Filter Banks (779443), страница 6
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. . ,P e l v + ( ~ - ~ ) ( - z ~ ’ )(6.138)},Qe(z2’) = diag{Qelv+(lv-l)(-z2’), . . . , Q ~ N + I ( - ~ ~ Qelv(-z2’)}.’),(6.139)The superscript ( p ) indicates the oversampling factor. Requiring)(.&))(.=.-P(6.140)for perfect reconstruction yields [86](6.141)and9 + e l v ( z ) Q~+k+elv(z)- P ~ + k + e l v ( zQk+elv(z))L0for L = 0,.. . ,N- 1;l = 0,.(6.142). . , p - 1.
The delay qp) is related to S asqp) = 2ps+ 2p(6.143)- 1,and the overall delay amounts toq=N-l+qt)N.(6.144)As we see, these conditions offer increased design freedom for an increasingoversampling rate. This is further discussed in [86], where solutions based ona nullspace approach are presented.184BanksChapter 6. FilterIf we restrict an oversampled cosine-modulated filter bank to be paraunitary, that is, d p ) ( z ) E ( p ) ( z )= I N , we get the following constraints on theprototype P ( z ) [85, 861:Interestingly, for pQ ( z ) such that>1, we still may choose different prototypes P(,) andwithExample.
We consider a 16-bandfilter bank with linear-phase prototype andan overall delay of 255 samples. Figure 6.27 shows a comparison of frequencyresponses for the critically sampled and the oversampled case. It turns outthat the PR prototype for the oversampled filter bank has a much higherstopband attenuation. This demonstrates theincreased design freedom in theoversampled case.6.6.4Pseudo-QMF BanksIn pseudo-QMF banks, one no longer seeks perfect reconstruction, but nearlyperfect reconstruction.
Designing a pseudo-QMF bank is done as follows [127].One ensures that the aliasing components of adjacent channels compensateexactly. This requires power complementarity of frequency shifted versions ofthe prototype, as illustratedin Figure 6.28. Furthermore, one triesto suppressthe remaining aliasing components by using filters with very high stopbandattenuation.
Through filter optimization the linear distortions are kept assmall as possible. Anefficientdesignmethod was proposed in [166].Sincethe constraints on the prototype are less restrictive than in the PR case, theprototypes typically have a higher stopband attenuation than the P R ones.1856.6. Cosine-Modulated Filter Banks0 ,-20 -40 -60 -g-80 -100 -\::::I........................................ . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .-120 -140 -160 I0I0.60.20.4n/rC0.8I1(40. . . . . . :. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . ......................................00.60.20.4n/rC0.8.41(b)Figure 6.27. Frequencyresponses of 16-channel prototypes. (a) critical subsampling; (b) oversampling by p = 2.0'HiMFigure 6.28. Design of pseudo-QMF banks.1866.7BanksChapter 6. FilterLapped OrthogonalTransformsLapped orthogonal transforms (LOTs)were introduced in [21] and have beenfurther studied in [93, 97, 21.
Unlike block transforms, they have overlappingbasis functions, which better allow us to smooth out blocking artifacts incoding applications. LOTs may also be seen as a special kind of criticallysubsampled paraunitary filter banks. Typically, an overlap of one blockisconsidered, which means that the basis functions are of length L = 2M whenthe number of channels is M. Longer transforms have been designed in [l181and are called generalized lapped orthogonal transforms (GenLOTs). Morerecently, biorthogonal lapped transforms have also been proposed [96, 1441.Figure 6.29. Transform matrix of a lapped orthogonal transform.Figure 6.29 illustrates the structure of the transform matrix of a lappedorthogonal transform.Like in an M-channelfilter bank with length-2Mfilters,2M input samples are combined in order to form M transform coefficients.We will first consider the constraints on the M X M submatrices POand P I .From the condition of orthogonality,T ~ T = T T ~ = I(6.146)it follows thatP,TPo+PTP1=PoPoT+PIPT=IMxM(6.147)P; P1 = PO PT = O M x M .(6.148)and+Now let B = PO P I .
Note that B is orthogonal if PO and P1 satisfythe above conditions. Moreover, POP: and P I P : are orthogonalprojectionsonto two subspaces that are orthogonal to one another. Define A = POP:and verify that PlPT = I - A. Thus,PO = A B ,P1 = [ I - A ] B .(6.149)Orthogonal6.7. LappedTransforms187The most general way of constructing LOTS is to start with two matricesA and B , where A is a projection and B is orthogonal.The desiredmatrices P O and P1 are then foundfrom(6.149).Thismethod,however,does not automatically yield linear-phase filters, which are desired in manyapplications.In [98], a fast linear-phase LOT based on the DCT was presented, whichwill be briefly explained in the following. For this, let D e and Do be matricesthat contain the rows of the transposed DCT-I1 matrix with even and oddsymmetry, respectively.
Then,is a LOT matrix that already satisfies the above conditions. J is the counteridentity matrix with entriesJi,k = & + - l ,i = 0,1, . . . ,N - 1. In anexpression of the form X J , it flips the columns of X from left to right.Due to the application of J in (6.150), the first M / 2 rows of Q(') have evenand the last M / 2 rows have odd symmetry. A transform matrix with betterproperties (e.g.
for coding) can be obtained by rotating the columns of Q(')such thatQ = 2 Q('),(6.151)where 2 is unitary. For the fast LOT, 2 is chosen to contain only three planerotations, which help to improve the performance, but do not significantlyincrease the complexity. The matrix Q(o)already has a fast implementationbased on the fast DCT. See Figure 6.30 for an illustration of the fast LOT.The angles proposed by Malvar are O1 = 0 . 1 3 ~O2~= 0 . 1 6 ~and~ O3 = 0 .
1 3 ~ .0123456lFigure 6.30. The fast lapped orthogonal transform for M = 8 based on the DCTand three plane rotations.1886.8BanksChapter 6. FilterSubband Coding of ImagesTwo-dimensional filter banks for the decomposition of images can be realizedas separable and non-separable filter banks.For the sake of simplicity, we willrestrict ourselves to the separable case. Information on non-separable filterbanks and the corresponding filter design methods is given in [l,1541.In separable filter banks,the rows and columns of the input signal (image)are filtered successively.
The procedure is illustrated in Figure 6.31 for anoctave-band decomposition basedon cascades of one-dimensional two-channelfilter banks. In Figure6.32 an example of such an octave-band decomposition is given. Note that this decomposition schemeis also known as thediscrete wavelet transform; see Chapter 8. In Figure 6.32(b) we observe thatmost information is contained in the lower subbands. Moreover, local highfrequency information is kept locally within the subbands. These propertiesmake such filter banksvery attractive for image coding applications. In orderto achievehighcompressionratios, one quantizes the decomposed image,either by scalar quantization, orusing a technique known as embedded zerotreecoding [131, 1281; see also Section8.9.
The codewords describing the quantizedvalues are usually further compressed in a lossless wayby arithmetic orHuffman coding [76,63]. To demonstratethe characteristics of subband codingwith octave-band filter banks, Figures 6.32(c) and (d) show coding results atdifferent bit rates.Fl*FI*Fl*Fl*FLHHHLHHHLHHHvertical...Figure 6.31. Separable two-dimensional octave-band filter bank.6.9. Processing of Finite-Length Signals189Figure 6.32. Examples of subband coding; (a) original image of size 512 X 512; (b)ten-band octave decomposition; (c) coding at 0.2 bits per pixel; (d) coding at 0.1bits per pixel.6.9Processing of Finite-LengthSignalsThe term “critical sampling”, used in the previous sections, was used underthe assumption of infinitely long signals.
This assumption is justified withsufficient accuracy for audioand speech coding. However, ifwewant todecompose an image by means of a critically subsampled filter bank, we seethat thenumber of subband samplesis larger than thenumber of input values.Figure 6.33 gives an example. Ifwe simply truncate the number of subbandsamples to the number of input values - which would be desirable for coding- then PR is not possible any longer.
Solutions to this problem that yield PR190Chapter 6. Filter BanksFigure 6.33. Two-channel decomposition of a finite-length signal.with a minimum number of subband samples are discussed in the following.Circular Convolution. Assuming thatthelengthof the signal to beprocessed is a multiple of the number of channels, the problem mentionedabove can be solved by circular convolution. In this method, the input signalis extended periodically prior to decomposition [165], which yields periodicsubband signals of which only one period has to be stored or transmitted.Figures 6.34(a) and 6.34(c) give an illustration. Synthesis is performed byextendingthesubbandsignals according to theirsymmetry, filtering theextended signals, and extracting the required part of the output signal. Adrawback of circular convolution is the occurrence of discontinuities at thesignal boundaries, which may lead to annoying artifacts after reconstructionfrom quantized subband signals.Symmetric Reflection.
In this method, the input signal is extended periodically by reflection at the boundaries as indicated in Figures 6.34(b) and6.34(d), [136, 16, 23, 61. Again, we get periodic subband signals, but the periodis twice as long as with circular convolution. However, only half a period ofthe subband signals is required if linear-phase filters are used, because theylead to symmetry in the subbands. By comparing Figures 6.34(a) and (b) (or6.34(c) and (d))we see that symmetric reflection leads to smoother transitionsat the boundaries than circular convolution does. Thus, when quantizing thesubband signals, this has the effect of less severe boundary distortions.The exact procedure depends on the filter bank in use and on the signal6.9.
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