Wavelet Transform (779450), страница 5
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. . , l] with eigenvalue one.+8.7. Wavelet247and we see that the eigenvalue 1 exists. The eigenvalue problem we have tosolve is given by[E](8.167)WaveletFamilies8.7Various wavelet families are defined in the literature. We will only considera few of those constructed from filter banks. For further design methods thereader is referred to [36, 1541.8.7.1Design of BiorthogonalLinear-PhaseWaveletsIn this section, we consider the design of linear-phase biorthogonal waveletsaccording to Cohen, Daubechies and Feauveau [28].We start the discussionwith the first equation in (8.113),which is the PR condition for two-channelfilter banks without delay. We consider an overall delay of 7, that isHo(z)Go(z)+ Ho(-z)Go(-z) = 2 2 7 .(8.168)On the unit circle, this meansH0 W) Go ( ej'" + Ho(ej('" + "1) Go(ej('" + "1) = 2 e-j"J7.(8.169)In order to yield linear-phase wavelets, both filter Ho(z)and Go(.) have to belinear-phase.
Furthermore, the filters need to satisfy the regularity conditionas outlined in Section 8.6.6in order to allow the construction of continuousscaling functions and wavelets.When expressing the linear-phase property, two types of symmetry haveto be considered, depending on whether the filter length is even or odd. Wewill outline these properties for the filter Ho(z)and start with odd-lengthfilters.
The second filter Go ( z ) ,which completes a perfect reconstruction pair,has the same type of symmetry.Odd-Length Filters. Odd-length linear-phase filters satisfyH o ( e j " ) = e-jwrhH;(COSW),where the delay ~h is aninteger. Assuming that HA(cosw) hasW = X , we may writeH;(eju) =Jz (cos El2'~(cosw).2(8.170)zeros at(8.171)248TransformChapter 8. WaveletIt is easily shown thatG o ( e J W )hasthe same type of factorization, so thath (cos E)"2Go(ej"') = e-jWTgwhere T = ~hQ(cosw),(8.172)+T ~ .Even-Length Filters. Symmetric even-length filters can be expressed asH o ( e j " ) = e--ju(Th +Wcos - H; (cos W ) ,2(8.173)and according to the above considerations, we may writeHo(ejw)=e-jw(Th+3) (cos :)2~+12p cos W(1,where it is again assumed that Ho(ejw) has l! zeros athas a factorization of the formG o ( e j w )=h e-j"(Tg+f)cos + (sin 2WW)= 7r.
Go(ej") thenw (cos -)2L+1 Q(cosw).2Filter Construction. SubstitutingthefactorizationsG o ( e j w ) into (8.169) yields(cos - ) 2 k2W-)2k(8.174)(8.175)for Ho(ejw) andM ( - cosw) = 1(8.176)withM(COSW ) = P(COSW ) &(COS+(8.177)W)+and lc = l! 2 if the filter length is odd and lc =l+1 if it is even. Thisexpression will now be reformulated by rewriting M(cosw) as a polynomialin (1 - cosw)/2 = sin2w/2, so that M(cosw) := F(sin2w/2). We getW(cos - ) 2 k F(sin2 w/2)2+ (sin -W2) 2 k cos^ w/2)or equivalently,(1- X)k F ( X )= 1,+ X k F(1- X) = 1(8.178)(8.179)with X = sin2w/2.
Hence,F(X) = (1 - X ) - k - X k (1- X ) - k F(1- X).(8.180)Using Bezout's theorem, one can show that this condition is satisfied by aunique polynomial F ( z ) with a degree of at most lc - 1 [28]. Based on thisproperty, the polynomial F ( z ) of maximum degree lc - 1 can be found by8.7. Waveletexpanding the right-hand side of (8.180) into a Taylor series where only thefirst k terms are needed. This gives(8.181)The general solution of higher degree can be written ask-lk+n-l(8.182)n=Owhere R ( z ) is an odd polynomial. Based onthis expression, filters can befound by factorizing a given F(sin2w/2) into P(cosw) and Q(cosw). GivenP(cosw) and Q(cosw) one easily finds Ho(ej") and Go(ej") from (8.170) (8.175).Spline Wavelets.
Spline wavelets based on odd-lengthfilters are constructedbychoosing R ( z ) 0 and(8.183)The corresponding analysis filter isEven-length filters are given by(8.185)and250TransformChapter 8. WaveletThe scaling function 4(t) constructed from Go(z) according to (8.183) is aB-spline centered around T ~ and, the one constructed from Go(z) accordingto (8.185) is a B-spline centered around T~+ i.Filters with Almost Equal Length. In the spline case, the lengthof & ( z )is typically much higher than the length of Go(z). In order to design filterswith almost equal length, one groupsthe zeros of F ( z ) into real zeros andpairs of conjugate complex zeros and rewrites F ( z ) asJI&'(X) = A~ ( -z xi) n ( z z- 2?Ji{~j} zi= 1+ Izjl).(8.187)j=1Any regrouping into two polynomials yields a PR filter pair.
This allows us tochoose filters with equal or almost equal length. For example, the 9-7 filtershavebeenfound this way [28]; they are knownfor their excellent codingperformance in wavelet-based image compression [155, 1341.Examples. Table 8.1 shows some examples of odd-length filters. While thecoefficients of the spline filters (5-3 and 9-3) are dyadic fractions, those ofthe 9-7 filters constructed from (8.187) are not even rational. This meansan implementation advantage for the spline filters in real-time applications.However, the 9-7 filters have superior coding performance.
For illustration,Figures 8.15 and 8.16 show the analysis and synthesis scaling functions andwavelets generated from the 9-3 and 9-7 filters in Table 8.1.Table 8.1.Linear-phase odd-length biorthogonal wavelet filters.5-3n4.go9-7-1262-16819-34 . h0- 16 . go12116 ' h0goho3-6-16389038-16-63-0.06453888265083-0.040689417586800.418092273510290.788485616892520.41809227351029-0.04068941758680-0.064538882650830.03782845543778-0.02384946495431-0.110624404011430.377402855547590.852698678333840.37740285554759-0.11062440401143-0.023849464954310.037828455437782518.7.
WaveletFamilies2.01.5I*(t)561.o0.5:l-1.50,,,,,,,1234567180-0.5-1.0-1.50123478-3.0~2.01.o0-1.011-1.o-2.0'0'''''''12345678Figure 8.15. Scaling functions and wavelets constructed from the 9-3 filters.2.01.5-_E1.o0.50-0.5-0.5-1.o-1.5 0-1.o1234567-1.52.01.51.o0.5l:-1.50,,,,,,,1234567180-0.5-1.0-1.5l0'''''''1234567I8Figure 8.16. Scaling functions and wavelets constructed from the 9-7 filters.2528.7.2TransformChapter 8. WaveletThe Orthonormal Daubechies WaveletsDaubechies designeda family of orthonormal wavelets with a maximal numberof vanishingmoments for a given support [34]. Inorder to control theregularity, the following factorization of H0 (ejw)is considered:(8.188)Because of orthonormality, thePR condition to be met by the prototypefilterwith~ ( c o s w=) IP(ejw)12.(8.191)Inserting (8.190) into (8.189) yields+W(cos2 -)k ~ ( c o s w ) (sin2 - ) k ~ ( - c o s w )= 1.22(8.192)Using the same arguments as in the last section, (8.192) can also be writtenasW(cos2 E)' F(sin2 w/2) (sin2 -1'w/2) = 1,(8.193)22or equivalently as(1 - X)k F ( X ) X k F(1- X) = 1(8.194)+cos^+with X = sin2 w/2.
This is essentially the same condition that occurred in thebiorthogonal case, but wenow have to satisfy F(sin2 w/2) 2 0 V W, becauseF(sin2 w / 2 ) = IP(eju)12.Daubechies proposed to choosewhere R(z) is an odd polynomialsuch that F ( x ) 2 0 for X E [0, l]. Thefamily of Daubechies wavelets is derived for R ( x ) 0 by spectral factorizationof F ( z ) into F ( z ) = P ( x ) P ( x - ' ) . For this, the zeros of F ( x ) have to becomputed and grouped intozeros inside and outside the unit circle. P(.) then8.7.
Wavelet253contains all zeros inside the unit circle. This factorization results in minimumphase scaling functions. For filters & ( z ) with at least eight coefficients,more symmetric factorizations are alsopossible. The magnitude frequencyresponses, however, are the same as for the minimum phase case.Figure 8.17 shows some Daubechies wavelets, the corresponding scalingfunctions and the frequencyresponses of the filters.
We observe that thescaling functions and wavelets become smoother with increasing filter length.For comparison, some Daubechies waveletswith maximal symmetry,known assymmlets, and thecorresponding scaling functions are depicted in Figure 8.18.The frequency responses are the same as in Figure 8.17. Recall that with afew exceptions (Haar and Shannonwavelets), perfect symmetry is impossible.8.7.3CoifletsThe orthonormal Daubechies wavelets have a maximum number of vanishingwavelet moments for a given support. Vanishing moments of the scalingfunction have not been considered. The idea behind the Coiflet wavelets isto trade off some of the vanishing wavelet moments to the scaling function.This can be expressed as001, for Ic = 00, forIc=1,2 ,..., l - 1(8.196)andL00tk$(t) dt = 0 for Ic = 0 , 1 , .
. . , l - 1.(8.197)Note that the 0th moments of a scaling function is still fixed to one. Furthernote that the same parameter l, called the order of the coiflet, is used for thewavelet and the scaling function.The frequency domain formulations of (8.196) and (8.197) are1, for Ic = 00, f o r I c = 1 , 2 ,..., l - 1(8.198)..., l - 1 .(8.199)and=OforIc=O,l,Condition (8.198) means for the filter Ho(ejw) that254TransformChapter 8. Wavelet-0.50L---1f-lI-01t L-0.50:-50 0-40-30Loh-f-llFigure 8.18. Frequency responses of the maximally symmetric Daubechies filtersand the correspondingscalingfunctionsandwavelets(the indicesindicatefilterlength; the frequency responses are equal to those in Figure 8.17).8.8. The Wavelet Transform of Discrete-Time255Signalsfor some U(ej").
From (8.199) it follows that HO(ej") can also be written inthe form (8.188)(8.201)For evene, solutions to this problem can be formulated as[361f(ej"),(8.202)where f ( e j " ) has to be found such that (8.189) is satisfied. This results in e/2quadratic equations for C/2 unknowns [36].8.8The WaveletTransformofSignalsDiscrete-TimeIn the previous sections we assumedcontinuous-time signals and waveletsthroughout.
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