Ten Lectures of Wavelets

*moducerotations as well as dilations and translations. In two dimensions, for i-w, we then define

34

CHAPTER 2

where a > 0, b f It2,and where

Re is the matrix cos6 -sin9 sin 8 cos 9

>

.

The admissibility condition then becomes

and the corresponding resolution of the identity is 4

A similar construction can be made in dimensions larger than 2. Theae wavelets with rotqtior angles were studied by Murenzi (1989), and applied by Argoul et d. (1989) in a study of DbA (diffusion-limlted aggregates) and other two. dimensional fractals. 2.7.

Ptlrallels with the continuow windowed Fourier transform.

The windowed Fourier transform of a function f is given by

(2.7.1)

v""'"f)(w, t ) = (f, gW't),

where gW9'(x)= ewzg(z - t ) . Argumentg completely similar to those in the proof of Proposition 2.4.1 show that, for ail fl,f2 E L2(W),

which can be rewritten as

There is no admissibility condition in this case: any window function g in E2 will do. A convenient normalization for g is 11911 = 1. (The absence of a9 &himibility condition is due to the unimodularity of the Weyl-Heisenberg grou-. Groesmann, Morlet, end Paul (1985).) The continuous windowed Fourier transform can again be viewed aa a map frodZZ(R)to an r.k.H.s.; the functions F E T*'"L'(R) are all in L~(R') and moreover satisfy

where K(w, t ; w', t r ) = (@"*', pi). (We assume 11g11 = 1 here.) Ag& there exist very special choices for g which reduce this r.k.H.8. to a space of analytic functions: for g(x) = n-lI4 exp(-x2/2), one finds -

THE CONTINUOUS WAVELEZ' TRANSFORM

35

where q5 is an entire function. The set of all. entire functions # which can be obtained in this way constitutes the Bargmann Hilbert space (Bargmann (1961)): The obtained from g(a) = go(%) = exp(-z2/2) are often d i e d the canonical coherent states (see the primer in Klauder and Skiigerstam (1985)); the associate continuous windowed Fourier transform is the canonical coherent state representation. It has many beautiful and useful properties, of which we will explain one that will be used in the next section. Applying the differential d2 operator H = -&I "x 1 t o go(x) leads to g"lt

. '

'

+

,,- i.e., go is an eigenfunction of H witheigenvalue 0. In quantum mechanics lanb

'

-

-:, m

e , H is the harmonic oscillator Hamiltonian operator, and go is its ground state. (Strictly speaking, H is really twzce thestandard harmonic oscillator H d t o n i a n . ) The other eigenfunctions of H are given by higher or&r Hermite iimctions,

,

, fr

$

*Ki

A

satisfy

H4, = 2n 4, .

(The standard and easiest way to derive (2.7.4) is t o write H = A'A, $ where A = x + and A* is its adjoint A* = z - $, and to show that Ago = 0, A(A*)n = (Am)"A+ 2n(A*)n-a, so that Hq5, = an A*A(A*)n go = 2n(A*)"" go = 2n &; the n o ~ i z a t i o a i,~ can be computed easily as well.) It is well known that the (4,; n E N) form an o r t h o n o d basis for L2(R);they const~tutetherefore a "complete set of eigenfunctionsn for 11. Let us now consider the one-parameter firmilies = exp(-iHs)+. These wa the solutions to the equation

6,

*,

i&$s

= HJr,

,

'

(2.7.5)

t h initial condition qlb = @. In the very special case where &(z) = &"(x) = euz exp[-(2 -- t ) 2 / 2 ] ,we find Jt. = ew* g,Y.lt*, where w. = w ecs b n 28, t. = w sin 2s t cm 23, and a. = f (wt - w.t.) (as can easily be verified m c i t computation). That is, a canonical coherent state, when "evolved" &r (2.7.51, remains a c.anonica.1cohore~tetate (up to a phase factor which will important to us); the label (w,,t,) of the new coherent state is obtained the initial ( w , t) by a simple rotation in the time-frequency plane.

-

+

i b

-

-

+"3& The continuous transforms as tools to build useful operators.

-

, %%-ed

l l t i o s of the identity (2.4.4), (2.7.2) can be rewritten in yet another

way:

(2.8. la)

CHAPTER 2

where (., &)$ stands for the operator on L2(IR)that sends f to ( f , 4)4; this is a rank one projection operator (i.e., its square and its adjoint are both identical to the o ~ e r a t o ritself, and its range is one-dimensional). Formulas (2.8.1) state that a "superposition," with equal weights, of the rank one p~ojectionoperators corresponding to a family of wavelets (or a family of windowed Fourier functions) is exactly the identity operator. (As before, the integrals in (2.8.1) have to be taken in the weak sense.) What h a ~ p e n sif we take similar superpositions, but give different weights t o the different rank one projection operators? If the weight furiction is a t all reasonable, we end up with a well-defined operator, different from the identity operator. If the weight function is bounded, then the corresponding operator is as well, but in many examples it is advantageous t o consider even unbounded weight functions, which may give rise to ~nbounded operators. We will review a few interesting examples (bounded and unbounded,) in this section. U7estart with the windowed Fourier case Let us rewrite (2.8.lb) in the p, q (momentum, position) notation customary in quantum mechanics (rather than the w , t notation for the frequency-time plane), and insert a weight function

If w 6 L"(w~), then W may be unbounded and hence not everywhere defined; as a domain for 1%' we can then take ( f ; S l d p dq I ~ ( p , ~ ) I(f, l ' ~ P V Q <) ~ oo), which is dense for reasonable w and g.

Two useful examples in quantum rnrchanics are (1) w(p.9) = pZ, which leads to W = -$ Cg Id, where Cg = S@ if2lg(t)l2, and ( 2 ) w(p,g) = v(q), for which W is a multiplicative potential operator: (Wf )(x) = Vg(z)f (x). with V,(z) = J dq v(q)lg(z - q)12. Readers familiar with the basics of quantum mechanics will notice that in both cases the operator W corresponds pretty well to the "quantized version" of the phase space function w(p,q) (in units such that A = l ) , with a slight twist: t h e extra constant C; in the first case, the substitution of v * lg12 for the potential function v in the second case. In fact both formulas were used in Lie4 (1981) to prove that Thomas-Fermi theory, a semiclassicaI theory for atoms and molecules, is "asymptoticallyn correct (for Z400,i-e., for very heavy atoms); it gives the leading order term of the much more complicated quantum mechanical model. Lieb's proof used the two examples above in three dimensions (rather than one); the operators he really wanted to consider were, of course, -A = -a2= I -a2l a -d2 ,23 and V ( x ) = [x: 2; SO that he had to chat% an appropriate g and deal with the extra constant C, and the difference between V and V * lg12 by some other means. Note that choosing v(q) with an integrable singularity (such as the three-dimensional Coulomb potential) always leads to a nonsingular Vg: operators of type (2.8.2) cannot represent such singularities. Many othel applications of operators of type (2.8.2) exist. In pure m a t h e rnatics they are sometimes called Toeplitz operators, and whole books have been

+ +x Z ] - ~ / ~ ,

+

~

THE CONTINUOUS WAVELET TRANSFORM

3'1

written on them. In quantum optics they are also called "operators of type P," and again there exists an extensive literature on the subject (see Klauder and Skagerstam (1981)). But let us go back t o signal analysis, and see how (2.8.2) car, be used to build time-frequency localization operato'rs. - Let S be any measurable subset of Ht2. Let us return to tirne-frequency notation, and dkfine, via (2.8.2), the operator Ls corresponding to the indicator fu~ctionof S, a(w, t) = 1 if (w, t) E S, 0 if (w, t) 4 S,

.

It follows immediately from the resolution of the identity that

on the other hand, obviously ( L sf , f ) 2 0. In other words,

I f S is a bounded set, then the operator Ls is trace-class (see Preliminaries), since, for any orthonormal basis ( u ~ )in, L2(R), ~ ~ '

<

+

(order of integral and summation may be inverted by Lebesgue's dominated convergence theorem)

*4 where IS( is the measure of S. It follows that there exists then a complete set of eigenvectors for Ls, with eigenvalues decreasing t o zero,

%%i*

(4,; n E N) orthonormal basis for L ~ ( w.)

P

. Such an operator Ls has a very natural interpretation. If the window function g reasonably well localized and centered around zero in both time and frequency, 4

J'

CHAPTER 2

38

then (f,g"st) gWlt can be viewed as an "elementary component" of f , localized in the time-frequency plane around (w, t). Summing all these components gives f again; Ls f is the sum of only those components for which (w, t) E 5. Therefore, Ls f corresponds to thecextractionfrom f of only that information that pertains to the region S in the time-frequency plane, and the construction from that l e calized information of a function that "lives" on S only (or very nearby). This is the essence of a timefrequency -tion operator such as we saw in $2.3! We can now, moreover, study Ls for sets S much more general than rectangles [-0,521 x [-T, TI.(Note, however, that even for S = 1-R, Cl] x I-T, T j , our o g erators Ls tire different from the QT QT considered ia 52.3.) Unfortunately, for most choices of S and g, the eigenfunctions and eigenvalues of Ls are hard to characterize, and this construction is of limited usefulness. However, there is one choice of g and one particular family of sets S for which everything is transparent. Take g(x) = g,(z) = T-'/' ex&--z2/2), and SR = {(w,t ) ; u2 t2 R2). Let us denote the corresponding l o d b t i o n operator by LR,

+ <

These operators LR commute with the harmonic oscillator Hamiltonha H= + z2 - 1 from 52.7, as ca.n be seen by the following argument. Since

-&

e -*Ha

&.t

=

g,~.,t,

,

with a, = (wt - w,t,)/2 E R, we have

hence

I

If we substitute w' = w-,, t' = t-,, then one easily checks (use the explicit formules for a,, w., to at the end of $2.7) that % s t = g$'t' = exp [- (dt' - wt)] e-lH0 g t t t l . On the other hand, the domain of integration is M 8 n t under the transformation (w, t)+(wr, t') (because this transformation is simply a rotation in time-frequency space!). so that

4

and LR commutes with H, as announced. It fobws that there exists an orthonormal basis in which both LB anb H are diagonal (see PrdMnaries). But,

THE CONTINUOUS WAVELET TRANSFORM "

39

since the eigenvalues of \H are all n o n d e g e n d , there exists only one basis that d&gonalizea H,namely the Hermite fundions (see 52.7). It follows that the Hermitt: functions-& are necessarily the ~ c t i o n ofs LR. The eigenvalues of LR can be computed from

(There are many ways to compute this eqmmsion. One way, via the Bargmann Hilbert space, is explained in Note 3 at the enci d this chapter.) We then have

+FJ Y 1

3;

*

.LR & = h(RMnr

2

with

-- 2s

//

dwdt

1 n! Zn

I

exp

d + t 2 j R2

-r *hi&

is a so-called incomplete r-function. h this explicit formula for X,(R), &litis now poeaible to study its behavior as a function of n and R. I will summarize --$ 4"k?pdythe results here (details can be found in Dauhechies (1988)); Figure 2.4 also &shows a plot of &(R) for three different d u e s of R. For every R, the A.(R) %: * > ; , decrease monotonically as n increases; for smatl n they are close to 1, for large t% n c1ow to zero. The threshold value around wbicb they make this 'plunge," as ii5**( e e d , for example, by n t h , = max{n; & 1/21, is nth, R2/2. Note that %r%&s is again equd to sIZ2/27r, i.e.., the area of the timefrequency localization P w o n SR multiplied by the Nyquist density, just as in 52.3. The width of the &. w g e region is wider than in 52.3; however, -b

-

r

>

-c.

-9 h

p

$iZ t >

*

6

,%

&%; 73

*

%@ -pared i

$.*. $:,

-

*7 it&

-

# {n; l - ~ > X , 3 c ) s C , R , to the logarithmic width in (2.3.2)), but it is still negligible, for

R* when compared with nth. Another striking difference with 52.3 is that in this *case are i n m e n t of the size of the region SR eigenhnctions

the prolate spheroidal wave funetio11~):the R-dependence is completely ,$dncentmted in the An(R).

Ij: @dike

..-

CHAPTER 2

FIG 2 1

The etgenvalues A, ( R ) for R = 3, 5, and 7

Examples silnilar to all of the above exist for the continuous wavelet transform We can again insert a non-constant function w(a, b) in the integral in (2.8. la), and construct operators W different from the iderrtity operator. An example is w(a,b) -- a2 in three dimensions, with a spherically symmetric 1C, (where the resolution of the identity is given by (2.6.2)), i.e., ( W f ) ( x )=

c;'

B

"ah

u./,

d 6 ~ a 2 ( ~ " j ) ( a , b ) ~ " ~ ' ( z ) ,(2.8.3)

c+

4(()

where = 4(IEl) and (?+ = ( 2 ~ l )r d~s s#(s). Because the three-dimensional Fourier transform of g(x) = is g([) = %&/(,hi [(I) (in the sense of distributions), one easily checks that Wj can also be written a s

so that (Wf , g) represents the interaction Coulomb potential energy for two charge distributions f and g. This formula was used ,in, e-g., the relativistic stability of matter paper by Fefferman and de la Llave (1986). N0t-t (W f ,g) becomes "diagonal" in the representation (2.8.3) (which, incidentally, is why it turned out to be useful in Feff-an and de la Llave (1986)). Note also that this diagonal wavelet representEStion completely captures the s-ity of

41

THE CONTINUOUS WAVELET TRANSFORM

the kerrlel in (2.8.4) iio "clipping off" of the singularity as in the windowed Fourier case. This is due to the fact that wavelets can zoom in on singularities (an extreme version of very short-lived high frequency features!), whereas the ,, windowed Fourier functions cannot (see 51.2 or 32.9). We can also, as in the wlndowed Fourier case, choose to restrict the integral in (2.8. l a ) t o a subset S of (a, b)-space, thus defining time-frequency localization operators Ls. These are well defined for measurable S , and 0 Ls 5 1. For compact S not containing any points with a = 0, Ls is a trace-class operator. For general S, the eigenfunct~orisand eigenvalues may agair? be hard to characterize, ,Qut there exist again special choices of ?/I and S so that the eigenfunctions and eigenvalues of Ls are known explicitly. Their anaiysis is similar t o the windowed . -- Fourier case, but a bit more tricky. We will only sketch the results here; for full detaib the reader should consult Paul (1985) or Daubechles and Paul (1988). One such special is $(<) = 2< e-t for J 2 0, 0 for 5 0; the associated ; resolution of the identity from which we st@ is (see (2.4 9))

<

<

$J

where

++ = 4-(<) = $(-()

The operators LC = Ls, we consider are glven

$J,

+

with Sc = {(a,b) E W+ x W; a 2 f b2 1 _< 2aC), and C 2 1 In the representation of (a, b)-space as the upper half eompiex plane (z = h + za), the Sc correspond t o the disks Iz -zCI2 < C2- 1. The role of the harmonic oscillator Hamiltonian is now played by the operator H defined by

H(f)^(<)=

dL

d

For this H wehave then

3' '

where

b ( t ) + za(t) = ~ ( t=)

+

L

cost +sint

cost

- zsint '

with z = b ia. One easily checks that the flow z+z(t) preserves all the circles ]z - iCI2 = C2 - 1, as illustrated by Figure 2.5. It follows that H and the LC wmmute, so that they can be diagonalized simultsneously. The eigenvalues of H d l have degeneracy 2; for every eigenvalue En = 3 2n we can find two gigenfunctions,

+

(%43A(€) =

+

2& [ ( n 2 ) ( n

+ I)]-'/~

[ LZ(2t)e-c

for t 2 0

,

fort 5 0 ,

41

THE CONTINUOUS WAVELET TRANSFORM

the kerrlel in (2.8.4) iio "clipping off" of the singularity as in the windowed Fourier case. This is due to the fact that wavelets can zoom in on singularities (an extreme version of very short-lived high frequency features!), whereas the ,, windowed Fourier functions cannot (see 51.2 or 32.9). We can also, as in the wlndowed Fourier case, choose to restrict the integral in (2.8. l a ) t o a subset S of (a, b)-space, thus defining time-frequency localization operators Ls. These are well defined for measurable S , and 0 5 Ls 5 1. For compact S not containing any points with a = 0, Ls is a trace-class operator. For general S, the eigenfunct~orisand eigenvalues may agair? be hard to characterize, ,Qut there exist again special choices of ?/I and S so that the eigenfunctions and eigenvalues of Ls are known explicitly. Their anaiysis is similar t o the windowed . -- Fourier case, but a bit more tricky. We will only sketch the results here; for full detaib the reader should consult Paul (1985) or Daubechles and Paul (1988). One such special $ is $(<) = 2< e-t for J 2 0, 0 for 5 0; the associated ; resolution of the identity from which we st@ is (see (2.4 9))

<

where

++ = $, 4-(<) = $(-()

The operators LC = Ls, we consider are glven

+

with Sc = {(a,b) E W+ x W; a 2 f b2 1 _< 2aC), and C 2 1 In the representation of (a, b)-space as the upper half eompiex plane (z = h + za), the Sc correspond t o the disks Iz -zCI2 5 C2- 1. The role of the harmonic oscillator Hamiltonian is now played by the operator H defined by

H(f)^(<)=

dL

d

For this H wehave then

3' '

where

b ( t ) + za(t) = ~ ( t=)

+

L

cost +sint

cost

- zsint '

with z = b ia. One easily checks that the flow z+z(t) preserves all the circles ]z - iCI2 = C2 - 1, as illustrated by Figure 2.5. It follows that H and the LC wmmute, so that they can be diagonalized simultsneously. The eigenvalues of H d l have degeneracy 2; for every eigenvalue En = 3 2n we can find two gigenfunctions,

+

(%43A(€) =

+

2& [ ( n 2 ) ( n

+ I)]-'/~

[ LZ(2t)e-c

for t 2 0

,

fort 5 0 ,

(-c).

and;)t( )" (C) = (+$IA Here L : is the Laguerre polynomial (2 is a superscript, not a power) as given by the general formula

Since the operators LC comrnutk with the parity operation (IT>)"(() = j(-t), it follows that $ ,: are eigenfunctions for the LC as well (because of the degeneracy of H,not every eigehmction of H is a priori an eigenfunction of LC!).The corresponding eigenvalues of LC are

+;

that in this cam every (This means that LC has the same degeneracy as H , eigenfunction of H is in fact an eigedimction of LC as well.) We can therefore slso use t g,= .ji($: ',+ ) and $g= ($A- - $; ,-) as eigenfunciions;these

+;

-

4

43

THE CONTlNUOUS WAVELET TRANSFORM

3%

have the advantage that they are real. Figure 2.6 shows the plots of the first few 2;- $J,: (e for even, o for odd). A graph of X,(C), for various values of C, is given 2 in Figure 2.7. For reasonably large C , the X,(C) behave as we by now expect of @ * the eigenvalues of a time-frequency localization operator: they are close to 1 for 7 n small, with Ao(C) = 1and they plunge to 0 for a larger, C-dependent value of n. More precisely, for any -y E (0, I), the value of n for which A,(C) crosses y isequal t o n =qC+0(1) (Clarge), with ~ j ( 2 + 7 7 - ~ ) ( 1 - 2 C ' ~ )=~y~ 2 , s6 or 2q - ln(1 + 29) = - In y + Q(C-I). This implies that

+;

-+

-L

f?. Tg

# {eigepfunctions; A,(C) L 7 ) = 2#

&A

{n; L ( C )

2 71

= Z C F - ~( - I P T ) + O ( I ) ,

where F ( t ) = 2t - ln(1

+ 2 t ) . In particular,

9

# {eigenfunctions; X,(C) 2 1/21 = 2C J"'(In2) + O(1) .

(2.8.5)

In ~ r d e to r compare this with the Nyquist density, we first need to find the area in timefrequency space corresponding to LC.To do this, we go back t s the

+zb.

w e have

-

*,&J

@

Hence

k3' .A9*

Sc = {(a, b) E R+

x

W; a2 + d

+ 1 5 2aC) corresponds to the time-

frequency set sc=((u,t)E~2; *

'

9 C ^ +gw2 -+IS--

")

.

*

14

This co~espondsto a low frequency as well as a high frequency cut-off; see Figure 2.8 for a comparison between this timefrequency localization qet and the disks ' of the windowed Fourier case. lo The area of & is l$l= 6x(C - 1). Combining ' this with (2.8.4) gives

I ,C

# {eigenfunctions; X,(C) 2 1/21 = 1 ~"(ln2)+0(~-')

t&i

3~

which is different from the Nyquist density! This contradiction is only apparent, sad is due to the fact that the width d the "plunge region" of the &(C) is - proportional to C, and thus to I&/. We have indeed, for c > 0,

CHAPTER

2

FIG. 2.6. Plots of the ergenfunctrons $&,$: for n = 0 , 1, ond 2.

n FIG 2 7

The ezgenvalues X,(C)

for dzfferenl values of C

i

e

in contrast to the prolate spheroidal wave case, where the analog of the expression between curly brackets tends to zero as IS1-' log IS1 for ISI-oo, and with the windowed Fourier case, where it behaves like JSI-1/2 for ISI-cm. The fact that in the present case the width of the plunge region is of the same order as lScl itself results from the non-uniform time-frequency localization of the @ $*: it is an indication that we have to be careful with time-frequeocy-density-based intuition when dealing with wavelets. We will come back to this in Chapter 4. 2.9. I

.'

,, E

**i

a

-

The continuous wavelet transform as a mathematical zoom: The characterization of local regularity.

This section is entirely borrowed from Holschneider and Tchaniitchian (1990), who developed these techniql~esin part to study local regularity properties of a nondifferentiable functivn proposed by Riemar~n. THEOREM 2.9.1. Suppose that l d x ( l + 1x1) 1@(2)1 < oo, and $(0) = 0. If a bounded fvnrtzon f as Holder contznuotes with ercponent a, 0 < cr 5 1, i.e., .

then its wavelet tmlwfbnn sutzsfies

CHAPTER 2

FIG 2 8 (a) The set

& = {(t,w),

t2

+ 5 + 1 5 8)for d r f f e d values of C

(b) A

cornpanson between the tame-frequency l d a z a t w n sets fur the wtndwed Fourier tmnsform (the drrk SR = { ( t ,w ) , t 2 + w2 5 R ~ at) irfl) arid for the wavelet t r u w j m (at nght)

P m f S~nceJ & $(x) = 0 we have ($a*b,

(9) [fw

f) = J dz I ~ I - ~ / ~

- f(b11 ;

hence

The following is a converse theorem. THEOREM 2.9.2. Suppose that rs oompcrctlgj s u p p o w . Suppose also' that f E L2(R)u bounded and continuous. U, for some'u E 10, I[, the wavelet tmnsfm o f f satisfies ~(fi 5 ~ [ a l * + ' /,~ (2.9.1)

+

b

the^ f u Holder conitnuow wth e q p e n t a. 'L

47

THE CONTINUOUS WAVELET TRANSFORM

1. Choose +2 compactly supported and continuously differentiable, with dx h ( x ) = 0. Normalize4& s f h a t C#,* = 1. Then, by Pr~pasition 2.4.2.

da (f,Pb) *;*b(x) -w

-0

We will split the Litegral over a ink, two parts, Ial < 1 and la1 2 1, and call the two terms fss(x) (small axles) and fLs(x) (largescales). 2. F~rstof all, note that

r

fLS

5 C Next,

we look at I fLs(x

is bounded uniformly in s:

lI+lIIL1

/

do 14-3/2 = C, < 00

.

(2.9.2)

laiLi

+ h) - fLs(x)J for Ihl < 1:

Since J&(r+ t ) - ?,b2(z)l5 Cltl, and since support for some R < m, we can bound this by

(2.9.3)

+, support tCrz C [-R, R]

This holds for all /hi _< 1; together with the bound (2.9.2), we conclude that IfLs(x h) - fLS(x)( < Clhl for all h, uniformly in 2. Note that we did not evpn me (2.9.1)in this estimate: fLs is always regular.

+

3. The small d

e part fss b also uniformly bounded:

CHAPTER 2

4. We thcrefor~again only have to check I fss (x + h) - fss(x)1 fo; small h, such as lhl 5 1 . Ifsing again 1$2(2 t) - 1Ln(z)15 Cltl, we have

+

It follows that f is Hijlder continuous with exponent a. Together, Theorems 2.9.1 and 2.9.2 show that the Holder continuity of a function t a n be characterized by the decay in u of the absolute value of its wavelet transform. (Except for a = 1, where we do not have complete quivalence. ) Note that we did not assrime any regularity for .tD itself: apart from decay conditions on @, b e only exploited that $& $(x) = 0. (Although this condition is not stated expliritly in Theorem 2.9.2, $ nevertheless satisfies it: the bound (2.9.1) cannot hold otherwise.) Higher order difkrentiability of f and Holder continuity of its highest order well-defined derivative can be characterized similarly by means of the decay of the wavelet coefficients if 11 has more moments zero: in order to characterize f E Cn and Halder continuity with exponent a of f(") we will need a wavelet SO that J dx xm+(x) = 0 for m = 0 , 1 , .. . ,n. For such a wavelet we have, for a E 10, I[,

+

f

E

m = 0, . . - , n bounded 'and square integHijlder continuous with exponent a

C" , with all the f ("1,

rable, and

f(")

I (f,+a'b) 1 L ~la1"+'/*+~,uniformly in a . Again, we require no regularity for $. What is most striking about all these characterizationi is that they only involve the absolute value of the wavelet transform. Note that one can also derive regularity of f from the decay in w of the absolute value of its windowed Fourier transform T"'"(w, t), if the window g is chosen sufficiently smooth. In most cases however the value for the Holder expoaent computed from (w, t)1 will not be optimal. To obtain a true characterization, the phase of T"in(w, t) should also be taken into consideration, for instance via Littiewood-Paley type estimates (see, e.g., Frazier, Jawerth, and Weiss (1991)).

"'FI

-

49

THE CONrPINIJOUS WAVELET TRANSFORM

The wavciet transfornr cart also be uscd to charact.ctcrizelocc~lregularity, sorncthing that carlrlot be achivvt.d, cven when phase inforr~~at~on 1s taker1 irito a ~ count, by the windowed Fourier transfornl. Tile fidlowing two tlicorcms are again l~orrowedfrom Halschrtcider and Tchamitchiark (1990) THEOREM 2.9.3. Suppos~that d~ ( 1 -I-Irl) [+(x)] < DC) u ~ r dJ dx $(x) = 0. If a bounded furartzon f $ Hiildrr contmaorrs zn ro, 1~1th c q ~ u r r ~a n tE 10,1], r.e.,

theri

l- &

I(",$ U V ~ I tJt1) I <'c'f~1''~ ((cL~'' s

Proof. Dy transiatirlg everything we can

6%

< - C

J d.r

J.rlcx

Ibl")

~ s s t ~ r n rf rl ~ i r t :r,) --

0. Bcxause

(-;x - b ) 1

-

(lal" -Ilbl") .

- C' L

THEUREM 2.9.4. Suppose that 1/, is cornpartly .qul,yor ted. Suj~posealso that f t- L2(R)zs bounded and contznvous. If, for sonrr 7 IJ crntl ru C ]O,l[,

L

"*

I(f ,

@"qb)

1 5 CtuIVk1/2

unzformlg, u r b,

and 3.

$6 ' &,

X

1

then f t s Hiilder contznuous

'

271

.co urzth ercpcment a.

Proof.

e

C

i

1. The proof starts exactly like t h e proof of Theorcrn 2 9 2, ol which the first three points carry over without change, with 7 taking over the role of u irr point 3.

2. We therefore only have t o check Ijss(xo + h) - fss(xo)l for small h. By translating everything, we can assume xo = 0, and we obtain

'

h-b ~hla/~~~a~
50

CHAPTER 2

+

/

lhlllalll

$/-

-00

&(lala

+

-)

1+2

h-b

(TJ.-h

(-:)I

where we have assumed a > 7. (If a 5 7,things become simpler.) Let us denote the four terms in the right-hand side of (2.9.4) by TI, Tz, T3, and

T'

4. In the second term, we ase support & C

T-R,

R] to derive

5. Similarly, for sufficiently s d l Ihl,

5 Cfhp . f

6. Finally,

S

i theorerne f ~ higher r order local regularity can be proved. These theo-

rems justify the name "mathematical micmkope," which is sometimes bestowed on the wavelet transforrh. In Hdschneider and Tchamitchian (1990) these and other reeults were exploited to study the differentiability properties of the function defined by the Fourier series n-' sin(n2%z), first studied by Riernann.

xr=,

51

THE CONTINUOUS WAVELET TRANSFORM

Notes. -

I

1. All these B e r m spaces can be further transformed, via a (standard)

conformal map, to Hilbert space8 of analytic functions on the unit disk. 2. My main refkence for this paragraph am the articles by Grossrnann, Morlet, and Paul (1985, 1986). Their results can in fact be generalized to reducible representations as well, as bng as they have a cyclic vector (A. Grossrnann and T. Paul, private c o n m m i ~ t i a n ) .This is useful for the higher dimensional case, where the representations of the uz &group are reducible but cyclic.

+

3. The operator Pie obtained by "transporting" H to $he Bargmann Hilbert is particularly sirnpie: space, via the unitary map yh, h

= erp

1 (-;(d +

1')

- -ut 12(w + it)&(w 2

)

+ it)] ,

or ( H s @ ) ( z ) = 2z #'(z). It is obvious that the eigenfunctimk of HB are the monomials u,(z) = (2" ni)-'j2 zn. The following argument shows that these are indeed the fupctions in the Bargmantr space corresponding to the Hermite functions. One easily computes

$

1

1 = exp (-?(J t 2 ) - -ut 2 (-i)(w

+

c it) +C + it) ,

so that 4, .= (2" n!)-'/*(A*)"g,corresponds to (2" n!)-1/2(-i)nan = (-i)" &(r) in the Bargmann s&. (We use the normdization Ilt$llimgm. = / & J d p e-i(Z3fy') [WZ 4 i4l2,so that go itself corresponds to the constant function 1 in the Bargmann space.) In particular, this means that

&

['

+

-

(& ,g"*t) = exp - (a2 t 2 )- i w t ] (-ijn (zn n!)-lI2(w

+ a). .

4. Not every unbounded &tion leads to an unbounded operator; m e bounded operators can only be represented in this way if an unbounded weigtrt h e t i o n ie wed. In fact, KIauder 11966) proved that even some tr-eh operatars require a non-tempered distribution as weight f&tion! '

5.

Far 4 functiom w, one requires that W should be esaentidly self-adjoint on -this *domain.

,

CHAPTER 2

52

6. Yet another application in quantum nlechanics can be found in Daubechies and Klauder (1985), where it is shown how t o write the (mathematically not well defined) path int.egral for exp (-ith-) a s a limit of bona fide Wiener integrals (as the diffusion constant of the underlying diffusion process tends t o oo), provided H is of the form (2.8.2), with a weight function w(p,q) that does not increase too drastically for p, q-too. A similar theorem can be proved in the wavelet case (Daubechies, Klauder, and Paul (1987)).

7. Exactly the same arguments hold for all operators W of type (2.8.2) for which u(w, t) is rotationally symmetric, even if it is not an indicator function. An example is W(Y, t ) = exp[-cr(w2 + t2)J,fqr which it was first shown in Gori and Cuattari (1985) that the Hermite functions are the eigenfunctions (irrespective of a; the eigenvalues depend on a, of course!). 8. It is no coincidence that Fefferman and de la Llave would use a representation of type (2.8.3) for the operator (2.8.4): after all, Calder6n7sformula (to - which (2.4.4) is essentially equivalel;t) is part of a toolbox developed precisely for the study of singular integral operators (long before wavelets!) so that it is well adapted for treating the smjlar kernel in (2.8.4). In this particular instance, (2.8.4) makes sense even for nonadmissible (C+ cancels out); in Fefferman and de la Llave (1986), 2/, was taKen t o be the indicator function of the unit ball (which is nonadmjssible, since its integral does not vanish).

9. If we make an extra transformation, mapping the upper half plane {b +*in; a 2 0) to the unit disk (by means bf a conformal mapping), then everything becohes more transparent: the flow r-+z(t) then corresponds to a simple rotation around the center of the disk, and H as well as its eigenfunctions are given by simple expressions. See Paul (1985) or Seip (1991).

10. There exist many other choices of~!,t for which this analysis works. For each choice the set sc in time-frequency space corresponding t o Sc in (a,b)-space takes on adifferent shape. Explicit computations and a figure illustrating these different shapes can be found in Daubechies and Paul (1988).

CHAPTER

3

Discrete Wavelet Transforms: Frames

In this, the longest chapter in this book, we discuss various aspects of nonorthonormal, discrete wavelet expansions, tdgether with some parallels with the windowed Fourier transform. The, "frames" of the chapter title are sets of nonindependent vectors; they can nevertheless be used to write a straightforward and completely explicit expansion for every vector~inthe space. We will discuss wavelet frames as well as frames for the windowed Fourier transform; in the latter case, the'approach can he viewed as "oversampled" with respect to the Nyquist density in time-frequency space. A lot of the material in this chapter has been taken from Daubecliies (1990), updated here and there. A very nicely written review of frames (and of the continuous transforms as well), with some additional original theorems, is Heil and Walnut (1989).

*

'

d

u G ~

,t 2.i

,

,

2.

3.1.

Discretizing the wavelet transform.

In the continuous wavelet transform, we consider the family I

$"J'(x)

=

Ial-'/2

$

('0" --

, where b E W,,a E R+ with a # 0, and Q is admissible. For convenience, in ;". the discretization we restrict a to positive values only, so thal the admissibility '* 3 - condition becomes

q" I,

'

(see $2.4.) We would like to restrict a, b to discrete values only. The dlsaetiza-

* ' {ion of the dilation parameter seems natural: we choose o = a r l where m E 2, h d the dilation step a0 # 1 is fixed. For convenience we will assume ao > 1 @though it does not matter, since we take negative as well as positive powers m). For m = 0, it seems natural as well to discretize b by taking only . +he integer (positive and negative) multiples of dne fixed bo (we arbitrarily fix

$ > O), where 60 is appropriately chosen so that the $(z - nb) 'cover" the whole line (in a sense t o be made precise below). For different values of rn, the

-

CHAPTER 3

-

width of aim'2 + ( a L m ~is ) a F times the width of *(z) las measured, e.6, by width (f) = [ J d x x21f ( x ) [ ~ ] ' /where ~, we assume that $dz xi f = 0)) so that the choice b = nh aom will ensure that the discretized wavelets at level rn "cover" the line in the same way that the $(x - nb) do. Thus we choose a r aom,b = nboar, where m,n range over Z, and > 1, bo > 0 are fixed; the appropriate choices for ao, bo depend, of course, on the wavelet 1/, (see below). This corresponds to

(%)I2

We can now ask two questions: \

completely characterize f? Or, stronger, can we reconstruct f in a numerically stable way from the

(1) Do the discrete wavelet coefficients {f,Q,,,)

(f +m.n>? 1

(2) Can any function f be written as a superposition of "elenlentary building Can we write an easy algorithm to find the coefficients in blocks" I),,,?' such a superposition?

In fact, these questions are dual aspects of only one problem. We will see below that, for reasonable 11 and appropriate ao, bo, there exist $J, so that the answer CI

to the reconstruckion question is simply

It then follows that, for any g E

L2(R)

-

or g = En,,,(g,l/,m,n)I)m,n, at least in the weak sense;-this is effectively a prescription for the computat~onof the coefficients in s miperposition of ~ m . , leading to g. We will mostly focus on the first set of que~tionshere; for a more detailed discussion of the duality between (I) and (2), see Grijchenig (1991). In the case of the continuous wavelet transform, both quttstbns were answered immediately by the resolution of the identity, at least if was admissible. In the present discrete case there is no analog of the resolution of the identity,2 so we have to attack the problem some other way. We c8n aku, wonder whether there exists a "discrete admissibility condition," and wh& it is. Let us first give some mathematical content to the questions in (1). We will restrict ourselves mmtly to functions f E La(R), although dtrcrete families of wa&leta, l i i their continuously labelled cousins, can be used in many other h c t i o n spaces as well.

+

DISCRETE WAVELET TRANSFORMS. FRAMES

55

Functions can then be "characterized by means of their "wavelet coefficients" (f,$,,,) if it is true that *

(fit 1Clm,n) = (fi,

h,n)

for dlmtn E

implies fi s fi

,

or, equivakntly, if

,

(1,+m,n)=O

forallrn,n~Z=+ f =O.

A

But we want more than characterizabirity! we want b%@able t o reconstruct f in a numericdly stable way from the (f, Ilr,,,).In ord& for such an algorithm to exist, we must be sure that if the sequence ((fi, &,,n)),,nez is "close" to 4(f2, $m,ni)m,nEZcthe0 necessarily fi and f 2 were "close" as 4.In order to make thig,precise, we need topologies on the function spa& and qa the sequence space. P t h e function space L 2 ( R )we already have its Hilbert space topology; on tb-uence space we will choose a similar t2-topology, in which the distance sequences c1 = (c&,,),,,,,~~and 2 = (c$,,),,,~~ is measured by

1 4

"

- xe This implicitly assumm that the sequences ((f, $m,n))m,nEzare in C(Z2)thema

,"

mives, i.e., that Ern,,((f, &,,)I2 < oo for all f E L2(R). Ln practice, this is . bo problem. 'AS we will see below, any reasonable wavelet (which means that bas some decay in both time and frequency, and that J & * ( x ) = O ) , and any ' choice for a0 > 1, 4 > 0 leads to **

3

,<*bf

C I(II

< B IIfI12 .

@m.n)P

(3.1.2)

m,n

We will assume (without specifying any reetrictions yet on the $,,,; we will come back t a these later) that (3.1.2) holds. With the t2(z2) interpretation of ucloseness," the stability requirement means that if Em,,I(f , +,,,)I2 L small, n Uf 112 should be small. In particular, there should exist a < oo so that ,n [(f, $,n,n)I2 5 1 implies 11f ]I2 6 a. Now take arbitrary f E L~(W),and

< a. But thii means -1

$5 fbr dome A = a-I > 0. On the other hand, if (3.1.3) holds for all f, then the *+ distance Ilh -fzII emnot bear bit rail^ 1~ if Ern,,l(fi, - (k+ (Im,n)Ia

57

DlSCRETE WAVELET TRANSFORMS FRAMES <

I t follows that {el, e2, e s ) is a tight frame, but definitely not an orthonarr~)aJ basis: the three+vectorsr 1 , e2, e j are clearl~not linearly ~ndependent o

Fw

31

T h e ~ elhne vectors an C' m t c t u t e a laghi @me

4

Note tb&-in thls exn~rlplethe frame bound A = gives t h "redundancy ~ ratie" (three vcctorv In a two-dimensional space). If this redundancy patlo, as measured by A, IS equal to 1, then the tight frame is an orthonormal bmls PROPOS!T~ON 3 2 1 If (pJ)3)3E J t8 a hght h m e , wath frame bound A = 1, r f IIpJ11 = 1 for all 3 E J , then the pJ conahlute an orthonormal basas i Proof $rice ( j , cp,) = O for all g E J ~rnpliesf = 0, the vJspan all of 3.1 It -+ &ains to check that they are orthonormal We have, for any j E J , %

3'EJ

Since

Ih,n+12.

I(%, vJ~)I2 = ilnl14+

I I P ~ I I ~=

J'#J

a'€

llvJ11 = 1 , thls implies ( Q ) ,

fit)

= 0 for all 3'

J

# 3.

r

9

Formula (3 2.2) ~ v e as trivial way to recover f from the ( f , cp,), if the frame . Let us return to general frames, and see how things work there We troduce the frame opemtor. E F I N ~ O NIf ( g 7 1 ) I E J ~d a fMme SR 'F1, then tke b m e opero'w F as the JcJ < m) defined r opemtor jran X to t 2 ( J )= { c = ( ~ j ) ) ~[lcQ3 j ; =

b b

JEJ

.

( F f ) , = (f,9,) It follows from (3.2 1) that llFf 112 -< B 11f 112, i.e., F is bounded. The adjoint d F is easy to compute: *-

(F*c,f) = ( c . F f ) = C c , ( f , v , )

Ti

C *

f€J

= Cc,((p,,f), JEJ

90 that. 1 k

F*C

,-

=

C c,w

JEJ

,

6tl

DISCRETE WAVELET TRANSFORMS: FRAMES

g the operator (F0F)-I to the vectaq R y of vectors, which we denote by qj,

m W

q, = ( F * F ) - I

b

4"

$W %

-

i

8-'

I l f 112

l(fs$91'< A-I 4fHa -

5

(3.2.6)

3e J

i

%

vj .

The family turns out to be a frame aq well. PROPOS~TION 3.2.3. The ( @ , ) l E J c m t a t u l a ,fmme wth jhme m t a n t J B'landA-',

t. f

to an i n w h g W~

The assocsatsd frome operator : li --+ P(J), ( ~ f ) ,= dl, +,) satuficd P = F(F*F)-1, PP = ( F * F ) - l , PF = xd = F'P ~d FF FP' rs the mulogod pqstam opmtm, m P ( J ), onfo Ran ( F ) = Rr. (P) ,

'

.

17

1.

As an exercise, the reader can check that if a bounded operator S has a bounded inverse S-', and if S* = S, then ( $ - I ) *

= S"- It follows that

wce

v

51 G

- 2 f

l ( f l +,)I2 =

+

I((F*F)-'h

~ $ 1=~IIF(F*F)-'fli2

JG J

3E.J

= ( ( F ' F ) " ~ , F * F ( F F J - ~=~ () ( F * F ) - ' ~ j, )

.

13.2.7)

By (3.2.5),this implies (3.2.6); the 3, constitub a frame. Moreover, (3.2.7) implies dso that the frame operator @ atisfie P P = (F'Ff'l, i

2. (F(F*F)--'I),= ( ( F * F ) - i f ,9,)= V, @$) = (Pi),, P F= [ F ( F * F ) - * ] * F.;(F'F)-'F*F = M, F*F = F*F(FWF)-'= Id. also

QM

f

'2-

t-t +*\

-

+ ,&

+?tb

XI

bc

4

*

F

F F ' F ~= F ( F F ) - I

$ g.

.

(g

Ei

-

= F ( F * F ) - ~it, fnliows tha$ Ran c Ran (F). We have = P ( F * ~ ' hence ) ; Ran (F)C R8n (P). Consequently, F&n ( F ) Ran (F). Let P be the orthogonal projection operator onto Rgn (F).We wmt to prove that FF' = P, which is equivalent .to P F * ( F ~= ) Ff (i.e., BF* leaves elements of Ran (F) unchand) and PF'C = 0 for all c orthogonal t a Ran (F).Both assertJon% are easily cb&d

3. Since

:&cT

6

F * F= ~ ~f

*

ad cLRan(F) 3 (c, F f ) = O forall f E'H =+F'c=O #F*c=o.

60

CHAPTER 3

4

We will call J the dual frcrme of (cp,)J , J . It is easy to check that the dual frame of is the original frame ( c p 3 ) J c I hack again. We can rewrite some of the conclusions of Proposition 3.2 3 in a slightly less abstract form; P F = Id = F*F means that ,

This mems that we have a reconstruction formula for f from the (f, cp,)! At the same time we have also obtdned a recipe for writing f as a sttperposition of cp,, which demonstrates that the two sets of questions in $3.1 are indeed "dual." When given a frame ( P , ) ~ ~the~ only , thing we therefore need t o do, in order t o apply (3.2.8), is t o compute the cZr, = ( F * F ) - I rpj. We will come back to this soon. First we will address a question that often arises a t this point: I have stressed before that frames, even tight frames, are generally not (orthonormal) bases because the cp, are typically not linearly independent. This means that for a given f , there exist many different superpositions of the cp, which dl add up t o f . What then singles out the formula in the second half of (3.2.8) as especially interesting? We can get an inkling of the answer with a simple example. EXAMPLE. We revisit the simple example of Figure 3.1. We had there, for any v E c*,

3 Since C,=l el = 0 in this example, it follows that the following formulas are also t r u e

where a is arbitrary in C. (In this particular case, one can prove that (3.2.10) gives d l the possibie superposition formulas valid for arbitrary v . ) Somehuw, (3.2.9) seems more "economical" than (3 2.10) if a # 0. This intuitivestatement can be made mbre precise in the following way:

whereas

Likewise, the ( f , off into pj.

cii,) are the most

PROPOSITION 3.2.4. I f f af not alict eqwl ( f ,

@I),

"economical" coeEcients for a decompoeition

El,,

c, 9, f o r some c = (c,),~J E t ? ( ~ ) and , h n CjEJ Icj12 > CIEJ I(f, @j)I2=

, ",

61

DISCRETE WAVELET TRANSFOqMS FRAMES

1 Saying that f =

d

'

El,,c, cp, is equivalent to staying that f = F'c

2. Write c = a + b, where a E Ran (F)= Ran particular, a l. b; hence /(cl12= Ital12 Ifbf12.

+

(81,and b l. Ran ( F )

.,

In

+

.rl

3. Slnce a E Ran (F), there exists g E 31 so that a = pg,or c = Rg b. Hence f == F'c = F * ~ C JF'b + But b I Ran (F), so that F'b = 0, and F*F Id. It follows that f = g; hence c -i ~f b, and

+

,

4

-

CJEI( f, $,)12,

which is strictly larger than

unless 6 = 0 and c = Ff .

-.

This proposition can also be used t o see how the g3 play a special role m the fifit half of (3.2.8). We typically have nonuniqueness there as well: there may #st many other families ( u , ) , ~J SO that f = El, (f, cpl) u,. In our earl~er two-dimensional example, such other families are given by u, = +el a, where d is M arbitrary vector in c2.Since 3 e, = 0,we obv;ously have

+

r,=,

L,

3

2 )uJ = 3 ]=I

C(v,e])e] +

are "less economical" than the E,, in the sense that for 3

4)12

I(v,

+

%)I2

+3I(v,a)l2

,=I

2 Ilvl12 + 31(v,a)12 > 51tvl12 =

-

3

.,,I2 J=el

ilar inequality holds for every b e : if f =

CIE (f;

p,)ul, then J [(u,. g)I2 g)I2 f o r d l g E W , byPropasition3.2.4. , to the reconstruction issue. If we know $J = (F'F)-'q,, then (8.2 8) .'$@$, ~ie8hi.mhow to reconstruct f from the (f,v,). SOwe only need ti3 compute the +%@p+which involves the inversion of P F. If B and A are close to each other, i.e., B/A - 1 < 1, then (3.2.4) tells us that F*F is "close" to ~ db , that 1 W ~ F ) : is' UclOse"to Id, and @I "close" to ip3. More precisely,

tx,EI

*;,

%. .'3

#'t,; :".-

-a4 i

yr

-&

f=-

2

A+B

C (f,R)Vj + Rf , IEJ

(3.2.11)

CHAPTER 3

62

-m

Id.%This impliib where R = Id - 2 F'F; hence Id 5 R 5 B - A = &. If r is mall, we can drop the rest term Rf in (3.2.11), 1JRll I and we obtain a reconstruction formula for f which is accurate up .to an Laerror of 11f U. Even if r is not so small, we can mite an algorithm for the reconstruct~onof f with expoaeatial convergence. With the same definition of R, we have

+

A+B F'F = (Id - R) ;

-& -

2

<

hence (F*F)-' = (Id R)-'. Since \(RII & B < 1, t h e S e r b converges in norm, and its limit is (Id -R)-' . It followi that

Using only the zeroth order term in the reconstruction formula leads exactly to (3.2.1 1) with the rest term dropped, We obtain better appraximtions by truncating after N t e r n ,

with

as N increases, since &F which becomes exponentially the +? am be computed hy an iterative algorithfn,

< 1. In particular,

I

1

DISCRETE WAVELET' TRANSFORMS: FRAMES

with N

Q,!

2 =A + B 6cl +

N-1-

2 x

A+B

0;;'

( ~ m M , )

rnEJ

This may look daunting, but it is not so t e m t k in examples of practical interest, where maxiy (cp,, pc) are negligibly small.vThe same iterative technique can be applied directly to f :

f = ( F F ) " ' ( F ' F ) ~= N--.lim with

fN,

-

%OW that we have thoroughly explored abstract frame questions,-we return to

3.3. F'rames of wavelets. a numeridy stable reconstruction algom for f f r w the ( f , &,,), we require,iIlat $be $Jm,* constitute a frame. In we found an algorithm to reconstruct f from the (f, &,n) if the ,,$ , do ; for this-algorithm the r&io of the &frame bounds is important, k to ways of computing at least a bound on this ratio, later this section. First, however, we shm that the requirement that the ,,$ , saw in 83.1 that in order to have

nstitute a frame already imposes that qb is admissible

Admkeibility of the mother wavelet. --mi2 #(Grnx - nbo), m, n E Z, constrTHWREM 3.3.1. Jj the $,,,(z) =% fe a frome for L2(R)wtth frame bounds A, B, then y condition:

A9

1O"

# [-I

!$(F)\~< bolnao T B

(3.3.1)

(3.3.2) -w

64

CHAPTER 3

Proof. 1. We liave, for all f E LZ(R),

If we write (3 3 3) for f = uc, and add all the resulting inequalities, ~ 1 then ) ~ we weighted with coefficieiits cc 2 0 such that C, ~ ~ 1 1 <~ oo, obtain ..

In particular. then

if

C is any positive trace-class operator (see Preliminaries),

where the at are orthonormal, cc 2 0, and Cp,, cp = R C > 0. For any such operator, wc I ~ a v etherefore, by (3.3.4),

2 We now apply (3 3 5) to a very special operator C , constructed via the contznuous wavelet trallsforrn, with a dzflerent mother wave&. Take h to be any L~-functionsuch that support h c 10,oo), c& f-' fh(<)12< m, and define, as m Chapter 2, halb = a-'I2h (yb) for a, b E R, a > 0. If c(a, b) is a bounded, positive function, then

is a bounded, positive operator lsee 82.8). If, moreover, c(a,6) is integrable with respect to a-2 da db, then C is trace-class, and W d = f,"$ $-", @ c(a, b)llh112.6We will in particular choose c(a,b) = w(lbJ/a)if 1 <_ a 5 w, 0 otherwise, with w positive and integrable. We then have

db (-, hUvb) ha*' w and ds ~ ( J s 1hlj2 j ) = 2 In ao

):(

,

-

65

DISCRETE WAVELET TRANSFORMS FRAMES

3. The middle term In (3 3 5) becomes, far this C ,

z(c$rntn7 h , n ) = c m,n

m,n

/l

rnda

3 / O-mD d b w ( f )

(+mn,

h&)l2

But - 4 2

($~rn.n,h~'= ~)

=

I

a

-1/2

ak

@(U,~T

aom/2a - l I 2 Jdy +(y) h

(

- nbo) h y+nbo - bairn I

aim

Alter the change of var~ablesa' = airnu, Y = a i m b we thrhoSe obtain

2

2 2

Take now w(s) = X e-A " This function has only one local niax~rnurn and is monotone decreasing as Jslincreases. An elementary approxlni~tlol argument for integrals (the full details of which can be found in Daubechle (1990),Lemma 2 2) shows that for such.functions w and for any a,0 E R P > 0,

or, for our particular w ,

I

r;

with Ip(a, b)l

where

I w(0) = A. Consequently,

4

CHAPTER 3 which Ch as defined by (2.4.11. We can rewrite the first term in (3.3.7) a s

4. For the particular weight k t i o n w that we have chosen, we have dt w ( t ) = $,hence Tr C = llhIl2 1" ao. Substituting all our results in (3.3.9) we find

3

where IRI 5 X Ch l($l12. If we divide by llhllZ and let A tend to zero, then this proves (3 3 1). The negative frequency formula (5.3.2) is proved analogously. s

1 Formulas (33. I), (3.3.21 impme an a priori restriction on 11, namely that (-I < m and fwq1 ~ I - l < m. This is the aomc restrict~onas in the continuous case (see (2-4.6)).

JFe

d(<)12

14(<)/~

,+,

2 In defining the d~scretelylabelled we only took posrttue dilations a$' into consid6ratlon (the sign of rn affects whether af; is 2 3 or 1, but, a r > 0 for all m) Thii is tbe reason why formillas (3.3.1), (3.3.2) dlssoclate the positive and negative frequency domains. ff we had allowed negative d i i t e dilations as well+ then &hecondkion would b v i ? involved only 4 Kf-I M(t)12 (as bs easy to check by mimicking the above proof).

<

J-T

3. If the

,,+,

constitute a tight frame (A = B), then (3.3.1), (3.3.2) imply

constitute an orthonormal basis of L2(R)(such In particular, if the as the Haar basis, or other bases we will encounter), then

It is an easy exercise that the Baar basis does indied satisfy (3.3.8). Most of the orthon~rmalbases we will w i d e r are real, so that the h t equality. in (3.3.8) is trivially satisfied. J

4, A difierent proof of Propositioa 3.3.1 is given in Chui and Shi (1991). o

In all that fdm,sre wiU alwqys m u m e that fB is admiiible.

S

DISCRETE WAVELET TRANSFORMS- FRAMES

67

3.3.2. A sufficient condition and e s t b a t e e for the frame bounds, Not all choices for $, ao, 4 lead to frames of wavelets, even if $ is admisaible. In this subsection we derive some fairly general conditions on 9,m, bo under which we do ~ndeedobtain a fnune,$andwe estimate the corresponding frame bounds. To do this, we need to-estimate Em,, I ( j , tltm,n)12:

(by Plancherel's theorem for pertodic functions) 2.n = j ( t )f(< + 2 ~ k a ; ~ b o -i(go l) ~ ( 42nkb0'') t

I

I

1I ::, ir

c 114

60 m,&Z

+

Here Rest (f) is hounded by

I-.;".

(use Cauchy-Schwarz, and change vai-iables C = ( 2 n k b i 1 ~ "in the &nd facpx)

-

(~se Cauchy-Schwarz

on the swa over m)

.

CHAPTER 3

where B ( s ) = supc

CmEZ I&(GS)(t i ( a r ( + s)l. Putting (3.3.9) and (3.3.10)

together, we see that7

If the right-hand sides of (3.3 ll), (3.3 12) are strictly positive and bounded, then the 2hrn,- constitute a frame, and (3.3.11) givcs a lower bound for A, (3.3 12) an upper bound for 13. To make this work, we need that, for all 1 5 5 ao (other values o f t can be reduced to this range by multiplication with a suitable a?, except for 4 = 0;but this co~~stitutes a .set of measnre zero, and therefore does not matter), 0< < l r i ( ~ 021 B~< ; m€Z

xmEz l$(t$e))

moreover, ~$(~$e+s)lshould have sufficient decay at oo. "Sufficient" in this second condition means that [@( k ) P ( - $k)] ' I 2 converges, and that the sllm tends to 0 as tends to 0, ensuring that for small.enough 4 the first terms in (3.3.11), (3.3.12) dominate, so that the &,,, do indeed constitute a frame. In order t o ensure all this, it is sufficient t o require that the zeros of

x,;,

2

4 d o not "conspire," so that

for all ( # 0,

14((>(5 C(6la(1 +

with a > 0, 7 > a

1~(~)-7/~,

4

+ 1.'

(3.3.14)

These decay conditions on are very weak, and in practice k e will require much more! 1f is continuous, and decays at oo, then (3.3.13) is a necessary condition: if, for some b # 0, Xmfz\4(cb)12 5 c, then one can construct f E LZ(R), with I( f (1 = 1, so that (2n)-'h Ern,,((f, qm,n)I2 < Ze, implying A I 47r c/bo.'

4

69

DISCRETE WAVEZET TRANSFORMS FRAMES

If c can bechosen arbitrarily small, then there IS no finltc lower frame bound ( S e e also Chul and Shi (1991), where the stronger result A C, ~$(fl<)v< B is proved ) The following proposition summarizes our findings PILOPOSITION 3 3 2. If +, ag are 9uqh that

<

SUP

C

lSl
I$(G'<)I~ < ,

(3 3 15)

and zf A(.) = supr C, i$(ar()~ I$(@< + s)l d a y s at least as faqt as (1 + ls()-('+'), wtth c > 0, then there eztsb ( h l t h r > 0 such that the 1/,, constztute a jhme for all chosce,e < (bo),,,, For bo < (bOftM,the follounng ernresstons are fnrme bounds for the @,

( ~ = z{ '! A = %bo

-P

bo

2

~ < Iinf C I < ~ O m = - m ~ 4 ( a ~ k =~ oo~ ~2 (-

b ) ~ ( - ~ k ) ] ' " } ,

L f0

2

14~S0i2 +

I S I ~ I SmZ-00 ~W

2 b(3)

0( - ~ k ) ] ~ " }

k = -m

*#'J

+

The condztrons on 0 atad (3.3 15) are satsfied zf, e 9 , ,&<)I _< CI0,7>a+1 P m f U'e have dready carried out all the necessary estimates The decay of 0 ensures the existence of a (4)thc SO that Ek+o zk)]1/2 < inflll~lsao

x* I & ~ ~ c ) I ' lf

60 < (bo)thr.

[fl(3k)@(-

8

The moral of these technical estimates is simple. if y!~ is at dl "decent" (reasonable decay in time and frequency, J & + ( x ) = O), then there exists a whole range of %, so. that the corresponding $m,n constitute a frame Slnce our conditions on~!,t imply that -$is admissible in the sense of Chapter 2, this is not so surprising for values of ao, h~close to 1 , O respectively: we already know that the resolution of the identity (2.4.4) holds for such and it is reasonable to expect that a sufficiently fine discretization of the integration variables should not upset the reconstruction too much Surprisingly enough, for matly of practical interest, the range of "good" (m, 4) includes values which are quite 9v far from (1,O). We will see several examples below. But first we will look at ( t h e dual frame for a frame of wavelets, and discuss some variations on the basic scheme.

+,

J!+I

+

P

8.8.3.

~he'dualframe. As re saw ln 53.2, the dual lrame is defined by

CHAPTER 3

70

where F * F f = Ern,,(f,),$ ,, $m,n. We have an expucit formula for the inverse of F*F which converges exponentially fast, i.e., fke an,with a convergence ratio a proportional to ( f - 1). It is therefire usefol to have frame bounds A, B which are close to each other. Nevertheless, (3.2.8) necessitates, in principle, an infinite number of qb,,,,,to be computed. The situation is not quite as bad as one might expect: if we introduce the notation

EL,

-

then it is easy to check that, for all f E L2((R),

=

Jt follows that (F*F)-' and Dm cammute as well. In particular, since DmFg, ' = (F*F)-I D m P + = Dm (F*F)T' T n g ,

-

C

-

Unfortunately, F*Fand T" do not commute, so that we still have to compute, in principle, infinitely many &,,. In practice, one is interested only in functions "living" on a finite range of scales, on which F'F can be rearronably approxC (., +m,n) Ilm," (see the time-frequericy localization imated by CzL, section below, 53.5). l ~ ~ is an ~ integer, ~ tN =- a~'l-"'O,~ then one easily * checks that this truncated version of P F commutes with T ~so,that one only has to compute the N different q&, 0 < n N - 1 in this case. This number is still very large in many cases of practical interest, howewr. It is therefore especially advantageow to work with frames which are almost tight ("snug frames"), i.e., which have $ - 1 < 1: we can then stop after the zeroth order term of the reconstruction formula (3.2.1 l), avoid all the complications with the dud frame, and still have a high quality recomtruction of arbitrary f. On the uther hand, there exist very spacial choices of $, a.~,bo for wbich the &, are not c b t o a tight frame, but it so happens that all the . $ are generated by a single function, (3.3.17) +yn(2) = t,&m,n(~) = Q ~ ' $(O;"X ' - nf .

<

An example is provided by some of the biorthogonal bases that we will encounter in Chapter 8; andher example is given by the &transform of Frazier and Jawerth (1988) (see also Frazier, Jawerth, and Weiss (1991j).AC It is important t o realize that the &, and the $m,n may have very merent regularity properties, For instance, there exjst frameswhere $ itself is Cooand decays faster than any inverse polynomial, but where some of the are not in LP for d l p (implying that they have very slow decay). An example, due to P.,G. LemariB, is given in detail in Daubechiea (IM), pp, Q8&-!W.l0 Something similar may happen even if all the are generated by a six& b c t i o n 4: there exist examples where 3 E Ck (wit4 k arbitrarily large), but where is

Kn

$zm

4

DISCRETE WAVELET TRANSFORMS FRAMES

71

not continuous (The biorthogonal bases in Chapter 8 gtve examples where this happens; the first example was constructed by Tchamitchian (1989).) One can exclude such diiunilarities by imposing extra conditions on +,m,and bo (see Daubechies (19901, 5 II.D.2, pp. 991-992).

Some variations on the basic scheme. So far, we have not restricted the value of @, beyond a0 > 1. In practice, however, it is very conve nient to have a0 = 2. Going from one scale to the next then means doubling or halving the translation step, which is much more practical than if another ao is used. On the other hand, we have just seen th8t it is advantageous to use f r m with B / A - 1 < 1. Since our estimates (3,3.11), (3.3.12) for A, B give 3.3.4.

for all [ # 0, these two requirements together mply that CmEZ 14(2~<)1~ should be almost' constant for = 0, which is a very strong restriction on $, not generally satisfied. The Mexlcan hat funaion $(z) a (1 - x2) e-"I2, for instance, Lads to a frame with B / A close to 1, for ao 5 z ] / ~ but , certainly not for rn = 2 because the amplitude of the millations of 14~(2"())~ is too large. In order to remedy this situation, without having to gve up too much of our freedom in chooging a d its width in the frequency domam, we can adopt a method used by A Grossmann, R. Kronland-Martinet, and J. Morlet, and use different "voices" per octave. This amounts to wing several different wavelets, qbl, - - , and to look at the frame {@&,; m,n E 2, v = 1,. . . ,N). One can repeat the analysis of 53.3.2 (see, e.g., Daubechies (1990)), leading to the following estimates for the frame bounds of this rnultivoice frame:

XmEt

+

+*,

with

N

and

-

00

By choosing the $I, .,4n to have slightly staggered heqiiency localization centers, coupled with good decay at c6, one can achieve B / A - 1 << 1. (See the e$icamples in 93.3.5 below.) The time-fresmcy lrrt prresponding to d s m u i t i w i w 5 ~ a n es-Ettle ~ ~ werent b 4 + 1 . 4 q m exampla, with a

CHAPTER 3

FIG 3 2 The t ~ r n e : + ~ latttee for a scheme m t h four vorces I n thu w e It daflec-ent vorce waveleb ~ , l - - ~ , ore arsumed to be drlatrons of a mngle functwn JI, J s ( s ) 2 1 4 ( 2 ~ 1 sf ]g(t)l 4 ( w h d we assume to be even) peaks armrnd fwo, then M 143 1 unN be concenbmted anncnd f2 - ( ~ - ' ) / ' ~

,+'

four voices per octave, is ginren in Figure 3.2. Fof every dilation step, we find four different frequency levels (corresponding to the four differen4 frequency localizations of - - -,$4), all translated by the same translation step. Such a lattice can be viewed as the sujxqosition of four diierent lattices of the type in Figure 1.44 stretched by different &nts in the frequency direction. Each of these four sublattices has a Merent "density," which is reflected by the fact that typically the gDY have different Lz-norms. One choice favored by Grossm a n , Kronland-Martinet, and Morlet is to take "fractionallyn dilated versions of a single wavelet $:

(Note that these do indeed have different L2-norms!) In this cam N ~ simply Cz,=-, l i ( 2 m ' / N ~ ) ~and 2 ) this Cu=1 Cm=-ac 1 @ ( 2 ~ < ) 1bemrnes can easily be made to be almoet constant, by choosing N large enough.

DISCRETE WAVELET TRANSFORMS FRAMES

73

Fixing a0 = 2 allows also for a modification of the estimation techniques in $3.2,which may be useful in some instances. Let us go back to the estimate for Rest (f). We can rewrite k E Z, k # 0, as k = 2'(2k1 I), where P 2 0, k' E Z; the correspondence k-+(t', k') is one-teone. If ao = 2, then we can regroup different terms, and write

+

Rest (f) = -

c 2

jd~j~c~j~t+2+(2k~+i)~-!.~-m~)

ml.kttZ

~(zm'+fc)$/2t(2n1{ + 2

~ ( +2- l~) b 0 - l ) ] .

t-0

This leads to

where

These estimat'es are due to Ph. Tchamitchian. (Nl details of the derivation can be found in Daubechies (19901.) Note that &, unlike p, still takes the phases of into account; as a result, the estimates (3.3.21), (3.3.22) are often better than (3.3.11), (3.3.12) when is not a positive function. If is positive, then (3.3.11), (3.3.12) may be better. The estimates (3.3.21). (3.3.22) hold it we have one single voice per octave; they can of course also be extended to the multivoice

4

*vv

4

4

Case.

*

"4& ~

Y ."

A

3.3.5.

Examples.

d

A. Tit frames. The following construction (first proposed in Daubechies, G r m m n , and Meyer (1986)) leads to a family of tight wavelet frames. Let u bea Ck (or CQ))hnction from R to R that satisfies:

CHAPTER 3

74

(see Figure 3.3). An example of such a (C1)function v is

For arbitrary no > 1, bo > O we then define

4*(<)by e ~ t ~ t ? ~ ~ e ,

4+(6) = [In

- l l2

1

tIt
whcre 4 = 2n[h(ao2- I)]-', and $-(<) = G+(-(E) Figure 3.4 shows = 1 , and w as in (3 3.25). It is easy to check that

d+ for

erg = 2, 6"

where ~(o,,)

(c)

is the indicator function of

tfw, Q W I ~half

line (O,oo), i e.

x ( o , ~ ~ ) = 1 if 0 c < w, O otherwise. For any f E L2(R),one then has

hc. 3 3. The f i e v(a) defined by (3.3 25).

.

b+.

DISCRETE WAVELET TfUNSFORMS: FRAMES

If,

5,

= rn,n E Z, r = , ) J 2

i+*(.>l5 cnr a

%", I'

..$+ ?;? ;'ii c

J

4I ) .

It follows thet the + or -) is a tight frame for L'(w), with Frame bwnd One can uhe a variant to obtain a frame consisting of real wavelets: @ = Re $+ = I[@+ + and $2 = Cm @+ = - $-I gener&e 2 the tight 'frame {$&,, m, n E Z, A = 1 or 21. These frames h e not generated by translations and dilations of a single function; tflisis a naturel-consequence of the decoupling of positive and negative frequenctes in the construction. A more serious objection to their practical use is the fact that their Fourier transforms' are compactly supported, and that the sim of this support is relatrvely smdl (for reasonable ao;bo)- As a result, the decay of the wavelets is numerically rather alow: even though we may choose v to be Cm,so that the @* decay faster tban any inverse polynomial, , collection (q!$m,,;

Y

E P

.

$.

(1

+ t4)-

I

the value of CN turns out to be too large to be practicd. Note that we did nut introduce any restriction M w,bo in thii construction. ' \

B. The Mexican hat b c t i o n . The Mexican hat

function is the second derivative of the C a w i a n e-;'f2; if we normalize it so that its L'-norm hr 1, we obtain $(z)" =-1/4 (1 - z2)e-s2/2

-

f

43

.

Thb -hition (and dilated and translated versions d it) was plotted in Figure 1.a; if p u take one such plot, and imagine it .anMm n d its symmetry axb,

76

CHAPTER

3

then you obtain a shape similar to a Mexican hat. This function is popular in vision analysis (at least in theoretical expositions), where it was also christened. Table 3.1 gives the frame bounds for this function, as computed from (3.3.19), (3.3.20), with a0 = -2, for d i f f k n t values of bo and for a number df voices varying from 1 to 4. As soon a s we take 2 or more voices, the frame may be considered tight for all d~ 4 .75. Note that bo = .75 and ( w , ) , ~ ~ ~= , , 2'12 1.41 (intuitively corresponding to two voices per octave) are not small values fbr the Mexican hat function: the distance between the maximum of and its zeros is only 1, and the width of the positive frequency bump of (as measured by 1-23. For 4 (C -
4

-

[JF

+

-

C. A modulated Gaussian. This is the function most often used by R. Kronland-Martinet and J. Morlet. Its Fourier transform is a shifted Gaussian, adjusted siightly so that 4(0) = 0, -

-

Often &, is chosen so that the ratio of the highest and the second highdt maximum of is approximately i.e. 6 = ~ [ 2 In/ 2]$12 5.3364. ..; in practice ,one often takes Q = 5. For this value of to,the second term in (3.3.26) is so small that it can be neglected in practice. This Morlet-wavelet is complex, even though most applications in which it is used involve onIy real signals f . Often (see, e.g., Kronland-Martinet, MorIet, and Grossmann (1989)) the wavelet transform of a real signal withihis cornpiex wavelet is plotted in modul&phaseform, i.e., rather than (f, rC,m,n), Im (f, $rn,n), one plots l(f, $m,n)l and tan-' [Im (f, q!~~,,)/R.e(f, &,n)J; the phase plot is particularly suited to the detection of singularitis (Grossmann et al, (19879). For real f , one can exploit j(-6) = j(() to derive the foUowing frame bounds (this is analogous to what was done in $2.4 for real f):

i,

$J

A

IU,k , n ) 1 2 < B Jlf 11'

llf 11' 5

-

m,n€Z

for f red

I

I

DISCRETE WAVELET TRANSFORMS: FRAMES

77

Fhune boundr far wavelet f m m based on the Meziwn h t functwn +(z) = 2/fi n-'14(1 The dalatum p a m e t e r a0 = 2 in all we;N u the number of voicw.

x2)e-"'12-

8

%, These can, of c o r n , again be generaized to the multivoice case. Table 3.2gives ?-itbe tame bounds for Q = 2, several choices of 4, and number of voicos ranghg from 2 to 4. In practice, the number of voices ia often even higher.

r*

CHAPTER 3

F h m e boundr for wavelet * e s

-

b a d on the modulated Gawaaon, #(s) =n-'I4 (Cctoz T h e m i o n conatant oo = 2 in dl -; N i s the

e L ~ g / 2 fe - r a t a , wth €0 = *(2/ ln 2j1/l. nu& of wt-s.

-.

N =3 bo .5 1.

A 10,295 ' 5.147 1.5 3.366 2. 2.188 2.5 1.1% 3. .320

1 -

B

BIA

1.017 1.017 . 1.056 3 0 2 1.372 z.& 2.534 3.01 ' 8.024

113,467 5.234 3.W

D. A n example t h a t is easy to implement. So far we have not a d d r d the qucstion of how the wavelet coefficients (f,q,,,) are computed in practice. In real life, f is not given as a function, but in a sampled version. Computing the integrals Jdz f ( x ) $m,n(x) then requires some quadrature formulas. For the smallest scaleti (most negative m) of interest, this will not involve many samples of f, and one can do tbe computation quickly. For larger scales, however, one faces huge integrais, which might considerably slow down the ~01fiputatioaof the wavelet transform of any given function. Especially for o n - h e hpkmentations, orle should avoid having to compute these long integr&. A constructir7n achieving this is the so-called "algorithme a trous" (Hobcb&de;r et al. (1989)), which uses an interpolation technique to avoid lengthy cornputeftions (for details, I refer to their paper). Here I propose an analogous example (although it is not "Q trous"), by borrowing a leaf from multiresolution 84aapm and o r t h i o n d bases (to which we will come back), i.e., by i n t r o d ~ ~ d nang auxiliary Function 4. The basic idea is the fdlowing: suppose there exists a fundion q%80 that

DISCRETE WAVELET TRANSFORMS FRAMES

79

where in each case only finitely many coefficients are different from zero." (Such pairs of 4, abound; an example is given below. The "algorithme ti trown wnesponds to special 4 for which 1, all other even-indexed Q, = 0.) Here 4 does not have integral zero (but @ does!), and we will normalize 4 so that J& fix) = 1. Define, even though 4 is not a wavelet, &,+(x) = 2'*J2 &(2'*x - n);we take = 2 , 4 = 1. Then it is c h that

? r

$J

& problem (f, &,)

of findii the wavelet coefficients is reduced to mrnputing the (finite combinations of which will give the (f, $,)).* On the other

sa that the (f; b,,)can bc computed rec~rrsively,working from the smallest scale (where t& are easy to compute) to the largest wale Everything is done by simple, finite convolutions An example of a pair of functions satisfying (3.3.27), (3 3.28) is

,;* 1

L

r-

I #gi

& t ) = - @ - 21 r€

6

' $z ,

which corresponds to

$+*$b

@++

*R

6

&). = b

4

4

,* 5

e

ilfZ h

*

j

"fisrrs p

P

%

I

-21~5-1, + 2j3 , 4 - x2(1 + x / 2 ) , -1 f x 2 0 , $ - x2(1 - 1/21 , 0 < t < 1 ,

$

(2:

- 2)3

lo

9

l s ~ l 2 , otherwise .

a 110rmdbaWn constant chaten a, that JJ$J) = 1; one finds N = B&. 3.5s shows graph of 11; they are not unlike a Gaussian and its second

-$bimtive, piotted in Figure 32%for comparison. The function JI clearly satisfies &3.27) with do = N,d+l = -N/2, all other dk = 0, whereas **

CHAPTER 3

80 which implies

or Cg = 23 , C*l = 21, cf2 = i,allother ck = O For this$,ao = 2andbo = 1, the frame bounds are A = 73178, fl = 3.77107, corresponding to B / A = 2.42022; for a0 = 2, b~ = .5 we I~avcA = 2 33854, B = 2.66717, and B / A = 1.14053 (using bo = .5 means that the rt~urslonformula, linking the, ,@, with the 4,, and the cP,,, with the 4,- I ,,, , havc to be adapted, but this is easy). Here we have usecl only one voice. It is of course possible to choose severd different ,'Q corresppndirlg to different d i , that give rlse to a muftivoice scheme, closer *. to tight frames. This conciu(les our exarnplc se~ttoli,other examples are given in Daubechics (1990) (includ~ngune for which the ~ s t i n ~ a t e(3.3.21), s (3.3.22) outperform (3.3 1 1 , ( 312)) Many otlicr exanlples car) of course be consbructd. Tile wavclcts used in Mallat arid Zhong (19%)) are another examplc of the same tyyx as our last or1.e; in their case is choscn ta bo the first derivative of a fu~lction with non~zerointegral (so that S d x Q ( x ) = 0 but t @ ( z )# 0).

(b)

1

0

FIG 3.5.

JL] .-- -

.-

- --

-----

A n eurmpk

1

-

--

o& [l

-

---

.

-

L-

that w m y to cmplement: graphs of $,$I (ana), and a cornparwon

u n J a Cawstan and tts second denvatsue (an b).

3.4.

Frames for the windowed Fourier transform.

The windowed Fourier transform of Chapter 2 can also be discretized. The natural discretization for w , t in gWwt(x) = eWZ g(x - t ) is w = mq), t = nto, where wo, to > 0 are fixed, and m, n range over Z; the discretely labelled family is thus gm,n(x) = et-= g ( ~R~O)

.

.f

DISCRETE WAVELET TRANSFORMS: FRAMES

-

81

We can again seek answers to the sww questions as in the wavelet .case: for which choices of g, wo, to can a hetion be by the inner products (f, gmln);when is it possible to recopst~ctf in a numekally stable way from these inner products; can an e5cient &&ilBm be given to write f as a linear combination of the g,,,,,? The answers are ag&i pm*ided by the same abstract framework: stable numerical reconstruction of f from its windowed Fourier coefficients

~~

)r

i *

r

~7

h , n )

is only possible if the g,,, ,,sotilat

=

& f ( ~ eLamZ ) 9fz - e

lf

constitute a frame, i.e., if there erdst A

> 0, B < oo

*<

If the gm,, constitute a h e , then any function f E L 2 ( R ) can be written as

"

-"Whereg;nT. are the vectors in the dud fr-; (3.4.1) showsboth how to recover , f from the (f, g,,,,,) and how to write f as a superposition of the g,,,. The detailed analysis of frames of windowed Fourier functions brings out some features different from wavelet frames, due to the difkrences in their constmctions.

A necessary condition: Sufficiently h5gk t i m d k q u e n c - y denThe arguments in the proof of Theorem 3.3.1 can be wed for the windowed s@. burier case as well (with the obvious modifications),- 1 to the conclusion

* 3.4.1.

27r

A I- 11g1I2 L 3 woto

I

(3.4.2)

,,h any frame of windowed Fourier functions, with frame bounds A, B. This not impose any additional restrictions on g (we alwan wmme g G L2(R)). A ' consequence of (3.4.2) is that the kame bound for ory tight f r w equda $ 2 ~(u&)-' (if we choose g to have norm 1); in particular, if the g,,, constitute orthonormal basis, then woto = 2x. The t h e a c e of any owstmint on g in the inequalities (3.4.2) is similar to the

:-Fzwm ,s

I

b c e of an admissibility c o d t i o n for the continuous windowed Fourier tnubsk i n (see Chapter 2 ) , and quite amlike the constraint J 4 KJ-' 1 $ ( a 2 < oo on

&hemother wavelet, necessary for wavelet frames as well as for the continuous Wavelet transform. Another diEerence with the wavelet case is that the time and b x p n c y translation step and uij are constrained: there exist no windowed Pburier frames for pairs wo,to such that %to > 27r. Even more is true: if $d& > 2x, then for any choice of g E L 2 ( R ) there exists a corresponding T € L2(R) (f # 0 ) SO that f is orthogonal to dl the gm,, (z)= e i u Z g ( z -ntO). In this case the g,,, not only fail to constitute a frame, but the inner products

82

CHAPTER 3

Cf, ~ h n , ~are ) not even sufficient to determine f .

We are therefore restricted we even 'havetb chbose woto < 2?r. Note that no similar restriction w w,4 existed in the wavelet case! We wiM come back on these conditions in Cbpk,r 4, where we discuss in m e grester detail the role of time-hquency dm&y for wind d Fourier vemus wavelet frames; we postpone our prod of the neclessity of *to < 2r until then.

5 2n; in order to have good time- aud frequeacy-1&tbn,

to

A slrfaeient co&ti&

and estimates krs %beh m e bun*. do not necessarily tpmtituh a frame. An easy counterexampie is g(x) = 1 fm f) 5 x 5 1, o otherwe if I$ 7 1, then my func tion f supported in [I, to] will be orthogonal to all the h,,,bwever small uo is chosen. In this example essinf, C, (g(s - do)('= 0, and this is what stope the %+, frum being a frame. (Something M a r occurs fbr w a d ,see 53.3-1 Computations entirely similar to those in the wavelet csse shaw that

3.4.2

Even if woto

id

ft"fT

Ilf IId2

< %, the %,,

C I(f99mr)12

*

m.n

As in the wavelet ame, s&cientIy fast decay on g leads to d e w for P,so that ty choosing wo d en-, the s e e d terrrm in the rf&t-$and eides ofCf3.4.3), (3.4.4) caa be made aab&dlg smell- If C , Ig(x - nte)p is bomdd, and ltcmxkd b k o w by a strictly positive eamtmt (no "canspiring" of the -08 of g), then this implies that the h , constitu* a ~ E for W fdkbatly 4 UO, prieh &am&! bOtlllCts given by (3.4.3), (3.4.4). l%cpW*, ore b , t h e bllowiq prctpoaittan.

PROPOSITION 3.4.1. I f g ,

are suc-fr h t

DISCRETE WAVELET TRANSFORMS FRAMES

83

-

- nt01i b(z nto + $11 d a at hast ~ ad and rf P ( s ) = suposzg,, Cn fast as (1 + ~si)'('*'), mfh L > 0, then there ezwts (wO)thr > 0 SO that the gm,n(x) eCRUJb2g(z - nto) wnstttute a &me 11,heneuer < (@)thr For wo c ( w ~ ) the ~ ~nght-hand , ssdea of (3.4,3), (3.14) an? jhme bounds for the 9m,nThe conditions on and (3.4.5) are met if, e-g., Ig(z)l 5 C(1+ I X ~ ) - ~ with E

y

> 1. REMARK.The windowed Fourier case exhibits a symmetry under the Four~er

transform absent in the wavelet case. We have

$-

,

.

which impliea that (3.4 3), (3.4.4) still hold if we replace g , w, to by 8, to, wo, respectively, everywhere in the right-hand sides (including in the definit~onof p). Using this remark, we can therefore compute two est~mateseach for A and B, @ pick the highhest one for A, the lowest for B o 5.43.

5;

The dual frame. The dual frame 1s again defined by

F* F is now (F' F)f = C,,

(f, g,,,) g,,,,,. In this case one easily checks $hat F'F commutes mth translations by to as well as w~thmultiplications by i.e., if ( T f ) ( x )= f (s to), ( E f ) ( x )= e W xf ( z ) ,then *%ere

-

F*FT=TF'F,

F'FE=EFgF.

fdows that (F'F)" a h commutes with E and T, so that

-

gm,n = (F*F)" JiF = P T" (F'FL-I

g=(x)

T" g 9,

= elrnvozij(x - nto) = 4,,,(4

,

(F*Fj'lg. Unlike the generic wavelet case,the dud kame is dm* by a single function 4. This means that it is not aa important in the Fourier case that the frame be close to a tight frame: if B / A - I is ble, then one simply cumputts 3 to hi& precision, mce Bnd for all, with the two dual &ma.

dr~meawith compact support in time or hquency. The folmcthn, again &om Daubechiea, Grossmann, and Meyer (1986), arid to $3.3.6.A, leads to tight windowed Fourier W with arbitrarily

high regularity if wto< 2n. If support g C

[-$,

51,then

where we haw used that for any n, at most one value oft can contribute, because of the support property of g. Consequently,

and the frame is tight if and only if EnIg(x -nto)I2 = constant. For instance, if woto 2 n, then we can start again from a Ck or Cm-functionu satisfying (3.3.25) and define .

I

i

iI

I

sin

(

0

[

~

.

~

+ ( a ~ -. ~

~

o

- ) 5 ] 5, ~ 5 wo ~ - - t o ,

otherwise .

function (depending on the choice of u) with comp.et Then g is a Ck or support, Hgll = 1, and the g, constitute a tight frame with frame bound 27r(wohj)-' (as already followed from (3.4.2)). If woto < n, then this construction can easily be adapted. This construction gives a tight frame with compactly supported g. By taking its Fovxier transform, we obtain a frame for which the window function has compactly supported Fourier transf~rrn.'~

B. The Gaussian. In this case g ( t ) = n-li4 e - ~ ' / ~ Discrete . families of wiqdowed Fourier functions starting from a Gaussian window have been disc& ' extensively in the literature for many reasons. Gabor (1946) prop& their use for communication purposes (he proposed -to = 27r, however, which is inapprapriate: see below); because of the importance of the "canonical coherent states* in quantum mechanics (see Klauder and Skagemtarn (1985)) they are of interest to physicists; the link between Gaussian coherent states and the Bargrnann space of entire function makes it poesible to rewrite results concerning the gm,rn in

DISCRETE *AVELIW TRANSFOW.

7

.

85

FRAMES *

.

terms of sampling properties for the Bargmann space. Egploiting this lidc with entire functions, it was proved in Bargmann et al. (1971) and independently in Perelomov (1971) that the g, span aU of Gz(R) if and only if woto 5 2n; in Gacry, Grossmann, and Zak (1975) a different technique was used to show that if wotn = 2 7 ~ then .

even though the,,g , are "complete," in the sense that they span L~(IR).'~ (We will see ia Chapta 4 that this is a b t consequence of wo . to = 2n, and of the regularity of both g and g.) This is therefore an example of a family of g,,, where the inner products (f,gm,,) suffice t o characterize the function of f (if (fi, g,,,) = (fi, gm,n) for all n,n,then f i = f2), but where there is no n u m e r i d y stable reconstruction formula for f from the (f, g,,,). Bastiaans (1980,19%f)has constructed a d u d function 3 such that

f = C (f, 9m.n) b . n

=

-

1

(3.4 6 )

m.n

a#h 3m,n(~) e'mwo= g(z - nto), but comrergence of (3.4.6) holds only in a $'#wry weak (in the sense of distributions-see Jamsen (1981, 1984)), and wt even in the weak L2-sense; in fact, 3 itself is not in L*[w). to = 27r 1s thus completely undetstood, what happens if uoto < 27r? Table 3.3shows the values of the frame bounds A, B and of the ratio B/A, for " various values of wa.to, computed from (3.4-31, (3.4.4) and the analogous formulas uslng g. We find that the g,,, do constitute a frame, even for wo - to/(27r) = .95, &though B / A becomes very large so close to the 'Lcrit~cdl density. It turns out t0/(2n) = l / N , N E N, N > 1, the frame bounds can also be that when -computed via another technique, which leads to exact values (within the error tat ion) instead of lower, respectivelybupper,bounds for A, B.14For the to/(%) = f and Table 3.3 reveals these exact values as well; it surprising t o k e how close our bounds on A, B (which are, after d l , obtained hy-Schwarz inequality, and &&ht therefore be quite coarse) are t o ues Substituting these values for A, B into the aGroximation at the end of 83.2, we can compute 3 fixthese different choices of wo, to. e 3.6 shows plots of ij for the special case where wo = to = ( X ~ n ) l /with ~, X ues -25, .375, .5, -75, .95, and 1. Note that ~ a s t i k function ' 3, corresponds to X = 1 (lower right plot in Figure 3.6), has to be computed &ntly, since A = 0 for X = 1. FO; s m d A, the frame is very close t o tight, ij is close to g itxlf, as is illustrated by the near-Gaussian profile of ij for .25. As A increases, the frame becomes both less redundant (as reflected by ng maximum amplitude of 3) and less tight, causing 3 to defia$e more from a Gaussian. Because both 9 and j have (faster than) exponential , one can easily prove from the conve%ing series representation for 4 (see that 3 and 4 have exponential decay as well, if A > 0. It follows that good time-frequency locabation properties, for all the values of A < I in Figure 3.6, even though it is quite striking how ij tends to Bastiaans'

j.T

i,

C

I

86

CHAPTER 3 I

FIG 3 6 The dud /mme junciwn $'for G a w s ~ a ng and

= to =

(2r,4J1/2, &

X = 25,.375, 5 , 7 5 , 95, and 1. As A m m c l s e s , g h a t e s mom and mow fmm o Ga-n (rgectmng the rncseose of B / A ) , and its omplttude increases ar well ( k w e A + B deEnoses). For X = 1, g ta no longer square tntegmblc.

pathological as A increases. For A = 1 , dl-time-frequency lockization breaks down.15 The series of'plots in Figure 3-6suggests the conjecture, first formulated in Daubechies and Grossmann (1988), that, at teast for a u s s i a n g, the gmSn are a frame whenever woto .< 2 ~ .In Drtubechies (1990) it wes Shawn t h ~ thii t is indeed the case for u0to/(2?r) < .996. Using entire function methods, this conjecture has since been proved, by Lyubsrskii (1989) and independently by Seip and Wallsten (1990). There exist of course many other possible and popular choices for the window function g, but we will stop our list of examples here, and return to wavelets. 3.5.

I

I I

Time-frequency Iacalization.

One of our main motivations for studying wavelet transform (or windowed Fourier transforms) is that they provide a timefrequency picture, with, hopefully, good localization properties in both variables. We have awxted several times-that if q3 itself is well localbed in time and in frequency, then the frame Gnerated by will share that property. In this section we want to make this vague statement more precise. and 141 to be symmetric (true if, For the sake of convenience, we W u m e e.g., II,is real and symmetric-a good example is the Mexican hat f u n ~ t b n ) ' ~ , then 9 is centered around 0 in time 8nd near kto in frequency (with, e.g., b = E I $ K ) ~ ~ /4 [ Jl~j(c)v]). ~ If$ is well localized in time and frequency, then $, pjill similarly be well -1 wound comnbo in time and around fqmb , in frequency. Intuitively spadung, (f, b,,) then represents the "information contentn in f near time a?& d near the frequencies fq r n { o . If f itself is LLessentially localized" on two rectmgle9 in time-frequency space, meaning that, forsomeO<%
I

~

DISCRETE WAVELET TRANSMXUMS: FRAMES

87

TABLE 3.3 Values for the fmme boundP A , B and thew mtw B/A for the aue g(2) = r-'I4 exp(-z2/2), for drffereni d u e s of w, to For wolo = 1r/2 and x, the exoct d u e s can be computed ma the Zak tmruform (see Daubechtes and Gms8mann (1988)).

CHAPTER 3

where 6 is some small number, then this intuitive picture suggests that only those (f, +*,,)corresponding to m,n for which (aznbo, f iim w~thin or close to [-T, T] x (I-Q1,-&l l J [no,all)ate .needed t o reconstruct f to a very good approximation. The fobwing theorem itates that this is indeed trhe, thereby justifying our intuitive picture. THEOREM 3.5.1. Suppose t h t the $m,n(~)= a.-m/2 +(%-mx - &) wnstztute a jmme with frame bounds A, B, and suppose that

wzth a > 1, 0 > 0, 7 > 1. Then, for any c > 0 , there ezzsts a fintte set B, (Ro, Cll; T ) C z2 90 that, fix 011 f € L2(R),

1. If f satisfies (3.5.1) and (3.5.21, then the first two terms in the right-handA.

side of (3.5.3) are bounded by 26

11f 11;

choosing e = 6 then leads to

2. As €4, # Be(Oo, JZ1 ; T)doo (see proof, below):, infinite precision is only possible if infinitely many (f, &,n) are used. n Figure 3.7 gives a sketch ofthe set Bc(Oo, R1; T) far one particular value of e; the proof will show how we obtain this shape.

Pvf. 1. We define the set B, as

B,(Ro, R1 ; T) = {(m, n) E 2'; -

mo I m 5 m l ,

lnbol

5 aGmT+ t ) ,

where rno,rnl and t, to be dehed below,depend on & I , Q ~ TI , and e. The points (aomnbo, faornb)comqmnding to (m, n ) in such a set, do indeed fill a shape like Figure 3.7.

1

*

,

4

i

DISCRETE WAVELET TRANSFDaS: FRAMES

Ct

tr

t

L-----

FW. 3.7. Thc

set

-,---_I

Be(flo,R I , T ) of "wavelet io#ice p n b " needed for

raowtructron off sf f rs localrzed mostly an

2.

f-

C

(f, $m,n)

[-T,Tj

in time and an [-Cll,

appnmmate

an

-%I

U [Oo,fll] tn

+Zn

(m,n)EBt

=

SUP

=

s

SUP Uhll=l

7

1 C C

( f q

memo

,EZ

m>ml

[I(Pno.n,f.$m,n)I SUP fl'I1-l

h)

$m n ,$ ()*:,

(m,n)e~.

or

+

(f, +m,n)($xnv h)

(m,n)EB,

SUP #hH=l

C

(f h) -

Uhll=l

C

+

C

1((1*Pho,~1)f, +m,n)II ~($Ynv h)I

noSm<mr lntt,,l>.;-~+t

11( Q T +m,n) ~ ~ I + l((1- QTM,$m,n) I] I( $ x n j

'

h)I 9 (3.5.5)

where we have introduced (QT f )(z)= f(z)for 1x1 1 T, (QTf) (z)= 0

+=

<

otherwise, and (Pn,,n,f)"(O = if 00 It1 5 fh, (P~,,n,f )"(O = 0 otherwrse. since the constitute a kame with frame bounds B-I. A-'. y e have

=

[ /

1/2

e I~MII~]

(because Ilhtl = 1) .

I€I<%

Similarly,

It remains t o check that the other two terms in (3.5.5) can be bounded by

3. By the same Cauchy-Schwarz trick, we reduce the remaining two terms in

(3.5.5) t o

(3.5.6) It is therefore sufficient to show that for appropriate m, rnl, t each of the expressions between sqwm brackets is smaller than BE' llf 1!2/4.

4. We tackle the b t term in (3.5.6) by the same technique as in the-proof of

DISCRETE WAVELET TRANSFORMS FRAMES

Proposition 3.3.2:

where 0 < X < 1 will be fixed below. Since 11 + (u - s)~]-' (I + s2)[1 + (u + s ) ~ ] -is~ bounded uniformly in u and 8 , we have

Substituting this into (3.5.7) we find

The sum over t converge3 if yX > 1, ie., if X > 1/7; we can choose, e-g., A = i(1+ 7 - I ) . On the other hapd, for Ro L I 01,

CHAPTER 3

92

and

In all these estimates, the constants C, may depend on ao, bo, A, 8, and Ro, R1,no,and m l . Subdit~rtinginto (3.5.8), with the choice X = i ( 1 ? - l ) , leads to -y but are independent of

+

If rnl 2 (In m)-'((? - 1 ) - ' ln(4C7/Bt2) - In no]a d mo [r8-1(r - I)-' In (Bc2/4C7)- in Q l ) , then this leads to

'I (In Q)-'

,.

as desired.

5. The second term in (3.5.6) is easier to deal with. We have

C

l ( Q2I)~..,f+ .

JnhJ>a;"'T+t

+

The sum over n ~ p l i t sinto two parts, n 7 b;' (aCmT t ) , and n < - b ~ ' ( a & ~ Z ' - t ) . Let nl be the smallest integer larger then b;'(a;*T-tt).Then

+

,

-

(because JGmz6 1= n 4 - G m x > (n - ni)bo

t+Gm(T-x))

+

DISCRETE WAVELET TRANSFORMS F U M E S

The mlm over n < -6; that

93

'( a ~ ~ t)Tisf dealt with in the same way. It follows

wh~chcan be made srtlaller than B

c2((f 112/4 by

&wing

This concludes the proof

t The estimates for mo,m i , t that follow from this proof are very coarse; in practice, one can obtain much less coarse values ~f $ and hive faster decay than stated in the theorem (see, e.g., Daubcch~es(1990), p 996). For Iater reference, let us estimate # B,(Oo, R I ; T), as a function of !lo,i l l , T, and f We find

4

%

3

"

On the other hand, the area of the time-frequency region f-T, T] x (I-R1, Sto] U [h, a l l ) is 4T(F1 Qo) As Ro+O and T, ill -too, we find

-

ch is not independent of c. We will come back to this in Chapter 4. m 3.5.1 tells us that if tl, has reasonable decay in time and in ency, then frames generated by do indeed exhibit time-frequency 1 e ation features, at least with respect to time-frequency seb of the type T ] x ([-$I1, U [Q, $Il]). In practice, one is interested in l o c a l b on many other sets. A chirp signal,for instance, intuitively corresponds to nai region (possibly curved) in the timefrequency plane, and it should be to reconstruct it from only those +m,n for which ( a r n h , f%-mb) is in to this region. This turns out to be the case in practice (for chirp signals, others). It is harder to formulate this in a precise theorem, mainly one first has to agree on the meaning of "1ocdizationn on a prescribed uency set, when this set is not a union of rectangles as in Theorem 3.5. I.

+

CHAPTER 3

94

If we choose the interpretation in terms of the operators Ls defined i n $2.8 (i.e., f is mostly localized in S if II(1 - Ls) f 11 << I1 f It), then the theorem is almost trivial if the wavelets in the definition of Ls and in the frame both have good decay' properties. For any othQ timefrequency localization procedure (e.g., using Wigner distributions, or the sffine Wigner distributions of BeBertrand and Bertrand (1989)), we still expect results analogous to those in Theorem 3.5.1, but the proof will depend on the chosen localization procedure An entirely similar localization theorem holds for the windowed Fourier case. THEOREM 3.5.2. Suppose that the g,,,(x) = e'-~'g(r - nto) cowtotute a fmme mth fTnme bounds A, B, and suppose that

unth a > 1 . Then, for any f E L2(R), and all T, R > 0,

> 0,

c

there e m t t,,w,

.

> 0, such fhat, for dl

Proof. 1. By the ssme tricks as in points 2,3 of the proof of T b r e m 3.5.1, we have

IZ: \

+

mEZ IntoI>T+t,

liQ~f,*,.rl2

1'") . (3.5.12) = j(()for

,

where (QTj ) ( z ) = f (z) for Izl 5 T , 0 otherwise, and (Pnf)"(Z) 141 5 n, 0 otherwise. The theorem follows if we csn prove that the last tm tenns in (3.5.12) can be bounded by B1I2ci(f 11. Let us hst concentrah an the kust term.

DISCRETE WAVELET TRANSFORMS. FRAMES

%

d

> ;'+ :<

I '

$&

k ti ' p*a

*

+

95

One easily checks that the contribution for n > t ; ' ( ~ t,) is exactly equal to that for n < - t i l ( T t L ) ;we may restrict ourselves to negative n only, at the price of a factor 2. By reaefhhg y = E wo C if f! is positive, we see that ue may restrict owselves to negative C as well. Hence

+

sup

[l

-

+ (z+ nto)'~-~/*

I=l
CHAPTER 3

However, for any f i , v > 0, we have

I

It follows that

I

Let

7tl

be thc: smailcst integer larger than

00

<

I

I

[I

T + t,. Then

+ ( ( n - tc)toc t.)21-0+1/2

by the computation above. Putting it altogether, we have

where C3depends on UQ to, , a,and C,but not on T (or 0).

3. Similarly, one proves

Since cr > 1, it is clear that appropriate choices of t,, u, (independent of T or R!) make (3.5.15), (3.5.16) smaller than BE^ fl f 1i2/4. This concludes the proof. m

97

DISCRETE WAVELET TRANSFORMS FRAMES

<

Fgure 3 8 gives a sketch of the collect~or~ ( m , n )such that Iw( R + w e , Into( 5 T t,, as compared to the time-frequency rectangle [-T,T] x [-R,S2]. The "e-boxn has a d~fferentshape from that In F~gure3 7

+

FIG 3 8 The set of latirce pornts

Be needed

lor crppmnmote reconstructrota, vrcl Ule [-T, T]an hme and an 1-0, R]

wandowed Fmner tnirufonn, of a funcicon localued mosUv tn

compare agan the number of puiuts ;.the enlarged l k c Be = f.(m,n), lmwol<sZ+w,, InbJ
C/

-

1

3

f-l

Z T .I

6;s'+ % $;

+

*

..'

(n + u,)

+

2t;' (T t c ) --+ (woto)-' (3.5.17) 4TR In contrast t o the wavelet case, thts limit is independent of c We will come back bD the significance of this m Chapter 4. 5

*

-

# B. (T, n) - zw;

%

4TQ

3-8. Redundancy in frames: What does i t buy? yi-"h illustrated by the different tables of frame bounds, frames (wavelets or win' dowed Fourier functions) can be very redundant (as measured by, e.g.,

9

$ ifthe frame is cloae to tight, and ~f 811 the frame vectors are normalized). In ' applications (e.g., as in the work of the Marseille groups-see the papers of

-

@ tt*l i@

CHAPTER 3

98

i

Grossmann, Kronland-Martinet , Torresani) this redundancy is sought, because represent at ions cla3e to the continuous transform are wanted. It was noticed very early an by J Morlet (private communication, 1986) that this r e d y n b c y a1.w leads to robustness, in the sense that he could afford to store the wavdet coefficients (f, 11, ,) wlth low precision (only a couple of bits), and still reconstruct j with comparatively much higher precision. Intuitively, one can understand this phenomenon a.follows. Let (ql),EJ be a frame (not necyarily of wavelets or windowed ~ o d r i e rfunctions). If this frame is an orthonormal basis, then

p:

7f-.P(J1,

(Ff)]=(f,~.j) Is a unitary map, and the image of W under F is all of l?(J).If the frame is redundant, k c . , if the cp, are not independent, then the elements of F31 are sequencm with some correhtions bqilt into them, and F X = Ran (F) is a subspace of P2(.I), smaller than P2(J) itself. The more redundant the frame IS, the "smaller" Ran (F) will be. As shown in $3 2, the recoxlstnlction formula

involves a projection onto Ran ( F ) : i t can be rewritten as

and PC= 0 if c 1 Ran ( F ) . If the ( j , cp,) are adulterated by adding sump a, to each coefficiexlt (an example woutd be round-off error), the total e k t on the reconstructed h~nctionwould be Since p* includes a projection onto Ran (F),the component of the sequence a orthogonal t o Ran (F) does not contribute, and we expect 4 f - fappmx~~to be smaller than JIall. The effect should become more pronounced the "smaller" Ran ( F ) is, i.e , the more redundant the Frame is Let us make this more explicit with the twudimensiolial frame used as an example in 53.2, and by comparing it with an orthonormal basis. &fine 211 = 1 ( l , O ) , uz = (0, I), el = u*, ez = - p l , Q = $ul - iu,; (u,, 7.4,) constitutes an orthonormal basis for , 'c (el,ez,es) a tight frame with frame If we add q c to the coefficients (f, ul), where a, are independent bound random variables with mean zero and variance 1, then the expected error on the reconstruction will be

-921,

i.

If

add ale to the frame coefficients (f, e, ),, then we find

DISCRETE WAVELET TRANSPORMS. FRAMES

99

which represents a gain of over the orthonormal crrso! A similar argument can be applied to waveleior windowed Fourier frames. In order to confine ourselves to only Gnitefy #m,n or gmSn, we assume f to be LLessentiaEly localizedn on 1-T, T ] x ({-Q,, -%I U Po, Q,)) (wavelet case) x [S2J$ (windowed I, Fourief CBBB), so that there exists a finite set or 1-T, Be (see $3.5) fix wMch

+

.

i

-

(and similarly for the windowed Fourier case). Let us assume the frame is almost tight, #,,, 2: A-'tD,,,. Adding am,,6 t.u every (f, $m,n), under the assumptions E(a,,, a , t , , r ) 6,m~bnnt, E(a,,,) = 0, leads to

-

5 r2 ltf It2

+

@(# B,)A-'~

(3.6.1)

If we assume l(+rn,nI] = 1). If we "double the redundancyn by halving b,then tlte new frame would again be almost tight (see, e.g., (3.3.11), (3.3.12)), with a frame bound A' twice aa large; on the other hand, the new "c-box" B: would contain twice as many elements. It follows that

-21

$

i.e., doubling the redundancy leads to halving the dfed of adding errors to the wavelet coefficients. The same argument can be d e for the windowed Fourk - 'w. 31 As it stands, the argument above is rather heuristic. There are indications 6 %that it can be consideratly strengthened: the gain factor ohX?~ed by Morr kt was in fact larger than what would folbw from these arguments. hiore4 *$ ,%t ' Over, recently Munch (1992) showed'that for tight windowed Fourier frames ; with X = ( 2 ~ ) - ~ = wN ~"t ~, N E N, N > 1, tbe gaiin factor with respect to ' - the o r t h o n o d case i. in fact N-', and not N-', as would follow from our 0.-% I-. that A-I is integer in an essential way, but it is hard $ T e n t . His proof b believe tbat the same phenomenon would mt exist for nooninteger A*'; maybe holdsfor w&t frames! I put thia a$ a rbdlenge to the reader. . . . . it

' 6

*

s-"

'f"

.

loo 3.7.

CHAPTER 3

Some concluding remarks.

f n this chapter we have studied in some depth the reconstruction of

f from the

-m/2

sequence((f1 SDm,ti))m,nE~,whael(r,,n(~)=ao Il(qmz-nbo)(~dvariations thereof-see $3.3.4). We have seen that numerically stable reconstruction is only possible if the &,, constitute a frame, and we have derived a reconstruc, are a frame. One can, however, use other reconstruction tion formula if the,,$ formulas (provided the tlr,,, do constitute a frame: the necessity of that condition remains!) To conclude this chapter, let me sketch the appro& of S. Mallat, which addresses moreover the problem of shift-invariance. The discrete wavelet trausform, such as I have described it in this chapter, is highly non-invariant under translations. By thii I mean that two functions may be shifted verslons of each other, while their wavelet coefficients may be very different. This is already illustrated by the "hyperbolic lattice"" in Figure 1.4a, where the axis t = 0 plays a unique role. In practice one does not use an infinite number of scales, but cuts off very low and very high frequencies: only those m for which m l < rn 5 mo are used. The resulting truncated lattice is then invariant under translations by bo2"Q (choose a o = 2 for simplicity), which is, however, very large with respect to, e.g-, the sampling time step for f (in most applications, f is given in sampled form). If fl is a shifted version off obtained by a translat~on# nbo2w, then typically the hvelet c&cients of fl will be different from those of fo. Even if the shift is fibo2*, with ml m 5 m, then (fl, = (f, +m,n-zm-mii) if m 2 m, but no such formula can be written for m > m. For some applications (in particulat, all applications that involve "recognizing" f ) this can be a real problem. In a first approximation, the solution proposed by S. Mallat is the following:

<

Compute all the J & f ( r )+(2-"z - n2-"bo) = a,,,(/) (a special $Jsuch as in 93 3 5.D makes it-possible to compute these in C N log N operations, if j consists of N samples). This l i t of coefficients is invariant for shifts of f by nbo. At every level m, retain only those crm,,( f) that are local extrema (as a function of n). This effectively corresponds to a subsampling of the highly redundant am,,(f). In practice, the number of subsamples retained is proportional to 2'm times the original number; this is about the same number as one has in a not too redundant frame of the type described earlier, but the suhsampling is now adapted to f , and not imposed a priori by the hyperbolic lattice.

Together with this prescription for decomposition (here described in a simplified form) Maliat then p r o m a r d r u c t i o n algorithm, which works very well in practice (see Mallat (1991)).18 In M d a t and Zhong (1992), the procedure was extended to two dimensions, to treat images. One way to view Mailat's approach is to look upon the 2-m9(2-mx n2-mbo) as the underlying frame (note that the change in normalhtion caunters the larger number of frame vectors in every m-lwei) . Again, it is nemmuy that this family satisfy the frame condition

-

i

101

DISCRETE WAVELET TRANSFORMS: FRAMES

+

(3.1.4) for a stable reconstruction algorithm to exist, but once this condition is satisfied, several reconstruction algorithms can be proposed In this case, Mallat's extrema-algorithm is certainly more sophisticated than the standard frame inversion algorithm.

Notes. 1. If any f can be written as such a superposition, the

,

$m,n are ,also called "atoms," and the corresponding expansions "atomic decompositions." Atomic decompositions (for many spaces besides L2(R))have been studied and used in harmoruc analysis for many years: see, e.g., Coifmtp and Rochberg (1980) for atomic decompositions in entire function spaces.

2. This is true except for very special JI. If the qbm,, constitute an orthonormal h i s (see Chapters 4 and later), then the expansion with respect to this orthonormal basis provides a discrete "resolution of the identity."

3. The polarization identity recovers (f,g) from 11f f gll, 11f fig((:

That is, if (Jn)nENis any increasing sequence of fnite subsets of J, i.e., Jn c J,,, if n 5 m, tending to J as n tends to oo,i.e., U n E Jn ~ = J, then IIF'c - CIEJ., CI VJ11 0 for n-roo. The proof is in two steps:

-

If n2

2 nl

> no, then

I

whi& tends to 0 for w-oo. Hence the q,, = a Cauchy sequence, with limit 7 in L ~ ( R ) . For this 7 , and any f E L2(R),

This is proved

follows:

( ~ (+f91, f + 9 ) - (R(f - !I) f'- g ) (because R' = R) ;

cJpj constitute

,

CHAPTER 3

102 1-

(Rf,g) 5 2 B+*

1 1 1 1 1 I 2 + 11g1121 ; lllf + dl2+ (If - gl121= H

I(Rf,9)1 = ( R f , g ) ( R f , / I ( ~ f , g ) l

= (Rf *

i 31

(Rf ,9)9ll(RI,9)l)

B-A

jjg

1 6-A

55

Illf El2 + II(Rf,9)gll(Rf,g)l 1l21 Illf ll' + 11~1121;

IlRll = SUP#f,l=,,IgII=1 I(Rf,g)l 5

B-A rn .

6. Intuitively, % can be understood as a "supeiposition" of Jhe rank one trace-class operators (., ha*b)ha*b, with weights c(a, b). If c is integrable with respect to aT2da db, then the individual traces af {., halb)ha*b(which are all equal to I), weighted by the c ( a ,6 ) are "summable," so that the whole superposition has finite trace,

This handwaving argument can be made rigorous by approximation arguments. - 1

7. We use here the "essential infimum" (notation: ess inf) defined by

where IAJ stands for the Lebesgue measure of A c R. The difference between essinf, f ( x ) and inf, f(x) lies in the positive measure requirement: if f(0) = 0, f(x) = 1 for all x # 0, then inf, f(x) = 0, but ess inf, f(x) = 1, because f 2 1 except on a set of measure zero, which "does not count." In fact we could be pedantic, and replace inf or sup by essinf or-esssup in most of our conditions without invalidating them, but it is usually not worth it: in practice the expressions we are dealing with are continuous functions, for which inf and essinf coincide. In (3.3.11) the situation is different: even for very smooth 4 , the sum I$(U~F)~* is discontinuous at = 0, because 4 ( 0 ) = 0. For the Haar function, for instance, l$({)j = 4 ( 2 ~ ) - ' / ~ I ( ( - 'sin2 {/4, and CmeZ 1$(()12 = ( 2 ~ ) ~ ' ' if ( # 0, 0 if ( = 0. We therefore need to take the essential infimum; the infimum is zero.

8. This condition implies both the boundednas of decay of P(s):

zmEZ 14(ag6)12 and the

DISCRETE WAVELET TRANSFORh4S FRAMES

and

)

5 c2 sup

l
tg

aom (1

+ IaF2E + s12)-('-a)/2

=-a3

In the b t term we can use that, fm Is1 2 2, loF6 + ti > Is1 - 1 2 *, a(1 + + $I2)-' < 4{1 + /sI2)-I, for 1st 2, (I + aIgmt + s12)-I 5 1 < 5(1+ [sI2)-l. It &that the first term uro be bounded by C(1+ / s 1 ~ ) - ( 7 - ~as ) / soon ~ as a > 0, -y > a. For the second term that suPz,y,g (1 + y2)[1 + (z- y)2]-1[1 + (z3- v ) ~ ) - ' < 00 to bound the sum by C"(1 + Cz==a(l + loo*ft2)-c(~-Q)/2, whse b < E < 1 is wbitrary. Since 1 < 5 a,this can be bounded by if 7 > a. We have therefore, for 0 < p < y - a, Cm@-I-

lee

Em

If 6 is continuous and has decay at oo, then ili)(qf)l2 is mntinuous in c. errepr at 5 = 0. There exists therefore a a,that ~ $ ( @ c ) I5 ~ fr if K - QJ5 a. Define, for a' < a,a fundion f by f([) = (2a')-1/2 if 1.f - 5 a', i(c)= 0 atherwise. Then

k-bt-'

(me Cauchy-Wmm on the integral)

CHAPTER 3

3

SC

4-

2 ~ Sup ' f

19t&a$'{)l2. m~it

If /d(t)/< C(l + 1((2)-7/2 with y > 1, then this infinite s w is uniformly bounded in <, and we can choose cr' so that the whole right h d - s i d e of the inequaiity is 5 2c. 10. Beware of a mistake in the example on pp. 988-989 of Daubechies (1990). The fornlula for (b)" should read (b)" =C T, iii+,*, and leads to the conclusion that h $ L P ( R ) for .small p. I would like to thank Chui and Shi (1991) for pointihg this out to me.

11. This is slightly different from multiresolution analysis, where (3.3.27) would also contain a scaling factor 2:

12. One can ELLSO construct tight frames where neither g nor i have compact support. It is, for instance, possible to construct a tight frame in which both g and g have exponential decay. One way of doing this is t o start .from any windowed Fourier frame, with window function g, and to define the function G = (F*F)-"~ g, where P F = gm,,)gmVn.The functions Gm,,(x) = eamO" G ( r - n t o ) (same wo, to as in the h , n ) then constitute a tight frame. One has indeed

C,,,(-,

The explicit computation of G can be carried out by a series expamion for ( F * F ) - ' / ~analogous to the series for (F*F)-' in $3.2. If g and 4 have exponential becay (in particular if g is Gaussian), then the resulting G and its Fourier transform have exponential decay as well. -For more details, plots of examples, and an interesting application, see Daubechies, Jsffard, and J o u d (1991). t

I

1

I

C

105

DISCRETE WAVELET TRANSFORMS FRAMES 2

13. The proof in Bacry, Grossma~~n, and Zak (1975) uses the Zak transform, which we introduce and use in Chapter 4. Full details of their argument are also given in Daubechies (1990). It is ~nterestingthat their proof can be extended to show that the g,,, still span all of L2(R) if one (aay one) of the g,,, is deleted, but not lf two funct~onsare deleted.

F

14. These exact formulas use again the Zak transform. Their derivation is given in Daubechles and Grossmann (1988); it 1s also revlewed in Daubechies

( 1990). 15. In sonie applications, 3astiaans1 result is ~nterpreted(correctly) to TIISL: that one should "oversample" (I e., ci~ooscwuto < 27r) m order to restore stability. Nevertheless, even in buch an "oversampled" reginle, somet~mes Bastiaans' pathological dual function is st111used (sac, f a instance, Porat can be split into two families, and Zeevi (1988)). If wto= x, then the g,., . gm,zn and gm,zn+l, each of which can be corlsidcrcd to be a fam~lyof Gaussian windowed Fourier functions with w ~ t o= 2x, one generated by g i b i f , the other by g(x - t o ) . For both families, the badly cdnvergent (nonconvergent in LZ)expansion (3.4.6) can be wr~tten,axid a functio~ican Lc viewed as the average of the two expansions Thls is of course true in the sense of distrlbutio&, and in reasonable convergence (probably due to cancellations) seems to be achieved (us~nga truncated version of Bwtiaans' g- private communication by Zecv~(1989)), but far better timefrequency localization, and I suspect better convergence, in practice, would be achievable by using the optimal dual function ij (corr~spond~ng to X = .5 in Figure 3.6 in this case).

A

C

16. This symmetry is certa~nlynot necessary.

: 17.

9

It is in fact a true hyperbolic lattice with respect to the hyperbolic geometry on both the positive and negative frequency half plane.

$ 18, Note, however, that Y. Meyer has proved recently that the a,,,(f) which are local extrema in the construction above do not suffice to characterize XI tr* 5 f completely.

?

-.---

CHAPTER

4

4,

Time-Frequency Density and Orthonormal Bases

This chapter splits naturally into two parts. The first sectjon discusses the role of time-frequency density in wavelet transforms versus windowed Fourier transforms, In particular, for the windowed Fourier tramdorm, orthonormal bases we possible only at the Nyquist density but no such restriction exists for the wavelet case This leads naturally to the secand wtion, which discusses dire&posaibilitiea for orthonormal bases in the two cases. 5

4.1. i

The role of time-frequency density in wavelet and windowed Fourier frames.

We start w ~ t hthe windowed Fourier case. We mentioned in $3.4~1that a family 4

s

offunctioas(g,,,;m,n~Z),

,

g m , n ( ~= ) earnyz g(2 - d o )

-

.

(4.1 1)

cannot be a frame, whatever the choice of g, if w~ to > 2%. In fact, for any of g E L2(R), one C ~ Dfind f E L2(8t)80 that f # 0 but gmen)= O for m,n E Z. ff, for instance, wo = 27r, to = 2, then 5u~hs funct;iofif is easy to ruct: ( f , g,,,) = 0 for all m,n E 2 I& to

o = =

J dz

/d

1

2mmz

U,

f ( z ) s(x - en1

e2rlmz

C 1(..+1)g ( 2 + 1 - 2 n ) , CEZ

ZGE +

+

that it is sufficient to find f # 0 for which f (z I ) g(z I - 2n) * 0. -sE3efine ~AQW,for 0 5 s < I , ! E 2, f [ x + t ) = (-1)l g(z-4-1). Ckculy,

,,, +

-

+

-

f ( X e) g(3: t - 2n) = CtEZ(-l)' g(2 t - 1) g(=+ t 2n), which hms into its negative upon the substitution C = 2n - C' 1, and therefore 61p~abzero. The same constfiction can be used for any other pair wo,to Mth product 47r; a generalization of this ;construction ex$te if wo h > 2?r * " ~( 2d~ ) - ' w o t o is rational (aee Da~ibecbies(1990), p. 978). If wk(2x)- is far^ t4an 1 but irrational, tbea I know of no explicit construction for f # 0, 107

-

108

CHAPTER 4

f I g,,,. The existence of such an f was proved in Rieffel (1981), using arguments involving von Neumann algebras.' If only "reasonably nice" g are considered (i.e., g that have some decay in time as well as in frequency), and if we are only interested in proving that the gmVncannot constitute a frame (which is weaker than proving that there exists f I g,,,), then the following very elegant argument by H, Landau does the trick. If Ig(z)l 5 C(l z ~ ) - " / ~ , 1i(c)1 <- C(1 Ad the g,, mnstitute &.@me, then Theahem 3.5.2 tells us that functions f which are "essentially localizedn in [-T, T] x [-a,R] in the time-frequency pIane can be reconstructed, up to a small error, by using only the (f, g,) with 1-1 2 R, lntol 5 T . More precisely, iff is bandlimited t (IfII, then to [-a,01 and if & If

+

+

[hII>T

<

According to thi formula, all such hnctions can therefore be mitten, up to arbitrarily small error, as a superposition of the ,#,, with lml 5 w;' (0+6), In1 -< toT1(T+6), where b depends on the error allowed, but not on R or T. However, a corollary of the work of Landau, Pollak, and Slepian (see 82.3) is that the space of functions baridlimited to [-0,Q) and satisfying Xz.,r & 1f (x)I2 S 7 1f JI2 (0< 7 < 1. 7 fixed) contains at least - O(log (Q?')) dierdliit orthogonal functions (the appropriate prolate spheroidal wave functions). All these different orthonormal fiinctions can only be approximated by linear combinations of a fjnite number of gm,, if the number of gm,, exceeds that of the orthonormal ~ for any R, T. functions, i.e., if 27r-lQT -O(log ( R T ) )5 b i l t o - ' ( 0 + b ) ( +6), Taking the limit as 12,T--roo leads to (2x)-' < (woto)-' or woto < 27r. (This is really only a sketch of the proof. For full technical details, see Landau (1989).) For a31 pkctical purposes we e v ~ nhave to limit ourselves t o wo , to < 27r (strict inequality) if we *ant good timefkquency localization: frames for the limit case wo. to = 27r have necessarily bad localization properties in either time or frequency (or even in both). This is the content of the following theorem. THEOREM 4.'1.1. (Balian-Low) If the gm,n(s) = e2"'wg(x - n ) constitute a frome for LZ(W),then either $& x2fg(x)12= oo or e2(8(()l2 = 00. Before proceeding to the.proof of this theorem, let us review its history, and add some iemarks. Originally, the theorem was stated for orthonormal bases (instead of frames), independently by Balian (1981) and Low (1985). Their proofs we very similar, but contain a small technical gap which was filled by R.Coifman and S. Semmes; this proof c m then be extended to frames as well, as reported in Daubechies (1990), pp. 976-977. ~ubsequentl~, a different, very elegant proof for orthonormal bases was found by Battle (1988), which was generalized to frame in Daubechies and Janssen (1988). (This is the proof we give below.) Two well-known examples of functions g for which the family e2w'm2g(z- n) constitutes an orthonormal basii are

9

Je

1, OLxll, 0, otherwise

TIMEFREQUENCY%ENSITY AND ORTHONORMAL BASES

-sib n

109

and g(x) = 7 . In the first case, Sq C21p({)1* = 00,in the-second case J dx z219(x)12= 00- It is shown in Jenseq, Hoholdt, and Justesen (1988) that g with slightly better tirn&fiequency localization: they construct one can chg such that b&h g and jj are integrable (i.e., Sdz 1g(z)1 < a,j' 4 (#(()I < oo), but their dewy is still rather slow, as dictated by Theorem 4.1.1. Note that the choice wo = 21r, to = 1 in our formulation of Theorem 4.1.1 is not a serious restriction: the conclusion hofds whenever wo . to = 27r. To see &is, it su£6cesto apply the unitary operator (Uj ) ( x ) = ( 2 . r-1 ~) 1/2 ~ 9(2nw~'x); applying U to g,,,(z) = e'm"Ozg(z-nto) one finds ( U I Q ~ , ~ = ) (eZmimf X ) (ug)(xn)To prwe Theorem 4.1.1, we will use the so-called Zak transform. This transform is defined by (4.1.2) ( ~ f ) ( s1, ) = e2*auf(s- -

.

x

o

(€2 . .

'

A priori, this is well defined only for f such that C If (s - t )1 converges for all s, in *icrrlar for / f ( z ) l5 C ( l + I X ~ ) - ( ' + ~ ) . It turns out, however, that this restrictive interpretation of Z can be extended t o a unitary map from L2(R)to L3([O,1j2). One way of seeing this is the following: e m , n ( ~=)e3nrmze(x- n), with e ( x ) = lofor0 5 x < 1, e ( x ) = 0 otherwise, constit utes an orthonormal basis for fi2(R).

i

-

i

-

, e2rrcms

e-2rrtn

(ZeI(5,t ) -

%a.

iw,

n;c

, '4.

1,

F I

;.

4'

fV

(Zerfs, t ) = 1 dmoet everywhere on fO,112

.

It foIlows that Z maps an orthonorma) basis of L3(R) to an orthonormal basis of ~ ~ ~ (112), 1 0 ,so that Z is ufiitary. We can extend the image of L2(R) under Z to a different space, isomorphic to L'([o, 1 j 2 ) . b m (4.1.2),we find, if we allow ( 5 , t) outside [o, 112,

k t us therefore define the space-2 by

2 = { F : it2+@;F(s, t + 1) = F(s,t ) , F ( s + 1 , t ) G e2"" ~ ( st ) ,

$8 s,;ithen Z is unitary between L2(R) and 2. The inverse map is easy as well: $$p snyF~Z, id**+

$"" @

&>

(z"F)(z) =

1

dt F ( z ,t ) ,

for

110

CHAPTER 4

if this integral 1s well defined (otherwise, we have to work with a limiting argument). The Zak transform has many beautiful and useful properties. As is often the case with beautiful and useful concepts, it was discovered several times, and it goes by many different names, according to the field in which one first learns of it. It is also known as the Weil-Brezin map, and it is claimed that even Gauss was aware of some of its pioperties. It srtfs also used by Gel'fand. J. Zak discovered it independently, and studied it systematically, first for applications to solid state physics, later in a wider setting. An interesting review article, geared mainly to applications in signal analysis, is Jamsen (1988). Only two of the many properties of the Zak transform will concern us here. x The first is that if g,,,(x) = e2"g(x - n),then

(as we already showed above, in the special case g = e). This implies

Equivalently, we have Z(F*F)Z-'= multiplication by f(Zg)(s,t)lzon 2, where F'F f = Em,,( j , g,,,) g., The second property we need concerns the relation between Z and the operators Q,P defined by (Qj)(x) = xf(x), ( Pf )(z)= -a f ' ( ~ )(or, more pmpxly, (Pf )" (6) = [ j ( ( ) )b. a e checks that Iz(Ql)l(s,t) = s ( Z f ) ( ~ , t)

1

@t(Zj)(s,t)

which means that dz x21f (zjP < QO, i-e.,Qf E L2(W), if and only if at(2f) E L2([0,112). Similarly, J @ t'fj(~)/~ < oo or P f E L2(R)if and only if %(Zj) E L2([0,112). We are now ready to attack the proof of Theorem 4.1.1.

Pmf of Theorem 4.1.1. 1, Assume that the g,,,

constitute a frame. Since

and since Z is unitary, this implies

TIMICTREQUENCY DENSITY A N D OEYTHONORMAL BASES

111

2 The dual frame vectors ij,,,,, are given by *

Pm,n

= (Fop)-'

gm,n

(see 53.2,$3 4 3). Since Z(FmF)Z-I = multiplication by JZg12,it follows that z3m.n = 1z91-2Z9m.n

which is in 2 hy (4.1.3) In particular, (4.1.4) implies that

with 2 4 = I/%. 3 S r t p p e now that J d x z2Ig(x)l2 < oo, Jcif <214(()12 < oo,i.e., that Qg, Pg E L ~ ( w ) .This will lead to a contrsdictbn, which will then prove the theorem Since Q g , Pg E L2(W), we hatie B),(Zg), &(Zg) E L2((0,1i2). Consequently,

are In L ~ ( [ D ,112);hence Qij, Pjj E

L2(R).

similarly,

5. Since Qg, Pjj f L2(&t),and since the dual frames,we have

(~rlr)mFEZ,

(3m,n)m,nE~ constitute

-

m,n

But (Qg, jj,,,,,)

= 1uk zg(x) e*2*inV = J d z g ( r ) c-

'(-

= (9-m,-n,

93).

3; - n) (z

-

- n ) ij(z - n ) (g,3m,n)

~mo~no)

CHAPTER 4

112

Similarly, (gm,n, PG) = (Pg, 9-,, )-,

. Consequently,

where the last term is again weU defined bmause Pg, Q$ € L2(R).

6. We have now reached aur cantradictian: (Qg, Pg) = (Pg, Qij) is impossible. For any two functions f i , j2 satisfyjng If'(x)l 5 C(l x2)-', l&(<)l 5 C(1 + t2)-*,-we have

+

0

1

2

= Jd~xf~(x)i/:o

On the other hand, since Pg. Qg E LZ(W), there exist gn satisfying Ign(x)l 5 C n ( l + x2)-', [&(<)I5 Cn(l+ t2)-' S U C ~that limn,, 9, = g, limn,, Pg, = Pg, lim,, Qgn = Q g . (Take for instance g, = ,C;, (9, H k ) H k , where Hk are the Herrnite functions.) A similar sequence Gn can be constructed for g: Then

Together with (4.1.6) this implies ( g , g ) = 0. However, from (4.1.5) we have (9, 3) = 1. This contradiction proves the theorem.* rn We can summarize o w findings so far as

wok = 2?r w o k < 27r

-

-

there exist frames, but they have bad timefrequency localization. frames (even tight frames) with excellent timefrequency localization are possible (see 53.4.4.A).

This is represented in Figure 4.1, &owing the three regions in the w, to-plane. As pointed out in 53.4.1, orthonormat bases are only possible in the ''border casen = 27r. In view of Theorem 4.1.1, this meana that all orthonofm81 bases of

TIMEFREQUENCY DENSITY AND ORTHONORMAL BASES

113

NO FRAMES FOR

w,t,

>2n

o0b = 2~

ONLY FRAMES WITH BADTIME FREWENCY

LOCALIZATION ARE POSSIBLE TIGHT FRAMES

POSSIBLE IF W b t O < 2x

+1

The mgons y l t o > 2u where m jhunu atr pussrble, and %to < 2n, &em t r m e - M e n c y laualtzakm errst, o n sepamted by the hyperbola woto = Z r , the only regton where orthononnal bas- 5re possrble

Fw.

hqht

frwnc?s trnlh ezcellent

the type (g,,,.,;

m,n E Z), with g,,,

as In (4.1.1), have bad time-frequency

localition. In fact, wo to is a measure for the Yime-frequency density" of the frame c6nstituted by the g,,,. We can for instance define this "density" as lim

A-00

#

4; tfw*nto)E AS) lxsl

9

(4 1 7)

where S is a "ressonable" set in R ' (with nonzero Lebesgue measure). This llmt is independent of S, and equal to (wo - to)-'. This "density" also emerged in the time-fresuencg localization discussion in $3.5; see (3.5.17). The restriction (wo - to)-' 2 (%)'I means that the time-freqmmcy $ensity of the Frame has to be at Ieast the Nyquist density (in its ugenaal&d" form; see $2.3). In fact, Theorem 4.1.1 tells us that we have to be s W 3 y a t m e the Nyqulst density if we want good timefrequency localization withr".r$ind& Fourier frames. Let us now turn to wavelets, where the situation is very different. It turns out that there is no "cleann definition of timefrequency density for wavelet exp i o n s . We already saw a first indication of this in the study of the localizatioxt operators in 52.8: for the windowed Fourier case, the number of eigenvalues in the transition region became negligible (as compared to the number of eigenvalues close to 1) when the area of the localizatiop region tended to infinity, whereas these two numbers were of the same order of magnitude in the wavelet case. This made it impossible t o make an accurate comparison with the Nyquist density. Something similar happene with discrete wavelet families. In the discussion of tbe-huency l d i t i o n with frames, in 33.5, we saw that a function that ia amentially concentrated in 1-T, x ((41, -no] x [Ro, QI]) in the time

\

I

i

114

CHAPTER 4

frequency plane can be approximated with good precision by a finite number of wavelets. Unlike the windowed Fourier case, the ratio of this number to the timefreqpency area 4T(R1-5)0) dependson the desired precision of the approximation (seeW(35.lI)), which makes it imjmsible (as in the continuous case) t o make a precise cowparison with the Nyquist density. On the other hand, if we try to define the andog of (4.1.7) for the hyperbolic lattices of Figure 1.4a, then we find that # ((mn);(WO, nto) f AS)

Rs(A) = -

lxsl

(where S is chosen so that the numerator is finite) does not tend to a limit as For S = [-T,T] x ([-2"', -2mo] u [2m0, 2m1j) and a0 = 2, for instance, Rs(X) oscillates between p ( l - 2m0-m1-1)/(1 - 2m-ml) and 2p(l - 2mo-ml-1 )/(I - 2mo-m1), where p depends on the chosen wavelet I). It might be this phenomenon, rather than the absence of an intrinsic time-frequency density for the frame, ,that causes the probkrn in counting the number of wavelets needed for time-frequency loealization. So let us probe a little deeper. As we mentioned before, there is no a priori restriction on the range of dllatlon and translation 'parameters in a wavelet frame: any choice of ao, 4 can be used t o define a tight frame with gpod localization in both time and fqquency for II, (see 53.3.5.A). In fact, froma (tight) frame with discretization parameters w, we can always constrkt a Merent (tight) frame with parameters m, boJ (same %) with b& arbitrary, by. simple d ~ a t b a It . ~is therefore not surprising , We can remove this dilational that we have no a priori restrictions on q-,4. freedom by fixing not only the ~ o r m a l i ~ t i oofn I), 11$11 = 1, but by also imFor r& ~ b we , could impose, for posing a fixed value for df El-' \$(<)t2. instance, E-' i & ~=i ~ I({-1 /&(<)12 = 1. A tight frame generated by a JI thus restricted would auhmatically have f'rame bound A = (see Theorem 3.3.1). A cornwith the formula A = for tight windowed Fourier frames suggesta thst, maybe (bo hao)-' w d d play the role of time-frequency density for the wavelet case. The following e p p l e destroys all hope in this direction. In the oe;rrt ssctian we will encounter @e Meyer wavelet 11; it has a compactly supportgd Fourier trqpsform 6 ' C (where k m a y be oo, as in 53.3.5.A; the two co-tiow w related) and the &,, = 2-"I2 e(2-"2 - n), m, n E Z 2morthonormal basis for L~(B).Let us, for this chapter only, define '.

&

&

4

are

&,n(z) =2-"I2 *(2-mx - nb)

,

where .(/, is the Meyer wavefet, and b > 0 is arbitrary. Consider the Mependent families F(b) = {q;, m,n E 2). As b changes, the "density" of the assbciated lattice changes as well. (Note that and 'r), are the s a n e for all the P(b)!) If any representation like Figure-4.1 held also for wavelets, then we woqld expect, since F(1) is an orthonormal baais f& Z2(Q that F(b) would not span dl of L2(R)if b > 1 ('hot enough" veekrrs), and that F(b) would not be linearly

TIMEFREQUENCY DENSITY AND 0ELTH90NORM#L BASES

115

independent ("too many" vectors) if b < 1. Yet one can prove (see Theorem 2.10 in Daubechies (1990),we also sketch this proof later irt this chapter) that for some c > 0, F(b) 1s a Riesz bass for L ~ ( R ) ,for any b E 11 - c, 1 €1. This example shows conclusively that lt is not always safe to apply "time-frequency space density intuition" to families of wave&+.

+

4.2.

Orthonormal bases.

4.2.1. Orthonormal wavelet bases. The conckusion of the last paragraph seems a rather negative point for wavelets no clean time-frequency density concept In thls section we emphasize a much more positive aspect: the existence of orthonormal wavelet bases with good t i m e - w n c y localization. Historically, the first orthonormal wavelet basis is the Haer basis, consfructed long before the term "wavelet" was coined. The basic wavelet is then, as we already saw in Chapter I,

3,

1, O < z < t -1, ij 5 x < 1,

0 otherwise

-

* W eshowed in $1 6 that the V,!J,,~(Z) = 2'"@~9(2-~2 n) constitute an or4honormal basls for L2(W).The Haar funcQon is not contmnuous, and its Fourier

KI-',

t;ransform decays only llke corresponding to bad frequencjr iodiz&ion. It may therefore seem that ttus basis IS no better than the winFoulier basis

$ 5


34 Z@ ,s,

3

, *a-

with 1, O ~ z < l ,

0, otherwise, which is alsb an orthonormal basis for L'[R). w, theHam W i already has advantages that this wlndowed Fourier k k s &oeswaC have. It twns out, for inStance, that the Ha& basis is an unconditional bwia for LP(R), f < p < m, whereas the windowed Fourier basis (4.2.2) is aat tf p J' We will come back to this in Chapter 9. For the anaiysis of s d ~ ( u n d i o n s the , dLmntinuous Hasi besis is ill suited An orthonormal wavelet basis with tim&fr-8queacyproperties complementary to the Haar basis is given by the L i t t l e d basis, ~ ~ %

C

.P &

7

%

ica = {

[2r)-l/2,

ol

r*

sg&52*,

( 8 k k k 2 -&xz)

I

I I

otpemise

a' @(x) = (TZ)-'

\

3s

I

.

It is easy to check that the &,n (z) = 2-*i3 $[2-"'~ - n) constitute indeed orthonormal basis for L2(R). We have H+,, ,1/ = 1for all rn,n E 2, sod

116

CHAPTER 4

-

By Proposition 3.2.1, this implies that {$;, m, n E 2) is an orthonormal basis for L2(R). The decay of $(x) is as bad ($(XI izl for x-+oo) as that of the orthonormd windowed Fourier h i s used in the Shannon expansion (2.1.1); both have excellent frequency localization, since their Fourier transforms are compactly supported. In the last ten years, Weral orthonormal wavelet bases for L2(R) have been constructed which share the best features of both the Haar basis and the Littlewood-Paley basis: these new constructions have excellent Iocalizatbn p r o p ertiedl in both time and frequency- The first constmction is due to Strornberg (1982); his wavelets have exponential decay and are in Ck ( k arbitrary but finite). Unfortunately, his construction was little noticed at the time. The next example is the Meyer basis mentioned above (Meyer (1985)), in which is compactly supported (hence E Cm)and Ck (k arbitrary, may be w). Unaware at that time of Stromberg's construction, Y. Meyer actually found this basis while trying to prove a wavelet equivalent of Theorem 4.1.1, which would have $imthe non-existence of these nice wavelet bases! Soon after, Tchaanitchian (1987) constructed the first example of what we shall call biorthogonal wavelet bases (see $8.3). The next year, Battle (1987) and LemariC (1988) used very &&rent methods to construct identical families of orthonormal wavelet beses with exponentially decaying $ E Ck (k arbitrary but finite). (Bsttle was inspired by t W q u e s in quantum field theory; LemariC reused some of Tchamitchian's computations.) Despite bving similar properties, the Battle-Jtemarj6 wavelets are diffemt from the Stmmberg wavelets. In the fall of 1986, S. Mallat and Y. Meyer developed the "muEtiresolution analysis" framework, which gave a satisfactory explanation for all these constructions, and provided a tool for the construction of yet other bases. But this is for later chapters. Before we get into multirdution analysis, let US review the construction of Meyer's wavelet baeis. The construction of is similar to the tight frame in 53.3.5.A.That f m e had redundancy 2 (twice Rtoo manyn vectors). To get rid of this redundancy, Meyer'e construction combines positive and negative frequencies (reducing a pair of functions to a single function). Ln order to achieve orthonormality, some clever

4

141

TIMEFREQUENCY DENSITY AND OfWHONORMAL BASES

tricks with phase factors are needed. More explicitly, we define

J! , I

4(c)=

{

(2~)-112edI2 sin [; v (&It] - I)]

,

$ 5 It)5

(2n)-'I2 ea'12 cos ,; v (&It,- I), ,

%f 5

0

otherwise,

5

117

by

7, ,

(4.2.3)

where v is a Ck or C* function satisfying (3.3.25), i.e.,

u(x) =

0 if x < O , 1 if x L 1 ,

(4.2.4)

with the additional property v(x) + v(1 - x) = 1 . The regularity of

4 is the same as that of v.

(4.2.5)

Figure 4.2 shows the shape of a

typical v and 141. In order to prove that the +m,,(x) = 2-m/2 $(2-mx - n) constitute an orthonormal basis, we only have t;o check that l/+(I = 1 and that the +rn,, make up a tight frame with frame constant 1 (see Proposition 3.2.1).

1

9

X

0

1

1

1

( 2 %)-Z

i

"I

i

0

i'i

$

h 1 Ls i

.

\

, -10

FIG. 4.2.

-5

0

5

10

I c ( , t ~ ~ t i o n r ~ g-gitmby(4.2.3)-(4.2.5). djr

i

L

CHAPTER 4

We have

But

(because v(s

+

11+(12

+ 1/2) = 1 - 4 1 1 2 - s) by (4.2.5))

= 1. To evaluate [(f,$-,,)I2, we use Tchamitchian's frame bound estimates (3.3.21), (3.3.22). We first prove that P1(2n(2k + 1)) = 0 for all k E 2,i.e., for all ( E R, hence

Em,,

I r O

Because of the support of 9, nonzero contributions to (4.2.6) are oniy poasible if IPS1 < !f m d I2'(( 2*(2k 1))l < implying 2'I2k+ 11 5 8/3. The only pairs (t,k) that do not violate this are (0,0), (0, -I), (1, O), and (1, -1). Let us check k = 0 (k = -1 is analogous). Then the left-hand side of (4.2.6) becorn*

+

+

9,

5

119

TIMEFREQUENCY DENSITY AND ORTHONORMAL BASES

-9 < < -%. For

One easily checks that both terms in (4.2.7) vani& 5 C within this interval, C = - + ?a with Q < a I1,We have

9

(4.2.7) = e-'(12 sin

-

- cm

:[

[;i

A

u ( l - a)] e1((+2z)A sin

u ( a ) ]sin

[i

Y(LI)]

+ Bin [X2 u(h)] ~ 1 [:6

~(a)]

On the other hand, one a i l y checks that for all # 0. It then follows from (3.3.21), (3.3.22) that Em1 3 ( 2 ~ [ )=1 ~ the &,, constitute a tight frahe with frame bound 1- (Similar computatio-zis can be used to prove that F(b) (see the end of 84-11 constitutes a R i a z bitsis for L2(R) if b is close to I . ~ ) This proof that the Meyer wavelets constitute an orthonormal basis relies on quasi-miraculous cancellations, using the interplay between the phase of and the special property (4.2.5) of v. Using multirec~oIutionanalysis, we will be able to explain away most of the miracle (see next chapter). Figure 4.3 shows a graph of @(z), with bhe C?L choice v ( z )= x4((35 - 8 - b+ 7 k 2 2 h 3 ) for 0 5 z I1. N4te tbt e w h if v E Cw,so that $ decays faster than any inverse polynomial, i.e., for'& N E N, tbere exists CN < do so that

This establishes (4.2.6).

4

-

J$(X)~

CN (1 + 1x12)- N ,

(4.2.8)

the numerical decay of .d, may be rather slow (i.e., inf (a; J+(x)J _< .001 JJ+IJL= for 1x1 > a) may be very large, reAecting a large CN iD (4.2.8)). The exponentially decaying wavelets of Stromberg or B a t t l e - M d W much faster numerical 'b decay, at the price of sacrificing regularity.

CHAPTER 4

In the matter of orthonormal bases then, wavelets seem to do quite a bit better than windowed Fourier functions: there are coastructions in which both gl, and have fast decay, in stark contrast with Theorem 4.1.1, which forbids simultaneous good decay for 9 and if g is a window function leading to an orthonormal basis. If I had written this chapter three years ago, this is probably where I would have stopped. But matters are not quite that simple: in the last few years, the windowed Fourier transform has led to a few surprises, which we will discuss briefly in the remainder of this chapter. %

4.2.2. The windowed Fourier transform revisited: 'Goodn orthonormat bases after all! One way in which one could try to generalize the windowed Fourier construction, so as to get round Theorem 4.1.'1, is to consider families 9,,,(2) that are not generated by a strict timefrequency lattice. This allows for a little leeway: Bourgain (1988) has constructed an orthonormal basis ( g j l j E J for t 2 ( R ) such that

6 [

uniformly in j E J, where. z, = dz ~19,(+)1~, = 4 (IJ,(E)(~.'(Note t-hat wavelet bases do not satisfy such a uniform bound. ) Giving up the lattice structure therefore permits better l d z a t i o n than allowed by the BalianLow theorem. However, Steger (private communication, 1986) proved that even slightly better localization then (4.2.9) is impossible: L ~ ( w does ) not admit an orthonormal basis ( g j ) j eJ satisfying

uniformly in j, if E > 0. This approach can therefore not lead to good timefrequency localization. There is another way in which we can try to break away fkm the lattice scheme (4.1.1). Note that in (4.2.9), (4.2.10), "time-frequency localization" stands for strong decay properties of the g,,, (gm,n)A away from the average values x,, This corresponds to a picture in which both gm,rn and (g,,,,)" have essentially one peak. Wilson (1987) proposes instead to construct orthonormal bases g,,, of the type

where

has two peaks, situated near

and

-9,

;

$

'3 a

4

4

?i

TIMEFREQUENCY DENSITY AND OFWHONORMAL BASES

121

with ,& ; 4, centered around 0. This ansata Fhangea the picture completely. W~lson(1987) proposes numerical evidence for the existence of such an orthonorma1 basis, with uniform exponential decay f& f, and dA, 4.; In his numerical construction he further "optimizes" the l d & i O n by requiring

Sullivan et al. (1'387) present arguments explaining both the existence of Wilson's basis and its exponential decay In both papers %ere are infinitely many functions-42; as rn tends to oo,the #$ tend to a Limit f u n c t b 42. The moral of Wllson's construction is that orthononnal b p r i t h good phase space localization seern possible after dl if &modal functions as in (4.2.13) are used. Note that many of our wavelet constructions, frames as well as the orthonormal bases we saw earlier, have these two peaks in frequency (one for > 0, one for < 0). In the case of frames, or for the continuous wavelet transform, the two frequency regions can be separ9ted (corresponding to one-frequency-peak functions; see 53.3.5.A or (2 4.9)), but this does not seem to be the case for orthonormal bases. We will see later that the two frequency peaks of $J need not be symmetric: there even exist examples with l]$b-2 JtSP 4 jJ;(<)12 arbitrarily small (but strictly positive!). However, there is no example, so far, of reasonably well-localized funct~ons$* wlth support c R* and such that the ($&,,; rn, R E Z, e = + or -) constitute an o r t h o r m a l basii lor L2(R),mrresponding to wavelet ba&s with only one "peak" ia f r m c y . (Equivalently, there is no example of a reasonably smooth h c t i o n 9 = $+ such that the functions 2m/2exp (2x2 2" ~) ~ ( 2 ~ m,n 0 , E 2, are an orthonormal basis of L2(~+).) It is believed, without proof so far, that qo sueh b&i exists.' But let us return to Wilson bases. If one giwa up the restriction (4.2.13) (if have exponential decay, then th& queditiea decay exponentially fast fm, in Im - m't, In - nfJ anyway), then Wilson's ansatz (4.2,11), (4.2.12) can be dramatically simplified. In Daubechies, Jaffard, and ~ o u r d(1991), * a construction is p r o p d that uses only one function 4 Explicitly, this conetruetion defines

<

<

? :

1

t k

,

(G)

A

gm,n(z) = fm(x - n),

m E N \ {0), n E

,

(4.2,14)

*

CHAPTER 4

etc -

.

with I E N, u = 0 or 1, and P = 0, o = O excluded. The result of all these phase factors and alternating signs is that

If we relabel Limp,,

in; (4.2.14) %y defining G,

, ~ r Et N,n f Z as

then

and for P > 0,

This comtruction {as'weU ae others mentimed below) shows therefore that the key to obtaining good time-fiequenq localization (4 can be chosen so that 4,

4

have exponential decay) and orthonormality in the windowed Fburier framework

is to use sines and cosines (alternated in an appropriate way) rather than complex exponentials. But let &I get back to (4.2.14), (4.2.15) and sliow how this construction can lead to an arthonormat basis. As 11wa1,we only need t o check lfgm,,,il = 1 and C:=l C n El(h, ~ qm,n)lZ = llh1I2. We immediately havesl1g~,nlf = IIfi II = IIbII, and for m > 1,

(weaseume q5 is red, for simplicity). Hence

/Jgrn,,lt= 1 for all m,n if

14 #(€I 4 ~ + 4 =~6 m0 . On the other hand,

TIMEFREQUENCY DENSITY AND OFCI'HONORMAL BASES

123

This equals ljhH2 if

A few simple manipulations lead to

f

If k is odd, k = 2k1+ 1, then this reduces to

which is zero, since the substitution 'P = - (e + 2k' + 1) transforms (4.2.21) in its negative. If k is even, k = 2k1,then (9.2.1 9) reduces to

The {g,,,; m E N \ {0), n E Z) therefore constitute an orthonormal basis if 4 is a real function satisfying (4.2.18) and (4.2.22). Nde that integratkmg(42.22) over t , between 0 and 2n, mttomatically leads to (4.2.18), so that we really have cmly the single condition (4.2.22) to satisfg. This %wmsout to be easy: we can take for i-ce support 4 C j-2a, 2x1, so th& (4.2.22) is automatically satisfied for k' # 0, and we only need t o check Q(f 2 ~ 4 = ) (21)-I~ This is true if, e.g.,

xez

+

otherwise ,* with u as in (4.2.4). if u is Cm,then the f,,, harve Becay faster thanany imretee polynomial, but, es for the Meyer basis, tkntmwical decay xnay be slow. Faster decay for the f, can be obtained with mmampctly supported #. To construct such a #, satisfying (4.2.221, we can again use the Zak transform, now nosPLalized so that ( Z h ) ( t l) ~ ,= (43r)'l2 rhiu h(4r(r - Q)

.

t€Z

'

CHAPTER 4

124

With this normalization, Z is again unitary from L2(R) to L2([0,1J2). It is not hard to check that (4.2.22) is equivalent to

(Full details are given in Daubechies, J d a r d , and J o d (1991).) This suggests the following technique for constructing $: %

Take any h such that

Define 4 by

If h. and both have exponential decay, then 4 turns out to have exponential decay as well. Figure 4.4 shows the graph of d, and 6 when h is a Gaussian. (Gaussians do indeed satisfy (4.2.24).) An interesting observatidn is that (4.2.23) is exactly equivalent to the requirement that the &,,(z) = e2*'4 (Z- $), or equivalently, the @,(<) = erint #J(( - m ) , with m, n E 2,comtitute s tight frame (with necessarily redundancy 2) for L2(R). The construction (4.2.25) can then be interpreted as the transition from a general frame, generated by h, to a tight frame, by application of ( F ' F ) - ' / ~ (see note 11 after Chapter 3, or Daubechies, Jaffard, and J o d (1991)). This W i h n basis can therefore be viewed as the result ef a clever "weeding" process on a (tight) frame with "twice too many" elements. Many variations on thi W i h n scheme are possible. Laeng (1990) has constructed an extension of the above scheme in which the frequency spacing need not be as regular as here. Auscher (1990) has reformulated the whole carnstruction: starting directly from (4.2.16), (4.2.17) as an ansatz, he derives all the results without use of the Fourier transform, and constructs different exampIes. In particular, he obtains examples where, in the notations of (4.2.17), the "windown 4 is compactly supported, which is very useful in applications. (The decay in frequency is less crucial, as long as it is 'C-reasonable.") These examples can also be viewed as the result of a "weeding" procedure on the tight frames with redundancy 2 obtained by taking wo& = K in 53.4.4.A. Other windowed Fourier bams using cosines and sines rather than complex exponentials, and leading to good timefrequency i d i t i o n , have been found by Malvar (1990) and Coifman and M e w (1990). Malvar's paper again uses alternating sines and cosines; he presents applications of his c o ~ c t i o nto rspeech coding. Coifman and Meyer's "localized sine basisn starts h m a partition of R in intervals, R = bj, a t t l l ,

U

JEL

;

; 7 7

i

h Y

j

TIMEFREQUENCY DENSITY AND OFWHONORMAL BASES

t

FIG.4.4. Tha functiow

t$

and

4 c o r n d i n g to (4.2.25)

af h(z) = 7r-l''

125

exp(-xa/2).

i

i

is

2~

< aj+l and limj,*, a, = foo. They then build window functions w, locatized around these I, = [a,, overlapping slightly with the neighboring with aj

'.

55 %b a:

w,(z) = 1 for

U, + t j

5 %5 @,+I -€,+I

,

>

8

where we assume that the e k satisfy a j +c, a,+l -€,+I for all j. Moreover, we require that w, and wj1 complement each other near aj: wj ( 2 ) = w,-~ (2aj -;z ) > %$ T:,.k' h and W ; ( X ) W;-~(Z) = 1 if 1x ajl 5 c,. (All this can be achieved with smooth ;. 4n, 3; one can take, for instance, w,(z) =sin(; v( 2-a&,+e i)] for Ix-a,l y, and l+C'ti)] for (X - ~ ~ + €j+l, ~ with l w satisfying (4.2.4) wl(z) = m[lv ( = - ~ ~ + ~ :Ti L~J., snd (4.2.5).) C o i and Meyer (1990) prwe that the family ( u , , ~ j, ; k E Z),

+

with

I

-

<

&

I

<

I

constitutes an orthonormal basis for L ~ ( R ) consisting , of compactly supported functions with fast decay in frequency. This basis has moreover a very interesting property: if for any j E Z we define Pj to be the orthogonal projection onto the space spanned by the {u,,~; k f 2) (Pjis "morally" the projection into [aj, aj+l]), then Pj P , + 1 is exactly the projection operator P,, associated ' to [aj, that we would have obtained if we had deleted the point a j + ~ from our "slicing" of R (i,e. if we had started with the sequence t i k , tik = ak if .k 5 j , iik = ak+l if k 1 j 1). This property makes it possible, to split and regroup intervals at will, adapted to the application one has in mind. A very nice discpion of this whole construction, with full details, is Auscher, W e b , and W i h a u s e r ( 1992). So there is, efter all, more to orthonormal windowed Fourier bases than was expected even only a few years ago. None of these bases, however, are unconditional bases for P ( R ) if p # 2. This is one point where wavelet bases have the advantage: they turn out to be unconditional bases for a much larger family of function spaces than even these "good" windowed Fourier bases. We will wme back to this in Chapter 9.

+

+

.

Notes. 1. Rieffel's proof does not produce an explicit f orthogonal to all the h,,. This is a challenge to the reader: find a (simple) corutruction of f I g,,, for all m, n, for arbitrary wo, to with %to > 2n.

2. For orthonormal bases the proof is much simpler. In this case we need not bother' with the Zak transform, which was only i n t r o d u d to prove that if Qg, Pg E L2, then Q j , Pfi E L3 as well. For orthonormal beses we can start directly with point 5, establishing (Qg, Pg) = (Pg,Qg), which is impoeeihk by point 6. This is the original elegant proof in Battle (1988).

3. If the $,,,(x) = ao-"I2 $ ( ~ - ~- nbo) x constitute a (tight) frBme, then 80.. do the $m,nn(x) = cro-m/2$#(--"x - &I), with @(x) = ( b ~ / b o ' ) ~ ~lr(boxlbo'). '~

4. To illustrate this, the following e;cample shows that the complex exponenexp (27rinz) do not constitute an unconditional basis for LP([O,lj) if p # 2. One can show (see Zygmmd (1959)) that

In both caees, x = 0 L the womt singularity, and the integrability of powers of these functions an (0, I] is determined by their behavior around

TIMEFREQUENCY DENSITY AND ORTHONORMAI, BASES

127

0. The first function is in LP for p < $, the second is not, even though the absolute values of their Fourier coeffic~entsare the same. This means that the functions exp (2rznx) do not constitute an unconditional basis for L ~ / ~ ( [I)), o, The Haar h i s adapted to the interval [0,11 cbnsists of { 4 ) ~ { ~ , 6m, ~ ,n~E; 2, m 5 0, 0 n 5 2Iml - 11, with r$(x) 3 1 on [O, 11. ThlS basis is orthonormal in L2((0,I]), and is an unconditional basis for LP([O, I]) if l
<

B

5. The following is a sketch of the proof that F(b) = {$kt,; m, n E Z], with $&,, as In (4 1 8), constitr~trsa Riesz basis (i.e., a IinearIy independent frame) for LZ(R) if b is close t o 1. First of all, we can still apply (3.3.21), (3.3.22) to find estimates for the frame bounds. For b # 1, &(27r(2k + l)/b) 0, but if b < 2, then only k = O* f l , lt2 1 4 to nonzero PI. In the cornputation of (4 2.6) (with (2k 1) replaced by (2k l)/b), only a finite number o f t contribute, so that this expression is continuous in b. I t follows that thc "rest terms" in (3 3.211, (3.3.22) are continuous in b as well; since (3.3.21)=(3.3 22)= 1 if b = 1, we have A > 0,B c oo lor b in "" a neighborhood of 1. It remains to pkvc* that the @&,, are independent. To do this, we construct the operator S(b),

+

-

i

+

+

!

s ( b ) j=

1( j *@k,n)&," . m,''

Clearly, S(h)$L,, = zl,&,,. To prove Independence of the ,$ ,: it is sufficient to prove that ((S(b) f((2 C({{ll, unifornlly in f E LZ(R), for some

i t

Using that for JHlkJ = J R k l Jwe , have

I+

*

*4

-

n

we obtain

128

CHAPTER 4

- Because of the support properties of 4, only m' = 0, f 1 contribute in this sum.If m' = 0 or 1, then any choice of n gives the same result; if m' = 1, then the sum may have one of two possible outcomes, dependingon whether - n is odd or even. On the other hand, wing the decay }$(x)t 5 CN(l ( ~ l ' ) - ~of ~,one easily checks that Cn,EZ I($&,, $:,,)I converges and . is continuous in b for m' = 0, f1. It follows that the coefficient of I} f)la in the right-hand side of (4.2.26) is coritinuous in b;-since it is 1 for b = 1, it is > 0 for b in a neighborhood of 1.

+

& They satisfy

instead.

7. After the first printing of this book, a proof was found by P. Auscher, to he published in the Colnptes Rendus dc llAcaddrnie Scientifique, Paris, under the title "I1 n'existe pas de basev d'ondelettes r6gui&res dans I ' c s p e de Hardy HZ(R)." Explicitly, he proves that it is inlpossible that 7 E C1and 1~[€)1+ }v1(5)15 C(1 + KI)-" with a > 1/2.

CHAPTER *

5

Orthonormal Bases of Wavelets and Multiresolution Analysis n

The first constructions of smooth ort honormal wavelet bases seemed a bit muaculous, a s illustrated by the proof in 54.2.A that the Meyer wavelets constitute " aa orthonormal basis. Thw situation changed with the advent of ~nultiresolution analysis, forlnulated in the fall of 1986 by Mallat and Meyer. Multiresolutlon d y s i s provides a natural framework for the understanding of wavelet bases, a& for the construction of new examples. The history of the formulation of m u ~ l u t i o analysis n is a beautiful exarnpte of applications stimulating theoretidl development. When he first learned about the Meyer'basis, Mallat was workingm&*age analysis, where the idea of studying images simultaneously a t different scales and comparing the results had been popular for many years (see, e.g., Witkin (1983) or Burt and Adelson (1983)) This stimulated him to view orthonormal wavelet bases as a tool to describe mathematically the "increment in informationn needed to go from a coarse epproximation to a bigher resolution approximation. This insight crystallized into multiresolution &pis (Mallat r) (1989), Meyer (1986)).

Ic t

I

.

5.1.

The basic idea.

A multiresolution analysis consists of a sequence of successive a-praximhtion spaces . More precisely, the closed subspaces VJ satisfy1

VJ

a .

- V2 C Vl

C

U4

Vo C V-l C = L2(a) ,

C

.

-'

(5.1.1)

.

r

(5.1.2)

JEZ

k

'

2

If we denote by P, the orthogonal projection operator onto 6 , then (5.1.2) ensures that lim,+-, P, f = f for all f E L2(R). There exist many ladders of spaces satisfying (5.1.1)-(5.1.3) that have nothing to do with Umultiresolutionn; the multiresolution aspect is a consequence of the additional requirement

CHAPTER 5

130

That is, all the spaces are scaled versions of the central space Vo. An example of spaces 5 satisfying (5.1.1)-(5.1.4) is

We will call this example the Haat murtiresolution cmalysis. (It ia i e a t e d with the Haar basis; see Chapter 1 or below.) Figure 5.1 shows what the projection of some f on the Haar spaces Vo,Vil might look like. This ex.ample d w exhibitf another feature that we require from a multiresolution analysis: invariance of fi tinder integer trabslations,

f

E

Vo

*

-

f (. n) E Vo for all n E Z .

Because of (5.1.4) this implies that if f €

1

(5.1.5)

I

5, then f (- - 2'n) E % for all n E 2.

Flnally, we requlre dso that there exiets t$ E Vo so that

i:

wher;, for dl j, n E Z. 4,,,(rJ= 2-3i2 B ( 2 - J t -n). ~ o ~ e t h e(5.1.6) i, and (5.1.4) imply that (d,,,; -nE Z) is orthonormal basis for 25 for all j E Z. This lest requirement (5.1.6) seems a bit more "cpatrived" than the other ones; we win see below that it can be relaxed considerably. In the example given a h , a possible choice for Q is the indicator filnctioa for [O, 11, $(a) =, 1 if Q 5 x 5 1, &(xi= 0 otherwise. We will often call 4 the "scaling function" of the multiresolution analysis The basic tenet of mdtiresolution analysis is that whenever rr collection of closed su€xigmessatisfies (5.1.I)-(5.1.6), then there exists an orthonormal wavelet basis {$J~,~; j, k E 2)of L~(R), $J,k(z) = Z - J / ~ + ( ~ - J X - k ) , such that, for all f in L 2 ( ~ ) ,

(Pfis the orthogonal projection onto %.) The wavelet J! ,T a, moreover, be constructed explicitly. Let us see how. For every 1 E Z, define W, to be the orthogonal complement of 6 in We have V,-I = V,.@ Wj (5.1.8) and

(If j > j', e.g., then W, c V j l l Wj,.)It follows that, for j < J ,

I

I

MUI,"lREsOLUTloN ANALYSIS

FIG 5 1

A functton j and ttr pro)eetrons onto V-1 and Vo

*

&

where dl thae aubepeces are orthogonal By virtue of (5 1.2) and (5.1 3), this implies

L ~ ( R =@ ) W, IEZ

.

(5 1-11)

a decompasition of ~ ~ into ( 1mutually ) orthogonal subspscee. Furtherwe, the '>W, spaad inherit the acdlng property (5.1.4) fmm the 6:

&rmula (5.1.7) is equivalent to saying that, for fixed 3, (GJSk; k E Z) constitutes 'an orthonormal basis for W, . Recsuse of (5.1.11) md (5.1 2), (5.1.3), this then atomatically implies that the whole collectim (+,,k; j, k E 2)is an orthouorma1 for L2(B).On the other hand, (5.1.12) ensures that if ($oScl k E Z) is an onormal basis for Wo, then k E 2) will likewme be an orthonormal for W,, for any j E 2. Our task thus reduces to finding 11, E Wosuch that (. k) constitute an orthonormad bash for Wo. To construct thii t,h, let us write out some interesting properties of 4 and Wo.

-

Since t$ E

c V-1,and the

are an orthonormal twb in V'l, we

h, 4-1,.

O= n

I

(5.1.13)

132

CHAPTER 5

with hn = (4,

(hn12= 1

and

$-I..)

.

(5.1.14)

nEZ

We can rewrite (5.1.13) as either

where convergence in either sum holds in La-sense. Formula (5.1.16) can be rewritten as = ~ ( [ / 2 &/2) ) (5.1.17) where 9

Equality in (5.1.17) holds pointwise almost everywhere. As (5.1.14) shows, mo is a 2lr-periodic function in

L2([0, 2x1).

2. The orthonormality of the $(, - k) leads to special properties for

no. We

have

implying

li(~ + 27r.f?)12= (2n)-l

a.e.

(5.1.19)

I

Substituting (5.1.17) leads to

-

(C = (12)

splitting the sum into even and odd t!, using the periodicity of q,and applying (5.1.19) once more gives

3. Let us now characterize Wo:f E Wois equivalent to f E Since f E V-1, we have

and f I Vo.

i

133

MULTIRESOLUTION ANALYSIS

I

q

with f,, = (f, &-I,,,).

This implies

where

mf is a 2n-periodic function in L2([0,2~]);convergence in (5.1.22) holds pointwise a.e. The constraint f L Vo implies f lhYk for all k, i.e.,

-

hence

where the series In (5 1.23) converges absolutely in L' (1-n,TI). Substituting (5.1.17) and (5 1 21), regrouping the sums for odd and even e (which we are allowed to do, because of the absolute convergence), and using (5.1.19) leads to

Since n o ( < ) and mo(C + n) cannot vanish together on a set of nonzero nieasure (because of (5 1.20)), this implies the existence of a 2n-periodic function X(C) so that

and This last equation c& be recast as A(<)

= e'C

2C)

,

.

(5.1.27)

where v is 2n-periodic. Substituting (5.1.27) and (5.1.25) into (3.1.21) gives f(() = rnO(t/2 + 4 40 4(
where v is 2n-periodic.

-

CHAPTER 5

134

4. The general form (5.1.28) for the Fourier transform of f E Wo suggests that we take (5.1.29) = ee'* m ( S / 2 4- r)&/2)

&a

as a candidate for our wavelet. (5.1.28) can indeed be written as

Disregarding convergence questions,

so that the $(- - n) are a good candidate for a bas~sof Wo. We need to verify that the Q0.k are indeed an orthonormal basis for Wo. First of all, the properties of q a d ensure that (5.1.29) defines indeed an L2-function E V-I and _L & (by the analysis above), so that ~+5 E Wo. Orthon~nnalityof the #o,k is easy to check:

4

Now

Ce IS(< + 2 ~ 4 =P Ct lmo(t~+ =! + r)12 I & c /+~~ J ) I = = Imo(C/2 + t2 C P(t12 + 2rn)l2 + I ~ o ( ( / ~C ) I ~li(c/2 + r + 2 d i 2 n

n

= (24-' [Imo([/2)12 = (%)-I a.e.

+ Jrno((/2 -t?r)I2] (by'(5.1.20))

8.e.

(by (5.1.19))

.

Hence J & $ ( x ) 1C,(z- k) = dm. In order to check that the &,k are indeed a basis for all of Wo, it then suffices to check that any f E Wocan be written as

17n]2 < 00, or

with

-

n

P
MULTfRESOLUTION ANALYSIS

135

with 1 In-periodic and E L2((0,2~1).Let us go back to (5.1 28). We have = U(F)S(C),~ t: $h @ I Y ( E )= I ~21gI dc I X ( C ) IBY ~ . (5 1 22).

i(o

I

On the other hand, by (5 1.25),

Hence ff'@ Iv(i$)P= 2z (1f}I2 square integrable 23-periodrc 7

< 00, and f u of the form (5 1 30) w t h

-

We have thus proved the following theorem. THEOREM 5.1 1 If a ladder of c l o d subspaces (&)IEZ zn L2(lZ) satwfles $he andzttons (5.1.1) -(5.1.6), then them ezuts an assoctated orthonormal wavelet b u (@$,*, 1,k E Z) for L ~ ( w such ) h t

$

k

One possrhlity for the wnstructton of the umvelet #J rs 4

*#

4(0 = eg'2

:3

T!

ma(f/'l+

4kU2) ,

((ath p,-, 0s defined by (5.1.18), (5.1.Id)), or egustldentlly

I

h:

$(z)

k

&"

=

h x(-l)*-'

h-n- 4(2z - n ) .

n

t @ h P

( m t h convergenee of the last senes tn ~~-gense).

.?-

Note that II, is not determined uniquely& the mdtiresolution analysis ladder and requirement (5.1.31). if $Jsatisfies(5.1.31), then so will any y3# of the type

6% "

*

= P(t1$(€I ,

(5.13 3 )

with p 2~-periodicand lp(c)( = 1ae."In partiah, we can choose p(c) = pc etmP wUh m E 2, /pol = I, which correspasda to a phase change and a shift by m for 3, We will use this freedom to define, - b h d of (5.1.32),

CHAPTER 5

136

or occasionally 9n =

h - n + l + 2 ~1

with appropriately chosen N E 2. Of course we can take more general p in (5.1.33), but we will generally stick to (5.1.34) or (5.1.35).4 Even though every orthonormal wavelet basis of practical interest, known t o this date, is associated with a rnultiresolution analysis, it is possible to construct "pathological" II, such that the $,,*(z) = 2-jI2 $(2-3s - k) constitute an ort honormal basis for L' (W) but are not derivable from a multiresolution analysis. The following example (due to J. L. Journd) is borrowed from Mdlat (1989). Define (27r)-'I2

if

9 5 Kt 5 R

or 4r 5

1 ~ 51

otherwise .

,

(5.1.36)

x,

We immediately have 11+,,t (1 = (1+11 = 1. FUhermore, 27r (4(2j[)(' = 1 a.e. By Tchamitchian's criterium (3.3.21)-(3.3.22) the Ijl3,k therefore also constitute a tight frame with frame constant 1, provided

4+

One easily checks that support n[support (2k + 1)2n2'] has zero measure for all t 2 0. k E Z, so that (5.1.37) is indeed'satisfied. It. then follows from Proposition 3.2.1 that the Ilj$ constitute an orthonormal basis for L2(R). If ?I, were associated with a multiremlution analysis, then (5.1.29)and (5.1.17) would hold for the corresponding scaling function t$ (with possibly an extrti p({), Ip(<)J= 1 a.e. in the formula for +-see (5.1.33)). It then follows from (5..1.20) that (5.1.38) /#(6)12+ ~ & t ) d= ~ 1&([/2)l2 t

4

which implies, for [ # 0 ,

One easily checks from (5.1.36) that this implies

(

0

otherwise

.

If there existed a 2~-periodic% so that (5.1.17) held for this 4, then we would have Imo(<)l= 1 for 0 5 5 47117. By periodicity this would imply Imo(c)l = 1

4

137

MULTlRESOLUTION ANALYSIS

- as well for 27r 5 E 5 18n/7; hence I?q,(()l I&[)I = ( 2 ~ ) - I / 2for 27r 5 ( < 16n/7, even thodgh 1&2[)1 = 0 on this interval. This contradiction proves that this orthonormal wavelet basis is not derivable from a multiresolution analysis. Nute that @ has very bad decay. It is an open problem whether this kind of "pathologyn c$n persist if some smoothness is imp& upon (i.e., decay on T+!J).~ For later convenience, we note that ia terms of the h,, equation (5.1.20) can be rewritten a s

4

_

+ -

(This follows easily from writing out the explicit Fourier series for lrno(C)[' Im(C+ r)I2.)

-.*

6.2.

+

Examples.

Let us see what the recipe (5.1.34) gives for the Haar multiresolutio~ianalysis. In that case, 4(x) = 1 for 0 5 x < 1,0oth-, hence h, = f i

J

- n) =

& &(x) 4(2x

1/& 0

if n = 0,1,

otherwise.

1 if O < z < $ ,

-1

if

4
0 otherwise.

This is the Haar basis, which is no surp* we already saw in 51.6 that this wavelet basii is associated with the Haar multiresolution analysis. i The Meyer basis also fits neatly into this scheme. To see this, define 9 by r

$K) =

[

MI1 2~13,

(2~)-l/~, (2*)-1/2

[:Y

($1~1-

I)] , 2 ~ / 35

It1 5 4 ~ 1 3 ,

otherwise,

0

4

is plotted in a smooth function satidjing (4.2.4) and (4.2.5). 5.2. It ia an easy consequence of (425) that EkEz1)((+27rk)12 = (2~)-' , nrhich is equident to torthonormality of the #(- - k), k E Z (see 55.2). We then define Vo to be the closed subspace epanrled by this orthonormal set. Similarly, is the closed space spanned by the k E Z. The V, satisfy (5.1.1) if and if 4 E V-l,i.e., if and only if there exists a 2x-periodic function m, square -able on [O, 2x1, so that

where

iw

Y i8

+,,

CHAPTER 5

FIG. 5.2 The d s n g functm 4 for the Meyer basas, unUI v(x) = x4(35-- 842

-+ 70z2 -

2oz3).

In this particular c&, ma(()

=

and

%

can be easily constructed from 6 itself: This is 2~-periodicand in h2([0,2 4 ) ,

& CLEZ #(2(< + 2x4)).

~ ( t 1 2 i ) ( ~ 2 )=

6 C &t+ 4

4

C€Z

\

at) &/2)

=

+

(because [support 4(./2)] and [support J(- Ire)] do not m r l a p if e # 0)

= ict) (beeausu 6#(t/2) = 1 for 6 E support 8).

I leave it as an (easy)exerc-ke for the reader to check that the V, also satisfy properties (5.1.2), (5.1.3) ((5.1.4) and (5.1.5) are trivially satisfied already; see also 55.3.2). Let us now apply the recipe (5.1.29) to find $:

4(€)

eela %K/2

=

+

&t/2)

Ji; E l 2 C #({ + a(u+ 1)) & ~ / 2 ) ?EZ

=

6eel2 [& +

i-

- 2x11

(for all other C, the supports of the two facton, do not o v e t w.

It is w y to check (see also F i i 5.3) that tbit, is equivalent to (4.2.3). The phaae factor eel2 which w w needed for the Umiraculous cancellatione" in 54.2 occurs here naturally as a w~se~uence of the general analysis in $5.1. Before we discuse other 'emmples, we need Co relax condition (5.1..6). "C

139

MULTIRESOLUTION ANALYSIS

I

Relaxing some of the conditions.

3

.3.1. Riesz bases of scaling functions. The orthonormality of the 4 ( . - k ) (5.1.6) can be relaxed: we only need to require that the g5(- - k) const~tute a Riesz baas. The following argument shows how to construct an orthonormal h i s ##(- - k) for Vo starting from a Riesz basis {gl( - k); k E Z} of b. Tb &(. - k) are a &esz basis for Vo if apd only if they span 'I/o and if, for all (cki&z E

I

e2(z),

t

where J > 0, B < oo are independent of the c, (see Prelimideries). *

&t

and

so that (5.3.1) is equivalent to

We can therefore define q5# E L2(W) by

x,

+

1J6 (( 27re)12 = (27r)" ax., which means that the q5#( - k) are Clearly, orthonormal. On the other hand, the space ~ b #spanned by the q5#(. - k ) is given by

{ f; j = v J# with v 27r- perlodic, v E L ~ ( { o2x1)) , = {f; j = q with yl 27r- perlodic, vl E L ~ ( [ o2,~ 1 ) ) (use (5.3.2) and (5.3.3))

=

4

= Vo

(since the .($t

- n ) are a Riesz basis for VO)

Using the scaling function as a starting point. As &scribed in $5.1, a multiresolution analysis consists of a ladder of spaces (V,),EZ and a special function C#J E Vo such that (5.1.1)-(5.1.6) are satisfied (with (5.1.6) possibly relaxed as in 55.3.1). One can also try to start the mnstrudion from , an appropriate choice for the scaling function 4: after all, Vo can be constructed from the q5(. -k), and from there, all the other V, can be generated. This strategy is followed in many examples. More precisely, we choose # such that 5.3.2.

where

C, Ih12 < oo, and

-wethen define V.to be the c l o d subspace spanned by the 4j,k, k E 2,with &j,k(z)

= 2

(

2 - k). The conditions (5.3.4) and (5.3.5) are nk E 2)is a Ricm basis in each 5,srid

sary and sufiicient to ensure that {#j,k;

141

MULTIRESOLUTION ANALYSIS

that the V, satisfy the "ladder property" (5.1.1). It follows that the Vj satisfy (5.1.1), (5.1.4), (5.1.5), and (5.1.6); in order to make sure that we have a rnultiresolution analysis we need to check whether (5.1.2) and-(5.1.3) hold. This is the purpose of the following two propositions. PROPOSITION 5.3.1. Suppose 4 E L2(lR) satisfies (5.3.5), and define V, = Span {&k; k E 2). m e n n,,~ 5 = (0).

' +

Proof. \

1. By (5.3.5), the t$O,k constitute a Riesz basis for Vo. In particular, they constitute a frame for Vb, i.e., there exist A > 0,B < oo so that, for all L?

f

E Vo,

A

1111 51C ~

~ ( f a, s ) ! '

sB II~II~

(5.3.6)

k€Z

(see Preliminaries). Since 4 and the 4J,kare the images of Vo and the &,k under the unitary map (Dl f)(x) = 2 - ] I 2 f ( 2 - J x ) , it follows that, for all f ~ % , A ilfl12 l l(fl h.r)12 6 B 11f1I2, (5.3.7)

k~z with the same A,

B as in (5.3.6). .

2. Now take f E n,,~ 4. Pick e > 0 arbitrarily small. There exists a compactly supported and continuous j so that 11f - j l l L a 5 c. If we denote by P, the orthogonal projection on V, , then

i-

-+

llf

-PJ~II

= ll<(f

- j)ll 5 llf - ill 5 e ;

hence

*$

Ilf 11 I + H ~ ~ IforI all j

-

.

(R choeen so that [-R, R] contains the compact support of

j)

(5.3.8)

CHAPTER 5

142

with SR,J= UkEz[k-2-JR, k + 2 - $ R ] , where we assume that j is l&ge enough so that 2-3 R

< 3.

4. We can rewrite (5.3.9) as

cI

JR

t i , #J.t)~zs 2 ~ 1 1 i 1 1 2 - d~ x,(B)I,(Y)I~

k

(5.3.10)

where X, is the indicator function of SR,, i.e., xJ(y) = 1 if 9 E SR,, x, (y) = 0 if y @ SR,. For y @ 2, we obviously have xJ(3)-+ 0 for j -+ 00. It therefore follows from the dominated convergence theor$.m that (5.3.10) tends to 0 for j + oo. In particular, there exists a j such that (5.3.9) < r 2 ~ Putting . this together with (5.3.8), we find 11f 11 5 2 ~ Since . r was arbitrarily small to start with, f = 0. This pr9ves that (5.1.3) is satisfied. For (5.1.2) we introduce the additional hypotheses that is bounded and that J dx 4(x) # 0. PROPOSITION 5.3.2. Suppose that 4 E LZ(R)satisfies (5.3.5) and that, moreover, &(<) M bounded for a11 and mtinuous near = 0, with &(o) # 0. Define 5 as above. Then UJEz 5 = L 2 ( ~ ) .

4

<

Proof. 1. We will use again that (5.3.7) holds, with A, B independent of j.

€1~; Ez6)'.Fix c > 0 mbitrarily small. There exists a compactly supported "Co" function j so that ( 1 f < e. Consequently, for all

2. Take f

jllL1

J=-jEZ

On the other hand, by-(5.3.7),

3. By standard manipulations (see Chapter 3) we have

with

@a %
$!

143

MULTIRESOLUTION ANALYSIS

Since

is Cm, we can find C so that

It then follows that

c -

c ldlli- C (1 + r2PPJ)-'I2 J4 (1 + 11l2)-' LZO

(

use sup (1

+ y2)[l + (z-y)2]-1[l + ( z + y)2]-1 < ba twice

)

z.tt~IP

- C"2-J. <

(5.3-15)

4. Putting (5.3.12), (5.3.13), (5.3.14), and (5.3.15) together, we find

2~

*

1q

lj(2-

JolzI~ C ) 5I ZB C +~ cfl2-

J

.

(5.3.16)

Since d(t) is uniform% bounded as well as continuous in ( = 0, the lefb hand side of (5.3.16) converges to 27r[&(0)1~1( f I(:, (by the dominated convergence theorem) for J --+ 00.It therefore follows that

-

I I ~ I rI t~ &~ o ) ~ - ' c ~ ,

with C independenof c. Combining (5.3.17) with Ilf obtain Ilf l l ~ aI + I l f H ~ aI(1 + cl8(fi)l-')~. Since c was arbitrarily small, f = 0.

(5.3.17)

-fll~, 5

e, we

RE~~ARKS.

ri

::$

f

+

,

1. If slightly stronger conditions are imposed on 4, then Propositions 5.3.1 and 5.3.2 can be proved with easier estimates. In Micchelli (1990), for example, the aame conclusiom are derived if t$ is continuo~wand satisfies Jq5(x)J5 C(1+ 1~1)-'-~,CtEz$(x - Q)= const. # 0, which implies both 4 E L1 and J 4 ( 4 # 0.

*

.1

\Q

P

2. The extra condition that # be continuous in 0 in Proposition 5.3.3 is not necesssry. The following is an example of a multiresolution analysis in which the scaling function is not absolutely integrable. Let KM,p, be, respectively, the multiresolution spaces, the s d i g funGion, and the wavelet for the Meyer wavelet basis, with v E Coo(see 55.2). Let H be

eM

I

CHAPTER 5

144

<

<

the Hilbert transform, (Hj)"(t) = f ( ~if) > 0, -f (0if < 0. Define V, = H y M ,4 = ~ 4 Because ~ the . H'ilbert transform is unitary and commutes with scaling and with translations (in x), the V, still constitute a multiresolution aadysis, and the &,k are an orthonormal basis in Vo. But 6 is not continuous in 0. Because 0 @ support ( G M ) , 4 = ( H @ M ) A is a Coo-functionwith compact support, so that @ itself is Cw with fast decay. This is therefore an example of a very smooth wavelet with good decay, associated to a rnultiresolution analysis with bad decay for 4.6 Note also that 4M and 4 satisfy (5.1.17) with the same w ,illustrating that the c, in (5.3.4), or equivafently ma, do not determine 4 uniquely, and that decay of the c, as (n(-+oodoes not ensure decay for 4.7

3. If 4 is bounded, and continuous in 0, then the condition J(0) # 0 is necessary in Proposition 5.3.3. This can be seen asfollows. Take f E L ~ ( R ) , ' j # 0, with support j c I-R,R], R < m. If Uz, V, = L2(W),then f = limJ,, P- J j. But

6

as in (5.3.13). Since is continuous, the first term tends to A-' 27r]4(0)l2 J]f1I2 for f-too, by the dominated convergence theorem. The second term can be bounded exactly as in (5.3.15), so that this term tends to zero for J-roo. It follows that

Since 11f (1

#

C

w

0, this implies 6(0) # 0.

4. The argument in points 3 and 4 of the proof can also be wed to prove 14(0)12 5 B/2n. We have indeed

where (R1 can be bounded by C2-J for nice f . The other term tends to 2?rl$(o)(?llf 1' (see 4). 'Ibgether with remark 3 above, this implies A/27r 5 ( & o ) ( ~5 B/27r. In particular, if the &,k are orthonormal, then A = B and I$(o) ( = ( 2 ~-'I2. )

c

LO",4(0) # 0 (with 4continuous in 0) imply certain

3

'5. The conditions

E

restrictions on the c, aa well. l3quation (5.3.4) can be rewritten as

1

P

145

MULTIRESOLUTION ANALYSIS

with -(<) = C, c, cant . In particular, b(0) = %(0) &o), which implies ~ ( 0 = ) 1 (since &o) # 0) or

Moreover, (5.3.18) implies that m,-, is continuous, except pomibly near the zeros of In particular, mo is continuous in = 0. If, furthermore, lJ(<)l 5 C ( l I
6.

<

+

I&< +

4#

c

.

I&<

4

t$#(x)/c,@(<) d(r)

n

Together with c, = 2, this implies C, qn= 1 = C, czn+l. This is consistent with the admissibility condition for $.8 Note also that C, c2, = 1= czn+~is equivalent with M i d (1990)'s condition Ct d(x-e) = const. # O if I#(x)I C (1 1x1)-'-' and i f 4 i s ~ o n t i n u o u s . ~o

Cn

,

x,

8.

<

+

k,

C

All this suggests the following strategy for the construction of new orthonorma1 wavelet bases:

.

4

Choose # so that (1) 4 and have reasonably decay, (2) (5.3.4) and (5.3.5) ere satisfied, (3) J& 4 ( 4 2 0

If necessary, perform the "orthonormalization trick"

.e

Finally,

[Cr

G(f)

+2

+

= e"/2mo# ([/2 r)4#(
with m$(<) =

+

$;

e-i4.

~ ( 6 )

146

-

5.4.

More examplea: The BattleLemariB family.

CHAPTER 5

The Battle-Lernarid wavelets are associated with muItiresolutiori analysis ladders consisting of spline function spaces; in each case we take a B-spline witb knots at the integers as the original scaling function. If we choose C#J to be the piecewise constant spline, 1, 0 5 x 5 1 , 0 otherwise,

4

then we.end up with the H e e ~h i s . is the piecewise h e a r spline, The next

1 - 1 4 1 0 5 1x1 5 1, otherwise,

!.

plotted in Figure 5.4a. This 4 satisfies

,?

see Figtire 5.4h. Its Fourier trmsform is

and 2z

x,,ZI$(€+ 2nP)I2

1 -

(b1

J+

msc = + ( I

+ 2 m 2{/2).lO

;.-dl e(2xf

51 *(a++ 1

1 e( 2x-1 1

0 -1

n ~ 5.4. . &(a- 1).

0

1

The piwetube tincar 8-spiine 4; it racisjiee 4(2) = $4)(20

+- 1) f d(22) + 4

MULTIRESOLUTION ANALYSIS

147

Both (5.3.4) and (5 3.5) are satisfied, q5 E 1;' and l &$(x) = 1 # 0. The V, constitute a rnultiresolution analysis (consisting of piecewise linear functiom with knots at 232). Since q5 is not orthogonal to its translates, we need to apply the orthogonalization trick (5.3.3)

Unlike q5 itself, 4# is not compactly supported; its graph is plotted in Figurh 5.5a. To plot @#, the easiest procedure is to compute (numerically) the Fourier caefficlents of [I*+ 2 w2t/2]-1/2,

and to write 4#(x) =

C,

c,

d(x - n). The corresponding m$ is

FIG. 5.5. The a d i n g funetton p and Lhc wavelet T(, for the hnear splrne Battk-Lmrorit anrshuctton.

148

and

CHAPTER 5

4is given by

Again we can compute the Fourier coefficients d,, of [(I - sin2
+

cos2 ~ / 2 ) - l ( l cos2
This function is plotted in Figure 5.54. In the next example q5 is a piecewise quadratic B-spline,

(

$ ( 2 +1)2 ,

( (I as plotted in F~gure5.6a. Now 4 satisfies =

-llx
otherwise ,

a 4 ( 2 +~ 1) + :4 ( 2 ~ )+ 2 #(25

-

1)

+

- 2)

(see Figure 5.6b); we have

g+g

a n d 2 ~G l i ( < + z ~ e ) l = ~ COSC+& C O S ~ [ = & + ? - ~ C O S E + $ C O S ~ C Again (5.3.4) and (5.3.5) are satisfied, and 4 E L1,with r d z )(XI # 0. TI. q5(. - k) are not orthonormal, and we need to apply the orthogonalization tric (5.3.3) to find q5# and rnf before we can construct 1C, Graphs of $# and ar given in Figure 5.7. In the general case, q5 is a B-spline of degree N, $J

where K

=.

0 if N is odd, K: = 1 if N is even. This q5 satisfies

#(x)

= 1 anc

if N = 2M is even

i f N = 2 M + 1 isodd.

MULTIRaSOLUTION ANALYSIS

5 6 The quadmtrc B-spltne 4, trarulated so that sotufies &(z) = +4(2z 1) :&(22) :4(22 - 1) + :4(22

Fw

+ +

+

t b knota am

at the tntqcra It

- 2)

tb(< +

Explicit formulas for El 27rt?)i2, for general N, can be found, e.g., in Chui (1992). In all cases, 4 satisfies (5.3.4), (5.3.5). For even N, 4 is symmetric for odd N, around x = 0. Except for N = 0, the r$(. - k) are around x = not orthonormal, and the orthogonalization trick (5.3.3) has to be applied. The result is that support 4# = W = support tl, for all the Battle-Lemarik wavelets. The "orthonormalized" $# has the same symmetry axis as $. The symmetry axis of .rjl always lies at x = (For N even, tl, is antisymmetric around this axis, for N odd, $ is symmetric.) Even though the supports of 4# and 9 "stretch out" over the whole line, 4# and $Jatill have very good (exponential) decay. To prove this, we need the following proposition. PROPOSITION 5.4.1. Assume that $ has ezponenttal decay, I$(z)l 5 C e-rlzl, and that, for some a 5 7 (a> O),

4,

i.

sup l(ep 4)"(<)I lalla

Assume also that 0 < a 5 El I$([ 27r1)12]-1/2. d(4)

+

< C (1 + l€o"-C

EL I ~ ( +C 27r&)12.

(5.4.1)

Def;ne 4# by J#(<) Then d# has ezponenttd decay as well.

=

I

CHAPTER 5

FIG.5 7. The scaL*ngfinctton 4 and the wavefet Q for tke quodmfic spline BotUc-Lcmohl conslructson.

1. The bound lt$(z)l 5 C eeal'l implies that 6([) Eas an analytic extamion i ( 2 ) E L2(R) for all It21 < a. The to the,strip IIm [I < a,and that^ same is true for =

6(€)

a-0.

2. For fixed b , define FtZ([,) =

and

+

+ it2)$(-(I

- B).Then

MULTIRESOLUTION ANALYSIS

(We have used that

xt If (x+2~4)15 (2n)-IJ d x If (x)l+ J dz I f ' ( ~ ) l . ' ~ ) +

Consequently, Ct Ft, ({I 2 4 converges absolutely if tIm t21 < 7. Similar bounds, combined with the dominated convergence theorem, show Q, (tl 2 4 is analytic in { = El 2f2 on the atrip lIrn [I < 7. that

+

+

+

'g %

F" ' i

$"

3. The function G(()= C, /J(E 2 ~ l )has ( ~ thus an-analytic extension t o JIm €1 < 7. Since G is periodic, with period 2n, and Gill > a > 0, this implies that there exists 6, p m i b l y smaller than 7 , sa that ReG(<)1 a/2 for IIm
4

4. On the other hand, (5.4.1) implies that

4

4#is analytic, and is

5 a. It follows that on tItn
i

Consequently,

5 C

,. (~OROLLARY 5.4.2.

; L

,+;

fi-

e-IZ2

,

for

< min ( 6 ,a) ,

rn

All the Battle-LemariC wcnrekb t,b and the corresponding dwnonncrl sding junctions $# have ezgonential decay.

153

MULTIHESOLUTION ANALYSIS

Regularity of orthonormal wavelet bases.

5.6.

For wavelet bases (orthonormel or not-see Chapter 8) there ki a link between the regularity of $J and the multiplicity of the aeto at ( = 0 of This is a cansequeace of the following theorem (stated sad proved in greater generality than needed here, for later convenience). THEOREM 5.5.1. Suppose f , j am two functions, not identicufly -tat, . ruch that (13,kl& ~ , v = ) 6jj&w

4.

= 2-3/2 f ( 2 - ? ~- k), Afk(x) = 2 - ~ / * f ( 2 -~ k). ~ S?~ppo8t!fkjt Ij ( ~ ) l5 C(I ~ x l ) - ~tkth , a > rn I, and SUppoJe ~ o f t E P , with j ( f )

with

+

+

bounded fur C I rn. Then

1. The idea of the proof is very simple. Choose 3, k, j', k' so that f,,&is rather spread out, and very much concentrated. (For this expwitory point ' . only,we assume that f has compact support.) On the tiny support of f,lbkt 4he slice of fic L'seenplby ~ , I can ~ I be replaced by its Taylor series, with as -many terms as are well defined. Since, however, $ & f,,k(z) f 3 ~ , k l( x ) = 0, ~ , I , ~ I

this impies that the integral of the product of j and a polynomial of order m is zero. We can then vary the beahions of BB given byk'. For each location the argument can be repeated, leading to a whole family of different polynomials of or& m which all give zero i n t w a l when multiplied with Thb leads ta $be dmired moment mudition. But let us be more precise as follows.

1.

2. We prwe (5.5.1) by induction on L. The following argument works for both the iriitid step and the inductive step. Assume J & znj(x) = 0 for n E N, n < e. (If e = 0, then this amounts to no assumption at all.) Since f(') is continuow (e < m), and since the dyadic rationals %-jk, ( j ,k f 33) are denee in It, there exist J,K eo that ~ ( ' ) ( z - ~ K#) 0. (Otherwise f (0 0 would follow, implying f I constant if t = 0 or 1, which we know not to be the case, or, if 2, f = polynomial of order t - 1 1 1, which would imply that f is not bounded and is therefore also excluded.) Moreover, for aryt c > 0 there exists d > 0 so thet

=

!>

I-

fa)

if

(E

C

-! n=O

f

(

-

2 ( x ~2

'

2-J KJ 5 6. Take now j > J , j > 0. Then

I<

9- 2 - J ~ 1 1

CHAPTER 5

Since $ dz x n j ( z ) = 0 for n

< P, the first term is equal to

)I \

(!!)-I

j(t)(z-J~)z-(r+l)~

(5.5.3)

4

1 1

1

Using the boiindedness of the fin), the second term can be bounded by

t

where we replaced the upper integration bound by oo in the first tern, And where we tiscd in the second tern1 that (1 2Jt)-l j (1 1 ) - I 2-1 ( 1 t ) - I for t 6. Note that C1,4 only depend on C, a,and e; they are independent of c, 6, and 3. Combining (5.5.21, (5.5.3), and (5.5.4)

+

&!& +

+

>

<

leads to

Here c can be made wbitrarily small, and for the corresponding b we can choose 3 sufficiently larp to make the second term arbitrarily small as well. It follows that J dx xCf (x) = 0.

When applied to orthononnal wavelet b, this theorem haa the following carollades: COROLLARY 5.5.2. If the + J , r ( ~ )= 2-5l3 ltr(2-f~ k ) constitute an orthononnal set in L~(R), with I+(x){ C (1 + IX~)-~-'-' ,$ € E ( R ) and boundedforesm, t h e n J d z d $ ( x ) = O f o r 4 ! = 0 , 1 , . - . , m .

<

-

Proof. Follows immediately from Theorem 5.5.1, with f = j= $.

m

155

MULTIRESOLUTION ANALYSIS

1. Other proofs can be found in Meyer (1990), Battle (1989). Both proofs *

6 &2j

'

work with the Fourier transform, unlike this one. Similar links between zero moments and regularity also constituted part of the "folk wisdom" among Calderbn-Zygmund theorists, prior to wavelets.

2. Note that we have not used multiresolution analysis to prove Corollary 5.5.2 or Theorem 5.5.1, nor even that the $,,t farm a basis: orthonormality is the only thing that matters. Battle's proof (which insplred this one) a h uses only orthonormelity; Meyer's proof wea the full framework of multiresolution analysis. o

"0 q=.

COROLLARY 5.5.3. Suppose the +,,k are orthononnd. Then st aa tmpossrble 9 has ezponential decay and that 3 E P , wafh all denvatrues bounded,

A?t

,t

that

i

d e s s @ E 0.

J ,

%3

%f. ?.a'

t 4 2

%

If $ E CaOwith bounded derivative, then by Thebrem 5.5.1, J dz zt+{x) = = 0 fm d l t E N. 0 for all t E N; hence $

$1

=o

2. If 11, has exponential decay, then 4 is analytic on some strip (Im
3b:

This is the trade-off announced at the end of the'last section: we have to for exponential (or faster) decay ip either time or frequency; we cannot ckave both. In practice, decay in z is often preferred over decay in [. A last consequence of Theorem 5.5.1 is the following factorization. ROLLARY 5.5.4. Assume that the @,,k m t t t u t e an orthononnal basts multsresolution analysas as described m $5.1. If and @ € C" w t h ~ ( ' 1 bounded for t 5 m, 1.14), factontm as c$ooge

m(€)= &&E

2

)

m+l

*L(O

9

(5.5.5)

L u 2n-penodic and E Om.

+1. ByCarollary5.5.2,

k

(

$41 c=o = O f o r e ~ m .

q .the other hand, G(()

+

4

= e - a c / 2 ~ ( < / 2 n ) 4((/2). Since both and are in P ,and $(0) # 0 (see Remark 3 at the end of §6.3.2), this means

4

I

1

CHWI'ER 5

166.

that mo,b m times Merentiable in = T , and

st'

=O

forl<m.

.

We will come back to the regularity of wavelet bases in Chapter 7. 6.6.

Camectian with subband fil&ring schemes.

Multireeolution atralysis leads nat"maUy to a hierarchical and fast.'scheme for tbe computation of the wavebt coefficients of a given function. Suppose that we have computed, or are given, the inner products of f with the at some given, fine d e . By nscaiing our ''unita" (or rescaling j ) we can assume that the b l of this fine d e is j = 0. It is then easy to compute the ( j , qJVk) for j 2 1. F i i of all, we have (see (5.1.34))

''

where gn = ($,

= (-1)" h-,+'.

Consequently,

Xt foUm that

(1,#l,k) = i.e., the

(f,#I,))

ete obtained

K=z (f. &,,'I

*

condvins the quence ((f,&,n>)rez with

(a-n)nez,and then retaining only the even eamples. Similarly, we have

which can be d to compute the (f, CpfC) by meana of the 8-tion* (convolution with & decimation by factor 2) from the (f, t$j'5-l,k), if these are

MUL-LUTION

ANALYSIS

known. But, by (5.1.15)

h-a 6j-~.n(z)

=

(5.6.3)

I

n 95

.*** whence

cf,

+

a*

..4,

h.k)

=

CK G

(f9

-

oj-1.n)

n

(5.6.4)

The procedure to follow is now cles: -ts from the (f, h,,),we compute tbe (f,$ l , k ) by (5.6.2),and the ( j , &,k) by (5.6.4). We can then apply (5.6.2), -" (5.6.4) again to compute the ( f , h C ) ? . : ( j , hL)from the ( j , 4l,n), etc.. . .: at of the c o r n > every step we compute not only the ~p~veW Coefficients ( j , *spending j-!evel, but also the ( j , C$j,k)for the same j-level, which are useful for computation of the next level wavelet coefficients. $a,:the The whole process can also be viewed as the computation of successively -.$*,i m m e r approximations of f , together with the difference in "information" be+ :.--., t-areen every two successive levels. In this vieolr we start out with a fine-scale : . t -5 -approximation to f , f" = Pof (recall that P, is the orthogonal projection onto , V ;.are will denote the orthogonal projection onto Wjby Q,), and we decompose P E V o = Vl@IVl into f O = f 1 + 6 1 , r b e r e f ' = = Plf is the w x t coarser approximation of f in the multiresolution analysis, and 6' = p - j' = QlfO = Ql f is what is "lost" in the transition f o -+ f l . In each of these V, , W,spaces we have the orthomrmal b d (41.k)kEZ, (+l,k)kEz, respectively, so that

., i

%

'

P = C c".,n, f1 ='C 41,*,P = C dt *I,,

% r.+w

n

n

F ' k r (56.2), (5.6.4)

-

n

give the effixt on-tha & e% okcthe ieni

ck =

h-lt

4,

9n-1L C: .

d: = n

a

ii = (dncZ and (Ab)k =

With the notation a = csn rewrite this as

cl=BcO,

basis

-+

.

(5.6.5)

Ena2*-n bn, we

d'=GcO.

The coarser approximation f1 E Vl = Vz GI W2dtk again be d e c o m p d Into f' = f 2 +a2, j2 E Vz, 62 E W2,with f2

=

C

:C h e n

n

@=x&

h,n

-

n

We again have c2=fjc1,

6=Gc1.

Schematically, all this can be represented as in F

i 5.8.

CHAPTER 3

FIG 5 8

Schem4kc rrpnsentokon of (5.6.5).

In practice, we will stop after a h i t e number of levels, which means we have rewritten the information in ((f, &,n))nE&= d ) as d l , dL,d3,. - - ,dJ and a find coarse approximation cJ, i.e., ((f,' @ j , k ) ) & ~ ,> = I , , J and ( ( j ', # ~ , l i ) ) b ~Since ~. all we have done is a succession of orthogonal basis transformations, the inverse operation is given by the adjoint matrices. Explicitly,

hence

(use (5.6,1), (5.6.3)) . In electrical engineering terms (5.6.5) and (5.6.6) are the analysis and synthesis steps of a subbandfiltenng scheme with exact reconstruction. In a twechgnnel subband filtering scheme, an incoming sequence is wnvolved with two different filters, one low-pass and one high-pass. The two resulting sequences are then subsampled, i.e., only the even (or only the odd) eatnee are retained. This is exactly what happens in (5-6.5). For readers unfamiliar with this "filtering" terminology, let me explain briefly what it means. Any square summable sequence ( c , , ) , ~can ~ be interpreted a& the sequence of sampled valw ~ ( nof) a bandlimited function 7 with suppod j. c [-?r, rr] (see Chapter 2),

(e)n,Z

sin r (x - ra) n

A filtering operation corresponds to the multiplication of j. with a 2~-periodic function, e.g.,

I

MULTIRESOLUTION ANALYSIS

The result is another bandlirnitd function, a * 7 ,

or

sin n ( x - n) n

is m d l y concentrated on (-n/2, n / 2 ] , hghThe filter is low-pass if pars if &ll-r,xl is mostly concentratad on {C; r / 2 5 I[) 5 n ) ; see Figure 5 9. The "idealn low-pass and high-pass filters a& lit(() = 1 if ](El < n/2, 0 if n / 2 < (
I

2

for n = 0 ,

for n = 2k, k # 0 ,

a,L =

- I k f ~ Rl = 2k 1)n

(2k

h ~5.9. . A low-rn

+

+1,

(sobd line) ad a high-pw filter (dashed kne)

%en the ideal law-pess filter Is apptied tr, 7 , the result is a bandlimited Eunction with support C [ - z / 2 , ?r/2j. Gnch a function is completely determined by its sampled values in 22, and we bavt (m'(2.1.2)) sin [ ~ (-x2n)/2] n(t-2n)/2 .

160

CHAPTER 5 \

Similarly, the result of applying the ideal h~gh-passfilter to 7 is a frequencyshifted version of a bandlimited fuaction with support c ( - r / 2 , r / 2 ] . Such a function is again completely determined by its sampled values on 22,

Since the even-indexed entries of the convolutions of the a:, a: with the cr suffice to characterize a~ * y and a~ * 7 completely, it makes sense to retain only them after the convolution. This is the rationale behind the decimation by a factor 2 in subband filtering, also called "downsampling." Reconstructing the orignal c, from the two filtered and decimated sequences,

is easy:

(since tiL + hH = 1)

/ I

Distinguishing between even and odd m, we find

This ctto also be rewritten as

This last operation can be seen ae the result of interleaving both the 4 and $.t with zeros (i.e., constructiog new sequences with m o odd ex@h, and with even entjes given by the consecutive

4, e);

convolving these interleaved (%peempledn)sequences with the mters aL, a*, respectively;

I

a

adding the two results.

MULTIRESOLUTION ANALYSIS

161

Schematically, (5.6.8) and (5.6.9) can be represented as in F i 5.10. The filter coefficients a:, a t for the ideal h - p a m and hii-pass filters aL, aH decay much too slowly to be useful. In practice, one prefers t% use the scheme in Figure 5.10 with filters ao,a', go,i ' with much faster decaying coefficients. This can only be achieved if the corresponding 2.rr-periodic functions aO,a', hO,iil are emoother than aL,an. Thii means aliasing can occur: jaul,lalllook like "rounded" versions of aL,dl (as in Figure 5.9), which means their support is larger than [-u/2, r / 2 J and {t; a12 I KI 5 n), respectively. Consequently, a" + 7, a' * 7 are not truly bandlimited with maxfmal frequency 7r/2, and sampling them as if they were leuds to aliasing, as explained in 52.1. This hrrs to be remedied in the rewmtrwtkm stage: iio, 5' need to match a0 and a' to get rid again of the aliasing present after the decomposition. And even this "matchingn is only possible if aa and ' a sre already matched in some way. To find the appropriate conditions on these filters, it is convenient to use the 'z-notation," in which s sequence ( G ) , , ~ is represented by the formal series a ( z ) = LEZ an zn. If z = e-'E is on the unit cirde, then this is nothmg but a Fourier series; sometimes it is convenient to consider general x E C rather than bI = 1. The decomposition stage of the subband filtering scheme in Figure 5.10 can then be written as

Here crO(z) 2) is the t-notation for the convolution of a' and c; is equal to t e formal sequente b, z2", i-e.,b(z) with all

"6

C,

removed. The mnstructitm stage is

Fs.

5.10. Schmotsc representdion of the h p d h a and nsamstructia &ages oerticol dashed lme) Jn a Nbbwd Ptaing scbtme. Every Ictkr (a0,a ] , .) m a boz npumL m v d a i o n rrnth the m n q m m h g quma; 2 1 stands for the duwmam-

(separated by UK

..

4the even 2 f for upmmphg by 2 (interknuang unth x m s ) . Inthe W " u u e l a O = a L , a* = a H , 6 0 = 2 a L , ~ ~ l = % H ; t h e / i n o l ~ w u l m t i d

plhg by 2 (ntoiRing tOthe~,S=c.

"?'

where d ( z 2 )is the z-notation for the upsampled version of ct (zero8 have bee4 interleaved: ,c9(r2)= Cn-cl, ban). The total effect is

In this expression, the second term contains the al'iasing effects: c(-z) comesponds to a shiftlng of the Fourier series C, & e-'"t by a, exactly what you would expect from aliesing due to sampling ;at half the N'jtq$st tat& In order to eliminate aliasing, we therefore need

The first subband coding schemes wjthout aliasing date back to Egteban and Galand (1977). Ln their work, as in most schemes that will be considered in these notes, the sequences are real; they choose

so that (5.6.11) is indeed satisfied, and (5.6.10) simplifies to

= 4, then al({) = Cn aie-"'f is the "mirror" of a0 with respect to the "half-band" value 6 = 7r/2, since &I(<) = a! e-'"€ = ao(z - <). Filters chosen as in (5.6.12) are therefore called "quadrature znirrw filters'' (QMF).In practice, one likes to work with FIR filters (FIR = finite impulse response; this means that only finitely many an are nonzero). Unhtunately, there exist no FIR a0 EO that ~ O ( Z-) ~ = 2, so that E cannot be identical to c in this scheme. It is nevertheless possible to find aO so that - ~ ~ ( - 2is )c~ h to 2, so that the output of the scheme is indeed dose to the input. There is by naw an immense literature on the design of various QMF; see the issues of IEEE IIFans. Acowt. Speech Sagnal Process. for the kt 15 years. There a h exist many generalizations to splitting into more than 2 bands (GQMF-eneralized QhlF). In Mintzer (1985), Smith and Bzmqpll(19@),and Vetterli (I-) p scheme different from (5.6.12) was propasf&

I f - ao is symmetric, a!,

En

a'

(E)

=w

a-1 aO(-z-'1

,

= aa(z-I) , iiE[r) = ttl(z-l) = zap(-z)

(6.6.M)

GO+)

.

It is easy t9 check that this W i d i t w again (5.6:11), end that (&&lQ)

becornea

MULTIRESOLUTION ANALYSIS

163

For t = eat and real a:, the expression between square brackets becomes &[la0(e-'c)12 t la0(-e-ac)12] = [lao(t)12-t laO(c =)I2]. There now exist FIR choices of a0 for which this is exactly 1, so that we have exact reconstruction in the subband filtering scheme. Smith and l3arnwell (1986) named filters chosen as in (5.6.13)13 "conjugate quadrature filters" (CQF), but the term has not become as generally popular as QMF. One last remark before we retilnr to wavelets. The whole purpose of subband filtering is of course not to just decompose and reconstruct: a simple wire, instead of the scheme in Figure 5.10, would be far cheaper and more efficient. T h e goal of the game is to do some compression or processing between the decomposition and reconstruction stages. For many applications (image analysis, for example), compression after subband filtering is more feasible than without filtering. Reconstruction after such' compression schemes (quantization) is then not perfect any more,14 but it is hoped that with specially designed filters, the distortion due to quantization can be kept small, although significant compression ratios are attained. We will come back to this (albeit briefly) in the next, chapter. Back to orthonormal wavelet bases. Formulas (5.6.5), (5.6.6) have exactly same structure as (5 6.8), (5 6.9), respectively. Going from one level in a &tiresolution analysis to the next coarser level and the corresponding level of~rrwlets,and then doing the reverse operation, can therefore be represented by B diagram similar to Figure 5.11. Here (h), = h-,, (g), = g_, (see above). If we assume that the h, are real, and if we also take into account that gn = (-l)"h-,+l, then we can identify Figure 5.11 with 5.10 with the choices

4

+

Up to a trivial sign change m a' and Zil, thie corresponds exactly with (5.6.13). This means-that every orthonormal wavelet bases associated with a multiresolution analysis gives rise to a pair of CQF filters, i.e., to a subband filtering &me with exact reconstruction. The reverse is not true: in an orthonormal besis construction we necessarily have aO(l) = C h, = 2'j2 (see Remark 5 at the end of §5.3.2), but there*exist CQF for which a (1) i~ close to, but not equal to, 2lI2. Moreover, all the examples of orthonormal bases we have see11 so far correspond to infinitely supported &#, and hence to non-bite aequencea h,; for sppiicationa, FIR filters are preferred. Js it possible to construct orthonormal wavelet baees corresponding to finite @tars? What does it mean for these filters to carrespond to, e.g., regular wavelets? Hoar can wavelets be uwfd in filtering contexts? All these are questions that will be addreseed in the n a t chapter,

a

Notes. 1. I chooee here the same neeting arder (the more negative the index, the the order kgtz the -):ee in the ladder of ~ e apacc& v Ttris is

.Q

that foltms naturally from'the notathi of non-orthogonal watteleta as iniSated by A. Grommann and J. MorIet. It is non-standard, however: Meyer (1990) uses the reverse ordering, more in accordance with establiihed practices in harmonic analysis. For applications in numerical analysis,Beytkin, Coifman, and Rokhlin (1989) find the ordering presented here the most practical.

I

2. We impose here no a priori regularity or decay on 4, udfke, e.g., Meyer (1990).

I

-

3. Equation (5.1.33) characterizes all the p d b l e Lemma 8.1.1 in Chapter 8.

$#.

This follow from

1. In case 4 has compact support, and we would like $ to have the seme compact support, (5.1.35) is the only poesible choice.

5. It is generally believed that time is no such 'pathologlcsl" example with continuous $. Another challenge for the reader! 6 (1991) has When this book wss in its last stages, I heard that M proved that if is compactly supported (continuous or not), then it is automatically ~ssociatedwith a multiresolution analysis. Thii aalves the open problem for one very important special case.

+

gk

bds

0. Note that there exins @ E l 4 ~a,tht the are an s r t h o w d for Vo and d# E L1(R),d i k e 4. It suflkes to take J#(f) = A(€) &), where X is 2n-periodic and A(() = sign(() . e g / a for KJ 5 x. Thh -4 : is again a Schmirtz function. The same Hilbert transfoh trick can tn applied to other multiresolution andyees, such aa the Battle-LmuuiC axbe or the constructions with compactly supported 9 in the next chap*.

.

7. If ue impoee that uniquely.

4 is continuow at ( = 0, then mo dbee determine 4

8. The continuity of 4,together with &(o)# 0, implim that ~ ( 0 ) 1 and that rn,, is costinuou m 6 = 0. It kUm t h t fe contincous in = 0.

<

MULTiRESOLUTION ANALYSIS

Since

+-

+ lrnt((+ *)I2 = 1, it f o l h that (+~L$#Iis contiWow Consequ~ntly,14(t)( = ~ r n f(~12) + r ) ]lt$((/2)l is continuwas

l4(~)l2

t

=m = 0 since J, h a to be admissible, this implies %#(lr) ~ ( x= )O This providm another derivation for (5 3.20).

in in

r'

,

ir

I

9. Proof We prove that C c z , = 1 = Cczn+l if 1q5(x)1 5 C(l IZ()-'-' and 4 is continuous

+

= 0; hence

Cr#(x - f ) = const. # 0

+ Define f (x) = Ct r$(x - 1). The conditions on 4 ensure that f is well defirretl and continuous We have

Hence f

IS continuous,

perlodic with period 1, and

It folfows that f a constant. e C, &(r 0 = c implies &%n) ' mo(tl2) t $ ( t / 2 ) ;hmce

-

= S , ~ ( ~ U ) - ' / ' c. But &(<) =

+

( n # ) 0,then J(m(2n 1)) = 0 would fdbm for all n E 2 , in contradiction with C t$(n .t 2m)12 > 0. Hence ~ ( m =) 0, or

If ~

C

~ 2 = n

1=C

*

~ ~ n + l .

10. An easy way to compute the Fourier awffieents of Et following:

-

i' $

4'

L

I

+2nZ)12 is the

For B-splme 4, these are easy to compute; see also Chui (19!42) for an explicit formula

* for 0 I y 5 2n,

CHAPTER 5

166

12. If f is given in "sampled" form, i.e., if we only know the f (n), then the inner products (f, can be computed.by a convolution (or filtering) operation, under the assumption that f e V,to start with (components of f orthogonal to I/o cannot be recovered). We have f = (f, h , k ) 40,k; hence f ( n ) = (f, 40,k) 4(n - k). Consequd1~,

xk

zk.

i.e., the (f,&,k)

(Emd(m) e-'mt)-l.

are the Fourier coefficienta of It follows that (f,

14

(Enf (n) e-'M)

C q k=) C,, ak-,, f (n), where

2*

a,,, =' ( 2 ~ ) - '

etM

0

13. For convenience, they chose a ' ( ~=) ~ ~ * - ' a ~ ( - z - Z0(z) ~ ) , = z 2 N aO(z-I), iil(2) = z ao(-z) rather than (5.6.13), with N E Z picked so that all the a', 6J are polynomials in z (no negative powers). One finds then E(z) = z~~ ~ ( a )which , corresponds t o a pure delay in the reconstruction.

14. This is an azgument used by fans of the Esteban-Galand-type QMF filters: these fail to give exact reconstruction from the start, but their deviation from exact reconstruction can be made small in comparison with digtottioas introduced by quantization.

i i 5

A1

-

Orthonormal Bases of Compactly

Supported Wavelets

Except for the Haar basis, all the examples of orthonormal wavelet bases in the previous chapter consisted of infinitely supported functions, as a result of the orthogonalization trick (5.3.3). To ccmstruct orthonormal examples in which # is compactly supported, it pays to start from q-(or, , equivalently, from the subband filtering schem55 6) rather thaa from 9 or the V,. Ia $6.1 we show how to construct so that (5.1.20J is saijsiled as well as (5.5.5) for some N > 0 (a necessary condition to have some regulGty for +). Not &verysuch is associated to an o r t h o n o d wavelet basis, hawever, an issue addressed in 556.2 and 6.3. The main results of these two sections are summarized in Theorem 6.3.6, at thi 4 of 56.3. Section 6.4 contains examples of crunpactly supported wavelets genemfhg orthonormal bases. The orthonormal wavelet bases thus obtained cannot, in general, be written in a closed d y t i c tom. Their graph can be computed with arbitrarily high precision, via ae algorithm that I cdl the * q a ~ cade algorithm," which is in fact a "rdinemt *en as ueed in c o r n aided design. All this is discussed in 56.5. A Iat of this material back to D a u W - (198Bb); for many of the results, better, simpler, or more general pro~&have hem found since, and I have given preference to these new ways of looking at thin&. These d i h n t a p prow$m are'borrowed mainly from MaUat (1989), Coben (1990),Lawton (1990, 1991), Meyer (1990), and Cohen, Daubechies, and F'eauveau (1992); for the link with rehemebt equations the referemxa are Cavtmtta, Dahmen, md Miccheili (1991) and Dyn and Levin (1990), as weli as earlier papers by these authara (see 56.5). .*

=

4

"

6.1. Construction of m.

t

In this chapter we are m & l y interested in constructing compactly supported wavelets $. The easiest way to ensure compact support for the wavelet $I is to Eamk the d i n g fundion $t with%* support (in its orthogona@d d o n ) . ft then f o U a &om tbe ddbRw6 ofthe k,

.e

T

168

CHAPTER 6

that only finitely many h, are nonzero, so that $ reduces to a finite linear combination oi compactly ~ilpportedfundions (we (5.1 34)), and therefore automaticallj~has wmpacf eugport itself. Choosing both $ and qb with- ampact support also has the advantage that the corresponding subband filtering scheme (see $5.6) uses only FIR filters. For compactly supported q5 the 2~-periodicfunction m,

becomes a trigonometric polynomial. As shown in Chapter 5 (see (5.1.20)), orthonormality of the b,,implies

where we have dropped the "almost everywherep became is necessarily cont tinuous, w that (6.1.1) has to hold for dl if it holds a.e. We are also interested in xm&g-+ and 4 reasonably regular. By Cerol1aty 5.5.4, this me& that rq-,should be of the form

with N 2 1, end C's trigonometric polynomial. Note %hateven without regularity constraint, we Beed (6.1.2) with M at least 1.' Putting (6.1.l), (6.1.2) tagether, it follows that we are W n g for

and

where L(c)= IL(()I2 is BfeO a polynomial in c a t . For our purpose it is connient to rewrite L(() aa a polynomial in ain2
In terms of P, the constraint (6.1.4) become8

which should hold for all y we use b u t ' s theorem.'

E

[0,I], hence for all y

E

R To mlw (6.1.7) for P

169

COMPACTLY SUFPOEYTED WAVELETS ,$

THEOREM 6.1.1. If p l , p2 aw two polyromiab, of degree nl, nz, TWPCCttvelg, with no common zeros, then them ezijt mique polyromiab q g , qz, of d e p e nz - 1, nl - 1 , mpectiuely, so that

; k-

=1.

m(z) 4r(z) f pz(z) + **

(6.1.a)

.

t

Proof. 1. We first prbve existence; uniqueness follows later. We can assume that nr 2 n2 (by renumbering, if necessary). Since degree ( p z ) Idegree (pl), we can find pcdyoomialti a2{z),b(z),with degree ( a z ) = degree (pl) -

I

-

2. Similarly, we can find a3(x), h ( x ) , With degree(a3) = degree(p2) -degree (h), degree (4)< degree (b), that pl(4 = a3(4

bq(4

+ b3(5)

.

rc

We keep going with this procedure, with bn- 1 taking the role of pl in this last equation, and b, the role of b, b,-l(x) = &I+*(xS bn(x) + b n + l ( ~ .)

Since degree (b,) is strictly decreasing, this has to stop at some point, which Ss only possible if bN+l = 0 for some N,with bN # 0,

3. since

+

b ~ =-ON~b ~ - 1 b~

:

h

it follows that bN divides bNd2 as weil. By induction bN divides all the previous b,,, and pz, so that bN divides both pl snd pa. Since pa a d pz have no zeros in cxunmon, it follows tbat bN is e constant dr&arent ftom zero.

4.

?+*

8F

,

W e have now 9

bN

8 f;

= bN-2 - ON bN-1 = bN-2

- U N ( ~ NL UN-1 J bN-2)

= (1 -k t 3 ( ~z N - I ) ~ N-- ~ ON b ~ - s etc.. . .

!

I.

By induction 0

-

brv = ii#,, b ~ - k+ CN,). b~ -&-I ,

with i ~= --ON, , ~ 6 ~ =~1,16 ~ ~ =k &V,L + ~ - h , r ) a ~ - k ,&,N.~+I = iiN,k. It follom, again by induction, that degree(6~~,~) = degree(bNir-l)

-

.

170

CHAPTER 6

degree ( b ~ , ~ ) degree(h~,k) , = degree (bnr-c) - degree (bnr-;). k = N 1, we find

-

For

with degree ( ~ N , -1) N = degree ( p l ) - degree ( b ~1)- < degree ( P I ) , degree ( ~ N , N 1) - = degree (pz) - degree ( b N - 1) < degree ( p z ) . (We have used that degree ( 6 ~ - 1 )I 1; if degree (bN- 1 ) were 0, then bN would be zero.) It follows that ql = ~ N , N - ~ / B92N = , iiNs-l/bN. solve (6:1.8) and wtisfy the desired degree constraints.

5. It remains to establish uniqueness. Suppow q ~ g2 , and dl, & are two solution pairs to (6.1.8), both satisfying the degree restrictions. Then

Since pt , pl have no %era6in common, this imples that every zero of m is a zero of ql - 41, with &t least the same multiplicity. If ql # #*, then this means degree (qt - 4,) > degree ( B ) ,which is impossible since degree (ql ), degree (ql ) < degree ( p z ) Hence gl = Ql. It then follows immediately that

92 = 42-

1. For later convenience (Chtspter 8), we have stated Baout's theorem in gxwter generality than needed in the present dwpter. In fact, it holds under even mow general conditions: if pl and pa have zeros in common, then (6.1,8) can still be solved if its right-hand side is divisible by the greatest common denominator (g.c.d.) of p l , f i . The proof is still the same, but bN is now the g.c.d. of pl, pz instead of a constant. The argument in the proof is nothing but the m~istructionof the g.c.d. by Euclid's algorithm; it works in many other ftmeworlcs than the polynomials (in any graded ring, in algebraic terminology).

2. It 1s clear from the construction of pl, n,that if pr and haves~fyr a t i c ~ nal coefficients, then so will -and qz. This wilt be useful in Chapter 8. o Let us now apply this So the problem at hand, i-e., (6.1.7). By Theorem 6.1.1 there exist unique polynomials qf, qz, of degree J N - 1, so that

Substituting 1 - y for y in (6.1.9) U to

the dquenese of 91, a thus implies q2(y) = q l ( 1 - 9). It foUows that P ( y ) = ql(y) Is a sdution of (6.1.7). In thii case, we can find the expiicit form of ql

without ewn using EucM'e dgoritbm:

1

v

where we have written out explicitly the k t N terms of the Taylor expansion for (1 - Y ) - ~ . Since degree(g2)5 N 1, q~ is &ual to its Taylor expansion truncated after N terms, or-

-

'.1(.)="i:

; 1

1

k=O

N +( k-l)

yk.

-

This gives an explicit solution to (6.1.7). (Fortunately, it is positive for y f [O, I], candidate for JL(t)J2.)It is the unique lowest degree solution, which we will denote by There exist however many solutioni of higher degree. For any such higher degree solution,.webave so it is a good

P N b 3

Thts a&eady implies that P - PN is divisible by y N ,

Moreover, P(p)+f3(1-y)=0, i.e., P is antisymmetric with respect to follows.

1. We q summarize all our findings as

PROPOSITION 6.1.2. A trigonometric polyonid

satisfies (6.1.1)

and only

of We fonn

L(<)= IL([)j2 a n be written acr

L(<) = p(mn2 t/2) with

P(v) = PN(v) + lN

and R is an add poly"omial, chosen such

,

~ ( - 94 )

,

that P(y) > 0 for y E (0,I].

This proposition completely characterha (m,,(t)12. For out purposes we need however rno itself, not (mo/2.So how do we "extract the square rootn from L? Here a lemma by Riesz (see Polya a d S+ (1971)) c o w to our help. LEMMA 6.1.3.Let A be a positive trigonometric plyrwmial invariant under the substzktton - (; A is necessarily of the form

<+

M

~ ( € ) = ~ 4 n m with~ a,,,eR, m=O

76,

Then t h e e d l s a higonometric polyromial B of onler k4, a. e.,

--

Pnwrf.

1. W; dan write A(€) ="pA(cos\l), wherem is a polgaomiaiof d real coefficients.This polynomial can be factored,

mM e h

M

I

I

where the zeros cj of pA appear either in complex duplets c,, Ej7 or in real singlets. We ccin also write

where PA is a p02ynomtai 01 deep.ee 2M. For Izf = 1, we have

I

the ttwo polynomids in the rigfrt aPd left-hand sides of (6.1.13) therefore 8grw on ail of C.

I

-

JG.

2. Ifc, is red, then the zem oft -cj2+)z2 are ej f For l g l Z 1, tbae w t~ d (d-Ynt. if C, = i l ) of the fonn rjl ~7'.F'o~ {cjl < 1, the two zeroe are compb conjugate and of absolute value 1, i.e., they nre of the form ehj, e-&j. Sbwe lcj 1 < 1, such zero6 correepond to ''physicaln zeros of A (i-e., to values of for which A(<) = 0). In order not to cause any contrsdiction with A 2 0, th& z;eros must haw even multiplicity.

<

I

I

T.

173

COMPACTLY SUPPOBWBD W A V E L m

+

'

3. If cj ia not real, then we consider it together with c& =

nojnial

4f

5. The p l y -

(f- q z + 43)(4 - 9% + ir2) has four zeros, el f ,/-

ds.

and

One easily chedrs that the fouo~eermare all difkent, and form a quadruplet 3 , 2'; , f , , 27 l .

4. We therefore have

. '"

*

*

P.(Z) =

I

>

1

J

-ZJ)(z

- Zj)(% - %yl)(Z -

where we have regrouped the two different kinds of zerwr.

A

,,

1

5 aM

5.

L

For z = e-e on the unit circle, we have

Consequently,

where

fs clearly a trigonometric polynomial of order M with real coefficients.

m

1. This proof is constructive. It uses factorization of a polynomial of degree M ,however, which has to be done numerically and may lead to problems if M is large and some zeros are close together. Note that in this proof we

need to factor a polynomial of degree only M, unlike some other procedures, which factor directly PA,a polynomial of degree 2M. 2. This procedure of "extracting the square root" is also called spectral factorization in the engineering literature.

3. The polynomial B is not unique! For M odd, for instance, PA may have quadruplets of complex zeros, and 1 pair of real zeme. In each 2 quadrupiet we can choose to retain either r,, Z j to make up 8 ,or ql,z ;'; in each duplet -we can choose either rt or r;'. This d m already for different choices for B. Moreover, we can always muftiply B with e'M, n arbitrary in Z. o

-

.

Together, Propasition 6.1.2 and Lemma 6.1.3 tell us how to construct aU the possible trigonometric polynomials mo satisfying (6.1.1) and (6.1.2). It is not yet clear, however, whether any such ?no leads to orthonormal wavelet basis. In fact, some do not. This will be discussed in the next two sections. Readers who would like to skip most of the technicalities cap find the main results summed up in Theorem 6.3.6 at the end of $6.3. 0.2.

Correspondence with orthonormal wavelet bases.

We start by deriving a-formula for a candidate waling function 4. Once this is done, we will check when this candidate defines indeed a hona fide multiresoiution analysis. If a trigonometric polynomial n o is associated with a m u l t i r ~ l u t i o nanalysis as in 55.1, and if thp corresponding scaling function @ is in L1(R), then we know that for all <, (6-2.1) = ~ ( < / 2&/2) ) .

J(c)

(See (5.1.17). Continuity of and mo dm us to drop the "8.e.") Moreover, we know from Remark 3 fobwing Proposition 5.3.2 that necesearily J(0) # 0, hence ~ ( 0 =) 1. Becaw of (6.1.1) thi4 in turn implitm ~ ( x =)0. It follows that, for all k E 2, k # 0, &kn)

= d(2 Y(2m + 1 ) n )

(for some C 1 0, rn E Z)

176

COMPACTLY SUPPORTED WAVELETS

xt

+

f ~ i n c e I&( 2n!)12 = (2n)-' (see (5.1.29)),this fixes the normalization of 4: ?, &o)l = (2u)"12, or I $& d(x)l = 1. It is convenient to choose the phase of 4 I. so that Jdz flz) = 1, Taking all this into account, it follows from (6.2.1) that

This infinite product makes sense: since %(() = 2-'12 hn e-("€ satisfies

Cn

Cn lhnl fnl < oo, and ~

( 0 =) 1,

hence w

fl 1 ~ ( 2 - ' < ) I5 exp )=I

? The infinite product

'

in the right hand side of (6.2.2) therefore converges abeoW i y and uniformly on compact sets.4 A4i thin =lies generally whenever 4 E L1,m d the h, have sdeient decay. is a trigonometric polynomial (only finitely many of the in present c e , bi&e different from zero), and we are looking for q5 with compact support. Tog&her with the obvious constraint 4 E L2, compact support for 4 means I#I E L', so that the above discussion applies. It follows that (6.2.2) is the I only possible candidate (up to a constant phase factor) for the scaling function - corresponding to a trigonometric poiynomial w constructed tat in $6.1. We now need to check that 4 satisfies some basic requirements for a scaling function. First of all, q5 is square integrable: LEMMA 6.2.1. (Mailat (1989)) I f mo w a 2n-pen'odac function sutufyrng (6.1.1), and tf (2r)-'I2 %(2-'C) amvnyes pmntwe a.e., then its hmit

!

nzl

ts

i n i 2 ( ~ ) , ' a n Ipllr2 d 5 1.

Proof. I. Define

fk(C)

=

X ~ - ~ , ~ ]=( C 1 if)

(2%)-'I2

[n:=l m0(2-j()]

ICI 5 T , 0 otherwise.

Then fk

--+

~ ~ - r , * ~ ( 2 - ~where (),

6 pointwiae a.e.

2. Moreover,

= (2r)-'

n k

2h+lr

d(

1=1

lm(2-%)I2

(by the 2%-periodicityof mo)

CHAPTER 6

3. It follows that, for all k , llfkl12

=

I1fk-11Ia

1 ;

- . = Hfol12 '= 1 .

Consequently, by Fatou's lemma,

/4I ~ ( c )5I ~lim k

-+

sup 00

J

~fk(<)l'

5 1.

Second, since nb is a trigonometric polynomial, the following k3-a borrowed from Deslauriers arid Dubuc (1987) provos that 4 haa compact support. LEMMA6.2.2. rg) = 1. ~ ~ with~ 6 7, = I, U~CR , r [ 2 - 2 4 ) za an entrm functzon of exponential type. In pmticulor, it ia the F o u w tmmform of a dutnbutzon with suppord in fN1, Nrf. hf, By the Pdey-Wiener theorem for distributions, it is sufficient to prove that. r(2-a ie an entire function of exponentid type with bounds

,n:

c,":~,

c::~,

u

I

nzl

n n rt2-'0

r(2-'C)

' ( N I llm (1) 5 4 ( 1 + 1 ~ 1 ) ~ exp

for Im F 2 0 ,

5 c2(1 + I E I ) ~ =exp (NZ~b 40

for~m (50

.

00

,

I

1

ji.1

for some Cl, C2, M I ,M2. We will only prove the h t bound; the m n d is entirely analogous. Define

Then

00

JJ r(2-30

=

fi

r1(2-$€) ,

m we only need to prove a polynomial bound for For Irn C I0 we have

nGl ra(2-j€)hb

2 0.

COMPACTLY SUPPOFU'ED WAVELETS

I

<

Take arbitrary, with Im I

If

(€1

> 0. If )I)5 1, then

2 1, then there exists j o

<

> 0 so that 2J0 < KI < ~ J o * ' , and

(1

+ C)l0+' eC

5 eC ( 1 + C) exp Iln(l+ C) In I<)/In 21

Combining (6.2.3) for polynomial bound. '

I[( 5

1 and (6.2.4) for

> 1 establishes the desired 17

So far, so good. All this is not sufficient, however, to define a bona fide ecding function. A counterexample is

This satisfies (6.1.1), as well as m(0)= 1. Substituting it into (6.2.2) leads to5

1

3 , 05x53, 0 otherwise.

This is not a "goodnscaling function: the &,,,(x) = &(z-n) are riot orthonormal, even though satisfies (6.1.1). Another way of looking at this ia to eee that

178

I

CHAPTER 6

(5.1.19) is not satisfied:

xt

+

that this m e s that ~ IT!)? = 0 for ( = ?f,so that 4kdi (11.3.2) is 'hob satisfied: the h , n are not even a Rieez basis for the space they span.= &k order to amid this kind of mishap, we have to impom extra conditions on to make sure that 4 generates a true m u l t h l u t i o n analpis. These dnditione ensure that

tor dl (. Once (6.2.5) is sstiafied, everything else works: the -s V, = Span(4>,,; n E 2) constitute a muftiresolution ~nalysis(by 85.3.2); in Vj , the ($j,n)nEZ constitute an orthonormal basis. We define t(, by i

+(x) = \/i

z(-l)n Q(ZI-

n);

(6.2.6)

n

thie is automirtically compa~tlysupported because 4 is and becauae only finitely many h,, differ from zero. The ($,,k)J,kEZ constitute then an orthonormal basis of compactly supported wavelets for L2[R). Before we go into the conditions on that ensure (6.2.5), it b interesting to remark that even if (6.2.5) is not satisfied, the function $ defined by (6.2.6) still generates a tight Frame, as p r d in Lawton (1990). PROPOSITION 6.2.3. Let mo be o trigonometric polyrornial satisbrag (6.1.1) and m(0)A 1, and let 4, $ be llre compacffp supported L2-functions defined by (6.2.2), (6.2.6). Define, usual, q j , k ( x ) = 2-]I2 $(2-lx - k). Then,for all f E L2(W),

be., the ($,,&; j, k E 2)h t i t u t e a tight ftnme for L2(R).

1. First reanember that (6.1.1) eau abm be written as

(see (5.1.39)). 2. 'XBke f compactly supported sad CQD.Then ,all j :

I (f, +j,k)lz

caaobrgee for

I

179

COMPACTLY SUPPOIZTED WA-

S" %

-

Choose K

80

+

thst 2-3 suppot(j ) n[2-3 suppwt(f) kj is empty d k 2 K.

Then

cl -c

&&

wE2-J .uppoP(f)

z,

5

g

we-'

/d* Id,

Qy Id(v - k)12 dy I~(Y - nlr~- C)I=

~~~port(l1

-

(because, for every C, the mts (2-'suppart(f) C PIK),~ do~not overlap)

*s"-

+ +

..

q$"

6 Simlarly,

11411'

CkI( f ,v&,k) I2

converges for all j

It ~seasy to check that the right-hand side of (6.2.9) is absolutely summable (use that only finitely many h,, ate muzero), ~o that we may jnvert the order of the summations. 4. If n, m are even, n 3 2r, m =P 28, we have

hb-p = 4,. = 6,,,l.

(substitute k = 8 + t 4')

=

hhl

Similarly, for n = 2r + 1, m

28

+ 1 both odd,

(by (6.2.7))

.

180 5.

CHAPTER 6

If n = 2r is even and m = 2s + 1 is odd, then

-&

' k

(substitute k = s

6.

-.

+ t - C)

This establishes

for all rn, n. Consequently,

By "telescoping," we have J

7. The same estimates as in points 3 and 4 of the proof of Proposition 5.3.1 show that, for fixed continuous and compactly supported f, ((f , # J , ~ l2 ) It: if J is large enough, with t: arbitrarily small (J d+ pending on f and c). Similarly, the estimate in point 3 of the proof of Proposition 5.3.2 leads to

xk

6

with IR1 5 c if J is sufficiently large. Since is continuous tit = 0, and b(0) = (2?).)-1/2,the fimt term in thb right-hand side of (6.2.11) converges to If({)12 for J-oo (by dominated convergence: ]&{)I 5 ( 2 ~ ) - ' I 2 for all <,because 15 1 by (6.1.1)). C o m b i i all this with (6.2.10), we have !(it d'j,k)12 = !If112 j,k&

for all compactly supported CODfunctions f. Since these form a dense set in LZ,the result extends to all of L2(R)by the standard density argument. s Without any extra conditions on mo, we therefore already have a tight frame with fiame wmtent 1. By Propaition 3.2.1, this frame is an orthonormal h i s

COMPA&I'LY

181

SUPPORTED WAVELETS

if and only if I($i(= 1 (use that tftkj,kU = lI+11 for all j,k E Z), or equivalently, if @(x) Q(z- k) = i5k,o for all k E 2.' This in turn is equivalent to & id(€+ 2741' = (2x1-l. Using 14(t)1=1mo(t/2+n)l16(
I&

'

S

+

with a(() = 2%C, I$(< 2rt)12. This is equivalent to

"E" rk

,

c::,

We have %(<) = h,, e-$*, with h ~#, 0 # h ~ =so, that lmo(<)12is a 2plynomia in urs C of degree N2- Nl.On the other hand, a(<)= a t e-"<, with at = ( 2 ~ ) - ' J & e"€ ($(€)/' = (2%)-'f & 4(z) &(z - I ) = 0 if L 1 N2 Nl, since support & c [NL,Nzj. Consequently, a*(<)- 1 is a poly1! nomial in cos C of degree N2 - Nl - 1. Howwer,by (6.2.13), a(()- 1 is zero '-. whenever ( n ~ ( ( ; )is( ~(Im(C) l2 and !%(C x)12 have no zera in common), so that this polpomial has s t least N2- Nl aenrs (wunting multiplicity). Since it L of degree N2- Nl - 1, it therefore has to vanish identically, i-e,, a(C)= 1, or -.-. +2zt)l2 = (ax)-'. This is another nsy to derive that (6.2.5) is necmmy ~~:m sufficient d for the $I~,~ to constitute an & w ) d basis. In the nonorthonormal example we ss(r above, with &(c) = i ( 1 e - 3 g ) , the recipe (6.2.6) for leads t o

zt

-

+

.zcl&c

*

+

~

h this case $ is indeed not nomdizzd, (I+(! = 3-'1'. If we define 4 = (J~tl-'$, then the $,,r are normaiized, and comtit\ae a tight frame with fiame mwtant 3: the "redundgncy factorn of the frame is 3. This is not so surprising once qne rrediees thbt the family (4jL)j,kEZ can be considered as the union of t h e e shifted copies of a ustretchedn Haar basis: I

3-

3

5-b

=

(4,,3*+1)(~) = 3 D I

+

2 =

03

w,

(& +,,k5) (x - 1/31 , (D3 $?) (2 - 2/31 3

#

with ,(D3f )(XI = 31/2f (32). But let us return to the condition (6.2.51,

$r it8

equivalent

(6.2.14)

182

CHAPTER 6

Several strategies have been developed, corresponding to conditions on m, to ensure that (6.2.5) or (6.2.14) hold. Most of these strategies involve proving that the truncated functions jk introduced in the proof of Lemma 6.2.1 (or some other truncated family) converge to not only pointwise, hut also in L'(R). Since it is not hard to show that, for every fixed k, the {fk(- n); n E Z)are orthonorma), this L2-convergence then automatically implies (6.2.14). Conditions on mo sufficient to easure this L2-convergenceare, e.g.,

4

inf

}m&)

1 >0

-

(Mailat (1989))

&1<*/3

. \

Neither of these conditions is necessuy' but both cover many interesting examples. Better bounds in ICl than (6.2-16) lesd ta regularity for @ and $; ws will come back to this in Chapter 7. Subtquently, newsmy and sufficientconditions on were found. We discuss these at length in the next section.

6.3. Necessary and sufiicient conditions for ortbonormality. Cohen (1990) identified a first necmary and sufictent condition on m)ensuring L2-con~ergence of the fk. Cohen's condition involves the structure of the zero-get of mo. Before starting his resilt, it is conwmbt to introduce a new &incept. DEFINITION. A compact set K is &led congncent to [-n,n] modulo 211 if

<

2. For at1 an [-a, r],them e ; . d r t s _ t ~ Z so t h a t ( + 2 t ? r ~K.

Typically, such a compact set K conwent to 1-r, u] can be viewed t& the result of some "cut-and-pastework" on 1-?r, u]- An example is given in Figure 6.1. We are now ready to state and pnwe Cohen's theorem. THEOREM 6.3.1. (Cohen (1990)) Assume that w a fritripondric pol& nomid satisfk,ng (6.1.1), with -(O) = t, and define 4 as in (6.2.2). Then the frdbwing am equivdent:

P,,Them ezish a compact set K congruent to [- r , r] madub On and wntarnmng a nesghborhood o/O so that inf inf

k>O eEK

{mo(2-kt)l > 0 .

I I

1

COMPACTLY SUPPOKFED WAVELETS

nc. e l x = f

set aonqrrcrnt to 1-T,

ii

b 4 ~ '

[ - ? f , - p w ] u [ - , - i ][ ~- f , P ] u [ 4 , ~ ] u [ ~ , U' Df -]W ~

s]modulo 21; at mn be v#cad as the rvuU of cuttang [ - r / 2 , - r / 4 ] and 15n/8,3n/4)a13 of I-*, sj and movnyf tht fint pia*r to ihe nght by 2m, the second b the Ieff

F

P k

r

REMARK. The condition (6.3.2) may seem a bit technfcal, and hard to verify in practice. Remember baweYer tbat K is compact, and is therefore bounded: K c [-R, R]. By the continuity of q and ~ ( 0 =) 1, it follows that lmo(2-k()l > un~formlyfor dl KJ 5 R, if k is larger than some ko. This means that (6.3 2) reduces to requiring that the ko functions ~ ( ( / 2 ) , ~ ( ( / 4 ) , - ., m0(2-*0[) have no, zero on K ,M equivalently, that m has no zero In K / 2 , K/4, - - , 2-&0K This is akeady much mote accessible! o

4,

Proof of lneorrem 6.3.1

I

proving (1) ~ = (2). t 1. We start Aaeurne that (6.3 1) holds, a equiudatly, &I&< Then, for ail E 1-n, n],there exists E N eo that

<

L

-.

7

E

k

c- . L

4' ?!

f? kP

+ 2nl)I2 = ( 2 r ) - ' .

z,r,
Since )is continuous, the finite sum id(*+ 2mL)12 is amtirmaus as well. Therefore there exists, for every in (-n, 4, a neighborhood (C; I< - €1 < Rt) so that, for all C in t b neighborhood,

2

a Gnite sub& of the coIlection of Since (-R, ?r] ie oompact, there intervals ((;I( -
C lkc + 2rl)i2 2

(&)-I

.

(6.3.3)

ItlSQ

t

2. I t W t h a t f o r ~ e r y ~ ~ [ - ~ , ~ , h e o t i s t s t ~ - & ~ d t ~ ~ that I&{+2 4 1 2 [8r(2& -t l)]-'P = C. Define now sets St, -4) < P 5 %

~

CHAPTER 6

for e # 0,

The St, - l o

5

e

t o form

I&O)1 '= ( 2 ~ ) - ' f 2> C , and since

4

a partition of I-r, KJ. Since is continuous, So contains a neigh-

borhood of 0. Now define

Then h' is dearly compact and congruent to I-n, a] modulo 213. By construction, I&(<)( > C on K ,and K contains a neighborhood of 0.

3. Next we show that K satisfies (6.3.2).As pointed out in the remark before the proof, we only need, to check that inftEK ( m o ( 2 - k ( ) l 0 for a finite number of k, 1 5 k 5 ko. For [ E K, we have that

is bounded below away from zero. Since 141 is also bounded, the first factor in the right-hand side of (6.3.4) has therefore no zeros on the compact set K . As a finite product of continuous functions it is itself continuous, so that

Since

5 1, we therefore have, for any k, 1 5 k 5 h,

This proves that (6.3.2) is satisfied, and biihes the proof (1) =+ (2). 4. We now prove the converse, (2)

Define pk(t) =

(2~)-I/2

* (1).

m0(2-j~)]' - X K ( ~ - ~ <whew ), XK is the

indicator function of K, XK(~)= 1 if ( E K, 0 otherwise. Since K contains a neighborhood of 0, pointwise for k-+m.

pk34

/

185

COMPACTLY SUPPORIBD WAVELETS

>

2 4

>

<

5. By assumption, I ~ ( 2 ' ~ , $ ) 1 C > O for k 1 and E K. On the other hand, we a h ha&, for any t , ]nso(C) mo(O)l C'IJI; hence Imo(E)I 2 1 - FIFI. Since K is bounded we can find4 so that ~-*c'IcI < ir( E K and k 2 b. Using 1 - x 2 e-2z Tor 0 5 x _< we find therefore, for

< E K,

-

< i,

4

We can rephrase this as

This implies

We can therefore apply the dominated convergence theorem and conclude

that j ~ ~ +in4 La.

6. The congruence of K with I-'17, A] aiSdu1o 27r means that for any 2 * - ~ e ~ d ifunction c f JtEX @ f (0= & f (C) = :J 4 f (F). 1"

particular,

=III

CHAPTER 6

. Since this implies J

Ipk(<)12e-'*

=

for all k. H e w

(because prc-+J pointwise, together with the dominance (6.3.5))

which is equivalent with (6.2.5) and therefore with (6.3.1). REMARK The "truncated" functions pk are not the same as the fk introduced in the proof of Lemma 6.2.1, but the following argument shows that L2-convergence of the pk implies L2-convergence of the fk. First of dl, K contains a neighborhood of 0, K > [-a,a] for some a, 0 <' a < n. Defille uk = (2n)-'I2 mo(2-'€1 X I - a , m l ( 2 - k ~ ) .Since XI-^,^^ .< X K , the same dominated convergence argument as for the pk applies, and vk+& in L2. Consequently, /(pk- ~ ~ 1 1 ~ 1fOr3 0~--+oo. Using the c o q p a x of K to [ - T , n] moddo 27r, one shows that (IpL- ~ k l l ~=a I(fk y l j L a - Hence I I -~illLa s llfk V ~ I I L Z 1luk - )NLZ-+O for LOO. Note that if Mdat's condition (6.2.15) ii satisfied,then we can simply take K = [ - T , n]; Cohen's condition is then trivially satisfied, and the are indeed orthonormal. The following C O gives ~another example of how to apply Cohen's condition. COROLLARY 6.3.2. (Cohen (1990)) Assme that r r ts ~ a trigonometric polynomial satzafpng (6.1.1), with ~ ( 0=)1, and dejine 4 as is (6.2.2). lf Q h no terns in [-n/3, 7r/3], then & A,, am orthonormal. P m f . We need only to construct a satisfactory compact eet X. Sine rca may have zeros in r / 3 < (€1 5 x/2, K = [-.rr, .rr] is no longer a g o d choice. But we start with thie choice as an anaatz, and "cut out" the raro8. More precisely, msume,that t h e r s a , o f m o h r / 3 < 5 n/2 are(: (They are nemsady finite in number, since ~ I O is a trigonometric polynomial.) Similarly, we have [t- 5 .. 5 ti for the zera d in - x / 2 I6 < -n/3. For every t &&we to be the bbmection with [-T, r] of a smaU open interval

n:=*

-

1

-

+

&,

<

<

--•

I

sc+.

I:

around$,dem~sotbat~~~~~wrb~aitherhotherorxitb I-r/3, */a] and so that JniojJ?< f . (If ',?= ir/2, ibm I; w i l l bc of the

COMPACTLY SUPPORTED WAVELETS

A very different approach to the derivation of conditions on

189 that ensure

(6.2.5) was initiated by Lawton (1990). Let us assume that n~ is of the form

i,e., h, = 0 for n < 0 or TL > N; m5 can always be brought into this forrn by multiplication by esNl<,corresponding to a shift of tb, by N1.Define ac = 1 N , and we J & b ( x ) #(x - Q). Since support($) C 10, N],at = 0 if < N into a (2N - 1)-dimensional vector can regroup the non-trivial at, ( ~ - N + I, - -,a&,' . . -, a ~ - l ) . Because $(x) = En hn 9(2r - n ) , the at satisfy

If follows that if we define the (2N - 1) x (2N - I ) matrix A by

where implicitly h, = 0 if m < 0 or m > N ,then

is

i.e., a is an eigenvector of A with eigenvalue 1. Note that 1 always an eigenvalue of A: if we defjne 0 by B = (0, - -.,0,1,0,. .- ,0) (1 in the central position), or Pc = 6c.0, then

by (6.2.71, i-e., AP = ,l? If the . eigenvalue 1 of A is nondegenerate, then a necessarily has to be a multiple of P, he., J & +(x) #(x - L) = ~ 4for,some ~ 7 E C. This implies I&(< 2nk)J2 = (27r)-'7; since 14(2nk)( = 0 for k # 0 (see the start of 56.2) a d j(0) = (2~)-')-l/~ by definition, it follows that 7 1, m that $ & +(x) +(x - Q)= 6[,0. We have thus a very simple sufficient criterion for orthonormality of the h,". THWREM 6.3.4. (Lawton (1990)) Assume that m~ is o trigonometric polynomial of the fonn f6.3.7), satismg (6.1.1) and m ( 0 ) = 1; define 4 as sn (6.2.2). If the eigenvalzre 1 of the (2N - 1) x (2N - I) m a t e A defined by (6.3.9) u nondagenemte, then the &,,_am orthonormal.

C,

+

-

I

CHAPTER 6 I

Orthonormality of the &,n can only fail if the characteristic equation for A has a multiple zero at 1. This indicates that among all the possible choicee for the h,n = 0,. . ,N (keep N fixed), the "badn choices (leading to nonorthonormal &,n) constitute a very "thin" set. (This statement is made more precise in Lawton (1990).) For N = 3, for instance, the only non-orthonormal choice (up to an overall phase factor) is ho = h3 = 1/2, hl = h2 = 0. Lawton's condition can be recast in tenns of trigolnometric polynomials. Define, as before, Ma(€) = Im(<)la, and d e h e the following operator Po,acting on 2~-periodicb c t i o n s f .

Clearly the constant polynomial 1 is invariant under Po by (6,l.l). everything out -in terms of Fourier coefficients, we have

(PO,), =

hn

k,n

h;~;

(

=

I ?

'

.

~h-y+m) h fm .

m

This is essentially the same expression as (6.3.8)! (We have not assumed that ,f = 0 for Iml > N, however, so it is not quite the same,) It bllowe that Lawton's condition is watisfied if we know that the only trigonometric polynornids invariant under Po are the constants. A priori it is not clear whether Lawton's condition is sufficient or not: it is conceivable that A has an eigenvector different from 0 with eigenvalue 1, but that a nemheless happens to be equal to B. However, in the spring of 1990 both Cohen end Lawton proved, indepsndently, that their two &ions are equivalent (a generabationagpemin Cobear, Daubechies, aad Pbauwu (1992) ae Theorem 4.3; see aleo Lawtan (1991)), implying the d c i e n c y of Lawton's condition. , THEOW6.3.5. Assunre that% is a trigonometniEpdynomicrl& that, (6.1.1) is satisfied and m&O) = 1. If there ezisb a cmnpad set K congruent to [-x, ~ rmoddo ] 2?r, cmtuining a neigbhood of 0, such that infklr ~ I I & ~ K lmo[2-"0l> 0,then &, only trigonmet& ~LyMmralrinw&&under A 8n the camstants.

I

I

1 4

[i !1

191

COMPACTLY SUPPOKIED WAVELETS

REMARK.This is sufficient to prow equivakmce. If we denote Lawton's original condition by (L), Cohen's condition by (C), Lawton's condition repbraad in $erms of Pi by (P)and the orthonormality of the&,, by (0),then we shady know (P) (A) (0) The implication (C) + (P)suffice8 to prim equivalence of a11 four conditions.

*

*

t

*

3

1. We will prove that the existence ofa noncomtant trigonometric polynomial f invariant for Po contradicts the existence of a compact set K with all the desired properties. Supf is such a oonconstant trigonometric polynomial, invariant for PO.Define f l ( 4 ) = f (<) - mint f (C), f2(t)= -f (6) + m a q f (0. Since f is not constant, at least one of fi, fi harc f3(0) # 0. Pick j so that f,(O) # 0, and define fo = f,. Then fo is nonnegative, fo(0)# 0, fa has a t least one zero, a d fo is invariant under Po.

I

i

i-

L

i li

2. Next we explore the zero set of fa, which turns out to have a very particular structure. If = 0 for 0 # ( E [O,2x[, then

fo(c)

Here Mo, fo are both nonnegative, and Mo(U2), Mo( 1 would lead to 41 = T ~ ' ~-
+

5:

k

Ii 6:

+

+

. c,-

this cycle. 3. If this cycle of zero8 does not exhaust the set of mos difkent from 0, then we can find 0 # C1 j = l , - . . , r - 1, for which fo((;) = 0. This can again be taken as a seed for a chain of zeros, & , G , . - - , G , . . .. Every element of this new chsin is neceesarily difierent from aU the 0, shyj~ = & would imply = ?'-I& = T'-~(,, i.e., would equal anw

+ c,,

'@ ' b

@

ct

&.By~he~unsupunaats.Ibova,C-Wthd~reafyckd-

i

CHAPTER 6

-192

for j , invariant under T , and disjoint from the first cycle. We can keep on constructing such cycles until exhaustion of the finite set of zeros of fo. The zero set of jo therefore oonsists of a union of finite invariant cycles for T . \

4. Now note that if fo(() = 0, then necessarily fof< -!- r ) # 0. indeed, since T< = T ( < n ) , both and C u would belong to the same cycle of zeros if fo(<) = 0 = fo(< x ) . If this cycle has length n, then it would follow that = rn<= = T"-'T(( n ) = ( ?r, which is impossible.

+

+

<

<

+ +

+

5. Finally, we remark that if fo(<) = 0 , then ' M ~ (+ C n) = 0. Indeed, for any [ so that jo(<)= 0, rf is also a zero for j o , and it follows that

+

Since fo(C) = 0 and f o ( t w ) # 0, this inlplies Mo(t + 17) = 0; hence mo(4 .r;) = 0. Therefore, the existence of jo implies the existence of a j = l , . . . , n - 1, cyclic set J l , . . . , < , , for T , with<] = = T < , , so that rno(<, A ) = 0 for all j. Since f o ( 0 ) # 0,we have # 0.

+

<,

+

6. We now show how these zeros t j + + for m0 are iacompatible with the = existence of K. Since T&+, = e l , T&, = t I , and, in particular, r n & , we have f, = 27rx1, where the x, c [O, I[ have the following binary representations: /

xi t2

= .dld2.-.tindl- - - d n d l . - . d n . - = .da-.-&dl*--dndl...d,.--

(d, =Oor 1)

G # 0, not all the dj are zero. Let us, for this point only, define -dSince = 1 - d for d = 0 or 1. Then <, + + = 27ry1 modulo 27r, with y, given by

We have q(27tyj) = 0, j = 1, -.- ,n. Suppose a compact set K existed with all the desired properties. Then there would be an integer t?, with a binary expansion with at moet a certain preassigned number L of digits (L depends only on the ~izeof K), so that 2 x 9 = 2 n ( 2 y l + l )has the property that r ~ t ~ ( 2 n 2 ' ~#y )0 for all k 10. We have

with ej = 1or 0 for j = 1 ,

..,L. We can also rewrite thia aa

193

COMPACTLY SUPPOKf'ED WAVELETS

t-

where el = 1 or 0 for j = I , . . . , L and e, = 0 if -J > L. The 2-ky are obtained by shifting the decimal point to the left. Since nq, is 2~-periodic, only the "tail," i.e., the part of the expansion of 2-'y to the right of the decimal point, decides whether ~ q , ( 2 r 2 - ' ~ )vanishes or not. If el = d l , then y/2 would have the same decimd part as 91, hence mo(27ry/2) = 0 would follow. Since mo(2-lry/2) # 0, we therefore have e 1 = d l . Similarly, we conclude ez = d,, ea = dn-1, etc It follows that e ~ + l ,. . ,e ~ + , are a h successively equal to dk, dk- 1, - . . ,dl, d,, . ,dk+ 1 for some k E {1,2,. .- ,n). Since the dl are not dl equal to 0, whereas e ~ =+ . .~- = e t + , = 0,this is a contradictiofi:'This finishes the proof.

t

:

With Theorem 6.3.5 we end orir discuosion of necessary and sufficient conditions on mo. The following theorem summarizes the main results of 586.2 and 6.3. THEOREM 6.3.6. Suppose mo rs a tngonomehc polynomial such that Imo(()12 Im([ + x)l2 = 1 and mo(0) = 1. Define 4 , @ by

+

i-

n

r

00

kt)

= (2~1-l'~

mo(2-'0 ,

Then 4, 11, are compactly supported Lz-functions, sattsfyng,

&

where h. u determined by ma via ~ ( t=) C, h, e-'"€. M o w e r , the T,~],~(x) = 2-]Iz $(2-f x - k), j , k E Z mnstitute a fight frame for L2(lR) with frame constant 1. This tight frame as an o r t b a d bcrsis if and only ifm-,. : satisfies one of the follovmzg equivalent conditions: e

There exists a compact set K, congruent to a neighborhood of 0, so that

[-T,a] modulo

ipf inf (rn.0(2-~.91>0

k>O (EK

k*

r

27r, containing

.

Them exists no nontrivial cycle {G, .t,,) in [0,2-/rl, invariant under [ w 2 ( modvlo27r, sueh t h a t r q ) ( ~ j + r ) = O f o r a l l j =1 , - - e n . The eigenvalue 1 of the [2(N2- Nl) - I] x mapix A &fined by

(where we assume h, = 0 for n < Nl, n

7:

[?(Nz - Nl)- 11-dimemional

> N2) j9 nondegenemte.

184

CHAPTER 6

From the point of view of a u b d filtering, this theorem tells us that, provided the high-pass filter has a null at DC (mo(lr)= 0, hence mo(O')= 1 with the appropriate phase choice), we &almostalways" have a corresponding orthonorma1 wavelet basis. The cormqmndence only fails "accidentally," as is illustrated by the last two equivalent necessary and sufficient conditions. In practice, one likes to work with filter pairs in which the low-pasa f3ter has no zeros in the band )(I 5 x / 2 , which is sufficient to ensure that the $&, are an orthonormal basis. But it is time to look at some examples! 6.4.

Examples of compactly supported wavelets generating an orthonormal basis.

All the examples we give in this section are obtained by apectrai factorization of (6.1.11), with different choices of N and R. Except for the Haar basis, we have no closed-form formula for t$(x), $(z);we- -will h-the next section how - - explain --the-plotg f o m . A f&t fmilly of examples, coastructed in Daubechies (1988b),corresponds to R -z 0 in (6.1 11). En the spectral factorization needed to extract C(<) from L(<) = pN (sin2 (/2), we retain systematically the zeros within the unit circle. For each N, the corresponding N- has 2N non-vanishing coefficients; we can Q that choose the phase of N ~ so

--/-----.

L

, N = 2 through 10. For fester implementation, Table 6.1 lists the ~ h for it makes sense to keep the factorization (6.1.10) explicit: the filter C is much shorter than m~(N t a p instead of 2N),and the filters are very easy to implement. Table 6.2 lists the coefficients of C(<), for N = 2 to 10. Figure 6.3 shows the plots of the correspondmg ~ 4 N+ , for N = 2,3,3,7, and 9. Both p,t#~ and Nt,$ have support width 2N - 1; their regulerity clearly increases with 6CpN, N. In fact, one can prove [see Chapter 7) that for w e N &, with C( e .2. Retaining systematicdIy the zeros within the unit circle in the spectrd facamounte to b i n g the 'minimum phase filter" nq,among torization all the possibilities once lmo12is fixed. Thiscorresponde to a very marked asymmetry in t$ and 9, as illustrated by Figure 6.3. Other choices may lead to i e s ~asymmetric 4,,)t although, as we will see in detail in Chapter 8, complete symmetry for 4, Q can not be achieved (except by the H w basis) within the hamework of compactly supported orthonormal wavelet bases. Table 6.3 lists the h,, for the "least asymmetricn 4, for N = 3 through 9, corresponding to $he same I;no12 as in Table 6.1, with a different "square root" m. We will come back in Chapter 8 on how this "lea& asymmetric square root" is determined. Figure 6.4 shows the corresponding Q and 9 functions. Figure 6.5 shows plots of I q t as functions of &, for the above examples, for N = 2, 6, and 10. These plota show that the subband filters for these

+,

r I

k

COMPACTLY SUPPORTED WAVELETS

2 %jikr ~ m&cunb ~ h (low-poaa n liUer) for the ~ p r ~ c ~ f l wavcletr unth e z b m d +sc and haghd number of tmnwhsng mommtr compuhbk! waih thar w p m wuith. The me & ~ c d 80 w ~ h= n

En

#

CHAPTER 6

TABLE 6.2

The coeficients

In

of fi C ( € )=

EnC,e-*4,

for N = 2 to 10. Normdirotwn:

EnCn = fi.

C O M P A ~ SUPPOFtTED Y WAVELETS

FIG.6.3. PYdr of the radag funduma ~4 tnul wavektu ~9 for the compcUt( wpported mmdttr tu& mccdmum number of wnhhing momenta for heir mpport width, and with the ezmmd phcrse choke, for N = 2,3,5,7, a d 9.

The low-pao filter cocficients for the " k t qpnmhicDmmpoetly nrppodd wovekb with N = 4 to 10. Luted hem a the CN,, = cN,n = 2. f i ~ . n ;a

maxtrnum number of vanishing momcntr, for

x,,

COMPACTLY SUPPOKTED WAVELETS

I

$ 8 -

g

FIO. 6.4. Plob of the rcdtng fincPian 4 and the wcrtrckt twmplly ruppwltd wa& and 10.

+

jar the "luwt -me&" with mazhwn number e j vanuhing momenh, for N = 4,6,8,

+(o)

COMPACTLY S U p p O m D WAVELETS

20 1

orthonormal bases are indeed very fiat at 0 and r , but very uround" in the transition region, near %/2. The filters can be made "steeper" in this transition region by a judicious choice of R in (6.1.1). Figure 6.6 shows the plot of lmol corresponding to N = 2 and R of degree 3 chosen such that I T T Q ( [ )has ~ ~ a zero * at t = 7n/9 (= 140°). This is much closer to a "realistic" subband coding filter. The corresponding "least asymmetricn function q5 is shown in Figure 6.7; it is less smooth than 4 4 (which has the same support width, but corresponds to N = 4 and R 3 O), but turna out to be smoother than 2r$ (for which has a zero of the same multiplicity, i.e., 2, at = T ) . In Chapter 7 we will come back 2 in greater detail to these regularity and flatness issues. The h, corresponding to .= Figure 6.7 are listed in Table 6.4., '

z ,z

$

f

3 S L

Rc. 6.7. The " h t aqmmetric" ading jiuaction 4 and wauekt w plotted in pioUn 6.6. ?g

+ cormqwd'ns to 1-1

All these examples correspond to real h,,, 4 and $, i-e., to I)( and 161symmet

,

k. ric arouad t = 0. It is also posaible t o construct (complex) examples with 161, ~ $ 1 @; p? t, fi

k

'

<

bncentrated much more on > 0 than on 4 < 0. Ttrlce for instance the m~ of the psevi~uaexaxn~~e, ahichsatisfea%(fq) = l,anddefioet~$([) = This obviously aatis6eS(6.l.l), since m~ d m , snd = 1. We can there fore construct tj# (4)= 4 ( 2 - j ( ) , @(() = c-*I2 & ( ~ 2 T ) @((/2);

4

nE1

+

m([-F).

202

CHAPTER 6

The coefictents for the low-pass filter r m s p o n d i n g to the sding.junctron in Figure 6.7.

these are compactly sap~~orterl L'-futlctions, and the @ f k , j, k E Z constitute a tight frarrie for iJ2(W), by Proposition 6.2.3. Moreover, since the only zeros of mu on 1-s,?rj are in = &%, * K , it follows that rn,#(<) = 0 only for = or - $. Coss 0 for 5 3 , and the ?+b:k rolistib~~tc an orttlonor~nalwavcllc~t.basis, by Corollary 6.3.2. Figure 6.8 plots [,no#(<)I, @(()I and l$#(()~;it is clear that J,-" 4 16#(<)l2is rnuch'larger than J!_ @ I$#( < ) I ' Note that the iicgativr frcyaency part of is much c l o s ~ rto thc origir~than thc ptmitive frcqi~encypart, as required by the liccessary condition @ /(I' I ) ' = 4 IS#(t)l2(n93.4). The existence of such ~ i in Cohen (1990); ill fact, for any c > 0 one "asymmetric." was first p o i n t ~ out can h i d an ortlionornial wavrlet basis such that 4 < f.

<

< &.

>

4#

s r

4

,:s

'

1G(()lZ

6.5.

The cascade algorithm: The link with subdivision or refinement schemes.

It can already be suspected frotn tlie figrlrcs in 36.4 that there is no closedforrrt analytic formula for the compactly supported (b(x),$(x) constructed here (except for the Haar case). Neverthclcss, we can, if 4 is continuous, compute $(x) with arbitrarily high precision for any given x; we also haye a fast algorithm to compute tlie plot of 4.' Let us see how this works. First of all, since 4 has compact support, and 4 E L1(R)with & #(x) = 1, we have PROPOSITION 6.5.1. I f f is a wntinu0u.g function on R, then, for all z E W,

5

I f f is uniformly continuow, then this pointwise convergence is uniform as well. If 'f is Holder continuous with exponent a ,

COMPACTLY SUPPORTED WAVELETS

T

-

'

I

'

'

I

FIG 6 8 i'lob of jmJ,(91, and 1$1 for an orthonormal wavelet bhvu whrw 1/, trnted m o w on posttrve than on negatloe frequencies

i

e

t h e n the convergence zs eqonentzally fast tn J

19

corlcrn

'

Proof. All the assertions follow from the fact that 2 3 4 4 2 5 ) is an "approx~mate bfunctlon" as 3 tends to m More precisely,

CHAPTER 6

(where we suppcwe support

9 C 1-8R]) .

If f is continuous, then this can be made arbitrarily small by choahg j 'sufficiently iarge. If f is uniformly continuous, bhen the choice of j can be made independently of x, and the convergence io uniform. If f is Holder continuous, then (6.5.2) foliaws immediately as well.

.

h u m e now that 4 itself is continuous, or even Hgder conthfbous with exponent a. (We will see many techniques to compute the Hiilder erlponent of 4 in the next chapter.) Take x to be any dyadic rational, z = 2-J K. Then Proposition 6.5.1 tells us that

1

Moreover, for j larger.+ than some jo,

l+($-rK) - 2312 (4,

#-j,2,-~~)(

C2-ja

I

(6.5.3)

where C,jo ard dependent on J or K. If 2 J - J is ~ integer, which is-automatically true if j 2 J, then the inner products (4, ) - j , v - ~ K ) are easy to compute. Under the assumption that the. &,, are orthonormal (which can be checked with any of the necessary and sufficient conditions on m,-, listed in Theorem 6.3.5), 4 is , the unique function f characterized by

(f (

h,n)

f

=

bo,n

j = 0

1

for j > O , k E 2 .

I

(6.5.4) (6.5.5)

f

We can we this ae input for the recaostruction algorithm of the subband 6ltering d a t e d with m~ (see 55.6). More specifically, we start with a low pess sequence cfl = &, and a highpass sequence dO, = 0, and we "crank the machine" to obtain

We then use d;;' = 0, to obtain, after another cranking, *

n I

etc. At every atage, the czf are equd to (4, 4-j,n). w e t h e r with (6.5.3), this means that we. have an algorithm with exponentially fast convergence to

I

compute the values of t$ at dyadic ratioads. W e can interpolate these k d ues and thus obtain a sequence of f p n c t i 5 approximating 4.9 We can, for instance, define $(z) to be the function, pi& constant on the intends 12-i(n- 1/21, 2-j(n 1/2)[, n E 2, such thbt #(2-jk) = 2jI2 (4, 4- j , k ) . Another possible choice is $ (z),p i e a r k linear ~n the [2-jn, 2-3 (n 1)]t a E it, so that 9;(2-Jk) = 2i/2{41 qLlSk). For both choices we have the following propodtion. PROPOSITION 6.5.2. If 4 & H oemtinwnu with ezponent a, then thew &hi:> 0 and jo E N so that?for j l y o ,

+

+

PmJ Take any z E R. For any j , choose n so that 2-in 5 z < 2-J(n I). By the definition of t$, $(z) is mmmarily a convex linear combination of 2112(4,9- ,,=) and 2iI2(+, 4- j,n+l), whether e = 0 or 1. On the other hand, if j is larger than some jo,

+

the same is true if we replace n by n + 1. It fdlm that e similar estimate holds for any convex combination, or Iq5(x) - $(%)I 5 C 2-la- Here C can be chosen independently of z,so that (6.5.8) fo110ws. This then is our fast aigorithm to compute approximate values of #(x) with arbitrariFy high precision:

PI Start with the sequence . - 0 - - .010- .- 0 - ., representing the r$(n), n G Z.

5F

F

b

2. Compute the qj (2-In),n E 2, by the machinen as in (6.5.7). At every step of this cascade,twice a s - n ~ values ~ ~ y are computed: values at "even pointsn 2-j(2k) are refined from the precious step,

-

and values at the "odd points" 2-j(2k

$(2-j(2k

+ 1))

+1) are domputed for the first*time,

h(k-wl

2

qJ-1

.

C

b

.

.

(6.5.10) : *

Both (6.5.9) and (6.5.10) can be viewed as convolutions.

3. Interpolate the G(2-jn) (piemwise constant if e = 0, piecewise linear if c = 1) to obtain r$(x) for non-dyadic z.

The whole algorithm was called the cascade algorithm in Daubechies and Lag&ias (1991), where 6 = 1 was chosen; in Daubechies (1988b), the choice c = 0 was made.1° All the plots of 4, $J in s6.4 and in later chapters are, in fact, plots of q j , with j = 7 or 8; at the resolution of these figures, the difference between r#~ and these T$ is imperceptible. A particularly attractive feature of the cascade algorithm is that it allows one to "zoom in" on particular features of 4. Suppose we already have c~mputedall the ~ g ( 2 - ~ n but ) , we would like to look at a blowup, with much better resolution, of 4 in the interval centered around 1. We could do this by computing all the ~ 7 ; ( 2 - ~for n ) very large J, and then plotting q;(x) only on the small interval of interest, corresponding to 25-4 . 15 5 n 5 2J-4 . 17 But we do not need to: by the "localn nature of (6.5.9), (6.5.10) much fewer computations suffice. Suppose h, = 0 for n < 0, n > 3. The computation of 7);(2TJn) only involves those rl;-, (~-~+'k)for which (n - 3)/2 5 k 5 n/2. Computation of these, jn tun, involves only the 77;-2(2-J+21) with (k - 3)/2 5 4' 5 k/2, or n/4 - 3/2 - 3/4 4' n/4. Working back to J = J - 4, we see that to compute 7$, on we only need the q ; ( Y 5 r n ) for 28 5 m < 34. We can therefore start the cascade from . . O . . .010 . . . 0 . . ., go five steps, select the seven values ~&(2-'rn), 28 5 m 5 34, use only these as the input for a new cascade, with four steps, and end up with a graph of q; on For larger blowups on even smaller intervals, we simply repeat the process; the blowup graphs in Chapter 7 have all been computed in this way. The arguments leading to the cascade algorithm have implicitly used the orthonormality of the @,,k, or equivalently (see $6.2, 6.3), of the &,,: we have characterized 4 as the unique function f satisfying (6.5.4), (6.5.5). The cascade algorithm can also be viewed differently, without emphasizing orthonormality at all, as a special case of a stationary subdivision or refinement scheme. Refinement schemes are used in computer graphics to design smooth curves or surfaces going through or passing near a discrete, often rather sparse, set of points. An excellent review is Cavaretba, Dahmen, and Micche!li (1991).'We will restrict ourselves, in this short discussion, to ohe-dimensional subdivision schemes.12 Suppose that we want a curve y = f (x)taking on the preassigned values f (n) = j,,. One possibility is simply to construct the piecewise linear graph through the points (n, f,); this graph has the peculiarity that, for all n,

12, 5 )

-

[s,gj< <

[E,31.

"

.

which gives a quick way to compute f a t half-integer points. The values of f at quarter-integer points can be computed similarly,

and so on for 2/4 + Z/8, etc. This provides a fast recursive algorithm for the computation of f at all dyadic rationals. If we choose to have a smoother spline interpolation than by piecewise linear splines (quadratic, cubic or even higher

,

207

COMPACTLY SUPPORTED WAVELETS

.,a

s#$

order splines), then the formulas analogous to (6.5.9), (6.5. lo), computing the f(2-Jn + 2-I-') from the f (2-Jk), would contain an infinite number of terms. It is possible t o opt for smoother than linear sprine approximation, with interpolation formulas of the type f(2-In

+ 2-I-')

= xakj(2-'(n

- k))

.

(6.5.13)

k

with only finitely many a k are nonzero; the resulting curves are no longer splines. An example is f ( 2 - J n +2-]-')

=

1 16 9 [f (2-In) 16

1)) + f(2-](n

+ 2))

-- [f(2-'(n

-

+-

+ f (2-3(n + 1))) .

(6.5.14)

This example was studied in detaiI in Dubuc (1986), Dyn, Gregory, and Levin (1987), and generalized in, for example, Deslauricrs and Dubuc (1989) and Dyn and L p i n (1989); it leads t o an almost C2-function f . {For details on methods to deterrrii~lethe regularity of j , .see Chapter 7.) Formula (6.5.14)describes an intcrpolat~onrefinement scheme, in which, a t every stage of the computation, the values computed earlier remain untouched, and only values a t intermediate points need to be computed. One can also consider schemes where a t every stage the values computed a t the previous stage are further "refined," corresponding t o a more general refinement scheme of t h e type fI+l(2-'-ln)

=

C

w n - 2 ~f1(2-'k)

.

(6.5.15)

k

Formula (6.5.15) corresponds in fact t o twoconvolution schemes (with two masks, in the terminology of the refinement literature), fJ+l(2-Jn) =

~ 2 ( n - k ) fJ(2-'k)

(6.5.16)

k

(the refinement of already computed values), and fJ+l(2-'n

+ 2-'-I)

~ 2 ( ~, I + I jj(2-Ik)

=1

(6.5.17)

k

(computation of values a t new intermediate points). In a sensible refinement scheme, the f, converge, as j tends t o oo, to a continuous (or smoother; see Chapter 7) function j,. Note that (6.5.15) defines the f, only on the discrete set 2-32. A precise statement of t h e "convergence" of j, to the continuous function f, is that -sm

m-+w

{

.-_ -- -

sup j20,k~Z

1ji12-m2-j*)

-

+

(6.5.18)

CHAPTER 6

208

where the superscript X indicates the initial data, f t ( n ) = A,. The refinement scheme is said to converge if (6.5.18) h& for all A E tm(Z); see Cavaretta, Dahmen, and Micchelli (1991). (It is also possible to rephrase (6.5.18) by first introducing continuous functions fj, interpolating the fj(2-jk); see below.) A general refinement scheme is an interpoiation scheme if mk = 6k,o, leading to fj+c(2-Jn) = fj(2-in). In both cases, general refinement scheme or more restrictive interpolation scheme, it is easy to see that the linearity of the procedure implies that the limit function f, (which we suppose c o n t i n u ~ u s is ~ ~given ) by

where F = F, is the "fundamental solution," obtained by the same refinement scheme from the initial data Fo(n) = bnp. This fundamental solution obeys a particular functional equation. To derive this equation, we first introduce functions fj(x) interpolating the discrete fj(2-ik): w

f3 (3)

=

(2-jk) u(2jx - k)

,

(6.5.20)

k

where w is a "reasonable" l4 function so that w(n) = bnS0. Two obvious choices w ( x ) = 1 for 5 x < 0 otherwise, or w(x) = 1 - 1x1 for 1x1 < 1, 0 otherwise. (These correspond to the two choicea in the exposition of the cascade algorithm above.) The convergence requirement (6.5.18) can then be rewritten as (1 - f&Il~-+0 for ,400. For the hndamental solution F,, we start from Fo(z)= w ( x ) . The next two appraxbating functions Fl,F2sat*

-;

are

1,

fix

Fl (x) =

FI( 4 2 ) w(2x - n)

x

(by (6.5.20)

n

=

urn u(2x-n)

(use (6.5.15) and Fo(n) = 6n,o)

n

This suggests that a aimilar formula should hold for all Fj, i.e.,

COMPACTLY SUPPORTED WAVELETS

Induction shows that this is indeed the case:

=

w,,-2k

tut

Fj-1(2-j+1~ - I ) w(2j+'z - n)

n,k,t

(by the induction hypothesis)

;>

Since F = F, = lim,,,

F, , (6.5.22) impties that the fundamental solution F

satisties

F(x) =

wk

F(2z - k)

.

(6.5.23)

k

It is now clear how our compactly supported settling functions 4 and the cascade algorithm fit into refinement schemes: on the one hand, 4 satisfies an equation of the type (6.5.23) (basically as a consequence of the multiresolution requirement Vo c V- and on the other hand the cascade algorithm corresponds exactly to (6.5.15), (6.5.20). Orthonormality in the underlying multiresolution framework made our life a little easier in the proof of Propxition 6.5.2, but similar results can be proved for refinement schemes, without orthonormality of the F(x - n). Some basic results for refinement schemes are: t

I

L

1b

If the refinement scheme (6.5.15) converges, then Enw2, = En Wzn+r = 1, and the associated functional equation (6.5.23) admits a unique continums solution of compact support (up to normalization). If (6.5.23) admits a continuous compactly supported solution F, and if the F ( z - n) are independent (i-e., the mapping P ( Z ) 3 A H &F(x - n) is one-to-one1'), then the subdivision algorithm converges.

C,

Micchelli (1991) and papers cited there. Note that the condition Enw2,, = EnW2n+l = 1 corresponds exactly to tbe requirements m(0)= 1,~ ( z=)0. 5 In a sense, constructions of compactly supported scaling functions and & wavelets can therefore be viewed an epecial cases of rehement schemes. I feel

CHAPTER 6

210

that there is a difference in emphasis, however. A general refinement scheme is associated with a scale of multiresolution spaces V, , generated by the F(2-3x - n), but typicdly no attention is paid a t all t o the complementary subspaces of the V, in c-1. Refining a sequence of data points in 1 steps corresponds t o finding a function in W-, of which the projection onto Vo, as gven by the adjoint of the refinement scheme (usually not an orthonormal projection), corresponds to the data sequence. There are many such functions in V-,, corresponding to the same data sequence, but the refinement scheme p i c b out the "minimal" ope. There is no interest in the study of the other non-minimal solutions in V - j , and how they differ from the unique refinement solution. This is natural: refine ment schemes are meant to build more "complicated" structures from simple ones (they go from Vo t o V-,). In contrast, wavelet analysis wants t o decompose arbztrary elements of V-, into building blocks in Vo and its complement. Here it is absolutely necessary to stress the importance of all the complement spaces Wt = Vt-l a Ve, and to have fast algorithms to compute the coefficients in those spaces a s well. This is where the wavelets enter, for which there is typically no analog in general refinement schemes. There is another, amusing link between orthononnal wavelet bases with compact support and refinement schemes: the mask associated with an orthonorma1 wavelet basis is always the "square root" of the mask of some interpolation scheme. More explicitly, define Ilfo(() = lmo(()12= En&, e-'*, i.e., hk+,. Then the urn are the mask coefficients for an interpolation urn = refinement scheme, since 2 ~ =2 ~ hk+2n= 6n,o(see (5.1.39)). In particular, as noticed by Shensa [1991), the interpolation refinement schemes obtained from the choice R r 0 in (6.1.11) are the so-called Lagrange interpolation schenes studied in detail by Deslauriers and Dubuc (1989),16of which (6.5.14) is an example. Note that it is impossible (except for the Raar case) for a finite orthonormal. wavelet filter mg t o be itself an interpolation filter as well: orthonormality implies jmo(t)I2 Imo(< r)I2= 1, while the interpolation requirement is equivalent t o hZn = -&6n,0, or mo(t) mo(C r ) = 1. If both requirements are met, then

z,

+

zk6

+

+

+

Assume that h, = 0 for n < Nl,n > Nz,and hN, # 0 # hN2. Then (6.5.24) already implies that either Nl = 0 or N2 = 0. Suppose Nl = 0 (N2= 0 is analogous); N2 is necessarily odd, N2= 2L 1. Take k = 2L in (6.5.24). Then

+

Since ho = 2-'I2, and h2L+l'# 0, this implies hl = 0. Similarly k = 2L - 2

COMPACTLY SUPPORTED WAVELETS

leads to

- which, together with hl = 0, hz, = 2-'/2~,,o implles hs = 0. It follows eventually that only ho and h2L+1are nonzero; they are both equal to 1/aso , that then forces the mask is a "staetched" Haar mask Orthonormality of the L = 0, or m,-,(O = ' i ( 1 + e - ' t ) , i.e., the Haar basis. If we lift the restriction that mo is a trigonometric polynomial, i.e., if 4, $ can be supported on the whole real line, then mo(<) + mo(< ?r) = 1 and Imo(<)12 I%(< ?r)I2 = 1 can be satisfied simultaneously by non-trivial mo; examples can be found in Evangelista (1992) or LemariCMalgouyres (1992).

+

+

+

Notes. 1. A compactly supported 4 E L2(R) is automatically in L1(w) It then follows from Remark 5 at the end of 35.3 that (0) = 1, mo(?r)= 0, i e , has a zero of multiplicity a t least I in ?r. that 2. In Daubechies (1988b), the solutions P of (6.1.7) were found via two corn-

binatorlal lemmas. The present more natural approach. using Bezout's theorem, was pointed out to me by Y. Meyer.

3. This formula for PN was already obtained in Hennann (1971), where maximally flat FIR filters were designed (without any perfect reconstruction schemes, however). 4. This convergence also holds if infinitely many h, are nonzero, but if they (hnl(l In\)' < oo for some c > 0. In that decay sufficiently fast so that case Isin nCI 5 ]n<~""('-") leads t o a similar bound.

+

5. We use here the classical formula sin x -= x

m

J-

m(2-Jz)

.

3=1

An easy proof uses sin 2a = 2 cos a sin cr to write 3

I-J cas(2-'x) 3=1

[

n J

=

j=l

sh(2-J+'s) 2 sin(2-I x)

-

sin x 2 ain(2- Jx) '

fer J+m. In Kac (1959) this formula is credited to which tends to Vieta, and used as a starting point for a delightful treatise on statistical independence.

1 h

kr

6.

This is true in general: if mo satisfies (6.1.1) and 4, as defined by (6.2.2),

generates a non-orthonormal family of translates h,,,then necessarily Cr 27rt)l2 = Q for m e 6. (See Cohen (1990b).)

I&( +

'

7.

The condition $ dx +(x) @(z - k ) = bi,o may seem stronger thsn 11311= 1, but eince the $jVr constitute a tight frame with frame constrcnt 1, the two are equimlent, by Propition 3.2.1.

8. Since +(XI ia a Bnite linear combination of translates of @x), fast alge rithrns td plot 4 lead to fast plots for 3. Throughout this section, we t restrict our attention to 9 only. 9. If 4 is not continuous, then the qj still converge to 4 iu Li (see $6.3). Moremet, they converge to & pointwise in every point wherei# is'contibuous.

101 The choice E = 1 was ueed in the proof of Propitdon 3.3 in Daubechiea (1988b), because the i)jl are abeolutely integrable, wherearr the qo are not. In Daubechia, (1988b) the convergence of the $ to 9 was actuahYproved first (using some extra technical conditions), and orthonormality of the h , n WEM then deduced from this convergence. 11. Note that there exiet many. other procedures for plotting graphs of wavelets. Instead of a mfinement cacpcade one can sleo stcrrt from appropriate #(n) and then compute the $(2-jk) directly from Nz) = h n 4 ( 2 ~- n). (In k t , when 9 is not continuoua, the M e 4gorithm may diverge, while this direct use of the 2 - d e equation with appropriate #(n)stilt converges. I would l i i to thank W. Sweldens for pointing this out to me.) This more direct aomputation can be done in a tree-like procedure; a different way of looking at this, avoiding the tree coe&ruction and leading to faster plots, uaes a dynamicel systems hamework, as developed by Bergei. and Wang (aee Berger (1992) for a review). The "zoom in* feature is loet, hawever.

fix,

12. Many experts on refinement or subdivieion schemes find the multidimenaional case much more interesting! 13. Thia is not a presentation with fullest generality! We merely suppose that the wk are such that there exists a continuoua limit. This already impliea Cwl,=Cwm+t=1.

14. For example, any compactly supported w with bounded variation wouId be "reasonablen here. 16. The following stretched Haar h c t i o n ahow8 how the F ( z - n) can fsil to be independent. W wo t w = 1,811other w, = 0. The solution to (6.5.23) is then (up to n o m d h f h ) F ( z ) = 1 for 0 5 z < 2 , 0 oth*. In this case the P-sequence A defined by A,, = (-1)" leads to EnAn F ( z -n) = 0 8.e. 'I

16. This hr no coinddenoe. If we Bx the length of the symmetric ater Mo = Iwf2,then the choice R r 0 means that Mo ia divisible by (1 coee) with the higheut poesible multiplicity compatible with its length arid the constraint Mo(e) Mo([ + r )= 1. On the other hand, Lagrange r e h e m a t

+

+

213

COMPACTLY SUPPORTED WAVELETS

schemes of order 2N - 1 are the interpolation schemea with the shortest length that reproduce all polynomials of order 2N - 1 (or less) exactly from'their integer samples. In terms of the filter W (c) = C, W, ei*, this mema

4

W ( 5 )+ W((-t- T ) = 1

(interpolation filter: w, = 6n,o)

and W(C) = 1 +

= 1

+ O((1 - ~

0 8 ~ ) ~ )

(see Cavaretta, Dahmen, and Mitxhelli

(1991), or Chapter 8).

+

The two requirements together mean that W(t n) has a zero of order 2N in E = 0, i.e., that W ( t + n) is divisible by (1 - coe E)*; hence W([) by (1 C W ~ It ) ~follows . that W = Mo.

+

-

CHAPTER

7

More About the Regularity of Compactly Supported Wavelets

The regular~tyof the Meyer or the Battle-LemariB wavelets is easy to assess: the Meyer wavelet has compact Fourier transform, so that it is Cw,and the BattleLemarik wavelets are spline functiorls, more precisely, piecewise polynomial of degree k, with (k - 1) continuous derivatives at the knots. The regularity of compactly supported orthonormal wavelets is harder to determine. Typically, they have a non-integer Holder exponent; moreover, they are more regular in some points than in others, as is already illustrated by Figure 6.3. This chapter presents a collection of tools that have been developed over the past few years to study the regularity of these wavelets. All of these techniques rely on the fact

.

C

(7.0.1) r(x) = cn o ~ -xn ) where only finitely many c,are nonzero; the wavelet $, BS a finite linear combination of translates of 4(2x), then inherits the same regularity properties. It follows that the techniques exposed in this chapter are not restricted t o wavelets alone; they apply as well to the Lasic tunctions in subdivision schemes (see $6.5). Some of the tools diussed here were in fact first dweloped for subdivision schemes, and not for wavelets. The different techniques fall into two groups: those that prove decay for the Fourier transform and those that work directly with 4 itself. We will illustrate each method by applying it to the family of examples ~ q constructed 5 in $6.4. It turns out that Fourier-based methods are better suited for asymptotic estimates (rate of regularity increase as N is increased in the examples, for instance); the second method gives more accurate local estimates, but is often harder to use. References for the results in this chapter are Daubechies (1988b) and Cohen (1990b) for 57.1.1; Cohen (1990b) and Cohen and Come (1992) for 57.1.2; Cohen and Daubechies (1991) for 87.1.3; hubechies and Lagarias (1991, 1992), Micchelli and Prautzsch (1989), Dyn and Levin (1989), and Rioul (1991) for $7.2; Daubechies (1990b) for 57.3.

6,

7.1. Fourier-based methods. he Fourier transform of equation (7.0.1) is

= nzo(tl2) 215

,

216

CHAPTER 7

3 x,

where m(<)= ~ e - ' * is a trigonometric polynomial. As we have seen many times before, (7.1.1) leads to I

where we have assumed ~ ( 0 = ) 1 and can be factorized as

J & +(x) = 1, as u ~ u d .Moreover,

1

where C is a trigonornet& polynomial as well; this leads to

I

A first method is based on a straightforward estimate of the growth of the infinite of the C(2-jt) as I(l-)oo.

I

I

7.11 Brute force methods. For 9 = n + P , n E N, 0 5 /3 < 1, we define Cato be the set of functions f which are n times continuously differentide and such that the nth derivative f is Holder continuous with exponent 0, i-e., ~f(") ( 2 )

-I(")( z i - t ) l 5 c ) t l P for a11 z,t .

It is well known and easy to check that if

I

then f € Ca.In partkuh, if 1/(015 C(1+ Itl)-l-Q-t ,then it follows that / E Ca.It follows that, if the growth for K]+oo of C(2-it) in (7.1.4) can

I

G,

be kept in check, then the factor ((1 - e-e)/y)Nensures 8mmthnss for LEMMA7.1.1. i / q = supt )t(c)(< 2N-a-11 then 4 E C".

Proof. 1. Since ~ ( 0=) 1, L(0)= f ;hence It(t)1

2. Now take any 6, with Hence m

L 1 + CKt. a

C$

~ u e n t l ~ ,

Kt 2 1. There exists J 3 1so that 2J-1 5 (€1 < 2J. J

w

.

MORE ABOUT COMPACTLY SUPPOKX'ED WAVELETS

Consequently, I ~ ( E )5) C"' (1 t

+ )[l)-n-l-t,and 4 E Ca.

217

Grouping together several L: leads to better ~stimates,as follows. LEMMA7.1.2. Define

i

r'

-

~

K

inf ECJ .

=

JGN

zjK < N - 1 - a , then&€ CP

Then K =lim,,,K,;

1. Take jz

> jl. Then j2 = njl + r 932

with 0

r

< r < jl, and

(93,)"

9;: . I

Consequently,

KJ, 5

nlogq,, j2

+ rlog91 5 K,, +

log 2

-

Cjlljz .

2. For any c > 0, there exists 30 so that K = inf, Kj > KJo - c. For j 2 jo we then have K1 5 K: c C j o / j K: c. Since c was arbitrary, it'

+ +

r

follows that

K

= lim,,,

J -W

+

KJ

3. if K: < N - 1 - a, then Kt < N - 1 - a for some t' E N. We can then repeat the argument in the proof of Lemmrr 7.1 . l , applying it to

n:;:

with &(() = 4 2 - I < ) , add with 2' plying the role of 2 in Lemma 7.1.1. This W to J&c)~5 ~ ( ItI)-N+"t;l l +5 C(1+ kl)-a-l-c, hence 4 E CP. a ;

The f 6 W n g lemma. &owe that in 'most cases, we will not be able to &sin

:

much better by the btute force method. LEMMA 7.1.3. There d t s a sequence

so

UIot

218

7

CHAPTER

1. By Theorem 6.3.1, the orthonormality of the &(- - n) implies the existence of a compact set K congruent to [ - A , T ] modulo 2a, such thht [&([)I 2 C > 0 for ( E K..Since K is congruent to I-n, lif ~ n LC d is periodic with period 2'+'n, we have

i.e., there exists Cc E 9 K so that (Lt(Ct)l = qr. Since K is compact, the 2-' Ct E K are uniformly bounded. We therefore have I

/

ICrl 5 2 (2

(7.1.7)

for 0 < C'.

2. Moreover, since

1 1

= Icos el21 5 1, we have for all ( r 2'K,

Putting it all together we find for tt = 2Ct

Since K = infr Kt, this is bounded below by a strictly positive constant. Let us now turn to the particular family of N@ Lonstructed in $6.4, and see how these estimates perform. We have

with

We start by establishing a few elementary properties of PN.

i

I

1

I

11

MORE ABOUT COMPACTLY SUPPOWTED WAVELETS i

LEMMA7.1.4. The polynomial PN ( x ) = il $ the followrng properties: 7F

05 x 5 y

A

05 x

$

c

<1

2 19

~~~~

=+ x - - ~ + 'P N ( x )2 p - N + l ~ N ( y ) , P N ( x )2 2 N - 1max ( 1 , 2 ~ ) ~ .- '

(7.1.8) .

(7.1.9)

F

Proof. 1 . I f 0 5 x 5 y, then

2. Recall (see 56.1) that

PN is the solution to

i, ~ ~ ( 3 )

On substituting z = it follows that PN(1/2) = 2 N - 1 . For x < we have P N ( z ) 5 = 2N-1 because PN is increasing. For x 2 3, applying (7.1.8)leads to PN( x ) 5 z N - 1 2 N - 1 P N ( i= ) 2 N - 1 ( 2 x ) N - 1 This . proves (7.1.9). m

i,

It is now easy to apply Lemmas 7.1.1 and 7.1.2. We have

Lemma 7.1.1 allows us to conclude that the Nt$ are continuous. In view of the graphs in Figure 6.3, which show the t$ to have increasing degree of regularity as N increases, this is clearly not optimal! Using 1C, for j > 1 immediately leads to sharper results. We have, for instance,

(because sin26 = 4 sin2€12 ( 1 - sin2€12)) .

4 I

4 + 4 (implying 4p(l - ,y) 5 f), then < p(N-l)by (7.1.9). In the remabiq window $ + 9 2

If eitha y 5 1/2 or y 2 [P~(u)P~(4y(1 y))] a,? $,wehave PN(v) PN(4y(l- Y))

<

'21(N-1)[ l d ( 1 - y)lN-'

C o ~ ~p, 5 l p(N-l) y 3-3(N-1)/2,and K2 5 (N 1)[2 - f 6 1 .It £ o U m that arymptoticdly, for huge N , N # E Q N with p = f - 1 u .1887. A sIightly better value can be obtained by estimating qr rather than p?; one finds then p N -1936. Note that y = is a fixed point for the map y w 4 y ( l - y), so that qk 1 [PN(3/4)Ikfor any k, leading to a lower bound on K: and an upper bound on the regularity of 4. In terms of <,y = sig $ = f co~~esponds to 4 = $;we already saw earlier that f play 8 epecial role because ( is an invariant cycle lor r n ~ l t ~ c a t i obyn 2 modulo 2n. In the next subsection, we will see how these invariant cycles can be used to derive decay estimates for +

9, T) 4.

9

7.1.2. Decay estimates from bvariaat cycles. The values of L: at an invariant cycle give riee to lower bounds for the decay of LEMMA 7.1.5. If f b , t l , - ., C [-n, .rr] is any non-trivial invariant cycle (z-e., b # 0) for the map T( = 2( (modulo 27r), uric tm= ~t,,,-~, m = l , - . - , N- 1, 7tMe1 = t o , then, for d l k N~,

-

where R =

kq~ IL(&,)l/(iW log2), and C

4.

> 0 is independent of k .

Proof. 1.

F i note that there existe Cl > 0 m that, for all k E N, Indeed, 2kMb = &I (mod %), ao that (7.1.10) f o U m if b # 0 or f T . We already know that to# 0; if b = fr , then = 0 (mod 2 ~ ahd ) hence & = 2 M - 1 =~ 0 (mod 2n), which ie impossible.

2. Now

L

MORE ABOUT COMPACTLY SUPPO-KI'ED WAWLETS

221

Since & ia a trigonometric polynomiai and L(0) = 1, there exists Cz so that I&(<)] 2 1 - C2 I,$/2 e-2 Ca for small enough. Hence, for r large enough, w

aD

Hence

We can apply this to the example at the en&of the last subsection: Lemma 7.1.5 ihpiies )&2"%)1 2 C(l + 1 2 " 9 1 ) - ~ +with ~ l? = log JL(F?j)L(-%)l/flogZ. If L has only real coeffic; 3"s (as is the case in most applications of practical interest), then It(-&)\= ": '?)I, and = log lC(F)(/ log 2. The next short 3, 3 ,-F, 2n 4~ 2n % invariant cyclea are {T, -7)) IT , ,- y6u) , etc.; each of them givea an upper bound for the decay exponent of 8. In some cases one of these upper bounds on a can be proved to be a lower bound as well. We firat prove the following lemma. LEMMA 7.1.6. Suppose that [-n,n] = Dl uD2 . . U D M ,and that thm ezicrts

,

q>Osothat

h

i? k

IL(0 t ( X ) . L(2""<)I Iq M [€DM. men 14(t)1 5 C(l+ ICI)-N+", with K = log p/ log 2.

,

:

Prw,f.

1. Let ua &hate. I

n I:

~ ( 2 - ~ < ) 1for , mme larjp but arbitrary j. Since C E Dmfor O T m me € {I, 2, M),

i;

We can now apply t,he same trick to 2-m<, and keep doing so until we cannot go on. At that point we have

with at most 111 - 1 different L-factors remaining (i.e.,r

< M - 1). Hence

with ql defined .as in (7.1.5). Consequently, with the definition (7.1.6),

and K = lim,,, Lemma 7.1.2.

K,

<

log q / log 2. The bound on

& now

follows horn

rn

In particular, one has the followirlg lemma. LEMMA 7.1.7. Suppose that r)

+

Then I&[)] 5 C(1 I
Of course, Lemma 7.1.7 is only applicable to very special t;in most cases (7.1.11) will not be satisfied: there even exist L for which C(%) = 0. Similar optimal bounds can be derived by using otherinvariant cycles as :reakpoints for a partition of [ - A , T T ] , and applying Lemma 7.1.6. Let us return to our "standardn example N ~ .In this case Cohen and Conze (1992) proved that try([) does indeed satisfy (7.1-1I), as follows. LEMMA7.1.8. For dl N E N, N 1, PN(y) =~~~~( N - '.+n)yn ~ntLjks

>

I

We start by proving yet another property of

PN.

223

MORE ABOUT COMPACTLY SUPPORTED WAVELETS

LEMMA 7.1 9. P ~ ( x )=

N

(PN(x) - P ~ ( ~ ) X ~ .- ' ] 1-x

(7.1.14)

Proof.

1.

3. Combining (7.1 15) and (7 1.16) gives

(1 - x)P',(x) Since PN(l) =

=N

N-1 En=., (N - ln+ n )

=

(

*

I ) , (7.1.14) follows. m

~

~

i

We now tackle the proof of Lemma 7.1.8. Proof of Lemma 7.1.8.

1. Since PN(y) is increasing on [O,l], we only need to prove (7.1.13). 2. Define f (y) = PN(y) PN(4y(l

- 3)). Applying Lemma 7.1.9 leads to

CHAPTER 7

224

3. Since 4y(l - y) 5 y for y

1 3/4,

we can apply (7.1.8) to derive

Substituting this into (7.1.17) leads to

The quantity in square brackets equals *(I - g)PN(y) 1 0 for y 5 1, so that g(y) 5 o for 5 y I It follows that PN(y) PN(4g(l - y)) is decreasing on which prove6 (7.1.13) for y _< 9.

i.

(i, I],

4. For

5 y 5 1 we follow a different strategy. PN(?) by Lenrma 7.1.4, it suffices to prove

(e)N'l

Since PN(y) 5

But PN(4y(l- y)) 5 [I - 4y(l - ~ ) I - N= ( 2 -~ 1)-2N ( b m (1 PN(t)= 1 x N P N ( -~ x ) I), and

-

<

(7.1.19) where we have @ Lemma 7.1.4 again, as well ae

I

To prove (7.1.18) it is therefore sufficient to prove

S i both (2y - 1)-2 and y(2y - I ) - ~are decmsb on to verify that (7.1.20) hold for y = i.e., that

1,

Thie ia true for N 2 13.

I

[i,11, it d c e a

I

225

MORE ABOUT COMPACTLY SUPPORTED WAVELETS B

8

5. It remains to prwe (7.1.13) for 5 y 5 1end 1 hostepa: y 5 po = and y yo. For y hence, again by Lemma 7.1.3,

r

9,

3

Similarly P N ( ~5)

v,

- y)IN-'PN (4) = [16y(l - 9))N-l .

P N ( ~ P-( v)) ~ 5

33

>

< N 5 12. We do tbis in < 4y(l -y) _> 4,

( ~ ) ~ PN($), - l

-

PN(u) PN( 4 ~ ( 1 Y))

<

);(

so that N-1

(i)

PN

5 (:)"-'PN(:)

I l d ( l - v)]N - l

,

(7.1.21)

-

[i,

b e c a w g ( l - y) is decreasing on 1 . One checks by numerical computation that (7.1.21) is indeed smaller then [PN(!)I2 for 1 5 N 5 12. 6. F o r u p = PN(9) 5 ($"l


P N ( ~ P ( ~))PN(V)

5

Y;~+'PN(W)(~V - 1)-2

3

[py-

zN PN(M),

lN-I (7.1.22)

where the last inequality uses that (2y - I)-' armd y(2y - 1)-2 are both decreasing in [yo, 11. One checks by numerical computation that (7.1.22) issmaller than [PN($)]2for 5 5 N < 12.

< y 5 1. For the& 7. It remains to prove (7.1.13) for 1 5 N 5 4 and small values of N the polynomial PN(y) PN(4y(l- y)) - PN(:)' ha8 degree at most 9, and its roots can be computed easily (numerically). One checks tw there am none in j i , I], which fini&q the proof, because (7.1.13) is &bfiediny=l. rn '

It f o l h from Lemmas 7.1.8 and 7.1.7 that we know the exact r a y q t o t i c decay of N&):

IN & < ) ~ 5 c ( l + l t ~ ) - N + l -

IJ'JW'~/~)V/"""~ ,

(7.1.23)

225

I

CHAPTER 7

For the first few values of N this translates into estimates for a:

Nc$

E Ca"

with the following

We can also use Lemma 7.1.7 to estimate the smoothness of Nq5 as N - y o . Since

1 3 " ~ '5 P N ( ] )< aN-'

(uee Lemma 7.1.4 for the upper bound, (7.1.19) for the lower bound),

'

9

h

logIp~(3/4)1r -log3 21og2 21og2 N[1-

o(N-'log A!)] ,

&

12 implying that asymptotically, for large N , N) E CpN with p ,2075.' One does not, in fact, need the full force of Lemma 7.1.8 to prove this asymptotic result: it is eufficient to prove that

PN(Y)5 C 3N-1 for y 5

(7.1.24)

PN(v) P N ( ~ Y-($1) ~ 5 c 32(N-1) for 3 5 y 5 1 ,

I I

,

(7.1.25)

with C independent of N . The asymptotic result then followa immedi8tely from Lemma 7.1.6. The estimate (7.1.24) ia immediate from PN(y) 5 PN(5) 5: 3N-' for y < Q; the estimate (7.1.25) follows easily h m Lemma 7.1.4 as followe. u 5y5 then PN (y) P N ( ~ P-( y)) ~ 5 (4ylN-' (ley($ u)IN-l = [W2(1 Y)IN-l < - 32(N-1) b m w y2(l - p) is decreasing on [$,I]; if y 5 1, thm PN(P) P N ( ~ v-( ~u)) 5 ( ~ u ) ~ - ' P N ( ) )= ( 8 ~ ) N - 1 < 4 32(N-1). This much simpler argument for the exact asymptotic decay of is due to Volkmer (1991), who derived it independently of Cohen and Conze's work.

3

-

v,

-

a<

6

761.5. Littlewood-Paley type estimates. The estimatea in this eubeection we L1 or L2-estimates for (1 K I ) Qrather ~ than poinhbe decay estimatea for itself. The basic ides is the usual Littlewood-Paky technique: the Fourier transform of the function gets broken up into dyadic piecea (i.e., roughly 2jC 5 MI 5 2j+'C) and the integral of each piece' estim a M . If $g
+

,I

227

MORE ABOUT COMPACTLY SUPPORTED WAVELETS

x;,

2JaU] < oo if o < - log A/ log 2, implying 4 E Ca.To obtain -ti; mates of this nature, we exploit the special structure of 4 ae the infinite product - ofmo(2-jc); the operator Po defined in 56.3 will be a basic tool in this derivation. we will first restrict ourselves to positive trigonometric po1ynomials Mo(<). (Later on,we will take 'Mo([)= Iw(t)12 to extend our results to non-poeitive m~.)As in $6.3 we define the operator Po acting on 27r-periodic functions by C[1+

,Po,,(<,

(f) + M o ( $ + r ) f

= ~ o ( f ) f

($+%)

.

This operator was studied by Come and Flaugi and several 6f, the results inthis subsection come from their work (Conze and Raugi (1990), Corn (1991)). Similar ideas were a h developed independently by E-irola (1991)and Villemoes (1992). A first useful lemma is the Wowing. LEMMA7.1.10.For all m > 0 and 411 2n-penodic junctiow f ,

Pmf. 1. By induction. For m = 1,

3nf2

*

.

= 2/_d2 MO,C,~,,, =

J2% -2u

4f(f)Mo

(f) .

CHAPTER 7

Since M' is a positive trigonometric polynomia3, it can be written as

+

'

One then finds that the (2J 1)-dimensional vector space of trigonometric polynomials defined by

is invariant for Po. The action of Po in VJ can be represented by a (25 (23 1) matrix which we will also denote by Po,

+

' (P0)kt = 2 a 2 k - t

with the convention a, = 0 if Irl

t

' - J s k , t?< J :

+ 1) x

/

> J . For Mo of the type

where L is a trigonometric polynomial such that L ( r ) # 0, the matrix Po has very special apectral properties. LEMMA7.1.11. The values 1, f ,- - .,2-2K+1 am eigenvduea for Po. The row vectors e, = (jk)i,-J, ..,J, k = 0, . .,2K- 1 genemte a subsplzce which is left invariant for Po. More precisely,

ekPo = 2-kek

+

linear combination of the en, n < k

.

I

I

MO-

I I

ABOUT CO&fPACnY SUPPOFWED WAVELETS

229

1. The factorization (7.1.28) is equivalent to "" I

a, jk(ll)l

- .

= 0 for k = O , - - - , 2 K 1

1.

This means that the Moreover, since Mo(0)= 1, C a 2 3 = %j+l = sum of each column in the matrix (7.1.27) is equal to 1; Q is thus a left eigenvedor of Po with eigenvalue 1.

it

: it.

2. For 0 < k 5 2K \

i

- 1, define gk = ekPo,i-e., (gk).. = 2 z j k oy-,

.

3

t For m even, m = 2!,

: 1

Hence

where

A,,, = Co?,(~j)"' I

F

= ~ 0 z J + 1 ( 2+j lIrn f

A amsequence of Lemma 7.1.11 is that the epacea Ek,

bi%b 1 < k 5 2K,are all right invariant hr Po. The main result of this subsection irr then the foflowing.

230

CHAPTER 7

THEOREM 7.1.12.Let X be the eigenvalue of POIEaK value. Define F,a by

with the largest absolute

-

If IXI < 1, then F € Ca" for dl c > 0.

2. The spectral radius p(Polh,)equals IX/. Since, for any 6 > 0, there exists C > 0 so that (IAnII I C(p(A)+ 6)n for ail n E N, it follows that

3. On the other hand, f(<) 2 1 for 5 < )(I< K. Together with the boundedneas of fl& ~ ~ ( 2 - j for t) 4 T ( d e d d as us& from IMo(t)I 1+ Cltt), t h ~mplies ~

<

(use Lemma 7.1.10)

L C'(jA)+ 6)"

.

By the arguhent at the start of this subsection, this implies P E Ca-'.

rn

In fact, a slightly stronger result can be p ? d . If we extend the definition of C" (n integer) to include 'all the functions for which the (n - 1)th derivative is in the Zypund class /

3 = {f; 0

i

If

(x + 9 )

+ f(x- 9 ) - 2f (41L Clvl

for all ~ 9 8 ,)

then it is also true that F E Ca if fiIhK is diagonal (i.e., we can drop the c in this case). Moreover, both this smoothness estimate and the estimate F E Ca-' for all E > 0 in Theorem 7.1.12 sre optimal if F has no eero on [-r,ri. For a proof, see Theorem 2.7 in Cohen and Daubechiea (1991). /

231

MORE ABOUT COMPACTLY SZFPPOR'IXD WAVELETS

REMARK. The same result can be derived via an equivalent technique, which uses an operator Pk dehed in the 88me way as Po, but with Mo(e) replaced by the factor L([)in (7.1.28). In this case we define X~ = ~I(P;),and we factorize F(() [2(sin[/2)flzK (2~)-'/' H z l L(2-JF) to obtain

-

so that F E Ca-' with a = 2K @. Thh J I U $ ~has ~ the advantage that we start directly with a smaller matrix P t , so that the computation of the spectral radius is simpler. The tp.o methods ate compkte1y equivalent, as shown by the following argument. If p is an eigmvalue'd Po, with eigenfunction f, €&*,

then j, can be w-ritten as K

f p ( 0 = (sin'

f)

sp(0

-

&placing Mo(() by its factorized form in

~ f J ' (= t , M.

(f)f, (d)

+Y

(f

+I)

fp

(g +

x)

9

+I)

9

we obtain after dividing by [sin2 f me2 f l N , 82zKg,(~) =

L (f)%

(f) + i(; +*) (f g,

so that the eigenvalues of Pk are exactly given by IcL = 22Kp.

o

will not be positive. (Indeed,in the framework of orthonormal In genera), wavelet bases, mo is never positive, except for the Haar basis; see Janssen (1992).) However, we can then define Mo = Irnot2; the same techniques lead to

'\ e

where

is the spectral radius for P;, with L(() = JC(<)12.Hence

I

232

CHAPTER 7

6. Consequently 4 C"" for a N + of resulting cr, for the first few

if a + < N + For the special

I

E

NC$

$6.4, the

.XL

- 4.'

I

.

1

values of N, are:

Theae are much better than the values obtained from the pointwise decay of 4 (see 47.1.2). The size of the matrix Pk increases with N (linearly), and I do not know of any way to determine the asyrnptotics of its spectral radius as N-+oo; for asymptotic estimates, pointwise decay of is the best method.

6

II I1 1

7.2.

A direct method.

e

The smoothness results obtained at the end of $7.1.3 for N4 with small N are still not optimal. Moreover, Fourier-based methods can only give information on the global Holder exponent, whereas it is clear from Figure 6.3 that 29, for instance, is smoother in some points than in others. In fact, we will see that there exists a whole hierarchy of (fractal) sets in which 2 4 has different Holder exponents, ranging from .55 to 1. Results such as these can be obtained by direct methods, not involving For the sake of simplicity, I will explain tbe setup in t4e general case, but expose the method in detail for the example 2i$ only, and (hen state the general theorems on global a+ local regularity without proof. Proofs can be found in Daubechies and Lagarias (1991, 1992). Similar results about the global regularity were also proved (independently, a ~ in d fact before Daubechies and Lagarias) in Micchelli and Prautzsch (1989), in the framework of subdivisiop schemes. The method is completely independent of wavelet theory. The starting point is the equation

6.

c:,

ck = 2, and we are interested in the compactly supported L1-solution with F, which: if it exists, is uniquely determined3 (up to its n o ~ s t i o n ) :since F E L1, F is continuous, and (7.2.1) implies

I

*

MORE ABOUT COMPAGTLY SUPPOKI'ED WAVELETS

233

4

with m(<) = ~ f = ~ = g qe-ik6, and Lemma 6.2.2 the. tells us that support F = (0, Kj. Equation (7.2.1) can be considered as a fixed point equation. Define, for t ' functions g supported on [O, K], Tg by K

(Tg)(x) =

C

Ck

9(2x - k) -

Then F solves (7.2.1) if TF = F. We will try to find this fixed point by the usual method: find a suitable Fa: define F, = T ~ F and ~ , prove that F, has a limit. To determine Fo, note first that (7.2.1) impoees a conatra.int on the values F(n), n E Z if F is continuous. Since support F = [0, K ] , we only need to determine the F(k), 1 5 k 5 K - 1; the other F(n) are zero. Substituting x = k, 1 _< k 5 K - 1 into (7.2.1) leads to K 1 linear equations for the K - 1 unknowns F(6); the system of equations, can also be read as the requirement that the -or (F(l), . .- ,F ( K 1)) be an eigmvector with eigenvalue 1 of a (K - 1) x (K - 1)-dimensional matrix derived from the c k . It turns out that, ,modulo some technical conditions (see below), this matrix does indeed have 1 as a nondegenerate eigenvalue, so that (F(l), - ..,F ( K - 1)) Can be fixed up to an overall multiplicative constant. Let us supthis is done. One can prove that F(k) # 0, k we can fix the mmnahation so that F(k) = 1. (All this will be illustrated by an exam& below.) Define now Fo(x) to be the piecewise linear function which takes exactly the values F(k) at the integers, i.e.,

-

-

xf=il

EL<'

Fo(z) = F(k)(k + 1 - x)

+ F(k + 1){x - k)

for k 5 x i k

+1 .

(7.2.2)

Successive applications of T define the Fj = TiPo, i.e.,

it eaeily follows that the Fj are piecewise linear with nodes at the 2-in E (0, K], n € M. To discuss whether the F, have a limit as j-+oo, and to study the regrilarity of this h i t , it is convenient to recad (7.2.3) in mother form. The key idea is to study the F,(x), Fj (z+l) Fj(z K - 1)simultaneously, for z E [O,lj. We define vj(z) E R~~ by

.

+

Fm 0 5 z 5 4, (7.2.3), together with support 4 c [O, K j , implies that F ~ + ~ ( x ) , Fj+l(z + 1),- ..,Fj+l(x + K - 1) are all linear combinations of F, (2x), Fj(2z + 1),.-.,4(2x + K - 1). More prec%ely,in i nof vj(x), q + l ( z ) = To vj(2x) for 0 5 z_<

,

(7.2.5)

where Tofe the K x K matrix defined by ( T ~ ) m n = %-n-1

15 m, n I K ,

(7.2.6)

'

234

CHAPTER 7

with the convention ck = 0 for k < 0 or k > K. Similarly, v,+i(s) = Ti ~ ~ ( 2 1) 2 - for

5 z5 1 ,

(7.2.7)

t

where v

(T1)mn =c2m-ny lSm,nIK. (7.2.8) Equations (7.2.5), (7.2.7) both apply for x = because of the special structure of TO,Ti -and vJ (in ~artkular,(TO)",= (TI), n + ~[v,(O)), = [v, (I)], for n = 2, - .. ,K) , the two equations are identical for x = $ . We can w m b i i (7.2.5) and (7.2.7) into a single vector equation as fallows. Every x e f0,1]can be represented by a binary sequence,

i:

with &(z) = 1 or 0 for all n. S t M y spea4ing, two possibilities exist for every dyadic rational x, i.e. every z of the type k 2-2 : we am replace the last digit 1 followed by all wos by a digit 0 followed by dl ones. This wili not cause a problem, but to be clear we h t b g u i s h these two sequences by a s u h i p t : q ( x ) for the sequence ending in zeros (the expansion "from above," ie., the expansion that will start by the same J - 1 digits as x 2- for J-*oo), ti; (z) for the sequepee ending in onea (expansion ''horn below"). For btance,

+

4

The two definition regions 0 5 z < and 3 < z 5 1 of (7.2.5) and (7.2.7) are completely-characterized by dl(z):dl(%) = 0 if z < 3, 1 if z > For every binmy sequence d = we alSO define its right shift r d by ( ~ d= ) ~&+%, n = 1,2,--..

4.

Itisthenclaarthatrd(x) = d(Zc)ifO_Cx< ;,rd(x)=d(&--1)if $ < z 5 1. (For z = we have two possib'Ilith 7d+ = d(O), rd= d(l).) Alth~ugh T is really defined on binary seq.llences, we will make a slight abuse of notation and write T X = y rather than 74s) = d(y). With this new notation, we can rewrite (7.2.5), (7.2.7) as the angle equation

4,

(4)

(i)

If the v, have a limit v, then this Pector-valued function u will therefore be a fixed point of the linear operator T'ddld by %

(Tw)(z) =

Tdl(t)

47x1 ;

T acte on all the wtor-valued functions w : [0,1] requirements = 0, [w(O)j, F(o)]i = 0, [ w ( ~ ) ] K

r [w(~)]L-I,

-

R~ t b t

k F= 2,.

-

satis& the t

- - ,N . -

17.2.10)

MORE: ABOUT COMPACIW SUPPOKMCD WAVELETS

235

(As a result of these conditions Tw is d&ed unambiguouely at tbe dyadic rationals: the two expansions leed to the aame reeutt.) What has all this recasting the equations into different forme done for us? Well, it follows from (7.2.9) that

which implies

-

In other words, information on the spectral properties of products of the Ts mairicee wiil help us to control the dierence v, - v,+t, so tpat we &n prove v, v , and derive smootbnesa for v. But let us turn to an example. For the function 3d (7.2.1) reads

Note thgt Q+C2

=

C I + C ~=

and 2c2 = Cl

+ 3c3 ,

1

~(e)

ck4

both of which are consequence8 of the divisibilityof = f cr e-&{ by (1 + e'e)2. The values 2@(1), 2#(2) are detenmined by the qmhm

Because of (7.1.13), the columns of M dl aum to 1, emuring that (1, 1) is s I& eigenvectnr of M with eigenvdue I. This eigenvAiue is nondegenemk; the right eigenvector for the same eigenvalue is therefore not o r t h o n o d to (1, I ) , which means it can -be normaliized so that the sum of it9 entrim is 1. Thie choke of (24(1), 2@)) lef&

The ~t~

To,TIare 3 r 8 matrim given by

236

CHAPTER 7

Becsuee of (7.2.13), To and Tl have a common left eigenvector el = (1, 1, 1) with eigewalue 1. Moreover, for all z E 10, I],

= (1

-

2 ) [2#(l)

+ 24(2)1 + 2 [24(1)+ od(2ll (use (7.2.2))

= 1.

I

It folious that, for all z E [O, 11, all j E N,

3

= el -vo(71z)

(because elTd= el. for d = 0, 1)

+

Coneequently, vo(y) - vc(y) E El = {w; el .w = wl w2 +US = 01, the space orthogonal to el. In view of (7.2.11), we therefore only need to study products of Td-matrices restricted to El in order to control the convergence of the uj. But more is true! Define es = (1, 2, 3). Then (7.2.14) implies

e2To = ie2 ezT1 = iez

+ aoel , + ale1 ,

eo+b-i

*tho. = = q . 0 1 = cl+2c3-+ = ei = ea - 2ao el, Cben (7.2.15) becomee

e! To =

3 e8

and e4 T~ =

3 e$ - 4 el,

(7.2.15)

v.

or ep T~ =

ifwedhe

4 e! - 3 tiel .

On the other hend, eq - ~ ( z = ) (I

- Z) C! - ~ ( 0 )+ z 4 - vo(l) =

-2

;

I; 1 ;

.T

1 'r

MORE ABOUT COMPACTLY SUPPORTED WAVELETS

I

237

t

It follows that e! - [vo(x)- u~(x)]= 0. This means that we only need to study

,

products of Td-matrices r&icted to &, the space spanned by el and eg, in is oneorder to control v, - v,+t. But, because this is a simple example, dimensional, and TdlE,,is simply multiplication by some constant, namely the third eigenvalue of Td, which is for To, for TI. Consequently,

9

9

1 ~ 1

where we have used that the v1 M uniformly b o ~ n d L b .Since ~ < 1, (7.2.16) implies Ilvj I%) vj+t(x)ll 5 C 2-a, , with a = I log((1 + a)/4)l/log 2 = .550. It follows that the v,- have a limit function v , which is continuous since all the v, are and since the convergence is uniform. Moreover v automatically satisfies (7.2.10), since all the v, do, so that it can be "unfolded" into a continuous function F on [O,3].This function solves (7.2.1),so that 24 = F, and it is uniformly approachable by piecewise linear spline functions F, with nodes at the k2-3,

-

11 2 4 - F J l l t - o 5 C 2-03

(7.2.17)

It follows from standard spline theory (see, e.g., Schumaker (1981))~that 29is Holder continuous with exponent a = .550. Note that thls is better than the best result in 57.1 (we found cr = .5 - c at the end of 37.1.3). This Hijlder exponent is optimal: from (7.2.12) we have

hence

124(2-l) - 240)l = C(2-j)" . But this matrix method can do even better than determine the optimal Holder exponent. Since v ( x ) = Tdl(z)v(rx), we have, for t small enough so that z and x + t have the same j digits in their binary expansion,

This can be studied in exactly the same way as v,(x) - V ~ + ~ ( above; Z ) we fhd .

.

e:! [v(x)- V(Z

+t)] = t .

Fbr the remainder, only tbe T d l s matter, and we fmd

1

CHAPTER 7

238

where t itself is of order 2-j. With the notation r j ( x ) = f can be rewritten as

xi=,d , ( z ) , (22.18)

where P = 1 log l ( 1 - f i ) / ( l + &)jJ/log2. Suppose q ( z )tends to a Limit r ( x ) forj+m. If r ( z ) < = .2368,thcn the second term in (7.2.19) dominates the first, and v , hence 2 4 , is Holder continuous with exponent a @r(x). If r(x)> then the first term, of order 2-1, dominates, and 2 4 is Lipchits In fact, one can'even prove that 2 4 is differentiable in these points, which constitute a set of full measure. This establishes a whole hierarchy of fractal sets (the sets on which r ( x ) takes some preassigned value) on which 2 4 has different Holder exponents.. And what happens at dyadic rationals? Well,there you can define r * ( s ) , depending on whether you come- "from above" (associated with d + ( z ) ) or "From'belown (d- (x)); r+(x) = 0, r , (z)= I. As a consquence, 24 is left differentiable at dyadic rationals, x, but has Holder exponent -550 when z is approached from the right. This is illustrated by Figure 7.1, which shows blowups of 24,exhibiting the characteristic lopsided peaks at even very fine scales.

9

+

9,

In this example, we had twcs Usum rules'' (7.2.13), (7.2.14), decting that &(() = f C1 ck e-'N rss divleible by ( ( 1 e-q)/2)'. In general, .m is divisible by ( ( 1 + e ~ Y ) / 2 ) ~ipd , we b v e N sum rules. The mrbtqmce Enr

+

I

MORE ABOUT COMPACTLY SUPPOFtTED WAVELETS

239

will, h o m e r , be more than one-dimensional, which complicates wtimates. The general theorem about global regularity is as follows. THEOREM 7.2.1. Assume that the c k , k = 0, ..,K, satssfy K c k = 2

Ck=-,

6nd

K

C ( - I ) ~k'ck

= O forI=O, 1 , - - - , L .

(7.2.20)

k=O

For every m = 1,: ,. ,L + 1 , define Em to be the subspace of ltN orthogonal to U M = Span {el, - .,e,,,), where el = (11-', 23-', . .- ,NJ-') . Assume that there ezist 1/2 5 X < 1, 0 <_ e < L (e E N) and C > 0 such that, for all binary sequences ( d j ) j E Nand , dl rn E N,

1 . there ezists a non-trivral contsnuws L1-solution F for the two-scale equation (7.2.1) assocrated with the c,,, ,

2. this solution F is t times continuously differentiable, and 3. if X >

$,

then the llh derivative F ( ~ of ) f is Holder continuous, with expment at least tn A ( / In 2; if A = 112, then the eth denvatiue F(()of F is almost Lapschitz: it satrsfies

REMARK.The restriction X 2 means only that we pick the largest possible integer O 5 L for which*(7.2.21)holds with A < 1. If t = L, then n e c W y X 2' (see Daubechies and Lagarias (1992)); if L < L and X < then we could replace L by L+ 1 and X by 2X, and (7.2.21) would hdd for a larger integer t!. o A similar general theorem can be formulated for the local regularity fluctuations exhibiM by the example of 24. For-a precise statement, more details mil pro&, I refer to Daubechies and Lagarias'(1991, 1992). When applied to the Nt$, these methods lead to the following optin~alHolder exponents:

4

4,

These are clearly better than what was kbtained in §7.1.3;'m~reover,we see to our surprise that s4 is continuously differentiable, even though its graph seems to have a "peakn at z = 1. ~low&s show that thb is deceptive: the true maximum lies a little to the right of x = 1, md everything is indeed smooth (W Figure 7.2). The derivative of 3d is ~031tim10~6, but has a very small IIclder e q ~ ~ ~ eas n i!lurrtrated t, by Figure 7.3.

CHAPTER 7

MORE ABOUT COMPACTLY SUPPORTED WAVELETS

241

Unfortunately, these matrix methods are too cumbersome to treat Iarge examples. Another, more recent "direct methodn has been developed in Dyn and Levin (1989)and Rioul (1991); when applied to the N4 with N = 2,3,4 it re produces .the a-values above; since it is computationally less heavy, it can sJBo taclde larger valuea of N with better results t h in $7.1.3 (see Rioul (1991)). , 1. Note the similarity of the matrices To, TI and Po in 57.1.3 (see (7.1.27))! Even the spectral analysis, with the nested invariant subspcrces, is the same. This shows that the result in Theorem 7.1.12 is indeed optimal: if iis the

spectral radius of 'polhX = . T I J then ~ ~ ,

so that X in (7.2.21) must be at least &, and the H6lder exponent is at most C Ilog XI/ bg 2 5 1 log i t / log 2. The difference between the two appro& is that the present method also givea optimal estimates if &([) is not positive, unlike $7.1.3.

+

2. The condition (7.2.21) suggests that infinitely many conditions on the To, TIhave to be checked before Theorem 7.2.1 can be applied. Fortunately, (7.2.21) can be reduced to equivalent conditions which can be checked in a finitetime computer search. For details, see Daubechies and Lagarias (1992). 3. In practice, it is not necessary to work with To,TI and restrict them to GK. One can also define directly the matrices Po, TI corresponding to the coefficients of m ( t ) / ( ( l + e-g) f 2)K; it turns out )that bounds on llTd, - - .TL]&, 11 are equivalent to bounds on ...f" 11 . 2-Lm (see Daubechiea and Lagarias (1992), 55). The matf'd are much smaller than Td((N K) x ( N - K) i d e a d of N x N). o

~fi,

-

Since this method works for any function satisfying an equation of the type (7.2.1), we can apply it to the basic functions in subdivision schemes. For the Lagpangian interpolation function cornspoxding to (6.5.14), a detailed @pis shma that F is "almoet" @: it ia C1, and P &isfie8

Thie had already been obtained previously by Dubuc (1986). But our matrix methods can do morel They can prove that F b ahalmost everywhere differentiable, and they can even compute F" where it ie well defined. For details, see again Daubechiee and Lagarias (1992). 7.3.

Compactly supported waveke with more regularity.

By Corollary 5.5.2, an o r t h o n o d bmb of *veleta can consist of CN-I wa~elete only if the basic wavelet $ has N Mniehing momenta. (We implicitly aesume

-243

CHAPTER 7

+

that $ stems from a multiresolution analysis and that 4, have sufficient decay; hoth.conditions are trivially satisfied for the compactly supported wavelet bases as constructed in Chapter 6.) This was our motivation to construct the Nq5, which lead to N~ with N vanishing moments. The asymptotie results in 57.1.2 show however that the ~ 4 ~ , q E5 CpN with p = .2. This means that 80% of the zero moments are "wasted," i-e., the same regularity could be achieved with only N / 5 vanishing moments. Something similar happens for small values of N. For instance, 29 is continuous but not C1, 34 is C1 but not 0, even though 3$ have, respectively, two and threb vanishing moments. We can therefore Usacrifice" in each of these t;wo cases one of the wishing moments end use the additional degree of W o r n to obtain 4 with a better Holder exponent than 2 9 or 3 4 4 have, with the same support width. This amounts to replacing Imo(() 1' = (cos2 $)Nfi(sin2 $ ) by lrnO(t)l2 = (m2$)N-1[~N-l(sin2f) a(sin2 f l Nc a d (see (6.1.11)), and to choose a so thkt the regderity of I$ is impmved. Examples for N = 21 3 are shown in Fivires 7.4 and 7.5; the corresponding h, are as follows:

+

These examples correspond to a choice of a such that mexl p (Tol~,),p (TI I )] is minimized; the eigenvalues of To,TI are then degenerate.8 One can prove that the Holder exponents of t b two functions are at leaat .5864,1.40198 respectively, and at moet .60017,1.4116; these last values are probably the true Hiilder exponents. For more detaile, see Daubechies (1990b). 7.4.

Regularity or d h i n g momeds?

The examples in the previoua section show that for fixed support width of #,I), or equivalently, for fixed length of the filters in the associated subband coding scheme, the choice of the h, that leads to maximum regularity is dieerent from the choice with maximum number N of mishing moments for @. The question

I

MORE ABOUT COMPACTLY SUPPOKI'ED WAVELETS

RG.7.4. The r d n g jbction & for the mo& ngJor

wavelet cmutruetum wat/i ouppmt

width 3.

FIG. 7.5. The sealing jhction 4 for

the m o s t regulw wauckt w ~ t r u c t a o nwih support

wdih 5.

then arises: what is more important, vanishing moments or regularity? The answer depends on the application, and is not always clear. Beylkin, Coifrnan, and RokhIin (1991) use compactly supported orthonormal wavelets to cmmpress large matrices, i.e., to reduce them to a sparse form. For the details of this a g plication, the reader should consult the original paper, or the chapter by Beylkin in R u d d et al. (1991); one of the things that make their method work is the number of vahihing moments. Suppase you want to decompose a function F(z) into wave1ets (strictly speaking, matrices should be modelled by a function of two mriabks, but the point is illustrated juet as well, and in a simpler way, with one variable). You compute all the wavelet M c i e n t s (F,qjCk), and to compress all that informtion, you throw away all the coefficieni;~smaller than some threshold r . Let us see what this meaw at some fine scale; j = -J , J € N and J "large." If F is CL-'and J, has L vanishing moments, then, for x near

244

CHAPTER 7

2- jk, we have

where R is bounded. If we rnultipiy this by $(2Jx - k ) and integrate,-then the @t L terms will not contribute because j & x C ~ ( x=) 0, 1 = 0, . . - ,L - 1. Consequently, (F,

=

j

I/&

(r- l-3k)L R ( x ) 2'12 ~ 1 ( 2 ~ kx)-

For J large, this Mil be negligibly smatl, u n k R is verp large near k2- J . After thresholding, we *ill therefore only reb*n fine-scale wawle3 coefficients near singularities of F or its derivatives. The e k t will be all the more pronounced if the number L of vanishing moments of q5 is largeg Note that the regularity of 1C, does not play a role at all in this argurnent3t seems that for Beylkin, Coifman, and Rokhlin-type applications the number of vanishing moments is far more important than the regularity of $. For other applications, regularity may be more relevant. Suppose you want to compress the information in an image. Again, you decompose into wavelets (twedimensional wavelets, e.g-, aseociated with a tensor product multiresolution analysis), and you throw away all the small coefficients. (This is a rather primitive procedure. In practice, one chooses to allocate more piecision to some coefficients than to others, by means of a quantization rule.) You end up with a representation of the type

where S is only a (small) subset of all the possible values, c h m n i11 function of I. The mistakes you have made will consist of multiples of the deleted $,,&. If these are very wild objecd, then the difference between I and f might well be much more perceptible than if 9 L%ploother. This is admittedly very much a band-waving argument, but it suggests that a t least some regulatity might be required. Some fire* experhe& rqmrkd in Antonini e t al. (1991) seem to confirm this, but more expdmWs am required for a convincing answer. The sum rules (7.2.20), egUidt%&to tbe divisibility of m(<) by (l+a"~)L+l, have another interesting m m m ~In. the example studied in detail, &, we saw that (7.2.13) and (7.2.14) impzied'tbt (we proved both for V j , so that they deo hold for v = lim,,,

of

Q rather than

v,

v,), or, in terms

245

MORE ABOUT COMPACTLY SUPPOFLTED WAVELETS

k

-

6

k dl z E p,11. h u s e support(&) = [O, 31, one essily checks that this implies, forallyER,

An polynomials of degree less than or equal to 1 can therefore be written as tinesr combinations of the #(z - n). Something similar happens in general: the conditions (7.2.20) ensure that all the polynomials of degree less than or egual to L can be generated by Linear combinations of the 4(x - n). (See Fix and Strang (1969), Cavaretta, Dahmen, ahd Micchelli (1991).) This can again be dt used to explain why the condition Fmlt=r = 0, C = 0, - .. ,L is useful in ~ u b b a n dfiltering schemes. Ideally, one wants the low frequency channel, after the filtering, to contain all the slowly changing features, and to find only true "high frequency" fehtures in the other channel. Polynomials of low degree are essentially slowly changing features, and the sum r u b (7.2.20) ensure that they (or their restrictions t o a large interval, to keep it all in LZ(R); we disregard border effects here) are in every V j , i.e , they are given completely by the low frequency channel. In the design of FIR filters for subband coding, it is not customaiy to pay much attention. to the number of vanishing moments of nao, which is reflected by the "flatness" of the filter at ( = T." What follows is yet another argument showing that in implementations where filters are cascaded, it is nevertheless important to have at least some zero moments. Suppax that we apply three successive low pass filtering decimation steps t o s signal. If we call the original signal f", with Fourier transform (6) = En2-e-'e, then the result of one !filtering decim~tionis the sequence fA-, where f '(4) = C, f: e-'* satisfies

+

t

+

The second term can be viewed as the result of aliasing, due to the lower sampling rate in the f Similarly, three such operations lead to

'.

-

r3i2

[ p (g)

(8) (t) ma (5) +

I.

7 "folding" tams

(7.4.2) It follows that the product mo(() mo(X) m(q)plays an important role. l?igure 7.6 shows what this product loob like for the ideal low pass filter, mo([) = 1 5 a/2,Ofor u/2 I for I a. If the low pass filter is not ideal, then it will uleak" a little into the high-paes region 1r/2 I I(( 5 A. It is then important \

'

0 FIG. 7.6.

n/4

nt2,

3~14

n

Plbtr of mdt), mo(2*), mo(_QC)and of their pnnfvtfor the W bw-porr

ma.

to contain this leakage, especially when the filters are c d e d : it contribuks to the "folding" terms in (7.4.1) a d can lead to audible or visible aliasing once quantization is introduced and perfect reconstructipn ie no longer attained. In the ideal case'in Figure 7.6, the "bumpn of ~ ( q for)4 iE [3r/4, ?r) is killed In the product mo(<) m(2<) because %(() = 0 in this interval. The same happens for extra "bumps" of ~ ( qresulting ), in ~ ( 6 ~ ) ( q ) = 1 if ( E [0, x / 8 [ ,= 0 if ( E ]n/8,T ] . In the non-ideal case, a eimilar effect can be achieved by imposing that mo have a zero of reasonable multiplicity at ( = x , which "killsn the maximum of rnomo(X)in a concatenation. This phenomenon is illustrated in Figure 7.7, where a wavelet filter ie compared with a wn-wavelet perfect reconstruction filter. In Figure 7.7a we see plots of Im~(()l for two orthonormal perfect reconstruction firters (i.e., [mo(6)12 Imo(( %)I2= I), each with eight taps; the filter on the left componds to the example constructed in 56.4, with two vanishing momente (i.e., bas a zero with double multiplicity at ( = r),and an extra zero at ( = 77r/9. The filter on the right is not a wavelet filter, since m(n)# O and hence ~ ( 0# )I; it is conetructed mom according to standard wisdom, with an lequi-ripple" deeign: in this ceae the location of the nodm is choeen so that the amplitude of the two ripples ia the same es in the one ripple in the wavelet filter on the left, while keeping the trausition band as

~(e)

~(e)

+

+

MORE ABOUT COMPACTLY SUPPORTED WAVELETS

247

narmw as poasible within this constraint. The resulting fi1ter.i~a little steeper than the wavelet filter (its first zero is at ( = . 7 6 ~rather than .78n for the wavelet example), and seems therefore closer t o the ideal fiiter. (Of course, both are quite far from the ideal case, but remember that we have used only eigFit taps!) Figure 7.TowplotsI ~ ( t ) m,-,(q)j for these two examples, and Figure 7.70 blows up these plots in the region 7r/2 5 6 A. It is ciear that in the second (non-wavelet) case the leakage into this high frequency region is more important than in the wavelet case; this is true in L2-senseas well as in amplitude (the highest peak at right is about 3 dB higher thap at left). This effect can become even more pronounced when larger filters are considered."

~ ( x )

FIG.7.7. ComparLon of --h

:

<

thme cmcatenotimrs of two 8-tap low-paaa filters wtth Me perfect

(4p(oh of Imo(Oll (bl plob of b o ( 0 mo(2€) mo(W1, (4b l o w p a

ofbfmn/25€5+.

Notes.

,

1. Incidentally, thh prow that the statement in Remark 3 on p. 983 of D a u W e s (1988b)is m n g , I made a mistake in the extraction of a numerical value for p from Meyer's proof.

2. Thia ie ementially what was done in the Appendix of Daubechiee (1988b). Beware of the typos in that Appendix, however! 3. If the restriction that F is a compactly supported L1-frmction is removed, then many other solutions are possible. On the other hand* if we insist on compact support and F E L1, then necessarily ~ f =, PI1 ~ for some m € N, and F ia the mth derivative of the compactly supported L1-eblution to the equation obtained after replacing the by 2-m~k;n o - d y is . laet by the restriction ck = 2. For proofs, see Daubchiea and Lagarias (1991).

4. In all the examples we will consider, it is not really neceasery p choose the particular Fo conatructed later: the ,aIgorithm would work with any Fo with integral one.

5. We implicitly assume that the c k are real. Everything still carries through' for complex c k , but then v(x) E cK. 6. Equipping E l in this special case with the norm 11 1(a,-a - b, b)Hi2 = a2 bZ (equivalent with the standard Euclidian norm on E l ) , one finds SUPUEE,IIITo=III/IIbIII != . ~ ~ ~ , S \ ~ PIflTlulll/lllulll UEE, -859, 60 that Hfvj+l(x)- V ~ ( Z ) ~ ~5I Ajlllvl(x) - ~ o ( ~ ) lwith l J , A = .859, by (7.2.11). Et immediately follows that

+

J

Il%(z)ll

Ilvj(z)ll

+ CZi l v k ( ~-) ~ k - l ( ~ ) i l I k=l

5.

-

l l ~ o ~ ~ (1 ~ l l 4+ -

f

I

C l I l ~ l t-~ %iz)lll )

9

s

which b uqifarmly bounded in z and j. 7. The following argument a h givea a direct proof. Suppose that 2-b+') 5 p - x 5 2-3. 'Then there exists e E N eo that one of the two following alternatives holds: (e - 1)2-j 5 z y 5 l2-1 or (I - 1)2'j x 5 C2-J 5 y < (C 1)2-J, We will only diiuae the eecaad w;the first is similar. We then have . -

+

<

<

by (7.2.17). Because of the choice of C, there exists k E N so that zf -= k and f2-j = C2-J - k are both in [0, 1). We cab moreover 'choose binary expansions for zf and t 2 - j with coinciding first j digita (chcme

3:

-

249

MORE ABOUT COMPACTLY SUPPOWED W A V E L m

the expaneion ending in ones for t'2'j, and if x' is dyadic, the expansion ending in zercm for x'). It foUowe that

where we have used ITd, - - Tk,Ian < C 2-"3, the boundof vo, and %(u) ~ ( u ' E) El for all u,u'. We can eimilarly bound If, (y) ff(@-$)I; putting it all together lead^ to

-

-

- CNIz- *la , If( 4 - f (v)l< C 2-& < which proves the Holder ~~ptiauity with exponent a. 8. This corrects a mistake in the first priding, where a too large value was given for the Holder exponent for N = 2. I thank L. Villemoea and C. H d for pointing out this mistake to me. Incidentally, the N = 2 example is an instance where the best H b l e X in (7.2.21) is strictly larger thaa max(~(T0i~l)v P(TI~E,I )n] .t b cue P ( T O ~ E I=) ~ ( ~ 1 = ~ 1#, )a d Q I P ( ~ o ~ ~ a )czJ 1.09946.. 1 ~ l s . > 1. 0. Of course Beylkin, Coifman, and Bokhlin (1991) contains much more than thh! For a large class of matrices it turns oat t U after an o r t h o n o d b d i tranefm using wavelets, dense N x N matrices reduce,to sparse

atructurea with only O ( N ) entriae larger than the threshold c. The total L2-error made by throwing away all the entrieo below c turns out to be O(t), whicb b 8 much deeper rmult than the "comprdhn explained here, it is maentially the T(1) theorem of h v i d and JmmC, the proof of which u& "hardn analysis.

-

10. The argument below also holds for tbs biorthogonal ar (me Chapter 81, where the flatness of l r r ~ at l = 0 and at e = n need not be the same; it @ is the multiplicity of the zero at = 7c that comt4.

.d

4

11. In Cohen and Johnston (1992), 6ltere et% constructed which optimize rxitwb that are a mixture of "atsadu