CHAITER positio are exactly given by the wavelet transform of f. Both points of view lead to interesting applications. The correspondence f(r)-,(/wav)(a, b)represents i one-variable function by a function of ables, into which lots of correlat built in Chapter 2). This redundancy of the represent at ion can be exploited a beautiful application is the concept of the skeleton"of a signal, extracted fron the con- tinuous wavelet transforIll, which can be used for nonlinear filtering(see, e. g Torresani(1991), Delprat et aL.(1992)) 1.3.2. The discrete but redundant wavelet transform-frames. In this case the dilation parameter a anl the translation parameter broth take only discrete values. For a we choose the integer (positive and negative)powers of one fixed dilation paraineter (o>l, ip, aa. As alreadly illustrated by Figure: 1.2, different values of n correspond to wavelets of different widths.It follows that the discretivat ion of the translation parameter b should depend on 77: narrow(high frequency) wavelets are translated by sil: ll st eps in order to cever the whole tinne range, while wider(lower fre quency)wavelets are translated by larger steps. Since the width of y(au"r)is proportional to ai,. we choose therefore to discret ize b by b. nmn ai, where Iup.U is fixed, atd IL (. Z. The corresponding discretely labelled wavelets are there fore (x)= w(o"(r tlu p) /'((u"r nln) (1.3.3) Figure 1.4a shows schematically the lattice of time - y localization centers corresponding to the 4'm, n. For a given function, the inner products in) then give exactly the discrete wavelet transfor Twa (/)is de fine in(1.2.2) (we assume again that y is real In the discrete case, there does not. exist, in general, a"re solntion of the identity"formula analogous to(1. 3. 1)for the cont inous case. Reconstruction of f from Twa(), if at all possible, Inust therefore be done by sone other ineans The following questions nat urally arise (1) Is it possible to characterize f completely by knowing TWH()? (2)Is it possible to reconstruct f in a nerically stable way from T wa()? These questions concern the recovery of f froln Its wavelet transforM. We can also consider the dual problem(sce$1.3.1), the possibility of expanding f into wavelets, which then leads to the dual questions (1)Can any function be written as a superposition of ym, n? (2) Is there a numerically stable algorithm to compute the coefficients for such Chapter 3 addresses these questions. As in the continuous case, these discrete wavelet transforms often provide a very redundant description of the original
THE WHAT, WHY, AND HOW OF WAVELETS (a r 【-1,1) -1o)【,2 0,o)(ot(2) 【,o 12) (b) 【,o (0.01 PIG. 1.4. The lattices of time-frequency localization for the wavelet transform and wn dowed Rower tranform.(a)The wavelet transform: tom, n u localized around ao'nbo n tume We aswe Aere that ll has two peaks n frequency, at #Eo(thu w the case, e.g,for the Merican hat wavelet p(t)=(1-22)e-t/); Iim, (e)I then peaks at ta Eor which are the on centers of wm, n in frequency. (b)The windowed Fourer transform: gm,n w localised around nto in tme, around mwo an frequency
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 bare essentials( such as in the image compression work of Mallat and Zhong(1992)). It is in this discrete form that the wavelet transform is closest to the " o-transform" of Frazier and Jawerth(1988) The choice of the wavelet yl used in the continuous wavelet transform or in frames of discretely labelled families of wavelets is essentially only restricted by the requirement that C+, as defined by(1.3. 2), is finite. For practical reasons one usually chooses w so that it is well concentrated in both the time and the frequency domain, but this still leaves a lot of freedom. In the next section we will see how giving up most of this freedom allows us to build orthonormal bases of wavelets 1.3.3. Orthonormal wavelet bases: Multiresolution analysis. For some very special choices of wl and ao, bo the wm, n constitute an orthonormal basis for L (R). In particular, if we choose a0=2, b0=1, then there exist po, with geod time-frequency localization properties, such that the vm,n(=)*2-m/ p(2-mz-n (134 constitute an orthonormal basis for L2(R).(For the time being, and until Che er 10, we restrict ourselves to o =2. The oldest example of a function y for which the wm, n defined by (1.3.4)constitute an ortHonormal basis for L(R)is the Haar function 10<x va)=-11≤z< 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 p(E) IEl-I for 4+oo, Nevertheless we will use it here for illustration purposes. What follows is a proof that the Haar family does indeed constitute an orthonormal basis. This proof is diferent from the one in most textbooks; in fact, it will use multiresolution analysis as a tool In order to prove that the vm, n(=)constitute an ort. to establish that (1 )the am, n are orthonormal (2 )any L2-function f can be approximated, up to arbitrarily amall precision, by a finite linear combination of the wm Orthonormality is easy to establish. Since support(m, m)=[2 n,2m(n+1)], it follows that two Haar wavelets of the same scale(same value of m)never overlap,so that (vm, n, *m, n)=5n, n,. Overlapping supports are possible if the two wavelets have diferent sizes, as in Figure 1.5. It is easy to check, however, that if m< m, then support m, m) lies wholly within a region where wm, n, is
THE WHAT, WHY, AND HOW OF WAVELETS constant(as on the figure). It follows that the inner product of wim, n and ym'n' is then proportional to the integral of wo itself, which is zero v30 FIG 1 5 To Haar wavelets, the support of the narrower"wavelet w completely con- n an interval where the“ wder waυ elet u constant We concentrate now on how well an arbitrary function f can be approximated by linear combinations of Haar wavelets. Any f in L( R)can be arbitrarily well approximated by a function with compact support which is piecewise constant on the[ 22-3,(e+1)-(it suffices to take the support and j large enough). We can therefore restrict ourselves to such piecewise constant functions only: aseume f to be supported on( 2, 2], and to be piecewise constant on the [c2-Jo (L+1)2-Jol where Ji and Jo can both be arbitrarily large(see Figure 1.6).Let ps denote the constant value of fo=f on[c2-3o, (e+1)2-o[ by fe. We now represent fo as a sum of two pieces, fo=f+8, where f is an approximation a which is piecewise constant over intervals twice as large as originally, i.e. 如+1、+12-h+≡ constant=升2, The values fk are given by the aver- Ages of the two corresponding constant values for f, R =,(%+fik+1)(see 】6). The function6i nt with same stepwidth 如 pe immediately has 品=-=-+ 嘴;+=+1-=数一房)=品 It follow E a linear combination of scaled and translated Ha sB Iun ∑品中2hkx-0 t=-21+J-1+1 We have therefore writtenfas 广+2 J+1,即J+1!;
12 CHAPTER I 2 叫 BLOW UP fb=(8+f9) 50 5b-181'-29:1})-6 Fa16(a) A function∫ unth support-24,2小 piecewise constant on the k2-如, (k+1)2-如(b) A blowup o∫ a portion of∫ On every par of ntervals,∫ us replaced by its average(→∫1); the difference between∫and∫lwb, a lnear combnation of Haar wavelets wheref is of the same type as fo, but with stepwidth twice as large. We can apply the same trick to f, so that ∫2=/2+∑c-h+1v-+, withf2 still supported on(-21, 2 11, but piecewise constant on the even larger intervals [k2-Jo+2, (k+ 1)2 2 We can kee keep going like this, until we have Jo+J m=-J+1! Here Jo+J consists of two constant Figure 1.7),with Jo+J, f6 Jo+J1 the average over间,21 +山124,≡f+ the average of∫ower-2h,0 Even though we have“ filled out” the whole support of∫, we can still going with our averaging trick: nothing stops us from widening our horizon k