Wavelet Transform (779450), страница 2

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Specifically, wewill look at integral reconstruction from the entire time-frequency plane andat a semi-discrete reconstruction.8.3.1IntegralReconstructionAs will be shown, the inner product of two signals ~ ( tand) y(t) is related tothe inner product of their wavelet transforms as218Chapter 8. WaveletTransformwith C, as in (8.2).Given the inner product (8.21), we obtain a synthesis equationby choosingy t ( t ' ) = d(t' - t ) ,(8.22)because then the following relationship holds:mf ( t ' ) d(t'(X7Yt)=J-m(8.23)- t ) dt' = z ( t ) .Substituting (8.22) into (8.21) givesFrom this we obtain the reconstruction formulaz ( t )='ScQc,/m(T)W z ( b 7 a )lal-; .JI t - bda db(8.24)-cQ -cQProof of (8.2) and (8.21). WithP,(W)= X ( W ) !P*(wa)(8.25)equation (8.7) can be written asW z ( b ,a ) = la13 27rUsing the correspondence P, ( W )-mP,(w) ejwbdw.(8.26)p,(b) we obtaint)(8.27)Similarly, for the wavelet transform of y ( t ) we get&,(W)= Y ( w )Q * ( w ~ )~a(b),(8.28)(8.29)8.3.

Integral and Semi-DiscreteReconstruction219Substituting (8.27) and (8.28) into the right term of (8.21) and rewriting theobtained expression by applying Parseval's relation yields(8.30)By substituting W = vu we can show that the inner integral in the last line of(8.30) is a constant, which only depends on $(t):da =[ldw.(8.31)IWIHence (8.30) isThis completes the proof of (8.2) and (8.21).8.3.20Semi-Discrete Dyadic WaveletsWe speak of semi-discrete dyadic wavelets if every signal z ( t ) E Lz(IR,) canbe reconstructed from semi-discrete values W , ( b , a m ) , where am, m E Z aredyadically arranged:a, = 2,.(8.33)That is, the wavelet transform is calculated solely along the lines W,(b, 2,):ccW , ( t 1 , 2 ~ )= 2 - tz ( t ) $*(2-,(t- b))dt.(8.34)220TransformChapter 8. WaveletThe center frequencies of the scaled wavelets are(8.35)withWOaccording to (8.9).

The radii of the frequency windows areA,- = 2-mm E Z.A,,(8.36)amIn order to ensure that neighboring frequency windowsanddo adjoin, we assumeWO=3Au.(8.37)This condition can easily be satisfied, because by modulating a given wavelet&(t)the center frequency can be varied freely. From (8.33), (8.35) and (8.37)we get for the center frequencies of the scaled wavelets:wm = 3 * 2-m A,,m E Z.(8.38)Synthesis. Consider the signal analysis and synthesis shown in Figure 8.5.Mathematically, we have the following synthesis approach using a dual (alsodyadic) wavelet ( t ):4ccc~ ( t=)cc2-4mW 3 C ( b , 2 m4(2-"(t)- b ) ) db.(8.39)m=-mIn orderto express the required dual wavelet 4(t)by t)(t),(8.39) is rearranged8.3.

Integral and Semi-Discrete Reconstruction221as~ ( t )=EE2-im/-00 W,(b, 2m)4(2-m(t - b ) ) dbm=--00=-cc2-:m(W“ ( . , 2 r n ) , 4 * ( 2 F ( t- . )))m=--00For the sum in the last row of (8.40)cq*(2mw)6(2mw)=1(8.41)m=-ccm=-ccIf two positive constants A and B with 0 < Ac5 B < cc exist such thatccA51Q(2mw)125 B(8.43)m=-ccwe achieve stability. Therefore, (8.43) is referred to as a stability condition.A wavelet +(t) which satisfies (8.43) is called a dyadic wavelet. Note thatbecause of (8.42), for the dual dyadic wavelet, we have:1B-cc1(8.44)m=-ccThus, for * ( W ) according to (8.42) we have stability, provided that (8.43) issatisfied.

Note that the dual wavelet is not necessarily unique [25]. One mayfind other duals that also satisfy the stability condition.222Chapter 8. Wavelet Transform2- T**(-t/z")IWx(t,2")2-3%~(t/~)Figure 8.5. Octave-band analysis and synthesis filter bank.Finally it will be shown that if condition (8.43) holds the admissibilitycondition (8.2) is also satisfied. Dividing (8.43) by W andintegratingtheobtained expression over the interval (1,2) yields:Wit h(8.46)we obtain the following result for the center term in (8.45):(8.47)Thus(8.48)Dividing (8.43) by-Wand integrating over (-1, -2) givesA In2 5LccWdw5 B ln2.(8.49)Thus the admissibility condition (8.2) is satisfied in any case, and reconstruction according to (8.24) is also possible.2238.4. Wavelet Series8.48.4.1Wavelet SeriesDyadic SamplingIn this section, we consider the reconstruction from discrete values of thewavelet transform.

The following dyadically arranged samplingpoints areused:a, = 2,,b,, = a, n T = 2,nT,(8.50)This yields the values W , (b,,,sampling grid.= W , (2,nT, 2,).a,)Figure 8.6 shows theUsing the abbreviation(8.51)-2 - f . $(2Trnt - n T ) ,we may write the wavelet analysis asThe values {W, (2,nT, 2,), m, n E Z} form the representation of z ( t ) withrespect to the wavelet $(t) and the chosen grid.Of course, we cannotassume that anyset lClmn(t),m, n E Z allowsreconstruction of all signals z ( t ) E L2(R). For this a dual set t+&,,(t),m, n E Zmust exist, and both sets must spanL2(R). The dual setneed not necessarilybe built from wavelets.However, we are only interested in the case whereqmn(t)is derived ast+&,,(t)= 2 - 7 *t+&(2-Y- n T ) ,m, n EZfrom a dual wavelet t+&(t).If both sets $ m n ( t )and Gmn(t)with m, n Ethe space L2(R), any z ( t ) E L2(R) may be written as(8.53)Z span(8.54)Alternatively, we may write z ( t ) as(8.55)224TransformChapter 8.

Wavelett. .WOlog a~..........................................................*..... ........*. ..*. . ...*.. .. . ......... ......................................... . . . ........"-...... .......................................................................................................... . . . . .

. . . . . . ...m=-2m=-lm=Om= 1m=2b -Figure 8.6. Dyadic sampling of t h e wavelet transform.For a given wavelet $(t),the possibility of perfect reconstruction is dependent on thesampling interval T . If T is chosen very small (oversampling), thevalues W , (2"nT, 2"), m, n E Z are highly redundant, and reconstruction isvery easy. Then the functions lClrnn(t),m,n E Z are linearly dependent, andan infinite number of dual sets qrnn(t)exists. The question of whether a dualset Gmn(t) exists at all can be answered by checking two frame bounds' Aand B . It can be shown that the existence of a dual set and the completenessare guaranteed if the stability conditionMM(8.56)with the frame bounds 0 < A I B < CO is satisfied [35]. In the case of atight frame, A = B, perfect reconstruction with Gmn(t) = lClrnn(t)is possible.This is also true if the samples W , (2"nT, 2") contain redundancy, that is, ifthe functions qmn(t),m, n E Z are linearly dependent.

The tighter theframebounds are, the smaller is the reconstruction error if the reconstruction iscarried out according toIf T is chosen just large enough that the samples W , (2"nT, 2"), m, n E Zcontain no redundancyat all (critical sampling), thefunctions $mn(t),m, n EZ are linearly independent. If (8.56) is also satisfied with 0 < A 5 B < CO,the functions tjrnn(t),m, n E Z form a basis for L2 (R).

Then the followingrelation, which is known as the biorthogonality condition, holds:(8.58)Wavelets that satisfy (8.58) are called biorthogonalwavelets.As a specialcase, we have the orthonormal wavelets. They are self-reciprocal and satisfy2The problem of calculating the frame bounds will be discussed at theend of this sectionin detail.2258.4. Wavelet Seriesthe orthonormality condition($mn,$lk)= &m1 b n k ,m,n,1, IC E Z.(8.59)m,n E Z can be usedThus, in the orthonormal case, the functions q!Imn(t),for both analysis and synthesis. Orthonormal bases always have the sameframe bounds (tight frame), because, in that case, (8.56) is a special form ofParseval’s relation.8.4.2Better Frequency ResolutionOctaves-Decomposition ofAn octave-band analysis is often insufficient.

Rather, we would prefer todecompose every octave into M subbands in order to improve the frequencyresolution by the factor M .We here consider the case where the same sampling rate is used for all Msubbands of an octave. This corresponds to a nesting of M dyadic waveletanalyses with the scaled waveletsq!I@)(t)= 2A q!I(2Xt),k = 0,1,...

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