Wavelet Transform (779450), страница 3
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, M-1.(8.60)Figure 8.7 shows the sampling grid of an analysis with three voices per octave.Sampling the wavelet transform can be further generalized by choosing thesampling gridam = a p ,with an arbitrary a0the waveletsb,,= am n T ,> 1. This corresponds to M-$(h)(t)= a$,$(a,$t)7m,n EZ(8.61)nested wavelet analyses withIC = 0 ~ 1 .,.., M- 1.(8.62)For this general case we will list the formulae for the frame bounds A and Bin (8.56) as derived by Daubechies [35]. The conditions for the validity of theformulae are:3cc(8.63)(8.64)and3By “ess inf” and “ess sup” we mean the essential infimum and supremum.Chapter 8.
WaveletTransform226...................................................................... . . . . . . . . . .. . .. .. .. .. .. .. .. .. .. ..log-.b -Figure 8.7. Sampling of the wavelet transform with three voices per octave.withIf (8.63) (8.65) are satisfied for all wavelets defined in (8.62), the framebounds A and B can be estimated on the basis of the quantities~(8.67)(8.68)(8.69)Provided the sampling interval T is chosen such that(8.70)we finally have the following estimates for A and B:(8.71)(8.72)8.5.
TheWaveletDiscrete8.5227Transform ( D W T )The Discrete WaveletTransform(DWT)In this section the idea of multiresolution analysisand the efficient realizationof the discrete wavelet transformbasedonmultiratefilter banks will beaddressed. This framework has mainly been developed by Meyer, Mallat andDaubechies for theorthonormal case [104, 91,90, 341. Since biorthogonalwavelets formally fit into the same framework [153, 361, the derivations willbe given for the more general biorthogonal case.8.5.1Multiresolution AnalysisIn the following we assume that the sets?)mn(t) = 2-f ?)(2-79- n),m,n E Z?jmn(t) = 2-f ? j ( 2 - T(8.73)- n),are bases for &(R)satisfying the biorthogonality condition (8.58). Note thatT = 1 is chosen in order to simplify notation.
We will mainly consider therepresentation (8.55) and write it aswithd,(n)q,,)= ~ , f ( 2 " n , 2 " ) = (2,,m , n E Z.(8.75)Since a basis consists of linearly independent functions, L 2 ( R ) may beunderstood as the direct sum of subspacesL2(R) = . . . @ W-1 €B WO @ W1 €B...(8.76)withW, = span {?)(2-"t- n), nE Z} ,mEZ.(8.77)Each subspace W, covers a certain frequency band. For the subband signalswe obtain from (8.74):(8.78)n=-mEvery signal z ( t ) E L2(R) can be represented as00(8.79),=-cc228TransformChapter 8.
WaveletNowwe define the subspaces V,m EZ as the direct sum of Vm+l andWm+1:V, = Vm+l CE Wrn+l.(8.80)Here we may assume that the subspaces V, contain lowpass signals and thatthe bandwidth of the signals contained in V, reduces with increasing m.From (8.77), (8.76), and (8.80) we derive the following properties:(i) We have a nested sequence of subspacesc v, c v,-l c .
. .. . . c V,+l(8.81)(ii) Scaling of z ( t ) by the factor two ( x ( t )+ x ( 2 t ) )makes the scaled signalz(2t) an element of the next larger subspace and vice versa:(iii) If we form a sequence of functions x,(t) by projection of x ( t ) E L2(R)onto the subspaces V, this sequence converges towards x ( t ) :z(t)lim x,(t) = x ( t ) ,,+-mE L2(R),z,(t)EV,.(8.83)Thus, any signal may be approximated with arbitrary precision.Because of the scaling property (8.82) we may assume that the subspacesV, are spanned by scaled and time-shifted versions of a single function $(t):V, = span {+(2-,t- n), nE Z} .(8.84)Thus, the subband signals z,(t) E V, are expressed asc00zrn(t)=c,(n)$mn(t)(8.85)- n).(8.86)n=-mwith$mn(t)= 2-%#j(2-,tThe function +(t)is called a scaling function.Orthonormal Wavelets.
If the functions ~ , n ( t )= 2-?~(2-"t-n), m, n EZ form an orthonormal basis for L z ( R ) , then L 2 ( R ) is decomposed into anorthogonal sum of subspaces:1L 2 ( R ) = . . .$ W-11$11WO €B W1 €B. ..(8.87)2298.5. The DiscreteWavelet Transform ( D W T )In this case (8.80) becomes an orthogonal decomposition:(8.88)If we assume11q511= 1, then the functions$mn(t)= 2-?4(2-,t- n),form orthonormal bases for the spaces V,,m,n EZ,(8.89)m E Z.Signal Decomposition. From (8.80) we derivex, (t) = 2,+1 (t) + Y,+1 (t).(8.90)If we assume that one of the signals x,(t), for example zo(t),is known, thissignal can be successively decomposed according to (8.90):The signals y1( t ) , yz(t), .
. . contain the high-frequency components of zo(t),z1 ( t ) ,etc., so that the decomposition is a successive lowpass filtering accompanied by separating bandpass signals. Since the successive lowpass filteringresults in an increasing loss of detail information, and since these details arecontained in y1 ( t ) ,y2 ( t ) ,. .
. we also speak of a multiresolution analysis (MRA).Assuming a known sequence { c o ( n ) } , the sequences {cm(.)} and {d,(n)}for m > 0 may also be derived directly according to the schemeIn the next section we will discuss this very efficient method in greater detail.Example: Haar Wavelets. The Haar function is the simplest exampleof an orthonormal wavelet:1-10for 0 5 t < 0.5for 0.5 5 t < 1otherwise.,230~, W ) ;>t.
. . . . . .Chapter 8. WaveletTransform'"I,+W-1). . . . . .;>t.Figure 8.8. Haar wavelet and scaling function.The corresponding scaling function is1, for 0 5 t < 10, otherwise.The functions +(t - n), n E Z span the subspace W O ,and the functionsE Z span WI. Furthermore, the functions $(t - n ) , n E Z spanV0 and the functions +( ft - n ) , n E Z span VI. The orthogonality among thebasis functions + ( 2 - T - n ) , m, n E Z and the orthogonalityof the functionstj(2-Y - n), m,n E Z and +(2-jt - n ) , j 2 m is obvious, see Figure 8.8.+ ( i t - n), nExample: Shannon Wavelets. The Shannon wavelets are impulse responses of ideal bandpass filters:+(t) =sin ;t3n7cos -t.2(8.91)In the frequency domain this is{1for n5IwIW J ) = 0 otherwise.5 2n,The scaling function that belongs to the Shannon waveletisresponse of the ideal lowpass:(8.92)the impulse(8.93)3:@(W)=(8.94){1 for 0 5 IwI0 otherwise.5 n,(8.95)2318.5.
The DiscreteWavelet Transform ( D W T )IIl-2n1-2nW1-nnFigure 8.9. Subspaces of Shannon wavelets.The coefficients c m ( n ) , m,n E Z in (8.85) can be understood as the samplevalues of the ideally lowpass-filtered signal. Figure 8.9 illustrates the decomposition of the signal space.The Shannon wavelets form an orthonormal basis for Lz(lR,). The orthogonality between different scales is easily seen, because the spectra do notoverlap.
For the inner product of translated versions of +(t)at thesame scale,we get00+(t- m)+*(t - n) =-2.rrS"-"@(w)@*(w)e-J'(m-n)wdw(8.96)by using Parseval's relation. The orthogonality of translated wavelets at thesame scale is shown using a similar derivation.A drawback of the Shannon waveletsis theirinfinite support and the232TransformChapter 8. Waveletpoor time resolution due to the slow decay. On the other hand, thefrequencyresolution is perfect.
For the Haarwavelets, we observed the oppositebehavior.They had perfect time, but unsatisfactory frequency resolution.8.5.2Wavelet Analysis by Multirate FilteringBecause of V0 = V1 @ W1 the functions $on@) = $(t - n ) E VO,n E Z canbe written as linear combinations of the basis functions for the spaces V1 andW1. With the coefficients h o ( 2 l - n) and hl(2l - n ) , l,n E Z the approach is4on(t) =C h o ( 2 l - n) $lt(t) + h1 ( 2 -~ n ) $lt(t).(8.97)eEquation (8.97) is knownas thedecomposition relation,for which the followingnotation is used as well:&i4 ( 2 t - n) = C h o ( 2 L - n) $(t - l) + h1(2c - n) $(t - l).(8.98)eWe now consider a known sequence { c o ( n ) } , and we substitute (8.97) into(8.85) for m = 0.
We getWe see that the sequences {crn+l(l)} and {d,+l(l)} occurwith half thesampling rate of {crn(.)}. Altogether, the decomposition(8.100) is equivalentto a two-channel filter bank analysis with the analysis filters h0 ( n ) and h1 (n).8.5. The DiscreteWavelet Transform ( D W T )233UFigure 8.10. Analysis filter bank for computing the DWT.If we assume that q,(t) is a sufficiently good approximation of ~ ( t )and,ifwe know the coefficients co(n), we are able to compute thecoefficientscm+1(n), &+l ( n ) , m > 0, and thus the values of the wavelet transformusing the discrete-time filter bank depicted in Figure 8.10. This is the mostefficient way of computing the DWT of a signal.8.5.3Wavelet Synthesis by Multirate FilteringLet us consider two sequences gO(n) and g1(n), which allow us to express thefunctions $lo(t) = 2-1/2$(t/2) E V1 and$lo(t) = 2-1/2$(t/2) E W1 aslinear combinations of $on(t) = $(t - n ) E VO, n E Z in the formor equivalently as(8.102)Equations (8.101) and (8.102), respectively, are referred to as the two-scalerelation.
For time-shifted functions the two-scale relation is(8.103)234Chapter 8. WaveletTransformFrom (8.103), (8.78), (8.85) and (8.90) we deriveThe sequences gO(n) and g1(n) may be understood as the impulse responsesof discrete-time filters, and (8.105) describes a discrete-time two-channelsynthesis filter bank. The filter bank is shown in Figure 8.11.8.5.4The Relationship between Filters andLet us consider the decomposition relation (8.97), that isWavelets8.5. TheWaveletDiscreteTransform ( D W T )235Taking the inner product of (8.106) with &(t) and qle(t) yields(8.108)(8.109)(8.110)Substituting (8.101) into (8.108) yields(8.111)(8.112)236TransformChapter 8. WaveletThe conditions (8.112) arenothingbutthePR conditions for criticallysubsampledtwo-channel filter banks,formulated in thetimedomain,cf.Section 6.2.By z-transform of (8.112) we obtain2 = Go(z) Ho(z) + Go(-z) Ho(-z),2 = Gl(z) HI(z)+ GI(-z) HI(-z),0 = Go(z) Hl(z)+ Go(-z)(8.113)Hl(-z),+0 = Gl(z) H o ( z ) G1(-z) H o ( - z ) .Orthonormal Wavelets.
If the sets $mn(t)and ~ m n ( tm,) , n E Z accordingto (8.51)and (8.89)are orthonormal bases for V, and W,, m E Z,(8.109)becomesh o w - n) = @On, 41,),(8.114)h 1 W - n) = @On, $1,).Substituting the two-scale relation (8.103)into (8.114)yieldsObserving(+On, + O k )= dnk, we deriveHo(z) = G o ( z ) ,ho(n) = g;(%)h1(n)= g;(%)t)(8.116)Hl(2) = Gl(2).Thus equations (8.112) and (8.113)become(8.117)n0 = C g 1 ( n ) go*(.
- 21)nand2 = Go(.) G o ( z ) +Go(-.)+Go(z) G ~ ( z+) Go(-z)Gl(z) G o ( z ) + GI(-z)Go(-z),2 = G l ( z ) G ~ ( z ) G~(-z) GI(-z),0 =0 =GI(-z),Go(-z).(8.118)237from8.6. WaveletsThese are nothing but the requirementsbanks, as derived in Chapter 6.8.68.6.1for paraunitary two-channel filterWaveletsfrom Filter BanksGeneral ProcedureIn the previous sections we assumed that the wavelets and scaling functionsare given. Due to theproperties of the wavelet transform we were able to showthe existence of sequences h0 ( n ) h1, ( n ) go, ( n ) ,and g1 ( n ) ,which allow us torealize the transform via a multirate filter bank. When constructing waveletsand scaling functions one often adopts the reverse strategy.
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