Vetterli M. And Le Gall D., "Perfect Reconstruction FIR Filter Banks : some properties and factorizations", IEEE Transaction on Acoustic, Speech and signal Processing, vol. ASSP-37, pp. 1057-1071, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....into blocks of M samples, the LOT has M basis functions of length 2M and therefore generates M coe#cients. Indeed, the LOT is a subclass of maximally decimated M channel finite impulse response (FIR) paraunitary (PU) filter banks (FB s) This fact was firstly pointed out by Vetterli and Le Gall [7]. From the viewpoint of PU FB s, the basis functions of the LOT corresponds to the analysis and the synthesis filters. Moreover, Malvar provided a linear phase (LP) solution for the LOT [1] which is essential for image processing and can be expressed by a lattice structure which o#ers fast ....

M. Vetterli and D. L. Gall, "Perfect reconstruction FIR filter banks: Some properties and factorizations, " IEEE Trans. Signal Processing, vol. 37, pp. 1057--1071, July 1989.

.... and Daubechies [5] Since then, it becomes very popular and has been introduced to still image compression [6] and video compression [1, 2, 7, 8] It can be mathematically related to the subband coding [9, 10] Laplacian pyramid coding [11] and the transform coding as reported in some literatures [5, 12, 13, 14]. This means that developed strategies for each coding technique can be introduced into the wavelet transform straightforwardly. A point lies in their skillful modification utilizing properties of the wavelet transform. There are a lot of works which treat the wavelet transform as the subband ....

M.Vetterli and D.L.Gall: "Perfect reconstruction FIR filter banks: some properties and factor- izations", IEEE Trans. Acoust., Speech, Signal Processing, Vol. ASSP-37, No. 7, pp.1057-1071, July. 1989.

....al. HIGH PERFORMANCE COMPRESSION OF VISUAL INFORMATION 979 Then Smith and Barnwell [31] solved the problem of perfect reconstruction filter banks for a 1 D multirate system. Following these studies, substantial research effort has been devoted to perfect reconstruction filter banks theory [32] [34], which was then extended to the two dimensional (2 D) case by Vetterli [35] Applications of SBC to images was introduced by Woods and O Neil [36] by means of 2 D separable quadrature mirror filter (QMF) banks. SBC consists of the following three steps: 1) subband decomposition; 2) quantization; ....

M. Vetterli and D. Le Gall, "Perfect reconstruction FIR filter banks: Some properties and factorization," IEEE Trans. Acoustics, Speech, Signal Processing, vol. 37, pp. 1057--1071, July 1989.

....blocks of M samples, the LOT has M basis functions of length 2M and therefore generates M coefficients. Indeed, the LOT is a subclass of maximally decimated M channel finite impulse response (FIR) paraunitary (PU) filter banks (FB s) This fact was firstly pointed out by Vetterli and Le Gall [7]. From the viewpoint of PU FB s, the basis functions of the LOT corresponds to the analysis and the synthesis filters. Moreover, Malvar provided a linear phase (LP) solution for the LOT [1] which is essential for image processing and can be expressed by a lattice structure which o#ers fast ....

M. Vetterli and D. L. Gall, "Perfect reconstruction FIR filter banks: Some properties and factorizations," IEEE Trans. Signal Processing, vol. 37, pp. 1057--1071, July 1989.

....of compactly supported wavelets originated both from mathematical analysis and the signal processing community. The roots of critically sampled wavelet transforms are actually older than the word wavelet and go back to the context of subband filters, or more precisely quadrature mirror filters [36, 37, 41, 51, 52, 53, 54, 58, 56, 59]. In mathematical analysis, wavelets were defined as translates and dilates of one fixed function and were used to both analyze and represent general functions. 15, 20, 25, 35, 24] In the late eighties the introduction of multiresolution analysis and the fast wavelet transform by Mallat and ....

M. Vetterli and D. Le Gall. Perfect reconstruction FIR filter banks: Some properties and factorizations. IEEE Trans. Acoust. Speech Signal Process., 37:1057--1071, 1989.

....and thus we divide them into two categories accordingly. The first category consists of schemes such those proposed by Smith and Barnwell [SI84, SI86] and Vaidyanathan et al. [VH88] that use FIR filters that are not linear phase. The second category consists of schemes such as those reported in [ABMD90, VH88, VD89, NV89, VH90, VG89, RV91, ABMD90] that use FIR linear phase filters that give rise to a non orthogonal wavelet expansion, because they do not satisfy power complementarity. Note that approximate power complementarity implies approximate orthonormality of the underlying wavelet basis. It is not clear that the filters described ....

M. Vetterli and D. Le Gall. Perfect reconstruction FIR filter banks: some properties and factorizations. IEEE Transactions ASSP, 37-7:pp. 1057--1071, July 1989.

....perfect reconstruction filter banks if they reconstruct perfectly the input signal at the output when processing is omitted between the downsampling in the decomposition and the upsampling in the reconstruction. The theory of the filter banks was developed in many previous works (e.g. 17] [18], 19] 10] 6] 24] 22] 23] 21] Here we present a new methodology allowing more flexibility in choosing perfect reconstruction filter banks. 1.2 Wavelet Decomposition Wavelet transform applied to signal and image processing results in a recursive filtering algorithm, in which a ....

....are given in Section 4. 4 2 Theory of Orthogonal Subband Perfect Reconstruction Filter Banks 2. 1 Background on the Zak Transform The Zak theory for signal processing was developed in [20] 16] 15] and [21] The Zak transform has the same relation to the polyphase representation (see [17] [18], 19] as the Fourier transform to the Z transform. In [21] we showed that the Zak transform is very suitable for treating the filter bank theory without having exceptions for undersampling and unstable cases. By undersampling we mean a case when the decimation factor is larger than the number of ....

M. Vetterli, D. Le Gall, "Perfect Reconstruction FIR Filter Banks: Some Properties and Factorizations", IEEE Trans. on Acoust., Speech, and Signal Processing, vol. 37, pp. 1057-- 1071, July 1989.

.... [10] Partial results on factorization of general PR filter banks have been recently reported where, although the proposed factorization characterizes a large class of PR filter banks, it lacks completeness [11] Such structures have also been proposed for the bi orthogonal linear phase filters [13]. With the use of ladder structures, it can be shown that the factorization of general M channel bi orthogonal PR filter banks is possible. This factorization even though is complete lacks uniqueness [4] In [14] Vetterli and Herley demonstrated the close relation between the continued fraction ....

....= fiz 2n Gamma1 : Define G(z) e H(z) Gamma ff fi H(z) It follows that H(z)G( Gammaz) Gamma H( Gammaz)G(z) 0, which implies that G(z) is at most of order n Gamma 2. A necessary condition for PR is that H(z) and H( Gammaz) be relatively prime (they cannot have common zero) [13] so that the only solutions to the above equation are either G(z) 0 and G(z) H(z) The latter is not possible since G(z) is at most of order n Gamma 2, and thus G(z) 0 and H(z) fi ff e H(z) End of Proof The statement of the above lemma is also evident from the Euclid algorithm and ....

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M. Vetterli and D. J. LeGall, " Perfect reconstruction FIR filter banks: Some properties and factorizations ", IEEE Trans. on ASSP, Vol. ASSP-37, No. 7, pp. 1057-1071, July 1989.

.... [1] Partial results on factorization of general PR filter banks have been recently reported where, although the proposed factorization characterizes a large class of PR filter banks, it lacks completeness [2] Such structures have also been proposed for the bi orthogonal linear phase filters [3]. With the use of ladder structures, it can be shown that the factorization of general M channel bi orthogonal PR filter banks is possible. This factorization even though is complete lacks uniqueness [4] In [5] Vetterli and Herley demonstrated the close relation between the continued fraction ....

M. Vetterli and D. J. LeGall, "Perfect reconstruction FIR filter banks: Some properties and factorizations," IEEE Trans. on ASSP, Vol. ASSP-37, No. 7, July 1989.

....fields regularized reconstruction [1] order statistics filters [2, 3] with median filters as a particular case [4, 5, 6] Those methods work in the pixel domain, with the general assumption that an image is made of a set of uniform regions and edges. In the particular case of subband images [7, 8] corrupted by channel noise, the salt and pepper noise appears on each subband coefficient, and translates into modulations of the filters basis functions at the synthesis stage, appearing randomly on the reconstructed image. It seems interesting to try to use this knowledge to correct the ....

Martin Vetterli and Didier Le Gall. Perfect reconstruction FIR filter banks : some properties and factorizations. IEEE Transactions on Acoustics, Speech and Signal Processing, 37(7):1057--1064, July 1989.

....and their biorthogonal counterparts being compactly supported. This corresponds to the fact that only a finite number of square blocks both in A and in A that generates C( A) C(A) Gamma1 are non zero. It is well known, particularly in the filter bank context (see, e.g. 12] [13]) that this happens if and only if there exist a non zero constant ff and an integer p such that det( X k2ZZ A k z Gammak ) ffz Gammap for any z 2 CI , z 6= 0. Because determinant is the product of singular values, the equation above implies that Y j oe j (A( fi for some ....

M. Vetterli and D. Le Gall, Perfect reconstruction FIR filter banks: Some properties and factorizations, IEEE Trans. on Accoustics, Speech and Signal Processing, ASSP-37 (1989), pp. 1057--1071.

....coefficients of refinement equations, respectively. The requirement of bandedness corresponds to both the analysis and synthesis filters being FIR (finite impulse response) in the case of wavelets this is necessary if bases consisting of functions with compact support are to be developed ( 1] 4] [5]) School of Information Science and Technology, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia. February 28, 1994 2 In the special case of orthogonal banded block circulant matrices several possible complete characterizations are known ( 2] 3] 4] 5] They are usually ....

....be developed ( 1] 4] 5] School of Information Science and Technology, Flinders University, GPO Box 2100, Adelaide, SA 5001, Australia. February 28, 1994 2 In the special case of orthogonal banded block circulant matrices several possible complete characterizations are known ( 2] 3] 4] [5]) They are usually based on factorizations of corresponding block sequences into products of short factors. This makes it possible to construct such matrices in an easy way and even to include additional constraints. A necessary and sufficient condition for the existence of banded inverse of a ....

[Article contains additional citation context not shown here]

M. Vetterli and D. Le Gall. Perfect reconstruction FIR filter banks: Some properties and factorizations. IEEE Trans. on Accoustics, Speech and Signal Processing, ASSP-37:1057--1071, 1989.

....filters. For example, the jth row of A i is (h j [2N Gamma 1 Gamma iN ] h j [N Gamma iN ] for i = 0; 1. Note that the filter length is twice the number of channels. For an orthogonal, perfect reconstruction solution, the matrix T has to be unitary, which is equivalent to the following [19]: A T 0 A 0 A T 1 A 1 = I; A T 1 A 0 = A T 0 A 1 = 0: The second set of conditions above are called the orthogonality of tails conditions [19] One more fact will be of use later. Call B i the blocks when no windowing is used, or w 0 [n] 1, n = 0; 2N Gamma 1. That is, ....

....number of channels. For an orthogonal, perfect reconstruction solution, the matrix T has to be unitary, which is equivalent to the following [19] A T 0 A 0 A T 1 A 1 = I; A T 1 A 0 = A T 0 A 1 = 0: The second set of conditions above are called the orthogonality of tails conditions [19]. One more fact will be of use later. Call B i the blocks when no windowing is used, or w 0 [n] 1, n = 0; 2N Gamma 1. That is, these blocks will just contain the cosines. Then A 0 = B 0 Delta W; A 1 = B 1 Delta JWJ; 3) where W is a diagonal matrix with window coefficients on the ....

M. Vetterli and D.J. LeGall, "Perfect reconstruction FIR filter banks: Some properties and factorizations," IEEE Trans. Acoust., Speech, and Signal Proc., vol. 37, no. 7, pp. 1057--1071, July 1989.

....Possible design objectives of filter banks and an illustration of the design objectives of a single low pass filter. 0; p ] is the pass band; s ; the stop band; p ; s ] the transition band. designs, and all existing PR LP designs in signal processing are based on real coefficients [1, 10, 24, 12]. In a general two band filter bank, the reconstructed signal is: X(z) X(z) 2 [H 0 (z)F 0 (z) H 1 (z)F 1 (z) 1) X( Gammaz) 2 [H 0 ( Gammaz)F 0 (z) H 1 ( Gammaz)F 1 (z) where X(z) is the original signal, and H i (z) and F i (z) are, respectively, the response of the analysis and ....

M. Vetterli and D. Le Gall. Perfect reconstruction FIR filter banks: Some properties and factorizations. IEEE Trans. on Acoutics, Speech, and Signal Processing, 37(7):1057--1071, July 1989.

....from the orthogonal wavelet space of rank 2 and genus g to the (g Gamma 1) dimensional real torus. In Section 4 we study a fast wavelet transform implementation based on factorization. We conclude the paper in Section 5. 2. A REVIEW OF PREVIOUS WORK Most of the material in this section is from [1, 16, 20, 26, 27, 29, 34, 35, 36]. We refer to these papers and books for more details. 2.1. Laurent Polynomial and Polyphase Decomposition For a given sequence a = fa k ; k 2 Zg which has only finite nonzero elements, the Laurent polynomial a(z) of a is defined by a(z) X k2Z a k z Gammak = k2 X k=k1 a k z Gammak ....

M. Vetterli and D. Le Gall. Perfect reconstruction FIR filter banks: Some properties and factorizations. IEEE Trans. ASSP, 37(7):1057--1071, July 1989.

....functions and their biorthogonal counterparts being compactly supported. This corresponds to the fact that only a finite number of square blocks both in A and in A that generate C( A) C(A) # ) 1 are nonzero. It is well known, particularly in the filter bank context (see, e.g. 12] [13]) that this happens if and only if there exist a nonzero constant # and an integer p such that det X k#Z A k z k = #z p 992 RADKA TURCAJOV A for any z # C, z #= 0. Because the determinant is the product of singular values, the equation above implies that Y j # j (A(#) # for ....

<F3.993e+05> M. Vetterli and D. Le<F4.039e+05> Gall,<F4.112e+05> Perfect reconstruction FIR filter banks: Some properties and<F4.039e+05> factorizations, IEEE Trans. Acoust., Speech Signal Process., 37 (1989), pp. 1057--1071.

....Zak Transform, Projection, Orthogonalization, Wavelets, Biorthogonality, Perfect Reconstruction. EDICS Categories: SP 2.4, SP 2.2.1, SP 2.2. 2 1 Introduction Considerable effort has been made to find good subband filtering algorithms that allow perfect reconstruction of signals, e.g. see [6] [7], 19] 13] 14] In recent years the development of wavelet theory gave rise to new types of perfect reconstruction filter banks. The wavelet basis function upon which the signals or images are decomposed can be orthogonal or biorthogonal. The orthogonality condition leads to tight restrictions ....

....We believe that the discrete Zak transform deserves no less attention than the discrete Fourier transform in signal analysis. There is a close similarity between the discrete Zak transform and the polyphase representation. For the definition and properties of the polyphase representation see [6] [7]. One can conceive of the Zak Transform as the polyphase representation in the case of periodic signals. We emphasize this similarity, and, with the help of the Zak Transform are acquiring new insights into the theory of filter banks. We present also a technique for finding biorthogonal ....

[Article contains additional citation context not shown here]

M. Vetterli, D. Le Gall, "Perfect Reconstruction FIR Filter Banks: Some Properties and Factorizations", IEEE Trans. on Acoust., Speech, and Signal Processing, vol. 37, pp. 1057-- 1071, July 1989.

....matrix. We may therefore also say that we deal with unitary banded block circulant matrices. Matrices of this kind are, explicitly or implicitly, used in many fields. In different contexts the rows of A can be time reversed impulse responses of an m channel paraunitary FIR filter bank ( 9] [10]) or, when subjected also to basic regularity conditions (sums of the elements of each row but the first vanish) A can be a matrix of coefficients of a discrete higher multiplicity (m band) orthogonal wavelet transform ( 2] 3] 4] Both these concepts are generalizations of the classical case ....

M. Vetterli and D. Le Gall. Perfect reconstruction FIR filter banks: Some properties and factorizations. IEEE Trans. on Accoustics, Speech and Signal Processing, ASSP-37:1057-- 1071, 1989.

....and their biorthogonal counterparts being compactly supported. This corresponds to the fact that only a finite number of square blocks both in A and in A that generates C( A) C(A) Gamma1 are non zero. It is well known, particularly in the filter bank context (see, e.g. 12] [13]) that this happens if and only if there exist a non zero constant ff and an integer p such that det( X k2ZZ A k z Gammak ) ffz Gammap for any z 2 CI , z 6= 0. Because determinant is the product of singular values, the equation above implies that Y j oe j (A( fi 12 R. TURCAJOV ....

M. Vetterli and D. Le Gall, Perfect reconstruction FIR filter banks: Some properties and factorizations, IEEE Trans. on Accoustics, Speech and Signal Processing, ASSP-37 (1989), pp. 1057--1071.

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Vetterli M. And Le Gall D., "Perfect Reconstruction FIR Filter Banks : some properties and factorizations", IEEE Transaction on Acoustic, Speech and signal Processing, vol. ASSP-37, pp. 1057-1071, 1989.

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M. Vetterli and D. Le Gall, "Perfect ReconstructionFIR Filter Banks: Some Properties and Factorizations," IEEE Trans. on Acoustics, Speech, and Signal Processing, pp. 1057--1071, July 1989.

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M. Vetterli and D. L. Gall, "Perfect reconstruction FIR filter banks: Some properties and factorizations," IEEE Trans. on Acoustic, Speech and Signal Processing 37, p. 1057, July 1989.

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M. Vetterli and D. L. Gall, "Perfect reconstruction FIR filterbanks: Some properties and factorizations," IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 1057--1071, July 1989.

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M. Vetterli and D. Le Gall, "Perfect reconstruction FIR filter banks: some properties and factorizations," IEEE Trans. on ASSP 37(7), pp. 1057--1071, 1989.

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M. Vetterli and D. Le Gall. "Perfect reconstruction FIR filter banks: Some properties and factorizations". IEEE Trans. on Accoustics, Speech and Signal Processing, ASSP-37:1057--1071, 1989.

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