CHAPTER 1

The What, Why, and How ofWavelets

The wavelet transform is a tool that cuts up data or functions or operators intodifferent frequency components, and then studies each component with a resolu-tion matched to its scale. Forerunners of this technique were invented indepen-dently in pure mathematics (Calderón's resolution of the identity in harmonicanalysis—see e.g., Calderón (1964)), physics (coherent states for the (ax + b)-group in quantum mechanics, first constructed by Aslaksen and Klauder (1968),and linked to the hydrogen atom Hamiltonian by Paul (1985)) and engineering(QMF filters by Esteban and Galland (1977), and later QMF filters with exactreconstruction property by Smith and Barnwell (1986), Vetterli (1986) in elec-trical engineering; wavelets were proposed for the analysis of seismic data byJ. Morlet (1983)). The last five years have seen a synthesis between all thesedifferent approaches, which has been very fertile for all the fields concerned.

Let us stay for a moment within the signal analysis framework. (The dis-cussion can easily be translated to other fields.) The wavelet transform of asignal evolving in time (e.g., the amplitude of the pressure on an eardrum, foracoustical applications) depends on two variables: scale (or frequency) and time;wavelets provide a tool for time-frequency localization. The first section tells uswhat time-frequency localization means and why it is of interest. The remainingsections describe different types of wavelets.

1.1. Time-frequency localization.

In many applications, given a signal f (t) (for the moment, we assume thatt is a continuous variable), one is interested in its frequency content locally intime. This is similar to music notation, for example, which tells the player whichnotes (= frequency information) to play at any given moment. The standardFourier transform,

(F f) (w) = 1 f dt e t

f (t)

also gives a representation of the frequency content of f, but information con-cerning time-localization of, e.g., high frequency bursts cannot be read off easilyfrom .F f . Time-localization can be achieved by first windowing the signal f, so

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CHAPTER 1

as to cut off only a well-localized slice of f, and then taking its Fourier transform:

(Twinf)(w, t) _ fds ƒ(s) g(s — (1.1.1)

This is the windowed Fourier transform, which is a standard technique for time-frequency localization. 1 It is even more familiar to signal analysts in its discreteversion, where t and w are assigned regularly spaced values: t = nto , w = mwo ,where m, n range over Z, and wo, to > 0 are fixed. Then (1.1.1) becomes

Twin (f) = f ds ƒ(s) g(s — nto) e—imw08 . (1.1.2)

This procedure is schematically represented in Figure 1.1: for fixed n, theT,w`n(f) correspond to the Fourier coefficients of f (•)g(• — nto). If, for instance,g is compactly supported, then it is clear that, with appropriately chosen wo,the Fourier coefficients T(f) are sufficient to characterize and, if need be,to reconstruct f (•)g(• — nto). Changing n amounts to shifting the "slices" bysteps of to and its multiples, allowing the recovery of all of f from the T,W^; (f).(We will discuss this in more mathematical detail in Chapter 3.) Many possiblechoices have been proposed for the window function g in signal analysis, mostof which have compact support and reasonable smoothness. In physics, (1.1.1)is related to coherent state representations; the g' ,t (s) = e i"sg(s — t) are thecoherent states associated to the Weyl—Heisenberg group (see, e.g., Klauder andSkagerstam (1985)). In this context, a very popular choice is a Gaussian g. In allapplications, g is supposed to be well concentrated in both time and frequency; ifg and g are both concentrated around zero, then (T"n f) (w, t) can be interpretedloosely as the "content" of f near time t and near frequency w. The windowedFourier transform provides thus a description of f in the time-frequency plane.

FIG. 1.1. The windowed Fourier transform: the function f (t) is rnultiplied with the window

function g(t), and the Fourier coefficients of the product f(t)g(t) are computed; the procedure

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THE WHAT, WHY, AND HOW OF WAVELETS

1.2. The wavelet transform: Analogies and differences with thewindowed Fourier transform.

The wavelet transform provides a similar time-frequency description, with afew important differences. The wavelet transform formulas analogous to (1.1.1)and (1.1.2) are

(TWa.. f)(a, b) = lal - 1 /2 J dt f (t) ?,b (t — )

(1.2.1)\ a J

and

T^ (f) = ao "`/a f dt f (t) b(a^ m t — nbo) . (1.2.2)

In both cases we assume that 0 satisfies

J dt &(t) = 0 (1.2.3)

(for reasons explained in Chapters 2 and 3).Formula (1.2.2) is again obtained from (1.2.1) by restricting a, b to only dis-

crete values: a = a, b = nboaó in this case, with m, n ranging over Z, andao > 1, bo > 0 fixed. One similarity between the wavelet and windowed Fouriertransforms is clear: both (1.1.1) and (1.2.1) take the inner products of f with afamily of functions indexed by two labels, g"°t(s) = ei" 8g(s - t) in (1.1.1), and

ab(8) = la l-1/2 1,(eab ) in (1.2.1). The functions ba ,b are called "wavelets";the function is sometimes called "mother wavelet." (Note that i/' and g areimplicitly assumed to be real, even though this is by no means essential; if theyare not, then complex conjugates have to be introduced in (1.1.1), (1.2.1).) Atypical choice for i,b is (t) = (1 — t2 ) exp(—t2 /2), the second derivative of theGaussian, sometimes called the mexican hat function because it resembles a crosssection of a Mexican hat. The mexican hat function is well localized in both timeand frequency, and satisfies (1.2.3). As a changes, the z/,a ,O(s) = la! -1 /2z1,(s/a)cover different frequency ranges (large values of the scaling parameter lal cor-respond to small frequencies, or large scale ba 0 ; small values of lat correspondto high frequencies or very fine scale ba ,0 ). Changing the parameter b as wellallows us to move the time localization center: each 1,ab(s) is localized arounds = b. It follows that (1.2.1), like (1.1.1), provides a time-frequency descriptionof f. The difference between the wavelet and windowed Fourier transforms liesin the shapes of the analyzing functions gw ,t and ,a ,b , as shown in Figure 1.2.The functions g"" t all consist of the same envelope function g, translated to theproper time location, and "filled in" with higher frequency oscillations. All theg" ,t , regardless of the value of w, have the same width. In contrast, the 0a ,b havetime-widths adapted to their frequency: high frequency lba ,b are very narrow,while low frequency oa ,b are much broader. As a result, the wavelet transformis better able than the windowed Fourier transform to "zoom in" on very short-lived high frequency phenomena, such as transients in signals (or singularitiesD

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CHAPTER 1

Wa,b with a> 1b<0

0 x

FiG. 1.2. Typical shapes of (a) windowed Fourier transform functions g", and

(b) wavelets zpa ,b . The gc ,t (x) = e'iwxg(x — t) can be viewed as translated envelopes g, `filledin" with higher frequencies; the a ,b are all copies of the same functions, translated and com-

pressed or stretched.

in functions or integral kernels). This is illustrated by Figure 1.3, which showswindowed Fourier transforms and the wavelet transform of the same signal fdefined by

f (t) = sin(21rvit) + sin(2irv2t) + ry[S(t — t 1 ) + S(t — t2)] .

In practice, this signal is not given by this continuous expression, but by samples,and adding a S-function is then approximated by adding a constant to one sampleonly. In sampled version, we have then

f (ni-) = sin(2irvinr) + sin(27rv2nr) + a[6n , l + Sn , 112 ]

For the example in Figure 1.3a, v1 = 500 Hz, v2 = 1 kHz, r = 1/8, 000 sec (i.e.,we have 8,000 samples per second), a = 1.5, and n2 — n1 = 32 (correspondingto 4 milliseconds between the two pulses). The three spectrograms (graphs of

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THE WRAT, WHY, AND HOW OF WAVELETS 5

FIG. 1.3. (a) The signal f (t). (b) Windowed Fourier transforms off with three different

window widths. These are so-called spectrograms: only Twin(f)1 is plotted (the phase is not

rendered on the graph), using grey levels (high values = black, zero = white, intermediate

grey levels are assigned proportional to log IT` '(f)I) in the t(abscissa), w(ordinate) plane.

(c) Wavelet transform of f. To make the comparison with (b) we have also plotted T ° (f)I,with the same grey level method, and a linear frequency axis (i.e., the ordinate corresponds

to a-1 ). (d) Comparison of the frequency resolution between the three spectrograms and the

wavelet transform. 1 would like to thank Oded Ghitza for generating this figure.

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CHAPTER 1

the modulus of the windowed Fourier transform) in Figure 1.3b use standardHamming windows, with widths 12.8, 6.4, and 3.2 milliseconds, respectively.(Time t varies horizontally, frequency w vertically, on these plots; the grey levelsindicate the value of ITW'n(f)I, with black standing for the highest value.) Asthe window width increases, the resolution of the two pure tones gets better,but it becomes harder or even impossible to resolve the two pulses. Figure 1.3cshows the modulus of the wavelet transform of f computed by means of the(complex) Morlet wavelet ip(t) = C e—t2/t2 (ei-t — e —'2„a /4), with a = 4. (Tomake comparison with the spectrograms easier, a linear frequency axis has beenused here; for wavelet transforms, a logarithmic frequency axis is more usual.)One already sees that the two impulses are resolved even better than with the3.2 msec Hamming window (right in Figure 1.3b), while the frequency resolu-tion for the two pure tones is comparable with that obtained with the 6.4 msecHamming window (middle in Figure 1.3b). This comparison of frequency resolu-tions is illustrated more clearly by Figure 1.3d: here sections of the spectrograms(i.e., plots of (Tw in f)(•, t)I with fixed t) and of the wavelet transform modulus(^ (T' f) (., b) with fixed b) are compared. The dynamic range (ratio betweenthe maxima and the "dip” between the two peaks) of the wavelet transform iscomparable to that of the 6.4 msec spectrogram. (Note that the flat horizontal"tail" for the wavelet transform in the graphs in Figure 1.3d is an artifact ofthe plotting package used, which set a rather high cut-off, as compared with thespectrogram plots; anyway, this cut-off is already at —24 dB.)

In fact, our ear uses a wavelet transform when analyzing sound, at least inthe very first stage. The pressure amplitude oscillations are transmitted fromthe eardrum to the basilar membrane, which extends over the whole length ofthe cochlea. The cochlea is rolled up as a spiral inside our inner ear; imagine itunrolled to a straight segment, so that the basilar membrane is also stretchedout. We can then introduce a coordinate y along this segment. Experiment andnumerical simulation show that a pressure wave which is a pure tone, f (t) =e t, leads to a response excitation along the basilar membrane which has thesame frequency in time, but with an envelope in y, F„,(t, y) = eit q„,(y). In afirst approximation, which turns out to be pretty good for frequenties w above500 Hz, the dependence on w of &, (y) corresponds to a shift by log w: there existsone function 0 so that Ø(Y) is very close to O(y—log w). For a general excitation

function f, f (t) = 2= f dw f (w)e i"t , it follows that the response function F(t, y)is given by the corresponding superposition of "elementary response functions,"

F(t, y) = 1 dw f (w) F. (t, y)27r

1

= dw J(w) e

tçb(y — log w)

If we now introduce a change of parameterization, by defining

^(e —^) = (27r) -1/2 0(x), G(a, t) = F(t, log a) ,

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THE WHAT, WHY, AND HOW OF WAVELETS

then it follows that

G(a, t) = J dt' f (t') (a(t — t')) ,

which (up to normalization) is exactly a wavelet transform. The dilation param-eter comes in, of course, because of the logarithmic shifts in frequency in the 0 ,.The occurrence of the wavelet transform in the first stage of our own biologicalacoustical analysis suggests that wavelet-based methods for acoustical analysishave a better chance than other methods to lead, e.g., to compression schemesundetectable by our ear.

1.3. Different types of wavelet transform.

There exist many different types of wavelet transform, all starting from thebasic formulas (1.2.1), (1.2.2). In these notes we will distinguish between

A. The continuous wavelet transform (1.2.1), and

B. The discrete wavelet transform (1.2.2).

Within the discrete wavelet transform we distinguish further between

B1. Redundant discrete systems (frames) and

B2. Orthonormal (and other) bases of wavelets.

1.3.1. The continuous wavelet transform. Here the dilation and trans-lation parameters a, b vary continuously over R (with the constraint a # 0). Thewavelet transform is given by formula (1.2.1); a function can be reconstructedfrom its wavelet transform by means of the "resolution of identity" formula

f _ c 1^°° / °° da db (.f, oa,b) a,b , (1.3.1)

1-00 a2

where oa" b (x) = IaL 1 / 2 0 (x-b ), and ( , ) denotes the L2-inner product. Theconstant C,, depends only on 0 and is given by

Cb = 2j I)I2 II 1 ; (1.3.2)

we assume Co < oo (otherwise (1.3.1) does not make sense). If 0 is in L'(R)

(this is the case in all examples of practical interest), then' is continuous, so

that C.p can be finite only if 1(0) = 0, i.e., f dx'(x) = 0. A proof for (1.3.1)will be given in Chapter 2. (Note that we have implicitly assumed that 0 is real;for complex b, we should use 1 instead of 0 in (1.2.1). In some applications,such complex i/i are useful.)

Formula (1.3.1) can be viewed in two different ways: (1) as a way of re-constructing f once its wavelet transform TWa° f is known, or (2) as a way to

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CHAPTER 1

write f as a superposition of wavelets oab ; the coefficients in this superpositionare exactly given by the wavelet transform of f. Both points of view lead tointeresting applications.

The correspondence f(x) —> (Twa" f)(a, b) represents a one-variable functionby a function of two variables, into which lots of correlations are built in (seeChapter 2). This redundancy of the representation can be exploited; a beautifulapplication is the concept of the "skeleton" of a signal, extracted from the con-tinuous wavelet transform, which can be used for nonlinear filtering (see, e.g.,Torrésani (1991), Delprat et al. (1992)).

1.3.2. The discrete but redundant wavelet transform-frames. In thiscase the dilation parameter a and the translation parameter both take onlydiscrete values. For a we choose the integer (positive and negative) powers ofone fixed dilation parameter ao > 1, i.e., a = as'. As already illustrated byFigure 1.2, different values of m correspond to wavelets of different widths. Itfollows that the discretization of the translation parameter b should depend onm: narrow (high frequency) wavelets are translated by small steps in order tocover the whole time range, while wider (lower frequency) wavelets are translatedby larger steps. Since the width of ip(a^ mx) is proportional to a, we choosetherefore to discretize b by b = nboaó , where bo > 0 is fixed, and n E Z. Thecorresponding discretely labelled wavelets are therefore

bm,n (x) = a^ m/a ,b(a^ m (x — nboaó ))

= ao m/z O(a^ mx — nbo) . (1.3.3)

Figure 1.4a shows schematically the lattice of time-frequency localization centerscorresponding to the i/^,,,,n . For a given function f, the inner products (f, z/i,, n )then give exactly the discrete wavelet transform TT' (f) as defined in (1.2.2)(we assume again that ' is real).

In the discrete case, there does not exist, in general, a "resolution of theidentity" formula analogous to (1.3.1) for the continuous case. Reconstructionof f from Tv (f), if at all possible, must therefore be done by some other means.The following questions naturally arise:

(1) Is it possible to characterize f completely by knowing T`°a"(f )?

(2) Is it possible to reconstruct f in a numerically stable way from TWa" (f )?

These questions concern the recovery of f from its wavelet transform. We canalso consider the dual problem (see §1.3.1), the possibility of expanding f intowavelets, which then leads to the dual questions:

(1') Can any function be written as a superposition of m ,n ?

(2') Is there a numerically stable algorithm to compute the coefficients for suchan expansion?

Chapter 3 addresses these questions. As in the continuous case, these discretewavelet transforms often provide a very redundant description of the original

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(b1

THE WHAT, WHY, AND HOW OF WAVELETS 9

(a)

aowo• . . . . • • . • • • • . • • . • . s .(-,,o) (-1,2)

(0,0) (0,1) (0,2)• . • . w0 . . . . . •

wo10 (bol(1,) (1,2)

• aob, .

r

. • . • . • . • • . .

. . . . . . . . ♦ . . . . . . . . . . . .

FIG. 1.4. The lattices of time-frequency localization for the wavelet transform and win-

dowed Fourier transform. (a) The wavelet transform: i,bm,n is localized around aó nbp in time.

We assume here that Iz 1 has two peaks in frequency, at ±o (this is the case, e.g., for the

Mexican hat wavelet 0(t) = (1 — t2 )e —t2 /2 ); F m , n (^)I then peaks at ±aó ^o, which are thetwo localization centers of ?/im ,n in frequency. (b) The windowed Fourier transform: g,,, is

localized around nto in time, around mwo in frequency.Dow

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10 CHAPTER 1

function. This redundancy can be exploited (it is, for instance, possible to com-pute the wavelet transform only approximately, while still obtaining reconstruc-tion of f with good precision), or eliminated to reduce the transform to its bareessentials (such as in the image compression work of Mallat and Zhong (1992)). Itis in this discrete form that the wavelet transform is closest to the "O-transform"of Frazier and Jawerth (1988).

The choice of the wavelet ?p used in the continuous wavelet transform or inframes of discretely labelled families of wavelets is essentially only restricted bythe requirement that C,,, as defined by (1.3.2), is finite. For practical reasons,one usually chooses 0 so that it is well concentrated in both the time and thefrequency domain, but this still leaves a lot of freedom. In the next section wewill see how giving up most of this freedom allows us to build orthonormal basesof wavelets.

1.3.3. Orthonormal wavelet bases: Multiresolution analysis. Forsome very special choices of 0 and ao, bo , the im , n constitute an orthonormalbasis for L2 (R). In particular, if we choose ao = 2, bo = 1, 2 then there exist b,with good time-frequency localization properties, such that the

1I'm,n(x) = 2--/2 0(2--x — n) (1.3.4)

constitute an orthonormal basis for L 2 (R). (For the time being, and until Chap-ter 10, we restrict ourselves to ao = 2.) The oldest example of a function 0 forwhich the,.m , n defined by (1.3.4) constitute an orthonormal basis for L 2 (R) isthe Haar function,

1 0<x<20(x)= —1 2 <x<1

0 otherwise .

The Haar basis has been known since Haar (1910). Note that the Haar func-tion does not have good time-frequency localization: its Fourier transform b(^)decays like for —* oo. Nevertheless we will use it here for illustrationpurposes. What follows is a proof that the Haar family does indeed constitutean orthonormal basis. This proof is different from the one in most textbooks; infact, it will use multiresolution analysis as a tool.

In order to prove that the m ,n (x) constitute an orthonormal basis, we needto establish that

(1) the ?,,,,,,n are orthonormal;

(2) any L2-function f can be approximated, up to arbitrarily small precision,by a finite linear combination of the 1,b,,,,, n .

Orthonormality is easy to establish. Since support (z/' m ,n) _ [2mn, 2m(n+1)],it follows that two Haar wavelets of the same scale (same value of m) neveroverlap, so that (,bm ,n ,1m ,n') = ,?Z'• Overlapping supports are possible if thetwo wavelets have different sizes, as in Figure 1.5. It is easy to check, however,that if m < m', then support (Lm , n ) lies wholly within a region where 1m ',nh is

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THE WHAT, WHY, AND HOW OF WAVELETS 11

constant (as on the figure). It follows that the inner product of ,bm,n and 1,bmi,n iis then proportional to the integral of '' itself, which is zero.

w

ii0 4 --8

w3,0L_JFIG. 1.5. Two Haar wavelets; the support of the "narrower" wavelet is completely con-

tained in an interval where the "wider" wavelet is constant.

We concentrate now on how well an arbitrary function f can be approximatedby linear combinations of Haar wavelets. Any f in L2 (R) can be arbitrarily wellapproximated by a function with compact support which is piecewise constanton the [P2-i, (9 + 1)2-'[ (it suffices to take the support and j large enough). Wecan therefore restrict ourselves to such piecewise constant functions only: assumef to be supported on [-2J1 , 2j1 ], and to be piecewise constant on the [12 -J0 ,(E+ 1)2-'o [, where J1 and J° can both be arbitrarily large (see Figure 1.6). Letus denote the constant value of f° = f on [e2- 'o, (t + 1)2-Jo [ by f2. We nowrepresent f0 as a sum of two pieces, f° = f1 + 6, where f1 is an approximationto f0 which is piecewise constant over intervals twice as large as originally, i.e.,fl 1[k2-'o+1,(k+1)2-Jo+1[ = constant = f,. The values fk are given by the aver-ages of the two corresponding constant values for f°, fkl = 2 (f2°k + ƒ2k+1) (seeFigure 1.6). The fimction 6l is piecewise constant with the same stepwidth asf0 ; one immediately has

6 e=ƒW — ƒé = á(ƒ21 — fz°e+1)

and

sát+1 = fze+1 — fi = 2 (ƒ2i+1 — f2°e) _ —bieIt follows that 61 is a linear combination of scaled and translated Haar functions:

2J1+J0 -1

Sl = 621w(2Jo-12 — t)

t=-2J1+J0-1 +1

We have therefore written f as

f= f° = f1 + c—.^o+l,c ^G—jo+l,eeD

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CHAPTER 1

2*

BLOW UP

t of02f o fs o

.. O.. ...1

S_

1 i

-• — + 00 0

fl = 2(fo+f°) S o 6 3020

2(0f - f 0)= 1- sFic. 1.6. (a) A function f with support [-2",2'1 1, piecewise constant on the [k2 — "o,

(k + 1)2 — JO [. (b) A blowup of a portion of f. On every pair of intervals, f is replaced by

its average (—t f1); the difference between f and f1 is 8 1 , a linear combination of Haar

wavelets.

where f' is of the same type as f0, but with stepwidth twice as large. We canapply the same trick to f1, so that

fl = f2 +C— Jo+2,l I— Jo+2,t

P

with f2 still supported on [-2JI ,2'], but piecewise constant on the even largerintervals [k2 -J0+2 , ( k + 1)2-Jo+2[. We can keep going like this, until we have

J1f = fJo+Ji /'

+ C7n. 'W,Z m,em=—Jo+1 Q

Here f Jo+JI consists of two constant pieces (see Figure 1.7), withf Jo+Ji I [0 2 J 1 [ fó o+Jl equal to the average of f over [0, 2J1 [, andfJo+Jl1 [-2J1,0[ = f+Ji the average of f over [- 2J',O[.

Even though we have "filled out" the whole support of f, we can still keepgoing with our averaging trick: nothing stops us from widening our horizon from

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THE WHAT, WHY, AND HOW OF WAVELETS

13

f^°+Jl fJ°+Jl0

-2J, 0 2Ji

12 f 1°+Jl 1 f JO+Jl2 o

2J1+1

2 J 1 +1

1fJo+J1W(2-J1-1x+1) 1f0i+JiW(2-J1-1x)2 -1 2 0

FiG. 1.7. The averages off on [0, 2J1 ] and [-2"1 , 0] can be "smeared" out over the bigger

interwals [0, 2J1+1], [-2J1+1, 0]; the difference is a linear combination of verg stretched out

Haar functions.

2''1 to 2J1+1 , and writing f J1+Jz = fJ1+J2+1 + 6J1+J2+1 where

fJi+Jz+ll[02J1+i[ = 2 fbi+Jz, fJi+Jz+l l[-2J1 +i o[ = 2 f-1+Jz

and6J1+J2 = Z.fó 1+J2ip(2—J1-1x) — Z fji+Jz (2

—Ji—l x + 1)

(see Figure 1.7). This can again be repeated, leading to

J1 +Kf = fJo+Jl+K + E E Cm,t bm,t ,

m=—Jo+1 t

where support (f Jo+J1+K) = [_2J1+K 2J1+K] and

:Jo+J1+KI [0 2Ji+x[ = 2—K foto+J1

,ƒJo+J1+K I [-2Jl+K 0[ = 2—K fJ1+Ji

It follows immediately that

.J1 +K II 22

— Cm,2 YOm,e = II f Jo+Ji+K II Lzm=—Jo+1 P 11 L2

= 2—K/2 - 2J1/2 {I fó °+Ji

I 2 + If Ji+J1 I 2 ] 1/2

which can be made arbitrarily small by taking sufficiently large K. As claimed,f can therefore be approximated to arbitrary precision by a finite linear combi-nation of Haar wavelets!

The argument we just saw has implicitly used a "multiresolution" approach:we have written successive coarser and coarser approximations to f (the fi,

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14 CHAPTER 1

averaging f over larger and larger intervals), and at every step we have writtenthe differente between the approximation with resolution 2j -1 , and the nextcoarser level, with resolution 2j, as a linear combination of the ij ,k. In fact, wehave introduced a ladder of spaces (Vj)jEz representing the successive resolutionlevels: in this particular case, V^ = {f E L2 (R); f piecewise constant on the[2j k, 2j (k + 1) [, k E Z}. These spaces have the following properties:

(1) ••• C V2 C V1 C Vo C V_1 C V_2 C ...

(2) fl 7z VJ = {0}, Ujcz Vj = L2 (R);

(3) f E Vj "-> f (2^.) E Vo;

(4) fEVo —> f(•—n)EVoforallnEZ.

Property 3 expresses that all the spaces are scaled versions of one space (the"multiresolution" aspect). In the Haar example we found then that there existsa function ,0 so that

Proj v,_ 1 f = Projv; f + (f, /j,k)j,k . (1.3.5)kEZ

The beauty of the multiresolution approach is that whenever a ladder of spacesVj satisfies the four properties above, together with

(5) 30 E Vo so that the O0,(x) = 0(x — n) constitute an orthonormalbasis for Vo,

then there exists 0 so that (1.3.5) holds. (In the Haar example above, wecan take q(x) = 1 if 0 <_ x < 1, «(x) = 0 otherwise.) The ?j ,k consti-tute automatically an orthonormal basis. It turn out that there are manyexamples of such "multiresolution analysis ladders," corresponding to many ex-amples of orthonormal wavelet bases. There exists an explicit recipe for theconstruction of 0: since 0 E Vo C V_ 1 , and the 0_ 1 ,„(x) = f 0(2x — n)constitute an orthonormal basis for V_ 1 (by (3) and (5) above), there existan = (O, '_ l ,n) so that 0(x) = z ç5(2x — n). It then suffices to take0(x) = E er (-1)na_.+ 1 «(2x—n). The function 0 is called a scaling functionofthe multiresolution analysis. The correspondente multiresolution analysis --> or-thonormal basis of wavelets will be explained in detail in Chapter 5, and furtherexplored in subsequent chapters. This multiresolution approach is also linkedwith subband filtering, as explained in §5.6 (Chapter 5).

Figure 1.8 shows some examples of pairs of functions 0, 0 corresponding todifferent multiresolution analyses which we will encounter in later chapters. TheMeyer wavelets (Chapters 4 and 5) have compactly supported Fourier transform;0 and 0 themselves are infinitely supported; they are shown in Figure 1.8a. TheBattle—Lemarié wavelets (Chapter 5) are spline functions (linear in Figure 1.8b,cubic in Figure 1.8c), with knots at Z for 0, at 2 Z for 0. Both 0 and 0 haveinfinite support, and exponential decay; their numerical decay is faster- thanfor the Meyer wavelets (for comparison, the horizontal scale is the same in (a),

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THE WHAT, WHY, AND HOW OF WAVELETS 15

(a) 2Meyer

0

5 0 5

(b) 1.5

1 OeL,1

0.5

0

-0.5-5 0 5

(c)

1 0BL,3

0.5

0

-0.5-5 0 5

(d) 1 0Haar

0

0 1

(e) 2

1 20

0

-1 0 1 2

(f)1 ^

0

5 0 5

1 WMeyer

0

-1-5 0 5

2

WBL,1

0

-1-5 0 5

1 WsL,3

0

5 0 5

1 T WHaar

0

0 1

21H( 2T0

_12

1 0 1 2

2

-5 0 5

FIG. 1.8. Some examples of orthonormal wavelet bases. For every ii in this figure, the

family 1,b k(x) = 2 -3/2 ^i(2 —ix — k), j, k E Z, constitutes an orthonormal basis of L 2 (R). The

figure plots 0 (the associated scaling function) and %b for different constructions which we will

encounter in later chapters. (a) The Meyer wavelets; (b) and (c) Battle—Lemarié wavelets;

(d) the Haar wavelet; (e) the nest member of the family of compactly supported wavelets, i;

(f) another compactly supported wavelet, with less asymmetry.Dow

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16 CHAPTER 1

(b), and (c) of Figure 1.8). The Haar wavelet, in Figure 1.8d, has been knownsince 1910. It can be viewed as the smallest degree Battle—Lemarié wavelet(bHaar = ,bBL ,o) or also as the first of a family of compactly supported waveletsconstructed in Chapter 6, bHaar = j.. Figure 1.8e plots the next member ofthe family of compactly supported wavelets NO; 20 and 20 both have supportwidth 3, and are continuous. In this family of NO (constructed in §6.4), theregularity increases linearly with the support width (Chapter 7). Finally, Figure1.8f shows another compactly supported wavelet, with support width 11, andless asymmetry (see Chapter 8).

Notes.

There exist other techniques for time-frequency localization than the win-dowed Fourier transform. A well-known example is the Wigner distribu-tion. (See, e.g., Boashash (1990) for a good review on the use of the Wignerdistribution for signal analysis.) The advantage of the Wigner distributionis that, unlike the windowed Fourier transform or the wavelet transform,it does not introduce a reference function (such as the window function,or the wavelet) against which \the signal has to be integrated. The disad-vantage is that the signal enters in the Wigner distribution in a quadraticrather than linear way, which is the cause of many interference phenom-ena. These may be useful in some applications, especially for, e.g., signalswhich have a very short time duration (an example is Janse and Kaiser(1983); Boashash (1990) contains references to many more examples); forsignals which last for a longer time, they make the Wigner distribution lessattractive. Flandrin (1989) shows how the absolute values of both the win-dowed Fourier transform and the wavelet transform of a function can alsobe obtained by "smoothing" its Wigner distribution in an appropriate way;the phase information is lost in this process however, and reconstruction isnot possible any more.

The restriction bo = 1, corresponding to (1.3.4), is not very serious: if(1.3.4) provides an orthonormal basis, then so do the lm , n (x) = 2 — mh2

0(2— mx — nbo), with 0(x) = Ibo I -1 / 2 ij'(b^ lx), where bo 0 0 is arbi-trary. The choice ao = 2 cannot be modified by scaling, and in fact ao

cannot be chosen arbitrarily. The general construction of orthonormalbases we will expose here can be made to work for all rational choices forao > 1, as shown in Auscher (1989), but the choice a o = 2 is the simplest.Different choices for ao correspond of course to different 0. Although theconstructive method for orthonormal wavelet bases, called multiresolutionanalysis, can work only if ao is rational, it is an open question whetherthere exist orthonormal wavelet bases (necessarily not associated with amultiresolution analysis), with good time-frequency localization, and withirrational ao.

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