 - Hxyz (x, y, z) | x/2 -y/2 -z/2

Hxyz (x, y, z) | -x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 -z/2 } -

(∂3 (h(x, y, z)) /∂x∂y∂z) * { Þx (Hxyz (x, y, z)) | x/2 -y/2 z/2 - Þx (Hxyz (x, y, z)) | x/2 -y/2 -z/2

Þx (Hxyz (x, y, z)) | -x/2 -y/2 z/2 + Þx (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } =

(∂2 (h(x, y, z)) /∂y∂z) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | x/2 y/2 -z/2

Hxyz (x, y, z) | -x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 +



Hxyz (x, y, z) | -x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { -Þx (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þx (Hxyz (x, y, z)) | x/2 y/2 -z/2 +

Þx (Hxyz (x, y, z)) | -x/2 y/2 z/2 - Þx (Hxyz (x, y, z)) | -x/2 y/2 -z/2 Þx (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þx (Hxyz (x, y, z)) | x/2 -y/2 -z/2 +

Þx (Hxyz (x, y, z)) | -x/2 -y/2 z/2 - Þx (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } (27)

The Energy of the Interpolated Image Summing equations (9), (18) and (27), reported below in sequential order, it is found that equation (1) of Chapter III can be written as: EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) = (∂2 (h(x, y, z)) /∂x∂y) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 +



Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

529

Appendix Section

(∂3 (h(x, y, z)) /∂x∂y∂z) * { - Þz (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þz (Hxyz (x, y, z)) | -x/2 y/2 z/2 +

Þz (Hxyz (x, y, z)) | x/2 y/2 -z/2 - Þz (Hxyz (x, y, z)) | -x/2 y/2 -z/2 Þz (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þz (Hxyz (x, y, z)) | -x/2 -y/2 z/2 + Þz (Hxyz (x, y, z)) | x/2 -y/2 -z/2 - Þz (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 } +

(∂2 (h(x, y, z)) /∂z∂x) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 +



Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { - Þy (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þy (Hxyz (x, y, z)) | -x/2 y/2 z/2 +

Þy (Hxyz (x, y, z)) | x/2 -y/2 z/2 - Þy (Hxyz (x, y, z)) | -x/2 -y/2 z/2 Þy (Hxyz (x, y, z)) | x/2 y/2 -z/2 + Þy (Hxyz (x, y, z)) | -x/2 y/2 -z/2 + Þy (Hxyz (x, y, z)) | x/2 -y/2 -z/2 - Þy (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 }

(∂2 (h(x, y, z)) /∂y∂z) * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | x/2 y/2 -z/2

Hxyz (x, y, z) | -x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 +



Hxyz (x, y, z) | -x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

(∂3 (h(x, y, z)) /∂x∂y∂z) * { -Þx (Hxyz (x, y, z)) | x/2 y/2 z/2 + Þx (Hxyz (x, y, z)) | x/2 y/2 -z/2 +

530

Þx (Hxyz (x, y, z)) | -x/2 y/2 z/2 - Þx (Hxyz (x, y, z)) | -x/2 y/2 -z/2 Þx (Hxyz (x, y, z)) | x/2 -y/2 z/2 + Þx (Hxyz (x, y, z)) | x/2 -y/2 -z/2 + Þx (Hxyz (x, y, z)) | -x/2 -y/2 z/2 - Þx (Hxyz (x, y, z)) | -x/2 -y/2 -z/2 }

=

Appendix Section

[ ∂2 (h(x, y, z)) /∂x∂y + ∂2 (h(x, y, z)) /∂z∂x + ∂2 (h(x, y, z)) /∂y∂z ] * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2

Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 +



Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

[ (∂3 (h(x, y, z)) /∂x∂y∂z) * z ] * [ ζz ×

{ - Hxyz (x, y, z) | x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 z/2 + Hxyz (x, y, z) | x/2 y/2 -z/2 - Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } ] +

[ (∂3 (h(x, y, z)) /∂x∂y∂z) * y ] * [ ζy × { - Hxyz (x, y, z) | x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 z/2 +

Hxyz (x, y, z) | x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 z/2 Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } ]

[ (∂3 (h(x, y, z)) /∂x∂y∂z) * x ] * [ ζx × { - Hxyz (x, y, z) | x/2 y/2 z/2 + Hxyz (x, y, z) | x/2 y/2 -z/2 +

Hxyz (x, y, z) | -x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 + Hxyz (x, y, z)) | -x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } ]

Where from Lemma III of Chapter III, ζx, ζy and ζz are arrays of constants.

531

Appendix Section

[ ∂2 (h(x, y, z)) /∂x∂y + ∂2 (h(x, y, z)) /∂z∂x + ∂2 (h(x, y, z)) /∂y∂z ] * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2





Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

{ (∂3 (h(x, y, z)) /∂x∂y∂z) * (x * ζx ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (y * ζy ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (z * ζz ×) } * {- Hxyz (x, y, z) | x/2 y/2 z/2 + Hxyz (x, y, z) | -x/2 y/2 z/2 + Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } + { (∂3 (h(x, y, z)) /∂x∂y∂z) * (x * ζx ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (z * ζz ×) } * { Hxyz (x, y, z) | x/2 y/2 -z/2 - Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 } (∂3 (h(x, y, z)) /∂x∂y∂z) * (y * ζy ×) * { Hxyz (x, y, z) | x/2 y/2 -z/2 - Hxyz (x, y, z) | -x/2 y/2 -z/2 Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 } = { [ ∂2 (h(x, y, z)) /∂x∂y + ∂2 (h(x, y, z)) /∂z∂x + ∂2 (h(x, y, z)) /∂y∂z ] [ (∂3 (h(x, y, z)) /∂x∂y∂z) * (x * ζx ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (y * ζy ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (z * ζz ×) ] } * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2 } + { [ ∂2 (h(x, y, z)) /∂x∂y + ∂2 (h(x, y, z)) /∂z∂x + ∂2 (h(x, y, z)) /∂y∂z ] + [ (∂3 (h(x, y, z)) /∂x∂y∂z) * (x * ζx ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (y * ζy ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (z * ζz ×) ] } * { Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

532

Appendix Section

{ [ ∂2 (h(x, y, z)) /∂x∂y + ∂2 (h(x, y, z)) /∂z∂x + ∂2 (h(x, y, z)) /∂y∂z ] + [ (∂3 (h(x, y, z)) /∂x∂y∂z) * (x * ζx ×) - (∂3 (h(x, y, z)) /∂x∂y∂z) * (y * ζy ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (z * ζz ×) ] } * { - Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 } + { [ ∂2 (h(x, y, z)) /∂x∂y + ∂2 (h(x, y, z)) /∂z∂x + ∂2 (h(x, y, z)) /∂y∂z ] [ (∂3 (h(x, y, z)) /∂x∂y∂z) * (x * ζx ×) - (∂3 (h(x, y, z)) /∂x∂y∂z) * (y * ζy ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (z * ζz ×) ] } * { - Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 }

(28)

Keeping in mind that: (∂2 (h(x, y, z)) /∂x∂y) = - θxy ωf + z ωf (∂2 (h(x, y, z)) /∂z∂x) = - θxz ωf + y ωf (∂2 (h(x, y, z)) /∂y∂z) = - θzy ωf + x ωf (∂3 (h(x, y, z)) /∂x∂y∂z) = ωf It follows that: [ ∂2 (h(x, y, z)) /∂x∂y + ∂2 (h(x, y, z)) /∂z∂x + ∂2 (h(x, y, z)) /∂y∂z ] = [- θxy ωf + z ωf - θxz ωf + y ωf - θzy ωf + x ωf ] = ωf * [ - θxy - θxz - θzy + x + y + z ] [ (∂3 (h(x, y, z)) /∂x∂y∂z) * (x * ζx ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (y * ζy ×) + (∂3 (h(x, y, z)) /∂x∂y∂z) * (z * ζz ×) ] = ς ωf [ x + y + z ] Where ς is a non zero constant depending on the values of ζx, ζy and ζz: ς = (ζx ×) + (ζy ×) + (ζz ×) Thus equation (28) can be written as: { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x + y + z ] } * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2 } + { ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x + y + z ] } * { Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } +

533

Appendix Section

{ ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x - y + z ] } * { - Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 } + { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x - y + z ] } * { - Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 }

(29)

Using the expression of Hxyz (x, y, z) reported in the Deduction I of Lemma II in Chapter III let us write the following relationships: Hxyz (x, y, z) | x/2 y/2 z/2 = f(0,0,0) x y z / 6 + x2/8 y/2 z/2 θx + x/2 y2/8 z/2 θy + x/2 y/2 z2/8 θz x2/8 y2/8 z/2 θxy ωf - x/2 y2/8 z2/8 θzy ωf - x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf Hxyz (x, y, z) | -x/2 y/2 z/2 = - f(0,0,0) x y z / 6 + x2/8 y/2 z/2 θx - x/2 y2/8 z/2 θy - x/2 y/2 z2/8 θz x2/8 y2/8 z/2 θxy ωf + x/2 y2/8 z2/8 θzy ωf - x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf Hxyz (x, y, z) | -x/2 y/2 z/2 - Hxyz (x, y, z) | x/2 y/2 z/2 = -2 f(0,0,0) x y z / 6 - 2 x/2 y2/8 z/2 θy - 2 x/2 y/2 z2/8 θz + 2 x/2 y2/8 z2/8 θzy ωf Hxyz (x, y, z) | x/2 -y/2 -z/2 = f(0,0,0) x y z / 6 + x2/8 y/2 z/2 θx - x/2 y2/8 z/2 θy - x/2 y/2 z2/8 θz + x2/8 y2/8 z/2 θxy ωf - x/2 y2/8 z2/8 θzy ωf + x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf Hxyz (x, y, z) | -x/2 -y/2 -z/2 = -f(0,0,0) x y z / 6 + x2/8 y/2 z/2 θx + x/2 y2/8 z/2 θy + x/2 y/2 z2/8 θz + x2/8 y2/8 z/2 θxy ωf + x/2 y2/8 z2/8 θzy ωf + x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf Hxyz (x, y, z) | -x/2 -y/2 -z/2 - Hxyz (x, y, z) | x/2 -y/2 -z/2 = -2 f(0,0,0) x y z / 6 + 2 x/2 y2/8 z/2 θy + 2 x/2 y/2 z2/8 θz + 2 x/2 y2/8 z2/8 θzy ωf Hxyz (x, y, z) | x/2 -y/2 z/2 = -f(0,0,0) x y z / 6 - x2/8 y/2 z/2 θx + x/2 y2/8 z/2 θy - x/2 y/2 z2/8 θz - x2/8 y2/8 z/2 θxy ωf - x/2 y2/8 z2/8 θzy ωf + x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf

534

Appendix Section

Hxyz (x, y, z) | -x/2 -y/2 z/2 = f(0,0,0) x y z / 6 - x2/8 y/2 z/2 θx - x/2 y2/8 z/2 θy + x/2 y/2 z2/8 θz - x2/8 y2/8 z/2 θxy ωf + x/2 y2/8 z2/8 θzy ωf + x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf Hxyz (x, y, z) | x/2 -y/2 z/2 - Hxyz (x, y, z) | -x/2 -y/2 z/2 = -2 f(0,0,0) x y z / 6 + 2 x/2 y2/8 z/2 θy - 2 x/2 y/2 z2/8 θz - 2 x/2 y2/8 z2/8 θzy ωf Hxyz (x, y, z) | x/2 y/2 -z/2 = -f(0,0,0) x y z / 6 - x2/8 y/2 z/2 θx - x/2 y2/8 z/2 θy + x/2 y/2 z2/8 θz + x2/8 y2/8 z/2 θxy ωf - x/2 y2/8 z2/8 θzy ωf - x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf Hxyz (x, y, z) | -x/2 y/2 -z/2 = f(0,0,0) x y z / 6 - x2/8 y/2 z/2 θx + x/2 y2/8 z/2 θy - x/2 y/2 z2/8 θz + x2/8 y2/8 z/2 θxy ωf + x/2 y2/8 z2/8 θzy ωf - x2/8 y/2 z2/8 θxz ωf + x2/8 y2/8 z2/8 ωf Hxyz (x, y, z) | x/2 y/2 -z/2 - Hxyz (x, y, z) | -x/2 y/2 -z/2 = -2 f(0,0,0) x y z / 6 - 2 x/2 y2/8 z/2 θy + 2 x/2 y/2 z2/8 θz - 2 x/2 y2/8 z2/8 θzy ωf Equation (29) can thus be written as: EIN = EIN(x, y, z) = EIN (Ψxy) + EIN (Ψzx) + EIN (Ψyz) = { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x + y + z ] } * { Hxyz (x, y, z) | x/2 y/2 z/2 - Hxyz (x, y, z) | -x/2 y/2 z/2 } + { ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x + y + z ] } * { Hxyz (x, y, z) | x/2 -y/2 -z/2 - Hxyz (x, y, z) | -x/2 -y/2 -z/2 } + { ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x - y + z ] } * { - Hxyz (x, y, z) | x/2 -y/2 z/2 + Hxyz (x, y, z) | -x/2 -y/2 z/2 } + { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x - y + z ] } * { - Hxyz (x, y, z) | x/2 y/2 -z/2 + Hxyz (x, y, z) | -x/2 y/2 -z/2 } = { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x + y + z ] } * - { -2 f(0,0,0) x y z / 6 - 2 x/2 y2/8 z/2 θy - 2 x/2 y/2 z2/8 θz + 2 x/2 y2/8 z2/8 θzy ωf } +

535

Appendix Section

{ ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x + y + z ] } * - { -2 f(0,0,0) x y z / 6 + 2 x/2 y2/8 z/2 θy + 2 x/2 y/2 z2/8 θz + 2 x/2 y2/8 z2/8 θzy ωf } + { ωf * [ - θxy - θxz - θzy + x + y + z ] + ς ωf [ x - y + z ] } * - { -2 f(0,0,0) x y z / 6 + 2 x/2 y2/8 z/2 θy - 2 x/2 y/2 z2/8 θz - 2 x/2 y2/8 z2/8 θzy ωf } + { ωf * [ - θxy - θxz - θzy + x + y + z ] - ς ωf [ x - y + z ] } * -{ -2 f(0,0,0) x y z / 6 - 2 x/2 y2/8 z/2 θy + 2 x/2 y/2 z2/8 θz - 2 x/2 y2/8 z2/8 θzy ωf } ≠ 0

(30)

for (x, y, z) ≠ 0

APPENDIX II. PROOFS TO THE MATHEMATICAL ASSERTIONS OF CHAPTER IV Study of the Intensity-Curvature Functional This math process starts from the calculation of the first order partial derivatives of the Intensity-Curvature Functional. For simplicity let us posit θxy + θxz + θyz = θxyz. (∂ ΔE(x, y, z) / ∂x) = { y z θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - x y z θxyz f(0, 0, 0) { Hyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζx ×) Hxyz (x, y, z) } } / { Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }2

(1)

(∂ ΔE(x, y, z) / ∂y) = { x z θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - x y z θxyz f(0, 0, 0) { Hxz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζy ×) Hxyz (x, y, z) } } / { Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }2 (∂ ΔE(x, y, z) / ∂z) =

536



(2)

Appendix Section

{ x y θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - x y z θxyz f(0, 0, 0) { Hxy (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζz ×) Hxyz (x, y, z) } } / { Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }2



(3)

In order to find the extremes of ΔE(x, y, z) let us posit the first order partial derivatives (1), (2), (3) equal to zero and solve by x, y and z the resulting system of equations. For (∂ ΔE(x, y, z) / ∂x) = 0, it follows from equation (1) that: y z θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = x y z θxyz f(0, 0, 0) { Hyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζx ×) Hxyz (x, y, z) }

(4)



For (∂ ΔE(x, y, z) / ∂y) = 0, it follows from equation (2) that: x z θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = x y z θxyz f(0, 0, 0) { Hxz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζy ×) Hxyz (x, y, z) }



(5)

For (∂ ΔE(x, y, z) / ∂z) = 0, it follows from equation (3) that: x y θxyz f(0, 0, 0) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = x y z θxyz f(0, 0, 0) { Hxy (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζz ×) Hxyz (x, y, z) }

(6)

Solution of Equation (4) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = x Hyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - x (1 - ζx ×) Hxyz (x, y, z) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] + x (1 - ζx ×) Hxyz (x, y, z) = x Hyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]

537

Appendix Section

Hxyz (x, y, z) [ θxyz - y (1 - ζy ×) - z (1 - ζz ×) ] = x Hyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] Hxyz (x, y, z) / Hyz (x, y, z) = x [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] / [ θxyz - y (1 - ζy ×) - z (1 - ζz ×) ]

(7)

To solve equation (7) let us look at the relationship between Hxyz (x, y, z) and Hyz (x, y, z). Hyz (x, y, z) = Hzy (x, y, z) = f(0,0,0) y z + x y z θx + y2/2 z θy + y z2/2 θz - x y2/2 z θxy ωf - y2/2 z2/2 θzy ωf - x y z2/2 θxz ωf + x y2/2 z2/2 ωf Based on Deduction I of Chapter III, it can be written that: Hxyz (x, y, z) = f(0,0,0) x y z + x2/2 y z θx + x y2/2 z θy + x y z2/2 θz - x2/2 y2/2 z θxy ωf - x y2/2 z2/2 θzy ωf - x2/2 z2/2 y θxz ωf + x2/2 y2/2 z2/2 ωf Let us posit: Ř1 (x, y, z) = [f(0,0,0) + y/2 θy + z/2 θz - y/2 z/2 θzy ωf ] Ř2 (x, y, z) = [θx - y/2 θxy ωf - z/2 θxz ωf + y/2 z/2 ωf ] f1 (x, y, z) = y z Ř1 (x, y, z) f2 (x, y, z) = y z Ř2 (x, y, z) It follows that: Hxyz (x, y, z) = x f1 (x, y, z) + x2/2 f2 (x, y, z) = [x y z Ř1 (x, y, z) + x2/2 y z Ř2 (x, y, z)] Hyz (x, y, z) = f1 (x, y, z) + x f2 (x, y, z) = [y z Ř1 (x, y, z) + x y z Ř2 (x, y, z)] Therefore equation (7) becomes: x z y [Ř1 (x, y, z) + x/2 Ř2 (x, y, z)] / y z [Ř1 (x, y, z) + x Ř2 (x, y, z)] = x [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] / [ θxyz - y (1 - ζy ×) - z (1 - ζz ×) ] [Ř1 (x, y, z) + x/2 Ř2 (x, y, z)] [ θxyz - y (1 - ζy ×) - z (1 - ζz ×) ] = [Ř1 (x, y, z) + x Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]

538

Appendix Section

2 * [Ř1 (x, y, z) + x Ř2 (x, y, z)] [ θxyz - y (1 - ζy ×) - z (1 - ζz ×) ] 2 * [Ř1 (x, y, z) + x Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = 0 2 * [Ř1 (x, y, z) + x Ř2 (x, y, z)] { x (1 - ζx ×) } = 0 x [Ř1 (x, y, z) + x Ř2 (x, y, z)] * 2 (1 - ζx ×) = 0, which is decomposed into: x = 0

(8.a)

[Ř1 (x, y, z) + x Ř2 (x, y, z)] = 0

(8.b)

and equation (8.b) can be written as: [f(0,0,0) + x θx + y/2 θy + z/2 θz - y/2 z/2 θzy ωf - x y/2 θxy ωf - x z/2 θxz ωf + x y/2 z/2 ωf ] = 0

Solution of Equation (5)

(8.c)

Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = y Hxz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - y (1 - ζy ×) Hxyz (x, y, z) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] + y (1 - ζy ×) Hxyz (x, y, z) = y Hxz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - z (1 - ζz ×) ] = y Hxz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] Hxyz (x, y, z) / Hxz (x, y, z) = y [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] / [ θxyz - x (1 - ζx ×) - z (1 - ζz ×) ]

(9)

To solve equation (9), let us look at the relationship between Hxyz (x, y, z) and Hxz (x, y, z). Based on Deduction I of Chapter III, it can be written that: Hxz (x, y, z) = Hzx (x, y, z) = f(0,0,0) x z + x2/2 z θx + x y z θy + x z2/2 θz - x2/2 y z θxy ωf - x y z2/2 θzy ωf - x2/2 z2/2 θxz ωf + x2/2 y z2/2 ωf

539

Appendix Section

Let us posit: Ř1 (x, y, z) = [f(0,0,0) + x/2 θx + z/2 θz - x/2 z/2 θxz ωf ] Ř2 (x, y, z) = [θy - x/2 θxy ωf - z/2 θzy ωf + x/2 z/2 ωf ] f1 (x, y, z) = x z Ř1 (x, y, z) f2 (x, y, z) = x z Ř2 (x, y, z) It follows that: Hxyz (x, y, z) = y f1 (x, y, z) + y2/2 f2 (x, y, z) = [x y z Ř1 (x, y, z) + x y2/2 z Ř2 (x, y, z)] Hxz (x, y, z) = f1 (x, y, z) + y f2 (x, y, z) = [x z Ř1 (x, y, z) + x y z Ř2 (x, y, z)] Therefore equation (9) becomes: x z y [Ř1 (x, y, z) + y/2 Ř2 (x, y, z)] / x z [Ř1 (x, y, z) + y Ř2 (x, y, z)] = y [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] / [ θxyz - x (1 - ζx ×) - z (1 - ζz ×) ] [Ř1 (x, y, z) + y/2 Ř2 (x, y, z)] / [Ř1 (x, y, z) + y Ř2 (x, y, z)] = [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] / [ θxyz - x (1 - ζx ×) - z (1 - ζz ×) ] [Ř1 (x, y, z) + y/2 Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - z (1 - ζz ×) ] = [Ř1 (x, y, z) + y Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] 2 * [Ř1 (x, y, z) + y Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - z (1 - ζz ×) ] 2 * [Ř1 (x, y, z) + y Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = 0 2 * [Ř1 (x, y, z) + y Ř2 (x, y, z)] { y (1 - ζy ×) } = 0 y [Ř1 (x, y, z) + y Ř2 (x, y, z)] * 2 (1 - ζy ×) = 0, which is decomposed into: y = 0

(10.a)

[Ř1 (x, y, z) + y Ř2 (x, y, z)] = 0

(10.b)

and equation (10.b) can be written as: [f(0,0,0) + x/2 θx + y θy + z/2 θz - x/2 z/2 θxz ωf - x/2 y θxy ωf - y z/2 θzy ωf + x/2 y z/2 ωf ] = 0 (10.c)

540

Appendix Section

Solution of Equation (6) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] = z { Hxy (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] - (1 - ζz ×) Hxyz (x, y, z) } Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] + z (1 - ζz ×) Hxyz (x, y, z) = z Hxy (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) ] = z Hxy (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] Hxyz (x, y, z) / Hxy (x, y, z) = z [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] / [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) ]

(11)

To solve equation (11), let us look at the relationship between Hxyz (x, y, z) and Hxy (x, y, z). Based on Deduction I and Lemma II and of Chapter III, it can be written that: Hxy (x, y, z) = Hyx (x, y, z) = f(0,0,0) x y + x2/2 y θx + x y2/2 θy + x y z θz - x2/2 y2/2 θxy ωf - x y2/2 z θzy ωf - x2/2 y z θxz ωf + x2/2 y2/2 z ωf Hxyz (x, y, z) = f(0,0,0) x y z + x2/2 y z θx + x y2/2 z θy + x y z2/2 θz - x2/2 y2/2 z θxy ωf - x y2/2 z2/2 θzy ωf - x2/2 z2/2 y θxz ωf + x2/2 y2/2 z2/2 ωf Let us posit: Ř1 (x, y, z) = [ f(0,0,0) + x/2 θx + y/2 θy - x/2 y/2 θxy ωf ] Ř2 (x, y, z) = [ θz - y/2 θzy ωf - x/2 θxz ωf + x/2 y/2 ωf ] f1 (x, y, z) = x y Ř1 (x, y, z) f2 (x, y, z) = x y Ř2 (x, y, z) It follows that: Hxyz (x, y, z) = z f1 (x, y, z) + z2/2 f2 (x, y, z) Hxy (x, y, z) = f1 (x, y, z) + z f2 (x, y, z)

541

Appendix Section

Equation (11) becomes: Hxyz (x, y, z) / Hxy (x, y, z) = [ x y z Ř1 (x, y, z) + x y z2/2 Ř2 (x, y, z)] / [ x y Ř1 (x, y, z) + x y z Ř2 (x, y, z) ] = [Ř1 (x, y, z) + z/2 Ř2 (x, y, z)] z / [Ř1 (x, y, z) + z Ř2 (x, y, z)] = z [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] / [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) ] It follows that: [Ř1 (x, y, z) + z/2 Ř2 (x, y, z)] z [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) ] = z [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] [Ř1 (x, y, z) + z Ř2 (x, y, z)] [Ř1 (x, y, z) + z/2 Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) ] = [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] [Ř1 (x, y, z) + z Ř2 (x, y, z)] 2 * [Ř1 (x, y, z) + z Ř2 (x, y, z)] [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) ] 2 * [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] [Ř1 (x, y, z) + z Ř2 (x, y, z)] = 0 2 * [Ř1 (x, y, z) + z Ř2 (x, y, z)] { z (1 - ζz ×) } = 0 2 * z [Ř1 (x, y, z) + z Ř2 (x, y, z)] * (1 - ζz ×) = 0, which is decomposed into: z = 0

(12.a)

[Ř1 (x, y, z) + z Ř2 (x, y, z)] = 0

(12.b)

and equation (12.b) can be written as: [f(0,0,0) + x/2 θx + y/2 θy - x/2 y/2 θxy ωf + z θz - y/2 z θzy ωf - x/2 z θxz ωf + x/2 y/2 z ωf ] = 0 (12.c)

A Possible Solution to the Polynomial System of Equations: (8.a), (10.a), (12.a), (8.c), (10.c) and (12.c) The Sub-pixel Efficacy Region is the spatial set of intra-voxel points that is found from the solution of the polynomial system of first order derivatives of the Intensity-Curvature Functional. Such polynomial

542

Appendix Section

system was studied in the previous paragraph and the resulting equations are herein reported. They are equations (8.a), (10.a), (12.a), (8.c), (10.c) and (12.c): x = 0



[f(0,0,0) + x θx + y/2 θy + z/2 θz - y/2 z/2 θzy ωf - x y/2 θxy ωf - x z/2 θxz ωf + x y/2 z/2 ωf ] = 0 y = 0 [f(0,0,0) + x/2 θx + y θy + z/2 θz - x/2 z/2 θxz ωf - x/2 y θxy ωf - y z/2 θzy ωf + x/2 y z/2 ωf ] = 0 z = 0



[f(0,0,0) + x/2 θx + y/2 θy - x/2 y/2 θxy ωf + z θz - y/2 z θzy ωf - x/2 z θxz ωf + x/2 y/2 z ωf ] = 0 From equation (8.c) it follows that: x [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] = [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] x = [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ]

(13)

From equation (10.c) it follows that the explicit form of z is given by: z/2 θz - x/2 z/2 θxz ωf - y z/2 θzy ωf + x/2 y z/2 ωf = - f(0,0,0) - x/2 θx - y θy + x/2 y θxy ωf z (θz/2 - x/4 θxz ωf - y/2 θzy ωf + x/2 y/2 ωf) = - f(0,0,0) - x/2 θx - y θy + x/2 y θxy ωf z = (- f(0,0,0) - x/2 θx - y θy + x/2 y θxy ωf ) / (θz/2 - x/4 θxz ωf - y/2 θzy ωf + x/2 y/2 ωf) z = [- f(0,0,0) - y θy - x (θx/2 - y/2 θxy ωf)] / [(θz/2 - y/2 θzy ωf - x (θxz/4 ωf - y/4 ωf)]

(14)

Equation (13) is placed into equation (14) to derive a subsequent explicit form of z which descends originally from equation (10.c) and that includes also the explicit form of x which was derived from equations (8.c). The result of this process is that of grouping the two equations (8.c) and (10.c) into a single equation which unknown is z and it is here reported: z = {- f(0,0,0) - y θy - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } / { θz/2 - y/2 θzy ωf - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] }

(15)

Equation (15) can be written as:

543

Appendix Section

{- f(0,0,0) - y θy - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf z/2 θxz ωf + y/4 z ωf ] } = { z θz/2 - y/2 z θzy ωf - (θxz/4 ωf - y/4 ωf) [-z f(0,0,0) - y/2 z θy - z2 θz + y/2 z2 θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } from which it follows that: { (z θz/2 - y/2 z θzy ωf ) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] - [-z f(0,0,0) - y/2 z θy - z2 θz + y/2 z2 θzy ωf ] (θxz/4 ωf - y/4 ωf) } = { (- f(0,0,0) - y θy ) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] – [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] (θx/2 - y/2 θxy ωf) }

(16)

The polynomial system consisting of equations (8.c), (10.c) and (12.c) is thus reduced to the system of equations (13) and (16) and its solution consists of: (i) equation (13), (ii) equation (16) solved in the variable z (equation (17) below) considering the variable y as if it was a constant, and (iii) equation (16) solved in the variable y considering the variable z as if it was a constant (equation (18) below).

Deduction of Equation (17) To undertake this process let us re-write equation (16) in the following form: { z (θz/2 - y/2 θzy ωf ) [ θx/2 - y/4 θxy ωf + z (y/4 ωf - θxz/2 ωf)] - [z2 (y/2 θzy ωf - θy) - z (f(0,0,0) + y/2 θy)] (θxz/4 ωf - y/4 ωf) } = { (- f(0,0,0) - y θy ) [ (θx/2 - y/4 θxy ωf ) + z (y/4 ωf - θxz/2 ωf) ] - [z (y/2 θzy ωf - θz) - (f(0,0,0) + y/2 θy )] (θx/2 - y/2 θxy ωf) } which can be written as: { z (θz/2 - y/2 θzy ωf ) (θx/2 - y/4 θxy ωf ) + z2 (θz/2 - y/2 θzy ωf ) (y/4 ωf - θxz/2 ωf) - z2 (y/2 θzy ωf θy) (θxz/4 ωf - y/4 ωf) + z (f(0,0,0) + y/2 θy) (θxz/4 ωf - y/4 ωf) } = { (- f(0,0,0) - y θy ) (θx/2 - y/4 θxy ωf ) + z (y/4 ωf - θxz/2 ωf) (- f(0,0,0) - y θy ) - z (y/2 θzy ωf - θz) (θx/2 - y/2 θxy ωf) + (f(0,0,0) + y/2 θy ) (θx/2 - y/2 θxy ωf) } and from which it is obtained the quadratic equation in the variable z that considers the variable y as a constant: { z2 [ (y/4 ωf - θxz/4 ωf ) (y/2 θzy ωf - θy) + (θz/2 - y/2 θzy ωf ) (y/4 ωf - θxz/2 ωf) ] + z [ (θz/2 - y/2 θzy ωf ) (θx/2 - y/4 θxy ωf ) + (f(0,0,0) + y/2 θy) (θxz/4 ωf - y/4 ωf) (y/4 ωf - θxz/2 ωf) (- f(0,0,0) - y θy ) + (y/2 θzy ωf - θz) (θx/2 - y/2 θxy ωf) ] -

544

Appendix Section

[(- f(0,0,0) - y θy ) (θx/2 - y/4 θxy ωf) + (f(0,0,0) + y/2 θy ) (θx/2 - y/2 θxy ωf) ] } = 0

(17)

Equation (17) is written as: z2 * zα+ z * zβ - zγ = 0 where: zα = [ (y/4 ωf - θxz/4 ωf ) (y/2 θzy ωf - θy) + (θz/2 - y/2 θzy ωf ) (y/4 ωf - θxz/2 ωf) ] zβ = [ (θz/2 - y/2 θzy ωf ) (θx/2 - y/4 θxy ωf ) + (f(0,0,0) + y/2 θy) (θxz/4 ωf - y/4 ωf) – (y/4 ωf - θxz/2 ωf) (- f(0,0,0) - y θy ) + (y/2 θzy ωf - θz) (θx/2 - y/2 θxy ωf) ] zγ = [(- f(0,0,0) - y θy ) (θx/2 - y/4 θxy ωf) + (f(0,0,0) + y/2 θy ) (θx/2 - y/2 θxy ωf) ]

Deduction of Equation (18) To undertake this process let us re-write equation (16) in the following form: { (z θz/2 - y/2 z θzy ωf ) [ (θx/2 - z/2 θxz ωf) + y (z/4 ωf - θxy/4 ωf )] - (θxz/4 ωf - y/4 ωf) [y (z2/2 θzy ωf - z/2 θy) - ( z f(0,0,0) + z2 θy )] } = { (- f(0,0,0) - y θy ) [(θx/2 - z/2 θxz ωf) + y (z/4 ωf - θxy/4 ωf)] - (θx/2 - y/2 θxy ωf) [(- f(0,0,0) - z θz ) + y (z/2 θzy ωf - θy/2) ]} this can be written as: { [z θz/2 (θx/2 - z/2 θxz ωf) - y/2 z ωf θzy (θx/2 - z/2 θxz ωf) + y (z/4 ωf - θxy/4 ωf) z θz/2 - y2 (z/4 ωf - θxy/4 ωf) z/2 ωf θzy ] - [θxz/4 ωf (z2 θzy ωf - z/2 θy) y - (z2/2 θzy ωf - z/2 θy) ωf /4 y2 - θxz/4 ωf (z f(0, 0, 0) + z2 θy ) + (z f(0, 0, 0) + z2 θy ) ωf /4 y] } = { [- f(0, 0, 0) (θx/2 - z/2 θxz ωf) - f(0 ,0, 0) (z/4 ωf - θxy/4 ωf ) y - (θx/2 - z/2 θxz ωf) θy y - y2 (z/4 ωf - θxy/4 ωf ) θy ] - [θx/2 (- f(0 ,0, 0) - z θz) + y (z/2 θyz ωf - θy/2) θx/2 - y/2 θxy ωf (-f(0, 0, 0) - z θz) - y2 (z/2 θyz ωf - θy/2) ωf θxy/2 ]} and from which it is obtained the quadratic equation in the variable y that considers the variable z as a constant:

545

Appendix Section

{ y2 [(z2/2 ωf θzy - z/2 θy) ωf /4 - (z ωf /4 - θxy ωf /4) z/2 θzy ωf + (z/4 ωf - θxy ωf /4) θy + (z/2 θzy ωf - θy/2) θxy/2 ωf] + y [(z ωf /4 - ωf /4 θxy) z θz/2 - (θx/2 - z/2 ωf θxz) z/2 ωf θzy - (z2/2 ωf θzy - z/2 θy) θxz ωf /4 + (z/4 ωf - θxy/4 ωf) f(0, 0, 0) + (θx /2 - z/2 θxz ωf) θy + (z/2 θzy ωf - θy/2) θx/2 - (-f(0, 0, 0) - z θz) ωf θxy/2 + (z f(0, 0, 0) + z2 θy) ωf /4 ] + [z θz/2 (θx/2 - z/2 ωf θxz) - ωf θxz/4 (z f(0,0,0) + z2 θy) + (θx/2 - z/2 θxz ωf) f(0, 0, 0) + (-f(0, 0, 0) - z θz) θx/2] = 0

(18)

Equation (18) can be written as: y2 * yα + y * yβ + yγ = 0 where: yα = [(z2/2 ωf θzy – z/2 θy) ωf /4 - (z ωf /4 – θxy ωf /4) z/2 θzy ωf + (z/4 ωf - θxy ωf /4) θy + (z/2 θzy ωf - θy/2) θxy/2 ωf] yβ = [(z ωf /4 - ωf /4 θxy) z θz/2 - (θx/2 - z/2 ωf θxz) z/2 ωf θzy – (z2/2 ωf θzy – z/2 θy) θxz ωf /4 + (z/4 ωf - θxy/4 ωf) f(0, 0, 0) + (θx /2 - z/2 θxz ωf) θy + (z/2 θzy ωf - θy/2) θx/2 – (-f(0, 0, 0) - z θz) ωf θxy/2 + (z f(0, 0, 0) + z2 θy) ωf /4 ] yγ = [z θz/2 (θx/2 - z/2 ωf θxz) - ωf θxz/4 (z f(0,0,0) + z2 θy) + (θx/2 - z/2 θxz ωf) f(0, 0, 0) + (-f(0, 0, 0) - z θz) θx/2] A possible solution to the polynomial system of equations (8.a), (10.a), (12.a), (8.c), (10.c) and (12.c) is therefore constituted by equations: (13), (17) and (18). However, such solution does not ensure uniqueness because equations (17) and (18), as stated above, are parametric respectively in the variables y and z which were considered as constants in these two equations. In the next paragraph the solution of the polynomial system of equations (8.a), (10.a), (12.a), (8.c), (10.c) and (12.c) is found such that grounds of uniqueness are ensured.

The Solution to the Polynomial System of Equations: (8.a), (10.a), (12.a), (8.c), (10.c) and (12.c) The starting equation is (13): x = [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ]

546

Appendix Section

The following equation is (15): z = {- f(0,0,0) - y θy - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } / { θz/2 - y/2 θzy ωf - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } and from such equation it needs to be derived the explicit form that is dependent on the variable y only. Hereto follow is reported the process of deduction. {- f(0,0,0) - y θy - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf z/2 θxz ωf + y/4 z ωf ] } = z { θz/2 - y/2 θzy ωf - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] = z { (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] } z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] = z { (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] } z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) {[-f(0,0,0) - y/2 θy ] + z [- θz + y/2 θzy ωf ]} = z { (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] } z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf)

547

Appendix Section

{[-f(0,0,0) - y/2 θy ] + z [- θz + y/2 θzy ωf ]} - z { (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] } = 0 z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) {[-f(0,0,0) - y/2 θy ] + z [- θz + y/2 θzy ωf ]} - z { - z (θz/2 - y/2 θzy ωf) θxz ωf/2 + (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] } = 0 z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) {[-f(0,0,0) - y/2 θy ] + z [- θz + y/2 θzy ωf ]} - z { - z (θz/2 - y/2 θzy ωf) θxz ωf/2 + (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] - z (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf) } = 0 z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ] + z (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] - { - z2 (θz/2 - y/2 θzy ωf) θxz ωf/2 + z (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - z (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] - z2 (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf) } = 0 z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ] + z (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] + z2 (θz/2 - y/2 θzy ωf) θxz ωf/2 - z (θz/2 - y/2 θzy ωf) [ θx/2 - y/4 θxy ωf + y/4 z ωf ] + z (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] + z2 (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf) = 0 z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ] + z (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] + z2 (θz/2 - y/2 θzy ωf) θxz ωf/2 - (z θz/2 - z y/2 θzy ωf) [ θx/2 - y/4 θxy ωf + y/4 z ωf ] + z (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] + z2 (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf) = 0

548

Appendix Section

z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf + y/4 z ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ] + z (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] + z2 (θz/2 - y/2 θzy ωf) θxz ωf/2 - (z θz/2 θx/2 - z y/2 θzy ωf θx/2 - z θz/2 y/4 θxy ωf + z y2/2 θzy ωf/4 θxy ωf + z2 θz/2 y/4 ωf - z2 y2/2 θzy ωf /4 ωf ) + z (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] + z2 (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf) = 0 z [ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] + [- f(0,0,0) - y θy ] y/4 z ωf (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ] + z (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] + z2 (θz/2 - y/2 θzy ωf) θxz ωf/2 - (z θz/2 θx/2 - z y/2 θzy ωf θx/2 - z θz/2 y/4 θxy ωf + z y2/2 θzy ωf/4 θxy ωf + z2 θz/2 y/4 ωf - z2 y2/2 θzy ωf /4 ωf ) + z (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] + z2 (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf) = 0 z2 {(θz/2 - y/2 θzy ωf) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf)} + z {[ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] y/4 ωf + (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] - θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] } + {[- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ]} = 0



(19)

Equation (19) furnishes the following two solutions: z1,2 = ( -b ± Δ1/2 ) / 2 a where: b = {[ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] y/4 ωf + (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] } a = {(θz/2 - y/2 θzy ωf) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf)}

549

Appendix Section

Δ = b2 - 4 a c c = { [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ] } It follows that the next equation is the one derived by placing equation (13) into equation (12.c) and deriving the explicit form of the variable y. Such form will be an equation that is dependent on the variable z only. Similarly by placing equation (13) into equation (10.c), it was obtained equation (15) which yields to equation (19), being the explicit form of the variable z and also being dependent on the variable y only. Let us undertake this process in the writings hereto follows. Equation (12.c) is reported here for convenience to the reader: [f(0,0,0) + x/2 θx + y/2 θy - x/2 y/2 θxy ωf + z θz - y/2 z θzy ωf - x/2 z θxz ωf + x/2 y/2 z ωf ] = 0 and placing equation (13) in equation (12.c) it follows: [f(0,0,0) + x/2 θx + y/2 θy - x/2 y/2 θxy ωf + z θz - y/2 z θzy ωf - x/2 z θxz ωf + x/2 y/2 z ωf ] = [f(0,0,0) + x/2 (θx - y/2 θxy ωf - z θxz ωf + y/2 z ωf) + y/2 θy + z θz - y/2 z θzy ωf] = 0 {f(0,0,0) + [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] / [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] (θx - y/2 θxy ωf - z θxz ωf + y/2 z ωf)/2 + y/2 θy + z θz - y/2 z θzy ωf} = 0 { f(0,0,0) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] + [-f(0,0,0) - y/2 θy - z θz + y/2 z θzy ωf ] (θx - y/2 θxy ωf - z θxz ωf + y/2 z ωf)/2 + (y/2 θy + z θz - y/2 z θzy ωf) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } = 0 { y [ -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 ] + f(0,0,0) [ θx/2 - z/2 θxz ωf ] + [(-f(0,0,0) - z θz) - y (θy/2 - z θzy ωf/2)] (θx - y/2 θxy ωf - z θxz ωf + y/2 z ωf)/2 + (y/2 θy + z θz - y/2 z θzy ωf) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } = 0 { y [ -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 ] + f(0,0,0) [ θx/2 - z/2 θxz ωf ] + [(-f(0,0,0) - z θz) - y (θy/2 - z θzy ωf/2)] [y (- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 + (y/2 θy + z θz - y/2 z θzy ωf) [ θx/2 - y/4 θxy ωf - z/2 θxz ωf + y/4 z ωf ] } = 0

550

Appendix Section

{ y [ -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 ] + f(0,0,0) [ θx/2 - z/2 θxz ωf ] + [(-f(0,0,0) - z θz) - y (θy/2 - z θzy ωf/2)] [y (- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 + [y (θy/2 - z θzy ωf/2) + z θz] [ y (-θxy ωf/4 + z ωf/4) + (θx/2 - z/2 θxz ωf) ]} = 0 { y [ -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 ] + f(0,0,0) [ θx/2 - z/2 θxz ωf ] + (-f(0,0,0) - z θz) [y (- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 - y (θy/2 - z θzy ωf/2) [y (- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 + [y (θy/2 - z θzy ωf/2) + z θz] [ y (-θxy ωf/4 + z ωf/4) + (θx/2 - z/2 θxz ωf) ]} = 0 { y [ -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 ] + f(0,0,0) [ θx/2 - z/2 θxz ωf ] + y (-f(0,0,0) - z θz) [ (- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 - [y2 (θy/2 - z θzy ωf/2) (- θxy ωf /2 + z ωf/2) + y (θy/2 - z θzy ωf/2) (θx - z θxz ωf)]/2 + [y2 (θy/2 - z θzy ωf/2) (-θxy ωf/4 + z ωf/4) + y (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf) + z θz y (-θxy ωf/4 + z ωf/4) + z θz (θx/2 - z/2 θxz ωf) ]} = 0 y2 { (θy/2 - z θzy ωf/2) [ ( θxy ωf /2 - z ωf/2) + (-θxy ωf/4 + z ωf/4)] } + y { -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + (-f(0,0,0) - z θz) [(- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf)/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf) + z θz (-θxy ωf/4 + z ωf/4) } + { f(0,0,0) [ θx/2 - z/2 θxz ωf ] + z θz (θx/2 - z/2 θxz ωf) } = 0

(20)

Equation (20) furnishes the following two solutions: y1,2 = ( -b ± Δ1/2 ) / 2 a where: b = { -f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + (-f(0,0,0) - z θz) [(- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf)/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf) + z θz (-θxy ωf/4 + z ωf/4) } a = { (θy/2 - z θzy ωf/2) [ ( θxy ωf /2 - z ωf/2) + (-θxy ωf/4 + z ωf/4)] } Δ = b2 - 4 a c

551

Appendix Section

c = { f(0,0,0) [ θx/2 - z/2 θxz ωf ] + z θz (θx/2 - z/2 θxz ωf) }

APPENDIX III. EQ UATIONS (8) and (9) OF CHAPTER V Equation (8) of Chapter V From equation (8) it follows that: b = { (-f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + (-f(0,0,0) - z θz) [(- θxy ωf /2 + z ωf/2) + (θx - z θxz ωf)]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf)/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf) + z θz (-θxy ωf/4 + z ωf/4) } = { (-f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + [(- θxy ωf /2 + z ωf/2) (-f(0,0,0) - z θz) + (θx - z θxz ωf) (-f(0,0,0) - z θz)]/2 + (θy/2 - z θzy ωf/2) (θx - z θxz ωf)/2 + (θy/2 - z θzy ωf/2) (θx/2 - z/2 θxz ωf) + (-θxy ωf/4 z θz + z2 θz ωf/4) } = { (-f(0,0,0) θxy ωf /4 + f(0,0,0) z ωf/4 + [( θxy ωf /2 z θz - z2 ωf/2 θz + f(0,0,0) θxy ωf /2 - f(0,0,0) z ωf/2) + (-f(0,0,0) θx + z f(0,0,0) θxz ωf - z θz θx + z2 θxz ωf θz)]/2 + (θy/2 θx - z θx θzy ωf/2 - z θy/2 θxz ωf + z2 θxz ωf θzy ωf/2)/2 + (θy/2 θx/2 - z θzy θx/2 ωf/2 - θy/2 z/2 θxz ωf + z2/2 θxz ωf θzy ωf/2) + (-θxy ωf/4 z θz + z2 θz ωf/4) } = {z2 [-ωf/4 θz + θxz/2 ωf θz + θxz ωf θzy ωf/4 + θxz ωf θzy ωf/4 + θz ωf/4] + z [f(0,0,0) ωf/4 + θxy ωf /4 θz - f(0,0,0) ωf/4 + f(0,0,0)/2 θxz ωf - θz/2 θx - θx θzy ωf/2 - θy/2 θxz ωf θzy θx/2 ωf/2 - θy/4 θxz ωf - θxy ωf/4 θz] + [-f(0,0,0) θxy ωf /4 + f(0,0,0) θxy ωf /2 - f(0,0,0) θx + θy/2 θx + θy θx/4]} Also that: a = { (θy/2 - z θzy ωf/2) [ ( θxy ωf /2 - z ωf/2) + (-θxy ωf/4 + z ωf/4)] } = [ ( θxy ωf /2 - z ωf/2) (θy/2 - z θzy ωf/2) + (-θxy ωf/4 + z ωf/4) (θy/2 - z θzy ωf/2)] =

552

(1)

Appendix Section

[ θxy ωf /4 θy - z ωf/4 θy - θxy ωf /4 z θzy ωf + z2 ωf/4 θzy ωf + (-θxy ωf/4 + z ωf/4) (θy/2 - z θzy ωf/2)] [ θxy ωf /4 θy - z ωf/4 θy - θxy ωf /4 z θzy ωf + z2 ωf/4 θzy ωf + (-θxy ωf/4 θy/2 + z ωf/4 θy/2 + θxy ωf/4 z θzy ωf/2 - z2 ωf/4 θzy ωf/2)] = z2 [ωf/4 θzy ωf - ωf/4 θzy ωf/2 ] + z [-ωf/4 θy - θxy ωf /4 θzy ωf + ωf/4 θy/2 + θxy ωf/4 θzy ωf/2 ] + [θxy ωf /4 θy - θxy ωf/4 θy/2]

(2)

Also that: c = { f(0,0,0) [ θx/2 - z/2 θxz ωf ] + z θz (θx/2 - z/2 θxz ωf) } = { [ θx/2 f(0,0,0) - z/2 θxz ωf f(0,0,0)] + (θx/2 z θz - z2/2 θz θxz ωf) } = - z2/2 θz θxz ωf + z (θx/2 θz - θxz ωf f(0,0,0)/2 ) + θx/2 f(0,0,0) (3)

Equation (9) of Chapter V From equation (9) it follows that: b = {[ f(0,0,0) + y θy ] θxz ωf/2 + [- f(0,0,0) - y θy ] y/4 ωf + (θx/2 - y/2 θxy ωf) [- θz + y/2 θzy ωf ] θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (θxz/4 ωf - y/4 ωf) [-f(0,0,0) - y/2 θy] } = {[ f(0,0,0) θxz ωf/2 + y θy θxz ωf/2] + [- f(0,0,0) y/4 ωf - y2 θy /4 ωf ] + (-θx/2 θz + y/2 θxy ωf θz + θx/2 y/2 θzy ωf - y2/4 θxy ωf θzy ωf) - θz/2 θx/2 + y/2 θzy ωf θx/2 + θz/2 y/4 θxy ωf - y2/2 θzy ωf/4 θxy ωf + (-f(0,0,0) θxz/4 ωf + f(0,0,0) y/4 ωf - θxz/4 y/2 θy + y2/8 ωf θy)} = { y2 [-θy /4 ωf - θxy/4 ωf θzy ωf - θzy ωf/8 θxy ωf + ωf θy/8] + y [θy θxz ωf/2 - f(0,0,0)/4 ωf + θxy/2 ωf θz + θx/4 θzy ωf + θzy ωf θx/4 + θz/8 θxy ωf + f(0,0,0)/4 ωf θxz/8 θy] + [f(0,0,0) θxz ωf/2 - θx/2 θz - θz/2 θx/2 -f(0,0,0) θxz/4 ωf ]}

(4)

553

Appendix Section

Also that: a = {(θz/2 - y/2 θzy ωf) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (θxz/4 ωf - y/4 ωf) (- θz + y/2 θzy ωf)} = {(θz/2 - y/2 θzy ωf) θxz ωf/2 - θz/2 y/4 ωf + y2/2 θzy ωf /4 ωf + (- θz θxz/4 ωf + y/2 θzy ωf θxz/4 ωf + y/4 ωf θz - y2/8 ωf θzy ωf)} = y2 [ θzy ωf /8 ωf - ωf θzy ωf/8 ] + y [ θzy ωf θxz ωf/4 - θz/8 ωf + θzy ωf θxz/8 ωf + ωf θz/4] + [θz/2 - θz θxz/4 ωf]

(5)

Also that: c = { [- f(0,0,0) - y θy ] [ θx/2 - y/4 θxy ωf ] - (θx/2 - y/2 θxy ωf) [-f(0,0,0) - y/2 θy ] } = {[- f(0,0,0) θx/2 + f(0,0,0) y/4 θxy ωf - y θy θx/2 + y2/4 θy θxy ωf ] - [-θx/2 f(0,0,0) - θx/2 y/2 θy + y/2 θxy ωf f(0,0,0) + y2/4 θxy ωf θy ] } = y2 [θy/4 θxy ωf + θxy ωf θy/4] + y [f(0,0,0)/4 θxy ωf - θy θx/2 - θx/4 θy + θxy ωf f(0,0,0)/2] + [- f(0,0,0) θx/2 + θx/2 f(0,0,0)]

(6)

APPENDIX IV. PROOFS TO THE MATHEMATICAL ASSERTIONS OF CHAPTER VII Calculation of the First Order Partial Derivatives of the Intensity-Curvature Functional Let us posit:

554

Appendix Section

θx = [f(1,0) - f(0,0)] = [θx’ - f(0,0)] θy = [f(0,1) - f(0,0)] = [θy’ - f(0,0)] ωf = [f(1,1) + f(0,0) - f(0,1) - f(1,0)] = [ωf’ + f(0,0)] [ f(1,1) + f(0,0) - θx’ - θy’ ] = ωf ΔE (x, y) = Eo / EIN = - f (0, 0) x y θxy / Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]

(1)

∂ (ΔE (x, y))/∂x = - f (0, 0) y θxy {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]} / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 + f (0, 0) x y θxy {Hy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]} / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2

(2)

∂ (ΔE (x, y))/∂y = - f (0, 0) x θxy {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]} / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 + f (0, 0) x y θxy {Hx (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]} / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2



(3)

Let us calculate ∂ (ΔE (x, y))/∂x, it follows that: - f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ]{ y Hxy (x, y)} - x y Hy (x, y) } = - f (0, 0) y θxy [ f(1,1) + f(0,0) - θx’ - θy’ ]{ Hxy (x, y)} - x Hy (x, y) } = - f (0, 0) y θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] * { [f(0, 0) x y + [θx’ - f(0,0)] x2/2 y + [θy’ - f(0,0)] y2/2 x + [ωf’ + f(0,0)] x2/2 y2/2] [f(0, 0) x y + [θx’ - f(0,0)] x2 y + [θy’ - f(0,0)] y2/2 x + [ωf’ + f(0,0)] x2 y2/2] } = - f (0, 0) y θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] * { - [θx’ - f(0,0)] x2/2 y - [ωf’ + f(0,0)] x2/2 y2/2 } = f (0, 0) x2/2 y2 θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] * { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } (4) It follows that: ∂ (ΔE (x, y))/∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 y2 { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2

(5)

555

Appendix Section

Let us calculate ∂ (ΔE (x, y))/∂y, it follows that: - f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] { x Hxy (x, y)} - x y Hx (x, y) } = - f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] { Hxy (x, y)} - y Hy (x, y) } = - f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] * { [f(0, 0) x y + [θx’ - f(0,0)] x2/2 y + [θy’ - f(0,0)] y2/2 x + [ωf’ + f(0,0)] x2/2 y2/2]

[f(0, 0) x y + [θx’ - f(0,0)] x2 y/2 + [θy’ - f(0,0)] y2 x + [ωf’ + f(0,0)] x2/2 y2] } =

- f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] * { - [θy’ - f(0,0)] x/2 y2 - [ωf’ + f(0,0)] x2/2 y2/2 } = f (0, 0) x2/2 y2 θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] * { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 }

(6)

It follows that: ∂ (ΔE (x, y))/∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 y2 { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 }/ {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2

(7)

CALCULATION OF THE SECOND ORDER PARTIAL DERIVATIVES OF THE INTENSITY-CURVATURE FUNCTIONAL ∂2 (ΔE (x, y))/∂x2 = { f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x y2 { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 y2 { [θx’ – f(0,0)] + [ωf’ + f(0,0)] y/2} * { 2 Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 Hy (x, y)} } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 ∂2 (ΔE (x, y))/∂x2 = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x y2 { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } * Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 { Hxy (x, y) - x Hy (x, y) } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4

(8)

∂2 (ΔE (x, y))/∂y2 = { f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2 y { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2

556

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} * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 y2 { [θy’ – f(0,0)] + [ωf’ + f(0,0)] x/2 } * {2 Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 Hx (x, y)} / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 ∂2 (ΔE (x, y))/∂y2 = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2 y { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 } * { Hxy (x, y) - y Hx (x, y) } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4

(9)

To prove correct equations (8) and (9) let us write equations (2) and (3) in the forms as follows: ∂ (ΔE (x, y))/∂x = - f (0, 0) y θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] { Hxy (x, y) - x Hy (x, y) } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2



(10)

∂ (ΔE (x, y))/∂y = - f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] {Hxy (x, y) - y Hx (x, y) } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2



(11)

and calculating the second order derivatives besed on equations (10) and (11), let us obtain the same exact result as per equations (8) and (9) respectively. ∂ (ΔE (x, y))/∂x as per equation (10) is derived with respect to the variable x: ∂2 (ΔE (x, y))/∂x2 = { - f (0, 0) y θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ θy’ ]}4 } * { Hy (x, y) - Hy (x, y) - x {y [θx’ - f(0,0)] + [ωf’ + f(0,0)] y2/2} * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - { Hxy (x, y) - x Hy (x, y) } * 2 [ f(1,1) + f(0,0) - θx’ - θy’ ]2 Hxy (x, y) Hy (x, y) } = { - f (0, 0) x y2 θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * { -{[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 (x y)-1 {[Hxy (x, y)]2 [Hy (x, y)] - x [Hy (x, y)]2 Hxy (x, y)} * 2 [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } =

557

Appendix Section

{ - f (0, 0) x y2 θxy [θx’ + θy’- 2 f(0,0)] [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 [Hxy (x, y)] / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * { -{[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} [Hxy (x, y)] 2 (x y)-1 {[Hxy (x, y)] [Hy (x, y)] - x [Hy (x, y)]2 } } = { - f (0, 0) x y2 θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] [ f(1,1) + f(0,0) - θx’ - θy’ ]2 [Hxy (x, y)] / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * { Ρxy }

(12)

Ρxy = { -{[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} Hxy (x, y) 2 Hy (x, y) (x y)-1 [Hxy (x, y) - x Hy (x, y) ] } = { -{[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} Hxy (x, y) - 2 Hy (x, y) (x y)-1 { f(0,0) xy + yx2/2 [θx’ - f(0,0)] + xy2/2 [θy’ - f(0,0)] + y2x2/4 [ωf’ + f(0,0)] - x [f(0,0) y + x y [θx’ - f(0,0)] + [θy’ - f(0,0)] y2/2 + [ωf’ + f(0,0)] xy2/2] } = { -{[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} Hxy (x, y) - 2 Hy (x, y) (x y)-1 {- yx2/2 [θx’ - f(0,0)] - [ωf’ + f(0,0)] x2y2/4 } } = { -{[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} Hxy (x, y) + Hy (x, y) {x [θx’ - f(0,0)] + [ωf’ + f(0,0)] xy/2}} = { -{[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} Hxy (x, y) + x Hy (x, y) { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2}} = { {[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} [-Hxy (x, y) + x Hy (x, y)] } Therefore equation (12) is written as: ∂2 (ΔE (x, y))/∂x2 = { - f (0, 0) x y2 θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] [ f(1,1) + f(0,0) - θx’ - θy’ ]2 [Hxy (x, y)] / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * { {[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} [-Hxy (x, y) + x Hy (x, y)] } =

{ f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x y2 {[θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2} * Hxy (x, y)

558

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[ f(1,1) + f(0,0) - θx’ - θy’ ]2{ [Hxy (x, y) - x Hy (x, y)] } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4

(13)

which proves the correctness of equation (8). Above is possible because: Hy (x, y) = ∂ (Hxy (x, y))/∂x = f(0,0) y + x y [θx’ - f(0,0)] + [θy’ - f(0,0)] y2/2 + [ωf’ + f(0,0)] xy2/2 ∂ (Hy (x, y))/∂x = y [θx’ - f(0,0)] + [ωf’ + f(0,0)] y2/2 Hx (x, y) = ∂ (Hxy (x, y))/∂y = f(0,0) x + [θx’ - f(0,0)] x2/2 + x y [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2y/2 ∂ (Hx (x, y))/∂y = x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2/2 Hyx (x, y) = Hxy (x, y) = f(0,0) xy + yx2/2 [θx’ - f(0,0)] + xy2/2 [θy’ - f(0,0)] + y2x2/4 [ωf’ + f(0,0)] ∂ (ΔE (x, y))/∂y as per equation (11) herein reported: ∂ (ΔE (x, y))/∂y = - f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] {Hxy (x, y) - y Hx (x, y) } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 is derived with respect to the variable y: ∂2 (ΔE (x, y))/∂y2 = { -f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] / {Hxy (x, y) [θx’ + θy’- 2 f(0,0)]}4 } * { [Hxy (x, y) - y Hx (x, y) ] * 2 [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 Hxy (x, y) Hx (x, y) {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 * {Hx (x, y) - Hx (x, y) - y{x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2/2} } = { -f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] [ f(1,1) + f(0,0) - θx’ - θy’ ]2 Hxy (x, y) / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * { [Hxy (x, y) - y Hx (x, y) ] * 2 Hx (x, y) + Hxy (x, y) y {x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2/2} } = {-f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] [ f(1,1) + f(0,0) - θx’ - θy’ ]2 Hxy (x, y) / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * { Ρxy }

(14)

559

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where: Ρxy = { [Hxy (x, y) - y Hx (x, y) ] * 2 Hx (x, y) + Hxy (x, y) y {x [θy‘ - f(0,0)] + [ωf’ + f(0,0)] x2/2} } Since: [Hxy (x, y) - y Hx (x, y) ] = { f(0,0) xy + yx2/2 [θx’ - f(0,0)] + xy2/2 [θy’ - f(0,0)] + y2x2/4 [ωf’ + f(0,0)] - y { f(0,0) x + [θx’ - f(0,0)] x2/2 + x y [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2y/2 } } = {- x y2/2 [θy’ - f(0,0)] - y2x2/4 [ωf’ + f(0,0)] } Thus according to what stated above: Ρxy = {- x y2/2 [θy’ - f(0,0)] - y2x2/4 [ωf’ + f(0,0)] } * 2 Hx (x, y) + Hxy (x, y) {x y [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2 y/2} } = {- x y [θy’ - f(0,0)] - x2 y/2 [ωf’ + f(0,0)] } * y Hx (x, y) + Hxy (x, y) {x y [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2 y/2} } = {-Hxy (x, y) + y Hx (x, y)} * {x y [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2 y/2} } = x y {-Hxy (x, y) + y Hx (x, y)} * { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2} }

(15)

Therefore equation (14): ∂2 (ΔE (x, y))/∂y2 = { -f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 Hxy (x, y) / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * { [Hxy (x, y) - y Hx (x, y) ] * 2 Hx (x, y) + Hxy (x, y) y {x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2/2} } = { -f (0, 0) x θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 Hxy (x, y) / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } * x y {-Hxy (x, y) + y Hx (x, y)} * { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2} } = { -f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2 y { [θy‘ - f(0,0)] + [ωf’ + f(0,0)] x/2} }

560

Appendix Section

{Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } * {-Hxy (x, y) + y Hx (x, y)} / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 } = { f (0, 0) θxy [ f(1,1) + f(0,0) - θx’- θy’ ] x2 y { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2} } {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } * {Hxy (x, y) - y Hx (x, y)} / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 }

(16)

which proves the correctness of equation (9). Let us proceed now with the calculation of the second order partial derivatives with respect to the variables x and y: ∂2 (ΔE (x, y))/∂x∂y, which is not the same as ∂2 (ΔE (x, y))/∂y∂x. This was found, and accordingly, there are reasons to believe that is true, since the Jacobian matrix of second order derivatives is not necessarily symmetric. To calculate ∂2 (ΔE (x, y))/∂x∂y let us calculate the first order partial derivative ∂ (ΔE (x, y))/∂x with respect to the variable y (equation (5)) herein reported: ∂ (ΔE (x, y))/∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 y2 { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 }/ { Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] }2 it follows that: ∂2 (ΔE (x, y))/∂x∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 * { {2y { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } + y2 [ωf’ + f(0,0)] /2 } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] }2 - y2 { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } * 2 Hxy (x, y) Hx (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] }4 Let us calculate: {2y { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } + y2 [ωf’ + f(0,0)] /2 } = 2y [θx’ - f(0,0)] + 3/2 y2 [ωf’ + f(0,0)] y2 { [θx’ - f(0,0)] + [ωf’ + f(0,0)] y/2 } * 2 Hx (x, y) = y { [θx’ - f(0,0)] 2y + [ωf’ + f(0,0)] y2 } * Hx (x, y)

561

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Therefore: ∂2 (ΔE (x, y))/∂x∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 * { {2y [θx’ - f(0,0)] + 3/2 y2 [ωf’ + f(0,0)]} * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - y { [θx’ - f(0,0)] 2y + [ωf’ + f(0,0)] y2 } * Hx (x, y) Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4

(17)

To obtain ∂2 (ΔE (x, y))/∂y∂x let us calculate the first order partial derivative ∂ (ΔE (x, y))/∂y with respect to the variable x (equation (7)) herein reported: ∂ (ΔE (x, y))/∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] x2/2 y2 { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 it follows that: ∂2 (ΔE (x, y))/∂y∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] y2/2 { {2 x { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 } + x2 [ωf’ + f(0,0)]/2 } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - x2 { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 } * 2 Hxy (x, y) Hy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 Let us calculate: { 2x { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 } + x2 [ωf’ + f(0,0)]/2 } = { 2x [θy’ - f(0,0)] + 3/2 x2 [ωf’ + f(0,0)] } x2 { [θy’ - f(0,0)] + [ωf’ + f(0,0)] x/2 } * 2 Hy (x, y) = x { 2x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2 } * Hy (x, y) Therefore: ∂2 (ΔE (x, y))/∂y∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] y2/2 { { 2x [θy’ - f(0,0)] + 3/2 x2 [ωf’ + f(0,0)] } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - x { 2x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2 } *

562

Appendix Section

Hy (x, y) Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 (18) From equation (17) it follows that: ∂2 (ΔE (x, y))/∂x∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx‘ - θy‘ ] x2/2 * { {2y [θx’ - f(0,0)] + 3/2 y2 [ωf’ + f(0,0)]} * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - y { [θx’ – f(0,0)] 2y + [ωf’ + f(0,0)] y2 } * Hx (x, y) Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 = ∂2 (ΔE (x, y))/∂x∂y = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] [ f(1,1) + f(0,0) - θx’ - θy’ ]2 Hxy (x, y) x2/2 * y { {2 [θx’ - f(0,0)] + 3/2 y [ωf’ + f(0,0)]} * Hxy (x, y) - { [θx’ - f(0,0)] 2y + [ωf’ + f(0,0)] y2 } * Hx (x, y) } } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4

(19)

From equation (18) it follows that: ∂2 (ΔE (x, y))/∂y∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] y2/2 { { 2x [θy’ - f(0,0)] + 3/2 x2 [ωf’ + f(0,0)] } * {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}2 - x { 2x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2 } * Hy (x, y) Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ] 2 } / {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4 =



∂2 (ΔE (x, y))/∂y∂x = f (0, 0) θxy [ f(1,1) + f(0,0) - θx’ - θy’ ] Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]2 y2/2 * x { { 2 [θy’ - f(0,0)] + 3/2 x [ωf’ + f(0,0)] } * Hxy (x, y) - { 2x [θy’ - f(0,0)] + [ωf’ + f(0,0)] x2 } * Hy (x, y) } }/ {Hxy (x, y) [ f(1,1) + f(0,0) - θx’ - θy’ ]}4

(20)

From equation (19): x2/2 * y { {2 [θx’ - f(0,0)] + 3/2 y [ωf’ + f(0,0)]} * Hxy (x, y) { 2y [θx’ - f(0,0)] + y2 [ωf’ + f(0,0)] } * Hx (x, y) } } = x2/2 * y { 2 [θx’ - f(0,0)] Hxy (x, y) + 3/2 y [ωf’ + f(0,0)] Hxy (x, y) 2y [θx’ - f(0,0)] Hx (x, y) - y2 [ωf’ + f(0,0)] Hx (x, y) } =

563

Appendix Section

x2/2 * y { 2 [θx’ - f(0,0)] [Hxy (x, y) - y Hx (x, y)] + [ωf’ + f(0,0)] [3/2 y Hxy (x, y) - y2 Hx (x, y)] } (21) From equation (20): y2/2 * x { { 2 [θy’ - f(0,0)] + 3/2 x [ωf’ + f(0,0)] } * Hxy (x, y) { 2x [θy’ - f(0,0)] + x2 [ωf’ + f(0,0)] } * Hy (x, y) } } = y2/2 * x { 2 [θy’ - f(0,0)] Hxy (x, y) + 3/2 x [ωf’ + f(0,0)] Hxy (x, y) 2x [θy’ - f(0,0)] Hy (x, y) - x2 [ωf’ + f(0,0)] Hy (x, y) } = y2/2 * x { 2 [θy’ - f(0,0)] [Hxy (x, y) - x Hy (x, y)] + [ωf’ + f(0,0)] [3/2 x Hxy (x, y) - x2 Hy (x, y)] } (22) Let us recall that: x Hy (x, y) = f(0,0) x y + [θx’ - f(0,0)] x2 y + [θy’ - f(0,0)] x y2/2 + [ωf’ + f(0,0)] x2y2/2 y Hx (x, y) = f(0,0) x y + [θx’ - f(0,0)] y x2/2 + [θy’ - f(0,0)] x y2 + [ωf’ + f(0,0)] x2y2/2 x2 Hy (x, y) = f(0,0) x2 y + [θx’ - f(0,0)] x3 y + [θy’ - f(0,0)] x2 y2/2 + [ωf’ + f(0,0)] x3y2/2 y2 Hx (x, y) = f(0,0) x y2 + [θx’ - f(0,0)] y2 x2/2 + [θy’ - f(0,0)] x y3 + [ωf’ + f(0,0)] x2y3/2 Hxy (x, y) = f(0,0) x y + [θx’ - f(0,0)] y x2/2+ [θy’ - f(0,0)] x y2/2+ [ωf’ + f(0,0)] x2y2/4 Hxy (x, y) - x Hy (x, y) = - x2/2 y [θx’ - f(0,0)] - x2y2/4 [ωf’ + f(0,0)] Hxy (x, y) - y Hx (x, y) = - x/2 y2 [θy’ - f(0,0)] - x2y2/4 [ωf’ + f(0,0)] 3/2 x Hxy (x, y) = 3/2 f(0,0) x2 y + [θx’ - f(0,0)] y x33/4 + [θy’ - f(0,0)] x2 y2 3/4 + [ωf’ + f(0,0)] x3y2 3/8 3/2 y Hxy (x, y) = 3/2 f(0,0) x y2 + [θx’ - f(0,0)] y2 x2 3/4 + [θy’ - f(0,0)] x y3 3/4 + [ωf’ + f(0,0)] x2y3 3/8 [3/2 y Hxy (x, y) - y2 Hx (x, y)] = 3/2 f(0,0) x y2 + [θx’ - f(0,0)] y2 x2 3/4 + [θy’ - f(0,0)] x y3 3/4 +

564



Appendix Section

[ωf’ + f(0,0)] x2y3 3/8 - f(0,0) x y2 - [θx’ - f(0,0)] y2 x2/2 - [θy’ - f(0,0)] x y3 - [ωf’ + f(0,0)] x2y3/2 = 1/2 f(0,0) x y2 - [θx’ - f(0,0)] y2 x2/4 - [θy’ - f(0,0)] x y3/4 - [ωf’ + f(0,0)] x2y3/4 [3/2 x Hxy (x, y) - x2 Hy (x, y)] = 3/2 f(0,0) x2 y + [θx’ - f(0,0)] y x33/4 + [θy’ - f(0,0)] x2 y2 3/4 + [ωf’ + f(0,0)] x3y2 3/8 - f(0,0) x2 y - [θx’ - f(0,0)] x3y - [θy’ - f(0,0)] x2 y2/2 - [ωf’ + f(0,0)] x3y2/2 = 1/2 f(0,0) x2 y - [θx‘ - f(0,0)] x3y /4 + [θy‘ - f(0,0)] x2 y2/4 - [ωf’ + f(0,0)] x3y2/4 Therefore equation (21) becomes: x2/2 * y { 2 [θx’ - f(0,0)] [Hxy (x, y) - y Hx (x, y)] + [ωf’ + f(0,0)] [3/2 y Hxy (x, y) - y2 Hx (x, y)] } = x2/2 * y { 2 [θx’ - f(0,0)] {- x/2 y2 [θy’ - f(0,0)] - x2y2/4 [ωf’ + f(0,0)] } + [ωf’ + f(0,0)] {1/2 f(0,0) x y2 - [θx’ - f(0,0)] y2 x2/4 - [θy’ - f(0,0)] x y3/4 - [ωf’ + f(0,0)] x2y3/4} } = x2/2 * y { - x y2 [θy’ - f(0,0)] [θx’ - f(0,0)] - x2y2/2 [ωf’ + f(0,0)] [θx’ - f(0,0)] + 1/2 f(0,0) x y2 [ωf’ + f(0,0)] - [θx’ - f(0,0)] [ωf’ + f(0,0)] y2 x2/4 - [θy’ - f(0,0)] [ωf’ + f(0,0)] x y3/4 - [ωf’ + f(0,0)]2 x2y3/4 } = x3/2 * y3 { - [θy’ - f(0,0)] [θx’ - f(0,0)] - [ωf’ + f(0,0)] [θx’ - f(0,0)] x/2 + 1/2 f(0,0) [ωf’ + f(0,0)] - [θx’ - f(0,0)] [ωf’ + f(0,0)] x/4 - [θy’ - f(0,0)] [ωf’ + f(0,0)] y/4 - [ωf’ + f(0,0)]2 x y/4 } = x3/2 * y3 { - [θy’ - f(0,0)] [θx’ - f(0,0)] + 1/2 f(0,0) [ωf’ + f(0,0)] - [θx’ - f(0,0)] [ωf’ + f(0,0)] x 3/4 [θy’ - f(0,0)] [ωf’ + f(0,0)] y/4 - [ωf’ + f(0,0)]2 x y/4 }

(23)

And equation (22) becomes: y2/2 * x { 2 [θy’ - f(0,0)] [Hxy (x, y) - x Hy (x, y)] + [ωf’ + f(0,0)] [3/2 x Hxy (x, y) - x2 Hy (x, y)] } = y2/2 * x { 2 [θy’ - f(0,0)] {- x2/2 y [θx’ - f(0,0)] - x2y2/4 [ωf’ + f(0,0)] } + [ωf’ + f(0,0)] { 1/2 f(0,0) x2 y - [θx’ - f(0,0)] x3y /4 + [θy’ - f(0,0)] x2 y2/4 - [ωf’ + f(0,0)] x3y2/4 } } =

565

Appendix Section

y2/2 * x { - x2 y [θx’ - f(0,0)] [θy’ - f(0,0)] - x2y2/2 [ωf’ + f(0,0)] [θy’ - f(0,0)] + 1/2 f(0,0) [ωf’ + f(0,0)] x2 y - [θx’ - f(0,0)] [ωf’ + f(0,0)] x3y /4 + [θy’ - f(0,0)] [ωf’ + f(0,0)] x2 y2/4 - [ωf’ + f(0,0)]2 x3y2/4} = y3/2 * x3 { - [θx’ - f(0,0)] [θy’ - f(0,0)] - [ωf’ + f(0,0)] [θy’ - f(0,0)] y/2 + 1/2 f(0,0) [ωf’ + f(0,0)] - [θx’ - f(0,0)] [ωf’ + f(0,0)] x/4 + [θy’ - f(0,0)] [ωf’ + f(0,0)] y/4 - [ωf’ + f(0,0)]2 xy/4} = y3/2 * x3 { - [θy’ - f(0,0)] [θx’ - f(0,0)] + 1/2 f(0,0) [ωf’ + f(0,0)] - [θx’ - f(0,0)] [ωf’ + f(0,0)] x/4 [θy’ - f(0,0)] [ωf’ + f(0,0)] y/4 - [ωf’ + f(0,0)]2 xy/4 } (24)



The conclusion is that since equation (23) and (24) are not exactly the same then it is true that the Jacobian of second order derivatives is not symmetric.

APPENDIX V. THE SECOND ORDER PARTIAL DERIVATIVES OF ΔE FOR THE TRIVARIATE LINEAR INTERPOLATION FUNCTION Second Order Partial Derivatives of the Intensity-Curvature Functional ΔE ∂2 [ Eo / EIN ] /∂x2 = 2 [ βxx α - βx ( ∂α /∂x ) ] / α2

(1)

α = { Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] }2

(2)

( ∂α /∂x ) = 2 Hyz (x, y, z) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]2 2* (1 - ζx ×) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] [ Hxyz (x, y, z) ]2



(3)

βx = x [ R1 (y, z) + x R2 (y, z) ]

(4)

βxx = ∂ βx / ∂x = ∂ [ x R1 (y, z) + x2 R2 (y, z) ] / ∂x = R1 (y, z) + 2 x R2 (y, z)

(5)

R1 (y, z) = [ f(0,0,0) + y/2 θy + z/2 θz - y/2 z/2 θzy ωf ]

(6)

R2 (y, z) = [ θx - y/2 θxy ωf - z/2 θxz ωf + y/2 z/2 ωf ]

(7)

566

Appendix Section

∂2 [ Eo / EIN ] /∂x∂y = 2 [ βxy α - βx ( ∂α /∂y ) ] / α2

(8)

βxy = ∂ [ x R1 (y, z) + x2 R2 (y, z) ] / ∂y = [ ( θy/2 - θzy ωf z/4 ) + (ωf z/4 - ωf θxy/2) x ] x

(9)

∂2 [ Eo / EIN ] /∂x∂z = 2 [ βxz α - βx ( ∂α /∂z ) ] / α2

(10)

βxz = ∂ [ x R1 (y, z) + x2 R2 (y, z) ] / ∂z = [ ( θz/2 - θzy ωf y/4 ) + ( ωf y/4 - ωf θxz/2 ) x ] x

(11)

( ∂α /∂z ) = 2 Hxy (x, y, z) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]2 2 * (1 - ζz ×) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] [ Hxyz (x, y, z) ]2



(12)

∂2 [ Eo / EIN ] /∂y∂x = 2 [ βyx α - βy ( ∂α /∂x ) ] / α2

(13)

βyx = ∂ [ y R1 (x, z) + y2 R2 (x, z) ] / ∂x = [ ( θx/2 - θxz ωf z/4 ) + ( ωf z/4 - ωf θxy/2 ) y ] y

(14)

R1 (x, z) = [ f(0,0,0) + x/2 θx + z/2 θz - x/2 z/2 θxz ωf ]

(15)

R2 (x, z) = [ θy - x/2 θxy ωf - z/2 θzy ωf + x/2 z/2 ωf ]

(16)

∂2 [ Eo / EIN ] /∂y2 = 2 [ βyy α - βy ( ∂α /∂y ) ] / α2

(17)

βy = y [R1 (x, z) + y R2 (x, z)]

(18)

βyy = ∂ βy /∂y = ∂ [ y R1 (x, z) + y2 R2 (x, z) ] / ∂y =

(19)

R1 (x, z) + 2 y R2 (x, z)

( ∂α /∂y ) = 2 Hxz (x, y, z) Hxyz (x, y, z) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ]2 2 * (1 - ζy ×) [ θxyz - x (1 - ζx ×) - y (1 - ζy ×) - z (1 - ζz ×) ] [ Hxyz (x, y, z) ]2



(20)

∂2 [ Eo / EIN ] /∂y∂z = 2 [ βyz α - βy ( ∂α /∂z ) ] / α2

(21)

βyz = ∂ [ y R1 (x, z) + y2 R2 (x, z) ] / ∂z = [ ( θz/2 - θxz ωf x/4 ) + ( ωf x/4 - ωf θzy/2 ) y ] y

(22)

∂2 [ Eo / EIN ] /∂z∂x = 2 [ βzx α - βz ( ∂α /∂x )] / α2

(23)

βz = z [ R1 (x, y) + z R2 (x, y) ]

(24)

βzz = ∂ βz /∂z = ∂ [ z R1 (x, y) + z2 R2 (x, y) ] / ∂z = R1 (x, y) + 2 z R2 (x, y)

(25)

R1 (x, y) = [ f(0,0,0) + x/2 θx + y/2 θy - x/2 y/2 θxy ωf ]



(26)

R2 (x, y) = [ θz - y/2 θzy ωf - x/2 θxz ωf + x/2 y/2 ωf ]

(27)

567

Appendix Section

βzx = ∂ [ z R1 (x, y) + z2 R2 (x, y) ] / ∂x = [ ( θx/2 - θxy ωf y/4 ) + ( ωf y/4 - ωf θxz/2 ) z ] z

(28)

∂2 [ Eo / EIN ] /∂z∂y = 2 [ βzy α - βz ( ∂α /∂y )] / α2

(29)

βzy = ∂ [ z R1 (x, y) + z2 R2 (x, y) ] / ∂y = [ ( θy/2 - θxy ωf x/4 ) + ( ωf x/4 - ωf θzy/2 ) z ] z

(30)

∂2 [ Eo / EIN ] /∂z2 = 2 [ βzz α - βz ( ∂α /∂z )] / α2

(31)

APPENDIX VI. PROOFS TO THE MATHEMATICAL ASSERTIONS OF CHAPTER X IV Study of the Intensity-Curvature Functional ΔE of the Quadratic B-Spline Let us calculate the first order derivative of the Intensity-Curvature Functional of the quadratic B-Spline h3(x): (∂ ( ∆E(x) ) /∂x) = { θ[a,0] * { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] (2a + 1/2)/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } θ[a,0] x { 3 x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - 2 x [ θ[a,3/2] (2a + 1/2)/2 ] +

[ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } } /

{ x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] (2a + 1/2)/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] }2

(1)

(∂ ( ∆E(x) ) /∂x) = 0 implies that the numerator of equation (1) is posed equal to zero and solved: { θ[a,0] * { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] (2a + 1/2)/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } θ[a,0] x { 3 x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - 2 x [ θ[a,3/2] (2a + 1/2)/2 ] +

[ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } } = 0

That is: { θ[a,0] * { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] (2a + 1/2)/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } =

568

Appendix Section

θ[a,0] x { 3 x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - 2 x [ θ[a,3/2] (2a + 1/2)/2 ] +

[ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] }

x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x [ θ[a,3/2] (2a + 1/2)/2 ] + [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] - 3 x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] + 2 x [ θ[a,3/2] (2a + 1/2)/2 ] - [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] = 0

(2)

Solution of a 4th Degree Polynomial (2008, Carlo Ciulla, IGI Global) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the quartic equation is: http://en.wikipedia.org/wiki/Quartic_equation Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. Let the polynomial: ψ 4 φ4 + ψ 3 φ3 + ψ 2 φ2 + ψ φ1 + φ0 = 0

(3)

be written as: ψ 4 + ψ 3 ( φ3 / φ4) + ψ 2 ( φ2 / φ4) + ψ ( φ1 / φ4) + ( φ0 / φ4) = 0

(4)

Let us posit ψ = ( x - ( φ3 / 4 φ4)) and employ the Ferrari’s method. ( x - ( φ3 / 4 φ4))2 ( x - ( φ3 / 4 φ4))2 + ( x - ( φ3 / 4 φ4)) ( x - ( φ3 / 4 φ4))2 ( φ3 / φ4) + ( x - ( φ3 / 4 φ4))2 ( φ2 / φ4) + ( x - ( φ3 / 4 φ4)) ( φ1 / φ4) + ( φ0 / φ4) = 0

(5)

( x - ( φ3 / 4 φ4))2 [( x - ( φ3 / 4 φ4))2 + ( x - ( φ3 / 4 φ4)) ( φ3 / φ4) + ( φ2 / φ4) ] + ( x - ( φ3 / 4 φ4)) ( φ1 / φ4) + ( φ0 / φ4) = 0

569

Appendix Section

( x2 + ( φ3 / 4 φ4)2 - 2 x ( φ3 / 4 φ4) ) [ ( x2 + ( φ3 / 4 φ4)2 - 2 x ( φ3 / 4 φ4) ) + x ( φ3 / φ4) ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ] + x ( φ1 / φ4) - ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) = 0 x4 + x2 ( φ3 / 4 φ4)2 - 2 x3 ( φ3 / 4 φ4) + x3 ( φ3 / φ4) - x2 ( φ3 / 4 φ4) ( φ3 / φ4) + x2 ( φ2 / φ4) + x2 ( φ3 / 4 φ4)2 + ( φ3 / 4 φ4)4 - 2 x ( φ3 / 4 φ4)3 + x ( φ3 / φ4) ( φ3 / 4 φ4)2 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 - 2 x3 ( φ3 / 4 φ4) - 2 x ( φ3 / 4 φ4)3 + 4 x2 ( φ3 / 4 φ4)2 - 2 x2 ( φ3 / 4 φ4) ( φ3 / φ4) + 2 x ( φ3 / 4 φ4)2 ( φ3 / φ4) - 2 x ( φ3 / 4 φ4) ( φ2 / φ4) + x ( φ1 / φ4) - ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) = 0 x4 + x2 [6 ( φ3 / 4 φ4)2 - 3 ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ] + x [ -4 ( φ3 / 4 φ4)3 + 3 ( φ3 / φ4) ( φ3 / 4 φ4)2 - 2 ( φ3 / 4 φ4) ( φ2 / φ4) + ( φ1 / φ4) ] + [ ( φ3 / 4 φ4)4 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 - ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) ] = 0

(6)

Let us posit: α = [6 ( φ3 / 4 φ4)2 - 3 ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ] β = [ -4 ( φ3 / 4 φ4)3 + 3 ( φ3 / φ4) ( φ3 / 4 φ4)2 - 2 ( φ3 / 4 φ4) ( φ2 / φ4) + ( φ1 / φ4) ] γ = [ ( φ3 / 4 φ4)4 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) ] Thus: x4 + α x2 + β x + γ = 0 Let us consider: ( x2 + α )2 - x4 - 2 α x2 = x4 + 2 α x2 + α2 - x4 - 2 α x2 = α2 x4 + 2 α x2 + α2 = α2 + x4 + 2 α x2 = ( x2 + α )2 x4 = ( x2 + α )2 - α2 - 2 α x2 ( x2 + α )2 - α2 - 2 α x2 + α x2 + β x + γ = 0 ( x2 + α )2 - α2 - α x2 + β x + γ = 0

570

(7)

Appendix Section

( x2 + α )2 + β x + γ = α2 + α x2

(8)

Let us consider the variable y within the two following propositions: ( x2 + α + y )2- ( x2 + α )2 = x4 + 2 x2 α + 2 x2 y + 2 y α + y2 + α2 - α2 - x4 - 2 α x2 = 2y ( α + x2 ) + y2 = 2 y α + 2 y x2 + y2 ( x2 + α + y )2- ( x2 + α )2 = 2 y α + 2 y x2 + y2



(9)

Also: ( α + 2y ) x2 = 2 y x2 + α x2 0 = ( α + 2y ) x2 - 2 y x2 - α x2

(10)

Adding the two propositions: ( x2 + α + y )2- ( x2 + α )2 = 2 y α + 2 y x2 + y2 + ( α + 2y ) x2 - 2 y x2 - α x2 = 2 y α + y2 + ( α + 2y ) x2 - α x2 Adding: ( x2 + α + y )2- ( x2 + α )2 = 2 y α + y2 + ( α + 2y ) x2 - α x2

(11)

To: ( x2 + α )2 + β x + γ = α2 + α x2

(12)

Results into: ( x2 + α + y )2- ( x2 + α )2 + ( x2 + α )2 + β x + γ = 2 y α + y2 + ( α + 2y ) x2 - α x2 + α2 + α x2 = ( α + 2y ) x2 + ( 2 y α + y2 + α2 ) ( x2 + α + y )2 + β x + γ = ( α + 2y ) x2 + ( 2 y α + y2 + α2 ) ( x2 + α + y )2 = ( α + 2y ) x2 - β x + ( 2 y α + y2 + α2 - γ )

(13)

Let us consider now the quadratic function: ( s x + t )2 = ( s x )2 + 2 s x t + t2 = ( s2 ) x2 + ( 2 s t ) x + t2

571

Appendix Section

( 2 s t )2 - 4 s2 t2 = 0 shows that: { ( s u ) + [ ( 2 s t ) / ( 2 s ) ] }2 = { s u + t }2 is a perfect square. The aim is to make a perfect square the right side of the equation (13): ( x2 + α + y )2 = ( α + 2y ) x2 - β x + ( 2 y α + y2 + α2 - γ ) Therefore let us solve: ( - β )2 - 4 ( α + 2y ) ( 2 y α + y2 + α2 - γ ) = 0 The equation: ( - β )2 - 4 ( α + 2y ) ( 2 y α + y2 + α2 - γ ) = 0 ( - β )2 = 4 ( α + 2y ) ( 2 y α + y2 + α2 - γ ) [ ( - β )2 / 4 ( α + 2y ) ] = ( 2 y α + y2 + α2 - γ ) Shows that: ( α + 2y ) x2 - β x + ( 2 y α + y2 + α2 - γ ) = ( α + 2y ) x2 - β x + [ ( - β )2 / 4 ( α + 2y ) ] = { ( α + 2y )1/2 x + [ ( - β ) / 2 ( α + 2y )1/2 ] }2 Therefore: ( x2 + α + y )2 = { ( α + 2y )1/2 x + [ ( - β ) / 2 ( α + 2y )1/2 ] }2

(14)

Also being: ( - β )2 - 4 ( α + 2y ) ( 2 y α + y2 + α2 - γ ) = 0 it follows that: β2 - 4 [ 2 y α2 + y2 α + (α3 - α γ ) + 2 y3 + 4y2 α + 2y α2 - 2 γ y ] = β2 - 4 [ 5 y2 α + (α3 - α γ ) + 2 y3 + 4y α2 - 2 γ y ] = β2 - 4 [ 5 y2 α + (α3 - α γ ) + 2 y3 + (4 α2 - 2 γ ) y ] = 2 y3 + 5 y2 α + (4 α2 - 2 γ ) y + (α3 - α γ - β2 /4 ) = 0 y3 + (5/2) y2 α + (2 α2 - γ ) y + (α3 - α γ - β2 /4 ) / 2 = 0

572

(15)

Appendix Section

And this is a cubic equation which solution will lead to the solution of the starting quartic equation. Let us posit y = ( v - (5/6) α ) and place it into the cubic equation (15) such that: ( v - (5/6) α )3 + ( v - (5/6) α )2 (5/2) α + (2 α2 - γ ) ( v - (5/6) α ) + (α3 - α γ - β2 /4 ) / 2 = 0 ( v - (5/6) α )2 ( v - (5/6) α ) + ( v - (5/6) α )2 (5/2) α + (2 α2 - γ ) ( v - (5/6) α ) + (α3 - α γ - β2 /4 ) / 2 = 0 ( v2 - (5/3) α v + ( (5/6) α )2) ( v - (5/6) α ) + ( v2 - (5/3) α v + ( (5/6) α )2) (5/2) α + (2 α2 - γ ) ( v - (5/6) α ) + (α3 - α γ - β2 /4 ) / 2 = 0 ( v3 - (5/6) α v2 - (5/3) α v2 + (5/6) (5/3) α2 v + ( (5/6) α )2 v - ( (5/6) α )3 ) + ( v2 (5/2) α - (5/3) α2 v (5/2) + ( (5/6) α )2 (5/2) α) + (2 α2 - γ ) v - (2 α2 - γ ) (5/6) α + (α3 - α γ - β2 /4 ) / 2 = 0 ( v3 + (5/6) (5/3) α2 v + ( (5/6) α )2 v - ( (5/6) α )3 ) + (- (5/3) α2 v (5/2) + ( (5/6) α )2 (5/2) α) + (2 α2 - γ ) v - (2 α2 - γ ) (5/6) α + (α3 - α γ - β2 /4 ) / 2 = 0 v3 + v [ (5/6) (5/3) α2 + ( (5/6) α )2 - (5/3) α2 (5/2) + (2 α2 - γ ) ] ( (5/6) α )3 + ( (5/6) α )2 (5/2) α - (2 α2 - γ ) (5/6) α + [ (α3 - α γ - β2 /4 ) / 2 ] = 0

(16)

Let us posit: p = [ (5/6) (5/3) α2 + ( (5/6) α )2 - (5/3) α2 (5/2) + (2 α2 - γ ) ] q = - ( (5/6) α )3 + ( (5/6) α )2 (5/2) α - (2 α2 - γ ) (5/6) α + [ (α3 - α γ - β2 /4 ) / 2 ] to obtain: v3 + v p + q = 0

(17)

Any of the three complex roots: u = { ( q /2 ) ± { ( q /2 )2 + ( p3 / 27) }1/2 }1/3 v=[(p/3u)-u] will solve the cubic equation (17): v3 + v p + q = 0

573

Appendix Section

Since it was posed that y = ( v - (5/6) α ), then the cubic equation in y is solved by: y = [ ( p / 3 u ) - u ] - (5/6) α Being: ( x2 + α + y )2 = { ( α + 2y )1/2 x + [ ( - β ) / 2 ( α + 2y )1/2 ] }2 it results into: ( x2 + α + y )2 = ± { ( α + 2y )1/2 x + [ ( - β ) / 2 ( α + 2y )1/2 ] } and this can be written as: x2 + { - ± ( α + 2y )1/2 } x + { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } = 0 and its solution is: x = ± { { ( α + 2y )1/2 } + { ( α + 2y ) - 4 * { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } }1/2 } / 2

(18)

and being ψ = ( x - ( φ3 / 4 φ4)) from the quartic equation as above, it results that the solutions to the quartic are: ψ = ± { { { ( α + 2y )1/2 } + { ( α + 2y ) - 4 * { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } }1/2 } / 2 } - ( φ3 / 4 φ4)) The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve the quartic equation. Also, the author wishes to acknowledge that the main author of the solution of the quartic equation is the Italian mathematician Lodovico Ferrari, also the main reference to the solution of the quartic equation as well as to the Galois Theory (Stewart, 2004) as reported at: http://en.wikipedia.org/wiki/Quartic_equation.

The Derivation of the Novel Re-sampling Locations of the Q uadratic B-Spline Let us recall from Chapter XIV that the Sub-pixel Efficacy Region of the quadratic B-Spline h3(x) is: - 2 x2 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] + x [ θ[a,3/2] (2a + 1/2)/2 ] = 0

(19)

x(1) = 0

(20.a)

x(2) = [ θ[a,3/2] (2a + 1/2)/2 ] / { 2 * [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] }

(20.b)

574

Appendix Section

To obtain the novel re-sampling locations that improve the interpolation error for the B-Spline h3 (x), based on xsre = xsre(i) i =1, 2 of equations (20.a) and (20.b), the following equation is employed: { h((xsre - xr0 )/ xsre) * (∂2 (h(x)) /∂x2) (xsre - xr0) } / { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { EIN (xsre-x0) / EIN (xsre) }

(21)

to determine its solution through the following math deduction. Let us calculate: h((xsre - xr0 )/ xsre) = { f (0) + [ f (1/2) + f (-1/2) ] * [ -2 a ((xsre - xr0 )/ xsre)2 + 1/2 ( a +1 ) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a ((xsre - xr0 )/ xsre)2 - (2a + 1/2) * ((xsre - xr0 )/ xsre) + 3/4 (a+1) ] }

(22)

It is worth noting that the value of ((xsre - xr0 )/ xsre) is employed in equation (22) with the specific purpose to obtain the numerical value of the novel re-sampling location x r0 as percentage increase (or decrease) with respect to the numerical value of the SRE location xsre. The effectiveness of such purpose was empirically validated through experimentation. h(xsre) = { f (0) + [ f (1/2) + f (-1/2) ] * [ -2 a (xsre)2 + 1/2 ( a +1 ) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a (xsre)2 - (2a + 1/2) * (xsre) + 3/4 (a+1) ] }

(23)

Where: (∂2 (h(x)) /∂x2) (xsre - xr0) = (∂2 (h(x)) /∂x2) (xsre) = (∂2 (h3(x)) /∂x2) = - 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] = [ θ[a,0] / f (0) ]

(24)

is furnished by: (∂2 ( h3(x) )/∂x2) = (∂ (A)) /∂x) = - 4 a * [ f (1/2) +f (-1/2) ] + 2 a * [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] = - 4 a * [ f (1/2) +f (-1/2) ] + 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] = - 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ]



(25)

575

Appendix Section

Where the first order derivative of h3(x) with respect to the variable x: (∂ ( h3(x) ) /∂x) = { -4 a x * [ f (1/2) +f (-1/2) ] + [ 2 a x - (2 a + 1/2 ) ] * [(f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] } = A where it is posited that: θ[a,0] = {- 2 a * [ f (1/2) + f (-1/2) ] + 2 a * [ f (-1) + f (1) + f (3/2) + f (-3/2) ] } * f (0)

(26)

Let us calculate ΔE* = EIN (xsre-x0) / EIN (xsre) based on the formulation given by equation (9) of Chapter XIV herein reported: ∆E(x) = θ[a,0] x / { x3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - x2 [ θ[a,3/2] (2a + 1/2)/2 ] + x [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } Thus: EIN (xsre-x0) = { (xsre-x0)3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - (xsre-x0)2 [ θ[a,3/2] (2a + 1/2)/2] + (xsre-x0) [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } and EIN (xsre) = { xsre3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - xsre2 [ θ[a,3/2] (2a + 1/2)/2 ] + xsre [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } Therefore: ΔE* = EIN (xsre-x0) / EIN (xsre) = { (xsre - x0)3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - (xsre-x0)2 [ θ[a,3/2] (2a + 1/2)/2 ] + (xsre - x0) [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] } / { xsre3 [ θ[a,3/2] a/3 - θ[a,1/2] 2a/3 ] - xsre2 [ θ[a,3/2] (2a + 1/2)/2 ] + xsre [ θ[a,0] + θ[a,1/2] ( a + 1)/2 + θ[a,3/2] 3( a + 1 )/4 ] }

(27)

Because from equation (24) it follows that: (∂2 (h(x)) /∂x2) (xsre - xr0) = (∂2 (h(x)) /∂x2) (xsre), based on equation (21) it can be written that: { h((xsre - xr0 )/ xsre) } = { EIN (xsre-x0) / EIN (xsre) } * { h(xsre) } Now, employing equations (22) through (24) it can be written that: { h((xsre - xr0 )/ xsre) * (∂2 (h(x)) /∂x2) (xsre - xr0) } = { f (0) + [ f (1/2) + f (-1/2) ] *

576

(28)

Appendix Section

[ -2 a ((xsre - xr0 )/ xsre)2 + 1/2 ( a +1 ) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a ((xsre - xr0 )/ xsre)2 - (2a + 1/2) * ((xsre - xr0 )/ xsre) + 3/4 (a+1) ] } * [ θ[a,0] / f (0) ] { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { f (0) + [ f (1/2) + f (-1/2) ] * [ -2 a (xsre)2 + 1/2 ( a +1 ) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a (xsre)2 - (2a + 1/2) * (xsre) + 3/4 (a+1) ] } * [ θ[a,0] / f (0) ] Thus: { h((xsre - xr0 )/ xsre) * (∂2 (h(x)) /∂x2) (xsre - xr0) } / { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { h((xsre - xr0 )/ xsre) } / { h(xsre) } = { f (0) + [ f (1/2) + f (-1/2) ] * [ -2 a ((xsre - xr0 )/ xsre)2 + 1/2 ( a +1 ) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a ((xsre - xr0 )/ xsre)2 - (2a + 1/2) * ((xsre - xr0 )/ xsre) + 3/4 (a+1) ] } / { f (0) + [ f (1/2) + f (-1/2) ] * [ -2 a (xsre)2 + 1/2 ( a +1 ) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a (xsre)2 - (2a + 1/2) * (xsre) + 3/4 (a+1) ] }

(29)

And from equations (22) and (28) it follows that: { f (0) + [ f (1/2) + f (-1/2) ] * [ -2 a ((xsre - xr0 )/ xsre)2 + 1/2 ( a +1 ) ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ] * [ a ((xsre - xr0 )/ xsre)2 - (2a + 1/2) * ((xsre - xr0 )/ xsre) + 3/4 (a+1) ] } = { EIN (xsre-x0) / EIN (xsre) } * { h(xsre) } Let us posit: f1 = [ f (1/2) + f (-1/2) ]; f2 = [ f (1/2) + f (-1/2) + f (-1) + f (1) + f (3/2) + f (-3/2) ]; Thus: { EIN (xsre-x0) / EIN (xsre) } * { h(xsre) } - f (0) = - f1 2 a ((xsre - xr0 )/ xsre)2 + f1 1/2 ( a + 1 ) + f2 a ((xsre - xr0 )/ xsre)2 - f2 (2a + 1/2) * ((xsre - xr0 )/ xsre) + f2 3/4 ( a + 1 )

577

Appendix Section

{ EIN (xsre-x0) / EIN (xsre) } * { h(xsre) } - f (0) = (- 2 f1 + f2) a ((xsre - xr0 )/ xsre)2 - f2 (2a + 1/2) * ((xsre - xr0 )/ xsre) + f1 1/2 ( a + 1 ) + f2 3/4 ( a + 1 ) Thus: { [ EIN (xsre-x0) / EIN (xsre) ] * [ h(xsre) ] - f (0) - f1 1/2 ( a +1 ) - f2 3/4 ( a + 1 ) } = (- 2 f1 + f2) a ((xsre - xr0 )/ xsre)2 - f2 (2a + 1/2) * ((xsre - xr0 )/ xsre) -{ [ EIN (xsre-x0) / EIN (xsre) ] * [ h(xsre) ] - f (0) - f1 1/2 ( a +1 ) - f2 3/4 ( a + 1 ) } + (- 2 f1 + f2) a ((xsre - xr0 )/ xsre)2 - f2 (2a + 1/2) * ((xsre - xr0 )/ xsre) = 0 (- 2 f1 + f2) a ((xsre - xr0 )/ xsre)2 - f2 (2a + 1/2) * ((xsre - xr0 )/ xsre) - { [ EIN (xsre-x0) / EIN (xsre) ] * [ h(xsre) ] - f (0) - f1 1/2 ( a +1 ) - f2 3/4 ( a + 1 ) } = 0 This is solved as: ((xsre - xr0 )/ xsre) = { f2 (2a + 1/2) ± [ Δq ]1/2 } / [ 2 * (- 2 f1 + f2) a ]

(30)

Where: Δq = { [ f2 (2a + 1/2) ]2 + 4 * (- 2 f1 + f2) a * { [ EIN (xsre-x0) / EIN (xsre) ] * [ h(xsre) ] - f (0) - f1 1/2 ( a +1 ) - f2 3/4 ( a + 1 ) } } Equation (30) furnishes the following 4 solutions: xr0(k, i) = (xsre(i) - ρ(k) xsre(i)) with i, k = 1, 2.

(31)

Where: ρ(k) = { f2 (2a + 1/2) ± [ Δq ]1/2 } / [ 2 * (- 2 f1 + f2) a ] is calculated employing xsre(k) ( k = 1, 2 ) dictacted by equations (20.a) and (20.b). Solutions of equation (31) are the four novel-resampling locations of the B-Spline h3(x).

Calculation of the Intensity-Curvature Terms Before and After Interpolation for the Cubic B-Spline Let us calculate the first order derivative of h4(x) with respect to the variable x: (∂ ( h4(x) ) /∂x) = { [ f (1/2) + f (-1/2) + f (1) + f (-1) ] * [ 3/2 x2 - 2 x ] + [ f (-1) + f (1) + f (3/2)

578

Appendix Section

+ f (-3/2) + f (2) + f (-2) ] * [ -3/6 x2 + 2 x - 2 ] } = B It follows that the second order derivative of h4(x) with respect to the variable x is calculated as: (∂2 ( h4(x) ) /∂x2) = (∂ (B)) /∂x) = { [ f (1/2) + f (-1/2) + f (1) + f (-1) ] * [ 3 x - 2 ] + [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] * [ - x + 2 ] }

(32)

It follows that: (∂2 ( h4(x) ) /∂x2) (0) = {- 2 * [ f (1/2) + f (-1/2) + f (1) + f (-1) ] + 2 * [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] } Let the intensity-curvature term before interpolation be defined as: x Eo = Eo (x) = ∫ f (0) (∂2 (h4(x))/∂x2) (0) dx =



0

x ∫ {- 2 * [ f (1/2) +f (-1/2) + f (1) + f (-1) ] + 0

2 * [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] } * f (0) dx = x ∫ θ[a,10] dx = x θ[a,10]

(33)

0

where it is posited that: θ[a,10] = (∂2 (h4(x))/∂x2) (0, 0) * f (0) = {- 2 * [ f (1/2) + f (-1/2) + f (1) + f (-1) ] + 2 * [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] } * f (0) Let the intensity-curvature term after interpolation be defined as: x EIN = EIN (x) = ∫ h4 (x) (∂2 (h4(x))/∂x2) dx =



0

x ∫ { f (0) + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 x3 - x2 + 2/3 ] + [f (3/2) + f (- 3/2) + 0

f (1) + f (-1) + f (2) + f (-2)] * [ -1/6 x3 + x2 - 2 x + 4/3 ] } * { [ f (1/2) + f (-1/2) + f (-1) + f (1) ] *

579

Appendix Section

[ 3 x - 2 ] + [f (3/2) + f (- 3/2) + f (1) + f (-1) + f (2) + f (-2)] * [ - x + 2 ] } dx = x ∫ f (0) [ f (1/2) +f (-1/2) + f (1) + f (-1) ] * [ 3 x - 2 ] + f (0) [f (3/2) + f (- 3/2) + f (1) + f (-1) + 0

f (2) + f (-2) ] * [ - x + 2 ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 x3 - x2 + 2/3 ] [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 3 x - 2 ] + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 x3 - x2 + 2/3 ] [f (3/2) + f (- 3/2) + f (1) + f (-1) + f (2) + f (-2) ] * [ - x + 2 ] + [f (3/2) + f (- 3/2) + f (1) + f (-1) + f (2) + f (-2)] * [ -1/6 x3 + x2 - 2 x + 4/3 ] [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 3 x - 2 ] + [f (3/2) + f (- 3/2) + f (1) +f (-1) +f (2) + f (-2)] * [ -1/6 x3 + x2 - 2 x + 4/3 ] [f (3/2) + f (- 3/2) + f (1) +f (-1) +f (2) + f (-2)]* [ - x + 2 ] dx

(34)

Let us posit that: θ[a,1] = f (0) [ f (1/2) +f (-1/2) + f (1) + f (-1) ] θ[a,2] = f (0) [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] θ[a,11] = [ f (1/2) + f (-1/2) + f (-1) + f (1) ]2 θ[a,12] = [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] θ[a,22] = [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] 2 Therefore equation (34) becomes: x ∫ θ[a,1] * [ 3 x - 2 ] + θ[a,2] * [ - x + 2 ] + θ[a,11] * [ 1/2 x3 - x2 + 2/3 ] * [ 3 x - 2 ] + θ[a,12] * { [ 1/2 x3 0

- x2 + 2/3 ] * [ - x + 2 ] + [ -1/6 x3 + x2 - 2 x + 4/3 ] * [ 3 x - 2 ] } + θ[a,22] * [ -1/6 x3 + x2 - 2 x + 4/3 ] * [ - x + 2 ] dx = x ∫ θ[a,1] * [ 3 x - 2 ] + θ[a,2] * [ - x + 2 ] + θ[a,11] * [ 3/2 x4 - 3 x3 + 2 x - x3 + 2 x2 - 4/3 ] + θ[a,12] * { [ 0

580

Appendix Section

-1/2 x4 + x3 - 2/3 x + x3 - 2 x2 + 4/3 ] + [ -1/2 x4 + 3 x3 - 6 x2 + 4 x + 1/3 x3 - 2 x2 + 4 x - 8/3] } + θ[a,22] * [ 1/6 x4 - x3 + 2 x2 - 4/3 x - 1/3 x3 + 2 x2 - 4 x + 8/3 ] dx =

x ∫ θ[a,1] * [ 3 x - 2 ] + θ[a,2] * [ - x + 2 ] + θ[a,11] * [ 3/2 x4 - 4 x3 + 2 x2 + 2 x - 4/3 ] + θ[a,12] * { [ 0

-1/2 x4 + 2 x3 - 2 x2 - 2/3 x + 4/3 ] + [ -1/2 x4 + 7/3 x3 - 8 x2 + 8 x - 8/3] } + θ[a,22] * [ 1/6 x4 - 4/3 x3 + 4 x2 - 16/3 x + 8/3 ] dx = x ∫ θ[a,1] * [ 3 x - 2 ] + θ[a,2] * [ - x + 2 ] + θ[a,11] * [ 3/2 x4 - 4 x3 + 2 x2 + 2 x - 4/3 ] + 0

θ[a,12] * [ - x4 + 13/3 x3 - 10 x2 + 22/3 x - 4/3 ] + θ[a,22] * [ 1/6 x4 - 4/3 x3 + 4 x2 - 16/3 x + 8/3 ] dx = θ[a,1] * [ 3/2 x2 - 2 x ] + θ[a,2] * [ - 1/2 x2 + 2 x ] + θ[a,11] * [ 3/10 x5 - x4 + 2/3 x3 + x2 - 4/3 x ] + θ[a,12] * [ - 1/5 x5 + 13/12 x4 – 10/3 x3 + 22/6 x2 - 4/3 x ] + θ[a,22] * [ 1/30 x5 - 4/12 x4 + 4/3 x3 16/6 x2 + 8/3 x ] Therefore:

x EIN = EIN (x) = ∫ h4 (x) (∂2 (h4(x))/∂x2) dx =



0

θ[a,1] * [ 3/2 x2 - 2 x ] + θ[a,2] * [ - 1/2 x2 + 2 x ] + θ[a,11] * [ 3/10 x5 - x4 + 2/3 x3 + x2 - 4/3 x ] + θ[a,12] * [ - 1/5 x5 + 13/12 x4 - 10/3 x3 + 22/6 x2 - 4/3 x ] + θ[a,22] * [ 1/30 x5 - 4/12 x4 + 4/3 x3 16/6 x2 + 8/3 x ]

Study of the Intensity-Curvature Functional ΔE of the Cubic B-Spline Let us calculate the first order derivative of the Intensity-Curvature Functional of the cubic B-Spline h4(x) from the equation: ∆E(x) = θ[a,10] x / { θ[a,1] * [ 3/2 x2 - 2 x ] + θ[a,2] * [ - 1/2 x2 + 2 x ] + θ[a,11] * [ 3/10 x5 - x4 + 2/3 x3 + x2 - 4/3 x ] + θ[a,12] * [ - 1/5 x5 + 13/12 x4 - 10/3 x3 + 22/6 x2 - 4/3 x ] + θ[a,22] * [ 1/30 x5 - 4/12

581

Appendix Section

x4 + 4/3 x3 - 16/6 x2 + 8/3 x ] } = θ[a,10] / { θ[a,1] * [ 3/2 x - 2 ] + θ[a,2] * [ - 1/2 x + 2 ] + θ[a,11] * [ 3/10 x4 - x3 + 2/3 x2 + x - 4/3 ] + θ[a,12] * [ - 1/5 x4 + 13/12 x3 - 10/3 x2 + 22/6 x - 4/3 ] + θ[a,22] * [ 1/30 x4 - 4/12 x3 + 4/3 x2 - 16/6 x + 8/3 ] }

(35)

Therefore: (∂ (∆E(x)) /∂x) = - θ[a,10] * C / { θ[a,1] * [ 3/2 x - 2 ] + θ[a,2] * [ - 1/2 x + 2 ] + θ[a,11] * [ 3/10 x4 - x3 + 2/3 x2 + x - 4/3 ] + θ[a,12] * [ - 1/5 x4 + 13/12 x3 - 10/3 x2 + 22/6 x - 4/3 ] + θ[a,22] * [ 1/30 x4 4/12 x3 + 4/3 x2 - 16/6 x + 8/3 ] }2 Where: C = (∂ { θ[a,1] * [ 3/2 x - 2 ] + θ[a,2] * [ - 1/2 x + 2 ] + θ[a,11] * [ 3/10 x4 - x3 + 2/3 x2 + x - 4/3 ] + θ[a,12] * [ - 1/5 x4 + 13/12 x3 - 10/3 x2 + 22/6 x - 4/3 ] + θ[a,22] * [ 1/30 x4 - 4/12 x3 + 4/3 x2 - 16/6 x + 8/3 ] } /∂x) = { θ[a,1] * [ 3/2 ] + θ[a,2] * [ - 1/2 ] + θ[a,11] * [ 12/10 x3 - 3 x2 + 4/3 x + 1 ] + θ[a,12] * [ - 4/5 x3 + 39/12 x2 - 20/3 x + 22/6 ] + θ[a,22] * [ 4/30 x3 - x2 + 8/3 x - 16/6 ] }

(36)

To be (∂ (∆E(x)) /∂x) = 0 it has to be C = 0 as given by equation (36) above. It follows that: { θ[a,1] * [ 3/2 ] + θ[a,2] * [ - 1/2 ] + θ[a,11] * [ 12/10 x3 - 3 x2 + 4/3 x + 1 ] + θ[a,12] * [ - 4/5 x3 + 39/12 x2 - 20/3 x + 22/6 ] + θ[a,22] * [ 4/30 x3 - x2 + 8/3 x - 16/6 ] } = 0 x3 { 12/10 θ[a,11] - 4/5 θ[a,12] + 4/30 θ[a,22] } + x2 {- 3 θ[a,11] + 39/12 θ[a,12] - θ[a,22]} + x { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } + { 3/2 θ[a,1] - 1/2 θ[a,2] + θ[a,11] + 22/6 θ[a,12] - 16/6 θ[a,22] } = 0

582

Appendix Section

x3 { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } + x2 {- 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22]} + x { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } + { 3/2 θ[a,1] - 1/2 θ[a,2] + θ[a,11] + 11/3 θ[a,12] - 8/3 θ[a,22] } = 0

(37)

The Sub-Pixel Efficacy Region of the Cubic B-Spline Equation (37) can be written as: x { x2 [ 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] ] + x [ - 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22] ] + [ 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] ] }+ { -3/2 θ[a,1] + 1/2 θ[a,2] - θ[a,11] - 11/3 θ[a,12] + 8/3 θ[a,22] } = 0 Let us posit: φ0 = { 3/2 θ[a,1] - 1/2 θ[a,2] + θ[a,11] + 11/3 θ[a,12] - 8/3 θ[a,22] } φ1 = { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } φ2 = {- 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22]} φ3 = { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } Equation (37) becomes: x3 φ3 + x2 φ2 + x φ1 + φ0 = 0 x3 + x2 (φ2 / φ3) + x (φ1 / φ3) + ( φ0 / φ3) = 0



Solution

(2008, Carlo Ciulla, IGI Global) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the cubic equation is: http://it.wikipedia.org/wiki/Equazione_di_terzo_grado Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with

583

Appendix Section

no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. x3 + x2 (φ2 / φ3) + x (φ1 / φ3) + ( φ0 / φ3) = 0 is solved by the method illustrated herein by rewriting the equation in the form: y3 + py + q = 0 with x = [ y - (φ2 / 3 φ3) ] [ y - (φ2 / 3 φ3) ] 3 + [ y - (φ2 / 3 φ3) ]2 (φ2 / φ3) + [ y - (φ2 / 3 φ3) ] (φ1 / φ3) + ( φ0 / φ3) = 0 [ y - (φ2 / 3 φ3) ] [ y - (φ2 / 3 φ3) ] 2 + [ y - (φ2 / 3 φ3) ]2 (φ2 / φ3) + [ y - (φ2 / 3 φ3) ] (φ1 / φ3) + ( φ0 / φ3) = 0



[ y - (φ2 / 3 φ3) ] [ y2 - 2 y (φ2 / 3 φ3) + (φ2 / 3 φ3)2 ] + [ y2 - 2 y (φ2 / 3 φ3) + (φ2 / 3 φ3)2 ] (φ2 / φ3) + [ y - (φ2 / 3 φ3) ] (φ1 / φ3) + ( φ0 / φ3) = 0 [ y2 - 2 y (φ2 / 3 φ3) + (φ2 / 3 φ3)2 ] [ y - (φ2 / 3 φ3) + (φ2 / φ3) ] + [ y - (φ2 / 3 φ3) ] (φ1 / φ3) + ( φ0 / φ3) = 0 [ y2 - 2 y (φ2 / 3 φ3) + (φ2 / 3 φ3)2 ] [ y + (2 φ2 / 3 φ3) ] + [ y - (φ2 / 3 φ3) ] (φ1 / φ3) + ( φ0 / φ3) = 0 y3 + (2 φ2 / 3 φ3) y2 - 2 y2 (φ2 / 3 φ3) - (4/9) y (φ2 / φ3)2 + (φ2 / 3 φ3)2 y + (2/27) (φ2 / φ3)2 + y (φ1 / φ3) - (φ2 / 3 φ3) (φ1 / φ3) + ( φ0 / φ3) = 0 y3 - y (φ2 / 9 φ3)2 + y (φ1 / φ3) + (2/27) (φ2 / φ3)2 - (φ2 / 3 φ3) (φ1 / φ3) + ( φ0 / φ3) = 0 y3 + y [ -(φ2 / 9 φ3)2 + (φ1 / φ3) ] + [ (2/27) (φ2 / φ3)2 - (φ2 / 3 φ3) (φ1 / φ3) + ( φ0 / φ3) ] = 0 now let y = z - ( p / 3z )

584

(38)

Appendix Section

Also let: [ - (φ2 / 9 φ3)2 + (φ1 / φ3) ] = p [ (2/27) (φ2 / φ3)2 - (φ2 / 3 φ3) (φ1 / φ3) + ( φ0 / φ3) ] = q ( z - (p / 3z) )3 + ( z - ( p / 3z ) ) p + q = 0 ( z - (p / 3z) ) ( z - (p / 3z) )2 + ( z - ( p / 3z ) ) p + q = 0 ( z - (p / 3z) ) ( z2 - 2 z (p / 3z) + (p / 3z)2 ) + ( z - ( p / 3z ) ) p + q = 0 ( z - (p / 3z) ) ( z2 - (2/3) p + (p / 3z)2 ) + ( z - ( p / 3z ) ) p + q = 0 z3- (2/3) p z + z ( p / 3z )2 - (p / 3z) z2 + (2/3) p ( p / 3z ) - ( p / 3z )3 + z p - ( p / 3z ) p + q = 0 z3 + (1/3) z p - (1/3) p (p / 3z) - (p / 3z)3 + z ( p / 3z )2 - (p / 3z) z2 + q = 0 Now multiply by 27 z3 to obtain: 27 z6 + 27 (1/3) z4 p - 27 z3 (1/3) p (p / 3z) - 27 z3 (p / 3z)3 + 27 z4 ( p / 3z )2 - 27 (p / 3z) z5 + 27 z3 q = 0 27 z6 + 9 z4 p - 9 z3 p (p / 3z) - p3 + 3 z2 p - 9 p z4 + 27 z3 q = 0 27 z6 + 9 z4 p - 3 z2 p2 - p3 + 3 z2 p2 - 9 p z4 + 27 z3 q = 0 27 z6 + 27 z3 q - p3 = 0

(39)

zi = { ( q / 2 ) ± { ( q / 2 )2 + ( p3 / 27 ) }1/2 }1/3

i = 1, 2, 3

(40)

yi = zi - ( p / 3zi )

i = 1, 2, 3

(41)

i = 1, 2, 3

(42)

with: xsre(i) = [ yi - (φ2 / 3 φ3) ] where: q = [ (2/27) (φ2 / φ3)2 - (φ2 / 3 φ3) (φ1 / φ3) + ( φ0 / φ3) ] = { (2/27) [ { - 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ]2 -

585

Appendix Section

[ { - 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22] } / 3 { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ] * [ { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] }] + [ { 3/2 θ[a,1] - 1/2 θ[a,2] + θ[a,11] + 11/3 θ[a,12] - 8/3 θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ] }

(43.a)

p = [ - (φ2 / 9 φ3)2 + (φ1 / φ3) ] = [ - [{ - 3 θ[a,11] + 13/4 θ[a,12] - θ[a,22] } / 9 { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ]2 + [ { 4/3 θ[a,11] - 20/3 θ[a,12] + 8/3 θ[a,22] } / { 6/5 θ[a,11] - 4/5 θ[a,12] + 2/15 θ[a,22] } ] ]

(43.b)

The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve the cubic equation. Also, the author wishes to acknowledge the main authors of the solution of the cubic equation, the following Italian mathematicians: Scipione del Ferro (1465 - 1526), Niccolò Fontana (Tartaglia) and Gerolamo Cardano, also the main reference (Boyer, 1968) to the solution of the cubic equation as reported at:http://it.wikipedia.org/wiki/Equazione_di_terzo_grado.

The Derivation of the Novel Re-Sampling Locations of the Cubic B-Spline To obtain the novel re-sampling locations of the B-Spline h4 (x) the values of xsre(i), i = 1, 2, 3; are employed as obtained from equation (42). The math process starts from the following equation: { h(xr0) * (∂2 (h(x)) /∂x2) (xr0) } / { h(xsre) * (∂2 (h(x)) /∂x2) (xsre) } = { EIN (xsre-x0) / EIN (xsre) } Let us calculate: h(xr0) = { f (0) + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 (xr0)3 - (xr0)2 + 2/3 ] + [f (3/2) + f (3/2) + f (1) + f (-1) +f (2) + f (-2)] * [ -1/6 (xr0)3 + (xr0)2 - 2 (xr0) + 4/3 ] }

(44)

h(xsre) = { f (0) + [ f (1/2) + f (-1/2) + f (-1) + f (1) ] * [ 1/2 (xsre)3 - (xsre)2 + 2/3 ] + [f (3/2) + f (- 3/2) + f (1) +f (-1) +f (2) + f (-2)] * [ -1/6 (xsre)3 + (xsre)2 - 2 (xsre) + 4/3 ] }

586

(45)

Appendix Section

From equation (32) it is found that: (∂2 (h4(x))/∂x2) = { [ f (1/2) + f (-1/2) + f (1) + f (-1) ] * [ 3 x - 2 ] + [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] * [ - x + 2 ] } Therefore: (∂2 (h4(x)) /∂x2) (xsre) = { [ f (1/2) + f (-1/2) + f (1) + f (-1) ] * [ 3 xsre - 2 ] + [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] * [ - xsre + 2 ] } (∂2 (h4(x)) /∂x2) (xr0) = { [ f (1/2) + f (-1/2) + f (1) + f (-1) ] * [ 3 (xr0) - 2 ] + [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] * [ - (xr0) + 2 ] } Let us posit: [ f (1/2) + f (-1/2) + f (1) + f (-1) ] = [ α / f (0) ] [ f (-1) + f (1) + f (3/2) + f (-3/2) + f (2) + f (-2) ] = [ β / f (0) ] Therefore: h(xr0) * (∂2 (h4(x)) /∂x2) (xr0) = f (0) [ α / f (0) ] [ 3 (xr0) - 2 ] + f (0) [ β / f (0) ] [ - (xr0) + 2 ] + [ α / f (0) ]2 * { 3/2 (xr0)4 - (xr0)3 – 3 (xr0)3 + 2 (xr0)2 + 2 (xr0) - 4/3 } + { α β / [ f (0) ]2 } * { -1/2 (xr0)4 + (xr0)3 - 2/3 (xr0) + (xr0)3 - 2 (xr0)2 + 4/3 } + { α β / [ f (0) ]2 } * { -1/2 (xr0)4 + 3 (xr0)3 - 6 (xr0)2 + 4 (xr0) + 1/3 (xr0)3 - 2 (xr0)2 + 4 (xr0) - 8/3 } + [ β / f (0) ]2 * { 1/6 (xr0)4 - (xr0)3 + 2 (xr0)2 - 4/3 (xr0) - 1/3 (xr0)3 + 2 (xr0)2 - 4 (xr0) + 8/3 } = (xr0)4 * { 3/2 [ α / f (0) ]2 - α β / [ f (0) ]2 + 1/6 [ β/ f (0) ]2 } + (xr0)3 * { -4 [ α / f (0) ]2 + (16/3) α β / [ f (0) ]2 - 4/3 [ β/ f (0) ]2 } + (xr0)2 * { 2 [ α / f (0) ]2 - 10 α β / [ f (0) ]2 + 4 [ β/ f (0) ]2 } +

587

Appendix Section

(xr0) * { 3 α - β + 2 [ α / f (0) ]2 - (22/3) α β / [ f (0) ]2 - (16/3) [ β/ f (0) ]2 } + { -2 α + 2 β - (4/3) [ α / f (0) ]2 - (4/3) α β / [ f (0) ]2 + (8/3) [ β/ f (0) ]2 }

(46)

The value of xr0 is the novel re-sampling location. Since: {h(xr0) * (∂2 (h(x)) /∂x2) (xr0)} = EIN (xsre-x0) / EIN (xsre) * {h(xsre) * (∂2 (h(x)) /∂x2) (xsre)} equation (46) is equated to EIN (xsre-x0) / EIN (xsre) * {h(xsre) * (∂2 (h(x)) /∂x2) (xsre)} to obtain the novel resampling locations of the h4(x). Based on the results of equation (35) let us calculate: EIN (xsre-x0) / EIN (xsre) = { θ[a,1] * [ 3/2 (xsre-x0) - 2 ] + θ[a,2] * [ - 1/2 (xsre-x0) + 2 ] + θ[a,11] * [ 3/10 (xsre-x0)4 - (xsre-x0)3 + 2/3 (xsre-x0)2 + (xsre-x0) - 4/3 ] + θ[a,12] * [ -1/5 (xsre-x0)4 + 13/12 (xsre-x0)3 - 10/3 (xsre-x0)2 + 22/6 (xsrex0) - 4/3 ] + θ[a,22] * [ 1/30 (xsre-x0)4 - 4/12 (xsre-x0)3 + 4/3 (xsre-x0)2 - 16/6 (xsre-x0) + 8/3 ] } / { θ[a,1] * [ 3/2 (xsre) - 2 ] + θ[a,2] * [ - 1/2 (xsre) + 2 ] + θ[a,11] * [ 3/10 (xsre)4 - (xsre)3 + 2/3 (xsre)2 + (xsre) - 4/3 ] + θ[a,12] * [ - 1/5 (xsre)4 + 13/12 (xsre)3 - 10/3 (xsre)2 + 22/6 (xsre) - 4/3 ] + θ[a,22] * [ 1/30 (xsre)4 - 4/12 (xsre)3 + 4/3 (xsre)2 - 16/6 (xsre) + 8/3 ] }



(47)

{ { EIN (xsre-x0) / EIN (xsre) } * {h(xsre) * (∂2 (h(x)) /∂x2) (xsre)} } - χ

(48)

Based on equation (46) let us proceed to posit: φ4 = { 3/2 [ α / f (0) ]2 - α β / [ f (0) ]2 + 1/6 [ β/ f (0) ]2 } φ3 = { -4 [ α / f (0) ]2 + (16/3) α β / [ f (0) ]2 - 4/3 [ β/ f (0) ]2 } φ2 = { 2 [ α / f (0) ]2 - 10 α β / [ f (0) ]2 + 4 [ β/ f (0) ]2 } φ1 = { 3 α - β + 2 [ α / f (0) ]2 - (22/3) α β / [ f (0) ]2 - (16/3) [ β/ f (0) ]2 } χ = { -2 α + 2 β - (4/3) [ α / f (0) ]2 - (4/3) α β / [ f (0) ]2 + (8/3) [ β/ f (0) ]2 } Therefore: h(xr0) * (∂2 (h4(x)) /∂x2) (xr0) = (xr0)4 * φ4 + (xr0)3 * φ3 + (xr0)2 * φ2 + (xr0) * φ1 =

588

Appendix Section

Furnishes the equation that has to be solved to obtain the novel re-sampling locations of the cubic h4(x) B-Spline, and such equation is: ψ 4 + ψ 3 ( φ3 / φ4) + ψ 2 ( φ2 / φ4) + ψ ( φ1 / φ4) + ( φ0 / φ4) = 0 Where it is posited that:

(49)

(xr0) = ψ and φ0 = - { { EIN (xsre-x0) / EIN (xsre) } * {h(xsre) * (∂2 (h(x)) /∂x2) (xsre)} } + χ

(50)

Based on the solution of the fourth order polynomial system outlined in the paragraph that solves the quartic equation, equation (49) furnishes the following solutions: ψ (k, i)) = ± { { { ( α + 2y )1/2 } + { ( α + 2y ) - 4 * { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } }1/2 } / 2 } - ( φ3 / 4 φ4)) (51) k = 1, 2, 3, 4; and where i = 1, 2…3 as per the degree of the SRE polynomial system as per equation (42). α = [6 ( φ3 / 4 φ4)2 - 3 ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ]

(52)

β = [ -4 ( φ3 / 4 φ4)3 + 3 ( φ3 / φ4) ( φ3 / 4 φ4)2 - 2 ( φ3 / 4 φ4) ( φ2 / φ4) + ( φ1 / φ4) ]

(53)

γ = [ ( φ3 / 4 φ4)4 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) ]

(54)

p = [ (5/6) (5/3) α2 + ( (5/6) α )2 - (5/3) α2 (5/2) + (2 α2 - γ ) ]

(55)

q = - ( (5/6) α )3 + ( (5/6) α )2 (5/2) α - (2 α2 - γ ) (5/6) α + [ (α3 - α γ - β2 /4 ) / 2 ]

(56)

u = { ( q /2 ) ± { ( q /2 )2 + ( p3 / 27) }1/2 }1/3



(57)

v = [ ( p / 3 u ) - u ]

(58)

y = v - (5/6) α

(59)



The total number of novel re-sampling locations is thus twelve for the cubic B-Spline h4(x) obtained from: xr0(i, k) = ψ (k, i)) i = 1, 2, 3; k = 1, 2, 3, 4.

(60)

589

Appendix Section

References Boyer, C. B. (1968). A History of Mathematics. New York, NY: John Wiley & Sons. Stewart, I. (2004). Galois theory. Boca Raton, Florida: Chapman & Hall/CRC.

APPENDIX VII. PROOFS TO THE MATHEMATICAL ASSERTIONS OF CHAPTER X VIII Calculation of the Lagrange Intensity-Curvature Terms Before and After Interpolation Let Eo and EIN be the intensity-curvature terms before and after interpolation. Let us proceed to their calculation. x Eo = Eo (x) = ∫ f (0) (∂2 ( LGR3(x) ) /∂x2) (0) dx =



0

x ∫ f(0) θ dx = f(0) θ x



0

x EIN = EIN (x) = ∫ LGR3(x) (∂2 ( LGR3(x) ) /∂x2) dx = 0

x ∫ { f (0) + θ(1) [ (1/2) x3 - x2 - (1/2) x + 1] + θ(2) [ -(1/6) x3 + x2 0

- (11/6) x + 1] } * { θ(1) [ 3 x - 2 ] + θ(2) [ - x + 2 ] } dx =

f (0) θ(1) [ (3/2) x2 - 2 x ] + f (0) θ(2) [ - (1/2) x2 + 2 x ] + x ∫ { θ(1) [ (1/2) x3 - x2 - (1/2) x + 1] * θ(1) [ 3 x - 2 ] } dx + 0

x ∫ { θ(1) [ (1/2) x3 - x2 - (1/2) x + 1] * θ(2) [ - x + 2 ] } dx +

0

x ∫ { θ(2) [ -(1/6) x3 + x2 - (11/6) x + 1] * θ(1) [ 3 x - 2 ] } dx +

0

x

590





(1)

Appendix Section

∫ { θ(2) [ -(1/6) x3 + x2 - (11/6) x + 1] * θ(2) [ - x + 2 ] } dx



(2)

0

It follows that: x ∫ { θ(1) [ (1/2) x3 - x2 - (1/2) x + 1] * θ(1) [ 3 x - 2 ] } dx = 0

x ∫ { θ(1) θ(1) [ (3/2) x4 - 3 x3 - (3/2) x2 + 3x - x3 + 2 x2 + x - 2] } =

0

x ∫ { θ(1) θ(1) [ (3/2) x4 - 4 x3 + (1/2) x2 + 4x - 2] } =

0

{ θ(1) θ(1) [ (3/10) x5 - x4 + (1/6) x3 + 2 x2 - 2 x] }

(3)

x ∫ { θ(1) [ (1/2) x3 - x2 - (1/2) x + 1] * θ(2) [ - x + 2 ] } dx = 0

x ∫ { θ(1) θ(2) [ -(1/2) x4 + x3 + (1/2) x2 - x + x3- 2 x2 - x + 2 ] } dx =

0

x ∫ { θ(1) θ(2) [ -(1/2) x4 + 2 x3 - (3/2) x2 - 2 x + 2 ] } dx =

0

{ θ(1) θ(2) [ -(1/10) x5 + (1/2) x4 - (1/2) x3 - x2 + 2 x ] }

(4)

x ∫ { θ(2) [ -(1/6) x3 + x2 - (11/6) x + 1] * θ(1) [ 3 x - 2 ] } dx = 0

x ∫ { θ(2) θ(1) [ -(1/2) x4 + 3 x3 - (33/6) x2 + 3 x + (1/3) x3 - 2 x2 + (22/6) x - 2 ] } dx =

0

x ∫ { θ(2) θ(1) [ -(1/2) x4 + (10/3) x3 - (45/6) x2 - (20/3) x - 2 ] } dx =

0

{ θ(2) θ(1) [ -(1/10) x5 + (5/6) x4 - (15/6) x3 - (20/9) x2 - 2 x ] } x ∫ { θ(2) [ -(1/6) x3 + x2 - (11/6) x + 1 ] * θ(2) [ - x + 2 ] } dx =

(5)



0

591

Appendix Section

x ∫ { θ(2) θ(2) [ (1/6) x4 - x3 + (11/6) x2 - x - (1/3) x3 + 2 x2 - (22/6) x + 2 ] } dx =



0

x ∫ { θ(2) θ(2) [ (1/6) x4 - (4/3) x3 + (23/6) x2 - (28/6) x + 2 ] } dx =



0

{ θ(2) θ(2) [ (1/30) x5 - (1/3) x4 + (23/18) x3 - (7/3) x2 + 2 x ] }

(6)

Thus equation (2) results to be the sum of equations (3), (4), (5) and (6) plus the term: f (0) θ(1) [ (3/2) x2 - 2 x ] + f (0) θ(2) [ - (1/2) x2 + 2 x ] Therefore:

x EIN = EIN (x) = ∫ LGR3(x) (∂2 ( LGR3(x) ) /∂x2) dx =





0

f (0) θ(1) [ (3/2) x2 - 2 x ] + f (0) θ(2) [ - (1/2) x2 + 2 x ] + { θ(1) θ(1) [ (3/10) x5 - x4 + (1/6) x3 + 2 x2 - 2 x ] } + { θ(1) θ(2) [ -(1/10) x5 + (1/2) x4 - (1/2) x3 - x2 + 2 x ] } + { θ(2) θ(1) [ -(1/10) x5 + (5/6) x4 - (15/6) x3 - (20/9) x2 - 2 x ] } + { θ(2) θ(2) [ (1/30) x5 - (1/3) x4 + (23/18) x3 - (7/3) x2 + 2 x ] } = x5 { (3/10) θ(1) θ(1) -(1/5) θ(1) θ(2) + (1/30) θ(2) θ(2) } + x4 { - θ(1) θ(1) + (1/2) θ(1) θ(2) + (5/6) θ(2) θ(1) - (1/3) θ(2) θ(2) } + x3 { (1/6) θ(1) θ(1) - (1/2) θ(1) θ(2) - (15/6) θ(2) θ(1) + (23/18) θ(2) θ(2) } + x2 { (3/2) f (0) θ(1) - (1/2) f (0) θ(2) + 2 θ(1) θ(1) - θ(1) θ(2) - (20/9) θ(2) θ(1) - (7/3) θ(2) θ(2) } + x { - 2 f (0) θ(1) + 2 f (0) θ(2) - 2 θ(1) θ(1) + 2 θ(2) θ(2) } Let us posit: θ5 = { (3/10) θ(1) θ(1) -(1/5) θ(1) θ(2) + (1/30) θ(2) θ(2) } θ4 = { - θ(1) θ(1) + (1/2) θ(1) θ(2) + (5/6) θ(2) θ(1) - (1/3) θ(2) θ(2) } θ3 = { (1/6) θ(1) θ(1) - (1/2) θ(1) θ(2) - (15/6) θ(2) θ(1) + (23/18) θ(2) θ(2) } θ2 = { (3/2) f (0) θ(1) - (1/2) f (0) θ(2) + 2 θ(1) θ(1) - θ(1) θ(2) - (20/9) θ(2) θ(1) - (7/3) θ(2) θ(2) } θ1 = { - 2 f (0) θ(1) + 2 f (0) θ(2) - 2 θ(1) θ(1) + 2 θ(2) θ(2) }

592

(7)

Appendix Section

Where above positions make the expression of Eo(x) and EIN(x) dependent on: (i) relationships between pixel intensity at the neighbourhood of f(0) and (ii) the curvature of the interpolation function. So it is for the Intensity-Curvature Functional ΔE, and the novel re-sampling locations given in the following. Therefore: EIN (x) = x5 * θ5 + x4 * θ4 + x3 * θ3 + x2 * θ2 + x * θ1

(8)

The Derivation of the Novel Re-Sampling Locations of the Lagrange Interpolation Function Given a misplacement x0, the novel re-sampling locations xr0 ≠ (xr0(1,i), xr(2,i), xr(3,i), xr(4,i)) are found by solving equation (16) after placing in it the xsre = xsre(i), i =1, 2, 3, 4, 5. { LGR3(xr0) * { (∂2 (LGR3 (x)) /∂x2) (xr0) } } / { LGR3(xsre) * { (∂2(LGR3 (x)) /∂x2) (xsre) } } = { EIN (xsre-x0) / EIN (xsre) }

(9)

Where: xsre(i) = [ yi - (φ2 / 3 φ3) ]

i = 1, 2, 3

(10)

xsre(4) = xsre(5) = 0

(11)

zi = { ( q / 2 ) ± { ( q / 2 )2 + ( p3 / 27 ) }1/2 }1/3



(12)

i = 1, 2, 3 yi = zi - ( p / 3zi ) q = [ (2/27) ( 3 θ4 / 4 θ5 )2 - ( 3 θ4 / 12 θ5 ) ( 2 θ3 / 4 θ5 ) + ( θ2 / 4 θ5 ) ]

(13)

i = 1, 2, 3

p = [ - ( 3 θ4 / 36 θ5 )2 + ( 2 θ3 / 4 θ5 ) ] Let us proceed to calculate: LGR3(xr0) = f (0) + [ f (1) + f (-1) ] * [ (1/2) (xr0)3 - (xr0)2 - 1/2 (xr0) + 1 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ -(1/6) (xr0)3 + (xr0)2 - (11/6) (xr0) + 1 ]

(14)

Like wise the B-Splines, the value of the function is calculated at the location xr0 as it can be seen in the left term of equation (14). From equation (4) of Chapter XVIII it follows that: (∂2 (LGR3 (x)) /∂x2) (xr0) = θ(1) * [ 3 (xr0) - 2 ] + θ(2) * [ - (xr0) + 2 ]



(15)

593

Appendix Section

LGR3(xsre) = f (0) + [ f (1) + f (-1) ] * [ (1/2) (xsre)3 - (xsre)2 - 1/2 (xsre) + 1 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ -(1/6) (xsre)3 + (xsre)2 - (11/6) (xsre) + 1 ] (∂2(LGR3 (x)) /∂x2) (xsre) = θ(1) * [ 3 (xsre) - 2 ] + θ(2) * [ - (xsre) + 2 ]

(16) (17)

LGR3(xr0) * { (∂2 (LGR3 (x)) /∂x2) (xr0) } = { f (0) + [ f (1) + f (-1) ] * [ (1/2) (xr0)3 - (xr0)2 - 1/2 (xr0) + 1 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ -(1/6) (xr0)3 + (xr0)2 - (11/6) (xr0) + 1 ] } * { [ f (1) + f (-1) ] * [ 3 (xr0) - 2 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ - (xr0) + 2 ] } = { f (0) + θ(1) * [ (1/2) (xr0)3 - (xr0)2 - 1/2 (xr0) + 1 ] + θ(2) * [ - (1/6) (xr0)3 + (xr0)2 - (11/6) (xr0) + 1 ] } * { θ(1) * [ 3 (xr0) - 2 ] + θ(2) * [ - (xr0) + 2 ] } = f (0) θ(1) * [ 3 (xr0) - 2 ] + f (0) θ(2) * [ - (xr0) + 2 ] + θ(1) θ(1) * [ (1/2) (xr0)3 - (xr0)2 - 1/2 (xr0) + 1 ] [3 (xr0) - 2 ] + θ(1) θ(2) * [ (1/2) (xr0)3 - (xr0)2 - 1/2 (xr0) + 1 ] [ - (xr0) + 2 ] + θ(1) θ(2) * [ -(1/6) (xr0)3 + (xr0)2 - (11/6) (xr0) + 1 ] [ 3 (xr0) - 2 ] + θ(2) θ(2) * [ -(1/6) (xr0)3 + (xr0)2 - (11/6) (xr0) + 1 ] [ - (xr0) +2]= f (0) θ(1) * [ 3 (xr0) - 2 ] + f (0) θ(2) * [ - (xr0) + 2 ] + θ(1) θ(1) * { (3/2) (xr0)4 - (xr0)3 - 3 (xr0)3 - 2 (xr0)2 - (3/2) (xr0)2 + (xr0) + 3 (xr0) - 2 } + θ(1) θ(2) * { -(1/2) (xr0)4 + (xr0)3 + (xr0)3 - 2 (xr0)2 + 1/2 (xr0)2 - (xr0) - (xr0) + 2 } + θ(1) θ(2) * { -(1/2) (xr0)4 + (1/3) (xr0)3 + 3 (xr0)3 - 2 (xr0)2 - (33/6) (xr0)2 + (22/6) (xr0) + 3 (xr0) - 2 } + θ(2) θ(2) * { (1/6) (xr0)4 - (1/3) (xr0)3 - (xr0)3 + 2 (xr0)2 + (11/6) (xr0)2 (22/6) (xr0) - (xr0) + 2 } = (xr0)4 { (3/2) θ(1) θ(1) - (1/2) θ(1) θ(2) - (1/2) θ(1) θ(2) + (1/6) θ(2) θ(2) } + (xr0)3 { - 4 θ(1) θ(1) + 2 θ(1) θ(2) + (10/3) θ(1) θ(2) - (4/3) θ(2) θ(2) } + (xr0)2 { - (7/2) θ(1) θ(1) - (3/2) θ(1) θ(2) - (15/2) θ(1) θ(2) + (23/6) θ(2) θ(2) } +

594

Appendix Section

(xr0) { 3 f (0) θ(1) - f (0) θ(2) + 4 θ(1) θ(1) - 2 θ(1) θ(2) + (20/3) θ(1) θ(2) - (14/3) θ(2) θ(2) } + { - 2 f (0) θ(1) + 2 f (0) θ(2) - 2 θ(1) θ(1) + 2 θ(2) θ(2) }

(18)

From equations (16) and (17) it follows that: LGR3(xsre) * (∂2(LGR3 (x)) /∂x2) (xsre) = { f (0) + [ f (1) + f (-1) ] * [ (1/2) (xsre)3 - (xsre)2 - 1/2 (xsre) + 1 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ -(1/6) (xsre)3 + (xsre)2 - (11/6) (xsre) + 1 ] } * { [ f (1) + f (-1) ] * [ 3 (xsre) - 2 ] + [ f (1) + f (2) + f (-1) + f (-2) ] * [ - (xsre) + 2 ] } = { f (0) + θ(1) * [ (1/2) (xsre)3 - (xsre)2 - 1/2 (xsre) + 1 ] + θ(2) * [ -(1/6) (xsre)3 + (xsre)2 - (11/6) (xsre) + 1 ] } * { θ(1) * [ 3 (xsre) - 2 ] + θ(2) * [ - (xsre) + 2 ] }

(19)

On the basis of equation (8), let us proceed with the calculation of: EIN (xsre-x0) / EIN (xsre) = { (xsre-x0)5 * θ5 + (xsre-x0)4 * θ4 + (xsre-x0)3 * θ3 + (xsre-x0)2 * θ2 + (xsre-x0) * θ1 } / { (xsre)5 * θ5 + (xsre)4 * θ4 + (xsre)3 * θ3 + (xsre)2 * θ2 + (xsre) * θ1 }

(20)

From equation (9) it follows that: LGR3(xr0) * { (∂2 (LGR3 (x)) /∂x2) (xr0) } { EIN (xsre-x0) / EIN (xsre) } * LGR3(xsre) * { (∂2(LGR3 (x)) /∂x2) (xsre) } = 0

(21)

The term { EIN (xsre-x0) / EIN (xsre) } * LGR3(xsre) * { (∂2(LGR3 (x)) /∂x2) (xsre) } can be calculated based on equations (19) and (20). Also, placing in equation (21) the expression of LGR3(xr0) * { (∂2 (LGR3 (x)) /∂x2) (xr0) } that was derived through equation (18) it is obtained that: (xr0)4 { (3/2) θ(1) θ(1) - (1/2) θ(1) θ(2) - (1/2) θ(1) θ(2) + (1/6) θ(2) θ(2) } + (xr0)3 { - 4 θ(1) θ(1) + 2 θ(1) θ(2) + (10/3) θ(1) θ(2) - (4/3) θ(2) θ(2) } + (xr0)2 { - (7/2) θ(1) θ(1) - (3/2) θ(1) θ(2) - (15/2) θ(1) θ(2) + (23/6) θ(2) θ(2) } + (xr0) { 3 f (0) θ(1) - f (0) θ(2) + 4 θ(1) θ(1) - 2 θ(1) θ(2) + (20/3) θ(1) θ(2) - (14/3) θ(2) θ(2) } + { - 2 f (0) θ(1) + 2 f (0) θ(2) - 2 θ(1) θ(1) + 2 θ(2) θ(2) } -

595

Appendix Section

{ EIN (xsre-x0) / EIN (xsre) } * LGR3(xsre) * { (∂2(LGR3 (x)) /∂x2) (xsre) } = 0



(22)

Let us posit: φ4 = { (3/2) θ(1) θ(1) - (1/2) θ(1) θ(2) - (1/2) θ(1) θ(2) + (1/6) θ(2) θ(2) } φ3 = { - 4 θ(1) θ(1) + 2 θ(1) θ(2) + (10/3) θ(1) θ(2) - (4/3) θ(2) θ(2) } φ2 = { - (7/2) θ(1) θ(1) - (3/2) θ(1) θ(2) - (15/2) θ(1) θ(2) + (23/6) θ(2) θ(2) } φ1 = { 3 f (0) θ(1) - f (0) θ(2) + 4 θ(1) θ(1) - 2 θ(1) θ(2) + (20/3) θ(1) θ(2) - (14/3) θ(2) θ(2) } φ0 = { [ - 2 f (0) θ(1) + 2 f (0) θ(2) - 2 θ(1) θ(1) + 2 θ(2) θ(2) ] [ EIN (xsre-x0) / EIN (xsre) ] * LGR3(xsre) * [ (∂2(LGR3 (x)) /∂x2) (xsre) ] } Therefore equation (22) becomes: ψ 4 φ4 + ψ 3 φ3 + ψ 2 φ2 + ψ φ1 + φ0 = 0

(22.a)

Solution of a 4th Degree Polynomial (2008, Carlo Ciulla, IGI Global) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the quartic equation is: http://en.wikipedia.org/wiki/Quartic_equation Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. Equation (22.a) is re-written as: ψ 4 + ψ 3 ( φ3 / φ4) + ψ 2 ( φ2 / φ4) + ψ ( φ1 / φ4) + ( φ0 / φ4) = 0

(22.b)

where ψ = (xr0) and it admits the following four solutions (k) for each (i): ψ (k, i)) = ± { { { ( α + 2y )1/2 } + { ( α + 2y ) - 4 * { α + y + [ ( ± β ) / 2 ( α + 2y )1/2 ] } }1/2 } / 2 } - ( φ3 / 4 φ4))

(23)

k = 1, 2, 3, 4; and i = 1, 2…5 as per the degree of the SRE polynomial system in equation (10) of Chapter XVIII. α = [6 ( φ3 / 4 φ4)2 - 3 ( φ3 / 4 φ4) ( φ3 / φ4) + ( φ2 / φ4) ]

(24)

β = [ -4 ( φ3 / 4 φ4)3 + 3 ( φ3 / φ4) ( φ3 / 4 φ4)2 - 2 ( φ3 / 4 φ4) ( φ2 / φ4) + ( φ1 / φ4) ]

(25)

596

Appendix Section

γ = [ ( φ3 / 4 φ4)4 - ( φ3 / 4 φ4)3 ( φ3 / φ4) + ( φ2 / φ4) ( φ3 / 4 φ4)2 ( φ3 / 4 φ4) ( φ1 / φ4) + ( φ0 / φ4) ]

(26)

p = [ (5/6) (5/3) α2 + ( (5/6) α )2 - (5/3) α2 (5/2) + (2 α2 - γ ) ]

(27.a)

q = - ( (5/6) α )3 + ( (5/6) α )2 (5/2) α - (2 α2 - γ ) (5/6) α + [ (α3 - α γ - β2 /4 ) / 2 ]

(27.b)

u = { ( q /2 ) ± { ( q /2 )2 + ( p3 / 27) }1/2 }1/3

(27.c)

v = [ ( p / 3 u ) - u ] (27.d) y = v - (5/6) α (28) The total number of novel re-sampling locations is thus twenty for the Lagrange interpolation function obtained from: xr0(i, k) = ψ (k, i)) i = 1, 2, 3, 4, 5; k = 1, 2, 3, 4.

(29)

The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve the quartic equation. Also, the author wishes to acknowledge that the main author of the solution of the quartic equation is the Italian mathematician Lodovico Ferrari, also the main reference to the solution of the quartic equation as well as to the Galois Theory (Stewart, 2004) as reported at: http://en.wikipedia.org/wiki/Quartic_equation.

The Intensity-Curvature Functional of Sinc Interpolation Let us proceed to calculate the second order derivative of the equation: M –1

(N – 1) / 2

hO(x) = Σn = - L f (n) (1/N) [ 1 + 2 Σm = 1 cos (2 π m x / N) ]

(30)

( ∂2 ( cos (2 π m x / N) ) / ∂x2 ) = { ∂ [ - sin (2 π m x / N) * (2 π m / N) ] / ∂x } = - (2 π m / N)2 cos (2 π m x / N) Therefore: M –1



(N – 1) / 2

( ∂2 hO(x) / ∂x2 ) = Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 cos (2 π m x / N)

(31)

Which for x = 0 assumes the form: M –1



(N – 1) / 2

( ∂2 hO(x) / ∂x2 ) (0) = Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 = θ0

(32)

597

Appendix Section

The Intensity Curvature Terms The intensity curvature terms are calculated as follows: x Eo = Eo (x) = ∫ f (0) ( ∂2 hO(x) / ∂x2 ) (0) dx = 0



x x M –1 (N – 1) / 2 = ∫ f (0) Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 dx = ∫ f (0) θ0 dx = f (0) θ0 x 0

(33)

0

From equation (30) it is derived: M –1

M –1

(N – 1) / 2

hO(x) = Σn = - L f (n) (1/N) + (2/N) Σn = - L f (n) Σm = 1 cos (2 π m x / N) x 2 EIN = EIN (x) = ∫ hO(x) ( ∂ hO(x) / ∂x2 ) dx =

(34)

0



x M –1 M –1 (N – 1) / 2 ∫ { Σn = - L f (n) (1/N) + (2/N) Σn = - L f (n) Σm = 1 cos (2 π m x / N) } * 0 M –1



(N – 1) / 2

{ Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 cos (2 π m x / N) } dx = x M –1 M –1 (N – 1) / 2 ∫ { Σn = - L f (n) (1/N) }* { Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 cos (2 π m x / N) } dx + 0

x M –1 (N – 1) / 2 ∫ { (2/N) Σn = - L f (n) Σm = 1 cos (2 π m x / N) } *

0 M –1



(N – 1) / 2

{ Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 cos (2 π m x / N) } dx =

M –1

Let be [ Σn = - L f (n) (2/N) ] = θ2 x M –1 (N – 1) / 2 ∫ { Σn = - L f (n) (1/N) } * { θ2 Σm = 1 - (2 π m / N)2 cos (2 π m x / N) } dx + 0

x (N – 1) / 2 (N – 1) / 2 ∫ { θ2 Σm = 1 cos (2 π m x / N) } * { θ2 Σm = 1 - (2 π m / N)2 cos (2 π m x / N) } dx 0

598

(35)

Appendix Section

Equation (35) can be decomposed as the sum of equations (36) and (37) below: x M –1 (N – 1) / 2 ∫ { Σn = - L f (n) (1/N) } * { θ2 Σm = 1 - (2 π m / N)2 cos (2 π m x / N) } dx = 0 M –1

(N – 1) / 2

- { Σn = - L f (n) (1/N) } * { θ2 Σm = 1 (2 π m / N)2 sin (2 π m x / N) }

(36)

For simplicity let us consider the following example: 2

2

Σm = 1 ai * Σm = 1 bi = (a1 + a2 ) * (b1 + b2 ) = a1 b1+ a1 b2 + a2 b1 + a2 b2 2

2

2

Σm = 1 ( Σm = 1 ai ) bi = Σm = 1 (a1 + a2 ) * bi = (a1 + a2 ) * b1+ (a1 + a2 ) * b2 = a1 b1+ a1 b2 + a2 b1 + a2 b2 Therefore: 2

2

2

2

Σm = 1 ai * Σm = 1 bi = Σm = 1 ( Σm = 1 ai ) bi Allows to write: x (N – 1) / 2 (N – 1) / 2 ∫ [ θ2 ]2 { Σm = 1 cos (2 π m x / N) } * { Σm = 1 - (2 π m / N)2 cos (2 π m x / N) } dx = 0

x (N – 1) / 2 (N – 1) / 2 ∫ - [ θ2 ]2 Σm = 1 { Σm = 1 cos (2 π m x / N) } * { (2 π m / N)2 cos (2 π m x / N) } dx

(37)

0

To solve the integral (37) let us posit: (2 π m / N) = λ and λ x = ω, it follows that: dω = λ dx dx = dω / λ Therefore equation (37) can be written as: ω (N – 1) / 2 (N – 1) / 2 ∫ - [ θ2 ]2 Σm = 1 { Σm = 1 cos ω } * { λ cos ω } dω = 0

599

Appendix Section

(N – 1) / 2

(N – 1) / 2

- ( [ θ2 ]2 ) Σm = 1 { Σm = 1 sin ω } * { λ cos ω } ω (N – 1) / 2 (N – 1) / 2 ( [ θ2 ] ) ∫ Σm = 1 { Σm = 1 sin ω } * { λ sin ω } dω = 2

0

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

( [ θ2 ]2 ) Σm = 1 { Σm = 1 sin ω } * { - λ cos ω } ( [ θ2 ]2 ) Σm = 1 { Σm = 1 - cos ω } * { λ sin ω } + ω (N – 1) / 2 (N – 1) / 2 ( [ θ2 ] ) ∫ Σm = 1 { Σm = 1 - cos ω } * { λ cos ω } dω



2

0

Now let us consider that: ω ω ω ∫ cos ω cos ω dω = sin ω cos ω + ∫ sin2 ω dω = sin ω cos ω + ∫ ( 1 - cos2 ω ) dω = 0

0

ω sin ω cos ω + ω - ∫ cos2 ω dω 0

Therefore: ω ∫ cos ω cos ω dω = (1/2) * ( sin ω cos ω + ω ) 0

Allows:

ω (N – 1) / 2 (N – 1) / 2 ( [ θ2 ]2 ) ∫ Σm = 1 { Σm = 1 - cos ω } * { λ cos ω } dω = 0

ω (N – 1) / 2 (N – 1) / 2 - ([ θ2 ]2 * λ ) ∫ Σm = 1 { Σm = 1 cos ω } * { cos ω } dω = 0 (N – 1) / 2

(N – 1) / 2

- ([ θ2 ]2 * λ ) (1/2) Σm = 1 { Σm = 1 sin ω cos ω + ω } To prove above let us calculate: (N – 1) / 2

(N – 1) / 2

∂ { - ([ θ2 ]2 * λ ) (1/2) Σm = 1 { Σm = 1 sin ω cos ω + ω } } / ∂ω =

600

0

(38)

Appendix Section

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

- ( [ θ2 ]2 * λ ) (1/2) Σm = 1 { Σm = 1 cos2 ω - sin2 ω + 1 } = - ( [ θ2 ]2 * λ ) (1/2) Σm = 1 { Σm = 1 cos2 ω - ( 1 - cos2 ω) + 1 } = (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

- ( [ θ2 ]2 * λ ) Σm = 1 { Σm = 1 cos2 ω } = - ( [ θ2 ]2 * λ ) Σm = 1{ Σm = 1 cos ω } * { cos ω } (N – 1) / 2

(N – 1) / 2

Therefore - ([ θ2 ]2 * λ ) (1/2) Σm = 1 { Σm = 1 sin ω cos ω + ω } is the primitive function of: (N – 1) / 2

(N – 1) / 2

- ( [ θ2 ]2 * λ ) Σm = 1{ Σm = 1 cos ω } * { cos ω } Therefore equation (38) becomes: (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

( [ θ2 ]2 ) Σm = 1 { Σm = 1 sin ω } * { - λ cos ω } ( [ θ2 ]2 ) Σm = 1 { Σm = 1 - cos ω } * { λ sin ω } (N – 1) / 2

(N – 1) / 2

- ( [ θ2 ]2 ) (1/2) Σm = 1 { Σm = 1 sin ω cos ω + ω } * λ



(39)

And given that (2 π m / N) = λ and λ x = ω, it follows that equation (39) is written as: (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 sin (2 π m x / N) } * { - (2 π m / N) cos (2 π m x / N) } { [ θ2 ]2 } Σm = 1 { Σm = 1 - cos (2 π m x / N) } * { (2 π m / N) sin (2 π m x / N) } - { [ θ2 ]2 } (1/2) * (N – 1) / 2

(N – 1) / 2

Σm = 1 { Σm = 1 sin (2 π m x / N) * cos (2 π m x / N) + (2 π m x / N) } * (2 π m / N)

(40)

Finally, equation (35) which is the expression of the intensity-curvature term after interpolation becomes the sum of equation (36) and equation (40): x 2 EIN = EIN (x) = ∫ hO(x) ( ∂ hO(x) / ∂x2 ) dx = 0

601

Appendix Section

M –1

(N – 1) / 2

- { Σn = - L f (n) (1/N) } * { θ2 Σm = 1 (2 π m / N)2 sin (2 π m x / N) } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 sin (2 π m x / N) } * { - (2 π m / N) cos (2 π m x / N) } { [ θ2 ]2 } Σm = 1 { Σm = 1 - cos (2 π m x / N) } * { (2 π m / N) sin(2 π m x / N) } - { [ θ2 ]2 } (1/2) * (N – 1) / 2

(N – 1) / 2

Σm = 1 { Σm = 1 sin (2 π m x / N) * cos (2 π m x / N) + (2 π m x / N) } * (2 π m / N)

(41)

Let us posit the following: (2 π m / N) = θ3m; sin( θ3m x ) = λ1m; (2 π m x / N) = θ6;

(2 π m / N)2 = θ4m; cos( θ3m x ) = λ2m;

M –1

- { Σn = - L f (n) (1/N) } = θ5;

it follows that: ∂ λ1m /∂x = θ3m cos( θ3m x ) = θ3m λ2m ∂ λ2m /∂x = - θ3m sin( θ3m x ) = - θ3m λ1m ∂ θ6 /∂x = θ3m The intensity-curvature term EIN (x) can be written as: x EIN = EIN (x) = ∫ hO(x) ( ∂2 hO(x) / ∂x2 ) dx = 0 (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } { [ θ2 ]2 } Σm = 1 { Σm = 1 - λ2m } * { θ3m λ1m } (N – 1) / 2

(N – 1) / 2

- { [ θ2 ]2 } (1/2) Σm = 1 { Σm = 1 λ1m * λ2m + θ6 } * θ3m



The Intensity Curvature Functional: ΔE(x) = Eo (x) / EIN (x) is therefore written as: ΔE(x) = f (0) θ0 x / {

602

(42)

Appendix Section

(N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } { [ θ2 ]2 } Σm = 1 { Σm = 1 - λ2m } * { θ3m λ1m } (N – 1) / 2

(N – 1) / 2

- { [ θ2 ]2 } (1/2) Σm = 1 { Σm = 1 λ1m * λ2m + θ6 } * θ3m } = f (0) θ0 x / { (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } { [ θ2 ]2 } Σm = 1 { Σm = 1 - λ2m } * { θ3m λ1m } (N – 1) / 2

(N – 1) / 2

- { [ θ2 ]2 } (1/2) Σm = 1 { Σm = 1 λ1m θ3m * λ2m + θ6 θ3m } } =

f (0) θ0 x / { (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } { Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } - { Σm = 1 - λ2m } * { θ3m λ1m } (N – 1) / 2

- { Σm = 1 (1/2) λ1m θ3m * λ2m + θ6 θ3m } } }

(43)

For simplicity let us consider the following example: 2

2

2

2

Σm = 1 ( Σm = 1 ai ) bi= a1 b1+ a1 b2 + a2 b1 + a2 b2 2

2

Σm = 1 ( Σm = 1 ai ) bi = Σm = 1 ( Σm = 1 bi ) ai = 2

Σm = 1 (b1 + b2 ) * ai = (b1 + b2 ) * a1+ (b1 + b2 ) * a2 = a1 b1+ a1 b2 + a2 b1 + a2 b2

603

Appendix Section

Equation (43) can thus be written as: ΔE(x) = f (0) θ0 x / { (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } { Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } + { Σm = 1 λ1m } * { θ3m λ2m } (N – 1) / 2

- (1/2) { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } } } = f (0) θ0 x / { (N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } { Σm = 1 { Σm = 1 λ1m } * { { - θ3m λ2m } + { θ3m λ2m } } (N – 1) / 2

- (1/2) { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } } } = f (0) θ0 x / { (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } - { [ θ2 ]2 /2} Σm = 1 { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } }



(44)

The Sub-Pixel Efficacy Region of Sinc Interpolation For the Intensity Curvature Functional: ΔE(x) = Eo (x) / EIN (x), the first order derivative with respect to x is calculated as follow and equated to zero to derive the Sub-pixel Efficacy Region: ( ∂ ΔE(x) / ∂x ) = ∂ [ f (0) θ0 x / { (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

θ5 * { θ2 Σm = 1 θ4m λ1m } - { [ θ2 ]2 /2} Σm = 1 { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } } ] / ∂x = { f (0) θ0 *

604

Appendix Section

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ θ5 * { θ2 Σm = 1 θ4m λ1m } - { [ θ2 ]2 /2} Σm = 1 { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } } (N – 1) / 2

f (0) θ0 x * { θ5 * { θ2 Σm = 1 θ4m θ3m λ2m } (N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 /2} Σm = 1 { Σm = 1 θ3m λ2m * θ3m λ2m - λ1m θ3m θ3m + θ3m θ3m } } } / [ EIN (x) ]2



(45)

( ∂ ΔE(x) / ∂x ) = 0 implies that: (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ f (0) θ0 * { θ5 * { θ2 Σm = 1 θ4m λ1m } - { [ θ2 ]2 /2 } Σm = 1 { Σm = 1 λ1m * θ3m λ2m + θ6 θ3m } } (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ x θ5 * { θ2 Σm = 1 θ4m θ3m λ2m } - x θ3m θ3m { [ θ2 ]2 /2 } Σm = 1 { Σm = 1 (λ2m)2 - λ1m + 1 } } }

(46)

(N – 1) / 2

{ f (0) θ0 * { θ5 * θ2 { Σm = 1 θ4m λ1m - x θ4m θ3m λ2m } ( N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 /2 } Σm = 1 { Σm = 1 λ1m * θ3m λ2m - θ6 θ3m + x θ3m θ3m [ (λ2m)2 - λ1m + 1] } } } = 0

(47)

In the following, equation (47) is solved for the case of N = 3 using Newton’s iterative method. The first step is to write the function f ( x ) as per equation (48): f ( x ) = θ5 * θ2 { θ4m λ1m - x θ4m θ3m λ2m } { [ θ2 ]2 /2 } * { λ1m * θ3m λ2m - θ6 θ3m + x θ3m θ3m [ (λ2m)2 - λ1m + 1] }

(48)

And to calculate the first order derivative with respect to the x variable: θ5 * θ2 { θ4m ∂ λ1m /∂x - θ4m θ3m λ2m - x ∂ [ θ4m θ3m λ2m ] /∂x } { [ θ2 ]2 /2 } * { θ3m λ2m ∂ λ1m /∂x + λ1m * θ3m ∂ λ2m /∂x - θ3m ∂ θ6 /∂x + θ3m θ3m [ (λ2m)2 - λ1m + 1] + x θ3m θ3m ∂ [ (λ2m)2 - λ1m + 1] /∂x } =

θ5 * θ2 { θ4m θ3m λ2m - θ4m θ3m λ2m + x θ3m λ1m θ4m θ3m } { [ θ2 ]2 /2 } * { θ3m λ2m θ3m λ2m - λ1m * θ3m θ3m λ1m

605

Appendix Section

- θ3m θ3m + θ3m θ3m [ (λ2m)2 - λ1m + 1] + x θ3m θ3m { 2 λ2m [ ∂ λ2m /∂x ] - [ ∂ λ1m /∂x ] } } =

θ5 * θ2 { θ4m θ3m λ2m - θ4m θ3m λ2m + x θ3m λ1m θ4m θ3m } { [ θ2 ]2 /2 } * { θ3m λ2m θ3m λ2m - λ1m * θ3m θ3m λ1m - θ3m θ3m + θ3m θ3m [ (λ2m)2 - λ1m + 1] + x θ3m θ3m { 2 (λ2m) [- θ3m λ1m ] - [ θ3m λ2m ] } }

(49)

Therefore: ∂ f ( x ) /∂x = θ5 * θ2 θ3m θ3m { ( θ4m λ2m / θ3m ) - ( θ4m λ2m / θ3m ) + ( x λ1m θ4m ) } { [ θ2 ]2 /2 } θ3m θ3m * { (λ2m)2 - (λ1m)2 - 1 + [ (λ2m)2 - λ1m + 1] + x [ -2 λ2m θ3m λ1m - θ3m λ2m ] } (50)

Solution of Equation (47) This piece of work is a derivative from Wikipedia and this paragraph is intended to state that such work is released under the GNU Free Documentation Licence (GFDL). The URL of the Document that has allowed the solution of the equation (47) is found at: http://en.wikipedia.org/wiki/Newton%27s_method Copyright (c) 2008 CARLO CIULLA. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. The Newton’s iterative method for the calculation of the root of equation (47) for N = 3 is: xn+1 = xn - { f ( xn ) / [ ∂ f ( xn ) /∂x ] }

(51)

Hence, starting from an initial guess which is set to be: IG = x0 * θ2 /100; where x0 is the re-sampling location, the first iteration furnishes: x1 = IG - { f ( IG ) / [ ∂ f ( IG ) /∂x ] }

(52)

The value of IG = x0 * θ2 /100 was found empirically through experimentation. Through a series of iterations, after achieving convergence the value of the xsre location is given by the following formulation: xsre = xsre (n+1) = xsre (n) - { f ( xsre (n) ) / [ ∂ f ( xsre (n) ) /∂x ] } =

606

Appendix Section

xsre (n) - { θ5 * θ2 { θ4m λ1m( xsre (n) ) - ( xsre (n) ) θ4m θ3m λ2m ( xsre (n) ) } { [ θ2 ]2 /2 } * { λ1m( xsre (n) ) * θ3m λ2m( xsre (n) ) - θ6( xsre (n) ) θ3m + ( xsre (n) ) θ3m θ3m [ λ2m( xsre (n) ) ]2 - λ1m( xsre (n) ) + 1] } } / { θ5 * θ2 θ3m θ3m { [ θ4m λ2m ( xsre (n) ) / θ3m ] - [ θ4m λ2m ( xsre (n) ) / θ3m ] + [ x λ1m( xsre (n) ) θ4m ] } - { [ θ2 ]2 /2 } θ3m θ3m * { [ λ2m( xsre (n) ) ]2 - [λ1m( xsre (n) )]2 - 1 + [ λ2m( xsre (n) ) ]2 - [ λ1m( xsre (n) ) ] + 1 + x [ -2 λ2m( xsre (n) ) θ3m λ1m( xsre (n) ) - θ3m λ2m( xsre (n) ) ] }} (53) Where: λ1m( xsre (n) ) = sin( θ3m xsre (n) ) λ2m ( xsre (n) ) = cos( θ3m xsre (n) ) θ6( xsre (n) ) = (2 π m xsre (n) / N) The author wishes to express the most sincere gratitude to Wikipedia for the knowledge provided in order to solve equation (47). Also, the author wishes to acknowledge the main author of the iterative method herein employed: ISAAC NEWTON, also the reference (Tjalling J. Ypma, 1995) to the Newton’s Iterative method as reported at: http://en.wikipedia.org/wiki/Newton%27s_method.

The Novel Re-Sampling Locations of Sinc Interpolation The novel re-sampling locations are calculated from the following equation: { hO(xsre - xr0) * { (∂2 (hO(x)) /∂x2) (xsre - xr0) } } / { hO(xsre) * { (∂2(hO(x)) /∂x2) (xsre) } } = { EIN (xsre-x0) / EIN (xsre) }



(54)

Let us proceed with the calculation of hO(xsre - xr0) based on equation (34): M –1

Let be [ Σn = - L f (n) (2/N) ] = θ2 M –1

M –1

(N – 1) / 2

hO(xsre - xr0) = Σn = - L f (n) (1/N) + (2/N) Σn = - L f (n) Σm = 1 cos (2 π m (xsre - xr0) / N) = (N – 1) / 2

θ2 /2 + θ2 Σm = 1 cos (2 π m (xsre - xr0) / N)

(55)

Let us posit that: Λ(xsre -x0) = { EIN (xsre-x0) / EIN (xsre) }* hO(xsre) * { (∂2(hO(x)) /∂x2) (xsre) }

(56)

607

Appendix Section

Based on equations (31), (34) and (42): M –1



(N – 1) / 2

{ (∂2(hO(x)) /∂x2) (xsre) } = Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 cos [ 2 π m (xsre) / N ] = (N – 1) / 2

θ2 Σm = 1 - (2 π m / N)2 cos [ 2 π m (xsre) / N ] M –1

M –1



(57)

(N – 1) / 2

hO(xsre) = = Σn = - L f (n) (1/N) + (2/N) Σn = - L f (n) Σm = 1 cos [ 2 π m (xsre) / N ] = (N – 1) / 2

θ2 /2 + θ2 Σm = 1 cos [ 2 π m (xsre ) / N ]

(58)

(N – 1) / 2

EIN (x) = θ5 * { θ2 Σm = 1 θ4m λ1m } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 λ1m } * { - θ3m λ2m } { [ θ2 ]2 } Σm = 1 { Σm = 1 - λ2m } * { θ3m λ1m } (N – 1) / 2

(N – 1) / 2

- { [ θ2 ]2 } (1/2) Σm = 1 { Σm = 1 λ1m * λ2m + θ6 } * θ3m = (N – 1) / 2

θ5 * { θ2 Σm = 1 (2 π m / N)2 sin( θ3m x ) } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 sin( θ3m x ) } * { - (2 π m / N) cos( θ3m x ) } { [ θ2 ]2 } Σm = 1 { Σm = 1 - cos( θ3m x ) } * { (2 π m / N) sin( θ3m x ) } (N – 1) / 2

(N – 1) / 2

- { [ θ2 ]2 } (1/2) Σm = 1 { Σm = 1 sin( θ3m x ) cos( θ3m x ) + (2 π m x / N) } * (2 π m / N) { EIN (xsre- x0) / EIN (xsre) } = (N – 1) / 2

{ θ5 * { θ2 Σm = 1 (2 π m / N)2 sin( θ3m (xsre- x0) ) } + (N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 sin( θ3m (xsre- x0) ) } * { - (2 π m / N) cos( θ3m (xsre- x0) ) } -

608

(59)

Appendix Section

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 - cos( θ3m (xsre- x0) ) } * { (2 π m / N) sin( θ3m (xsre- x0) ) } - { [ θ2 ]2 } (1/2) *

(N – 1) / 2

(N – 1) / 2

Σm = 1 { Σm = 1 sin( θ3m (xsre- x0) ) cos( θ3m (xsre- x0) ) + (2 π m (xsre- x0) / N) } * (2 π m / N) } /

(N – 1) / 2

{ θ5 * { θ2 Σm = 1 (2 π m / N)2 sin( θ3m (xsre) ) } + (N – 1) / 2

(N – 1) / 2

(N – 1) / 2

(N – 1) / 2

{ [ θ2 ]2 } Σm = 1 { Σm = 1 sin( θ3m (xsre) ) } * { - (2 π m / N) cos( θ3m (xsre) ) } -

{ [ θ2 ]2 } Σm = 1 { Σm = 1 - cos( θ3m (xsre) ) } * { (2 π m / N) sin( θ3m (xsre) ) } - { [ θ2 ]2 } (1/2) * (N – 1) / 2

(N – 1) / 2

Σm = 1 { Σm = 1 sin( θ3m (xsre) ) cos( θ3m (xsre) ) + (2 π m (xsre) / N) } * (2 π m / N) } Based on equation (31):

(60)

{ (∂2 (hO(x)) /∂x2) (xsre - xr0) } = M –1





(N – 1) / 2

Σn = - L f (n) (2/N) Σm = 1 - (2 π m / N)2 cos [ 2 π m (xsre - xr0) / N ] = (N – 1) / 2

- θ2 Σm = 1 θ4m cos [ 2 π m (xsre - xr0) / N ]

(61)

Based on equations (54), (55), (56) and (61): hO(xsre - xr0) * { (∂2 (hO(x)) /∂x2) (xsre - xr0) } - Λ(xsre -x0) = 0

(62)

Therefore: (N – 1) / 2

{ θ2 /2 + θ2 Σm = 1 cos (2 π m (xsre - xr0) / N) }

(N – 1) / 2

* { - θ2 Σm = 1 θ4m cos ( 2 π m (xsre - xr0) / N ) } -

Λ(xsre -x0) = 0 609

Appendix Section

Therefore: (N – 1) / 2

(N – 1) / 2

{ θ2 /2 + θ2 Σm = 1 cos (2 π m (xsre - xr0) / N) }

* { - θ2 Σm = 1 θ4m cos ( 2 π m (xsre - xr0) / N ) } -

Λ(xsre -x0) =

(N – 1) / 2

{ - [ θ2 ]2 / 2 * [ Σm = 1 θ4m cos (2 π m (xsre - xr0) / N) ] (N – 1) / 2

- [ θ2 ]2 * [ Σm = 1 θ4m cos (2 π m (xsre - xr0) / N) ] * (N – 1) / 2

[ Σm = 1 cos ( 2 π m (xsre - xr0) / N ) ] - Λ(xsre -x0) } = 0

(63)

Considering that: 2

2

2

2

Σm = 1 ai * Σm = 1 bi = Σm = 1 ( Σm = 1 ai ) bi Equation (63) can be written as: (N – 1) / 2

{ - [ θ2 ]2 / 2 * [ Σm = 1 θ4m cos (2 π m (xsre - xr0) / N) ] (N – 1) / 2 (N – 1) / 2

- [ θ2 ]2 * { Σm = 1 [ Σm = 1 θ4m cos (2 π m (xsre - xr0) / N) ] * cos ( 2 π m (xsre - xr0) / N ) } - Λ(xsre -x0) = (N – 1) / 2

{ - [ θ2 ]2 / 2 * [ Σm = 1 θ4m cos (2 π m (xsre - xr0) / N) ] (N – 1) / 2

- [ θ2 ]2 * [ Σm = 1 θ4m cos2 (2 π m (xsre - xr0) / N) ] - Λ(xsre -x0) } = 0

(64)

Let us posit: Θ4m = cos (2 π m (xsre - xr0) / N)

(65)

Thus, equation (64) is written as: (N – 1) / 2

(N – 1) / 2

- [ θ2 ]2 / 2 * { Σm = 1 θ4m Θ4m } - [ θ2 ]2 * { Σm = 1 θ4m [ Θ4m ]2 } - Λ(xsre -x0) = 0 This book presents results for N = 3. Equation (66) is thus written as:

- [ θ2 ]2 / 2 * (2 π / N)2 cos (2 π (xsre - xr0) / N)

610

(66)

Appendix Section

- [ θ2 ]2 * (2 π / N)2 [cos (2 π (xsre - xr0) / N) ]2 - Λ(xsre -x0) = [ θ2 ]2 * (2 π / N)2 [cos (2 π (xsre - xr0) / N) ]2 + [ θ2 ]2 / 2 * (2 π / N)2 cos (2 π (xsre - xr0) / N) + Λ(xsre -x0) = 0



(67)

Which solution is given by equations (68) and (69) as follow: cos (2 π (xsre - xr0) / N) = { - [ θ2 ]2 / 2 * (2 π / N)2 + { ( [ θ2 ]4 / 4) * (2 π / N)4 - 4 * [ θ2 ]2 * (2 π / N)2 * Λ(xsre -x0) }1/2 } / { 2 * [ θ2 ]2 * (2 π / N)2 } = Γ+

(68)

cos (2 π (xsre - xr0) / N) = { - [ θ2 ]2 / 2 * (2 π / N)2 - { ( [ θ2 ]4 / 4) * (2 π / N)4 - 4 * [ θ2 ]2 * (2 π / N)2 * Λ(xsre -x0) }1/2 } / { 2 * [ θ2 ]2 * (2 π / N)2 } = Γ-

(69)

(2 π (xsre - xr0) / N) = acos ( Γ+ )

(70)

(2 π (xsre - xr0) / N) = acos ( Γ- )

(71)

Therefore the novel re-sampling locations are the following two: xr1 = xsre - [ 3 * acos ( Γ+ ) / 2 π ]



(72)

xr2 = xsre - [ 3 * acos ( Γ- ) / 2 π ]

(73)

Where xsre is calculated from equation (53) with m = 1.

References Boyer, C. B. (1968). A History of Mathematics. New York, NY: John Wiley & Sons. Stewart, I. (2004). Galois theory. Boca Raton, Florida: Chapman & Hall/CRC. Tjalling J. Ypma. (1995). Historical development of the Newton-Raphson method. SIAM Review, 37(4), 531–551.

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About the Author

Carlo Ciulla holds the Laurea in management engineering (University of Palermo, Italy), the MS in information systems and the PhD in computer and information science (NJIT, USA). A researcher at NIBH, Japan (1995-1997), he worked with magnetoecephalography (MEG) studying the spontaneous alpha rhythm of the human brain. In the year 2000, he joined NJIT where he served as research and teaching assistant. Between the years 2002-2007 he was research associate at Yale University, postdoctoral fellow at the University of Iowa and Wayne State University. He worked on several aspects of signal processing in magnetic resonance imaging (MRI), functional MR imaging (fMRI), perfusion weighted imaging (PWI), and phase imaging. In 2007 joined Lane College (USA) as assistant professor of computer science. His research interests relate to mathematical computational engineering: artificial neural networks, image registration in fMRI, signal-image interpolation, MEG alpha rhythm. He has authored and co-authored 15 among journal papers and conference proceedings. Also, Carlo Ciulla is the author of the book AUTOALIGN: Methodology and Technology for the Alignment of Functional Magnetic Resonance Imaging Time Series. Image Registration: The Case of Functional MRI.

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

613

Index

A

I

adaptive linear interpolation 15 ANN98 13 approximation 338,  348 artifact 19 assess the truth 48

B-Splines 214,  223 Before and After Interpolation 206 bio-signals 17 bivariate linear interpolation function 72

image energy 32 initial intuition 52 INRIA Odyssee Lab 3,  8,  9 Intensity-Curvature Functional 31,  41 intensity-curvature functional 31,  39,  41 Intensity-Curvature Terms 206 interpolation 1 interpolation error 172 interpolation error improvement 11,  17 Interpolation Function 180 Interpolation Procedures 172 intuition 29,  48

C

K

concavity 11 convexity 11

k-space (Fourier domain) 188 Kornprobst, Pierre 3

D

L

Dartmouth College, USA 3,  8

logical reasoning 52

E

M

efficacy 338 efficiency 338 efficiency of algorithms 338 error bounds 172

magnetic resonance imaging 1,  3,  17,  18,  19,  20 magnetic resonance imaging (MRI) 1,  3 main innovation 348 Marcus, Daniel 3 motion corrected and interpolated 240

B

F Frameworks 338 functional MRI 12

H hyperbolic paraboloid 26

N Notion 52 novel re-sampling 16,  17,  18

Copyright © 2009, IGI Global, distributing in print or electronic forms without written permission of IGI Global is prohibited.

Index

O Open Access Series of Imaging Studies (OASIS) 3,  4,  5,  6,  7,  8,  20

P Penny, William 3 piecewise polynomial interpolation functions 214 pixel 3,  10,  11,  15,  16,  17,  18,  19,  20 polynomial interpolators 215 properties of the unifying theory 338

spatial extent 23 SRE 2 SRE-based interpolation functions 2,  56 sub-pixel efficacy region 16,  17,  18,  19,  25,  40 Sub-pixel Efficacy Region (SRE) 29,  40 Swenson, Rand S. 3

T

Q

T1-MRI 4,  5,  6,  7,  8 Trivariate Linear 180 trivariate linear interpolation function 39 trivariate liner interpolation function 40

Quadratic and Cubic 223

U

R

unifying theory 1

re-sampling 2,  15,  16,  17,  18 Resilient Interpolation 206 resolution 348 resulting magnitudes 12,  13 RMSE Ratio 73 root-mean-square-error (RSME) 17 root mean square error (RMSE) 59

V

S Signal-Image Processing 1 signal processing 17

614

voxel 3,  11

W Wellcome Trust Centre for Neuroimaging 3,  8,  9 www-sop.inria.fr/odyssee 3,  9 www.oasis-brains.org 3,  4,  5,  6,  7