A. A. C. Kalker and I. A. Shah, "Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets," in Proc. Visual Commun. Image Process., 1992, pp. 12--20.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to design boundary filters. These are filters that are used at the boundary regions of a signal with finite duration, e.g. as in images. These filters have no overlap beyond the boundaries of the signal. Observe that the basic structure is analogous to the lifting scheme or ladder structure in [31] [33] or [34] The zero delay matrices alone would only allow us to design filter banks with the minimum system delay. To obtain a more general formulation, which also includes Fig. 3. Structure of the maximum delay matrices. orthogonal filter banks, we could additionally use paraunitary ....

A. A. C. Kalker and I. A. Shah, "Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets," in Proc. Visual Commun. Image Process., 1992, pp. 12--20.

....In [16] Dahmen and Micchelli propose a construction of compactly supported wavelets that generate complementary spaces in a multiresolution analysis of univariate irregular knot splines. ffl There are also close similarities between lifting and so called ladder structures in filter bank design [7, 30]. After finishing this paper we found that in this context a factorization result, similar to the one presented in this paper, was obtained earlier by Kalker and Shah in an unpublished manuscript [29] While our work goes into more detail concerning the non uniqueness, implementation, and ....

....scheme. Acknowledgments. The authors would like to thank Peter Schroder and Boon Lock Yeo for many stimulating discussions and for their help in computing the factorizations in the example section, Jelena Kovacevi c and Martin Vetterli for drawing their attention to the references [29] and [30], Paul Van Dooren for pointing out the connection between the M band case and the Smith normal form, and Geert Uytterhoeven and Avraham Melkman for pointing out several typos in an earlier draft. Ingrid Daubechies would like to thank NSF (grant DMS 9401785) AFOSR (grant F4962095 1 0290) ONR ....

A. A. C. Kalker and I. A. Shah. Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. In Visual Communications, Boston, pages 711--722. Proc. SPIE, 1992.

....the literature for the fast implementation of convolution and decimation. FFT implementations [12] are useful for large lters (of length 64 or 128) but do not bring any improvement over a straightforward implementation for short and medium size lter [12] Recently, several authors [13] 14] [15], 16] have proposed ecient implementations of one dimensional (1 D) biorthogonal lters using a factorization of the lters into smaller lters. In [15] the authors show that all biorthogonal lters can be factored into a sequence of ladder steps. A ladder step maps every couple of even and odd ....

....do not bring any improvement over a straightforward implementation for short and medium size lter [12] Recently, several authors [13] 14] 15] 16] have proposed ecient implementations of one dimensional (1 D) biorthogonal lters using a factorization of the lters into smaller lters. In [15] the authors show that all biorthogonal lters can be factored into a sequence of ladder steps. A ladder step maps every couple of even and odd samples (x 2k ; x 2k 1 ) into (y 2k ; y 2k 1 ) y 2k = x 2k L(x 2k 1 ) y 2k 1 = x 2k 1 (5) The proof of the factorization of 1 D biorthogonal lters ....

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A.A.C. Kalker and I.A. Shah, \Ladder structures for multidimensional linear phase perfect reconstruction lter banks and wavelets," in Visual Com. and Image Process.'92, 1992, 12-20.

.... cost of a multiplication is similar to the cost of an addition (a reasonable assumption in terms of number of cycles on RISC, and non RISC architectures) then the fast running convolutions methods do not bring any improvement over a straightforward implementation [25] Recently, several authors [11, 15, 17, 20, 27] have proposed efficient implementations of one dimensional (1 D) biorthogonal filters (perfect reconstruction filter banks) using a factorization of the filters into smaller filters. In [17] the authors show that all biorthogonal filters can be factored into a sequence of elementary ladder steps. ....

....any improvement over a straightforward implementation [25] Recently, several authors [11, 15, 17, 20, 27] have proposed efficient implementations of one dimensional (1 D) biorthogonal filters (perfect reconstruction filter banks) using a factorization of the filters into smaller filters. In [17] the authors show that all biorthogonal filters can be factored into a sequence of elementary ladder steps. Each ladder step transforms a couple (x 2k ; x 2k 1 ) of even, and odd samples of a 1 D vector two dimensional vector as follows: 0 B x 2k x 2k 1 1 C A 0 B x 2k f (x 2k 1 ....

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A.A.C. Kalker and I.A. Shah. Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. In Visual Com. and Image Process.'92, pages 12--20, 1992.

....transforms computed by a QMF bank in type 3 4 polyphase form, but here we extend the concepts to general M band transforms and also consider both type 1 2 and type 3 4 polyphase decompositions. Some of the ideas presented here are intimately related to those discussed by Kalker and Shah in [23] and [24] In these works, ladder network realizations of polyphase matrices are studied in depth. Since the primary focus of these papers was not lifting and the construction of reversible transforms, there are still some new ideas in our treatment of the subject. 3. Reversible Transforms 41 In ....

A. A. C. Kalker and I. A. Shah. Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. In Proceedings of SPIE: Visual Communications and Image Processing, volume 1818, pages 12--20, Boston, MA, November 1992.

.... Lifting is a flexible technique that has been used in several different settings, for an easy construction and implementation of traditional wavelets [32] and of second generation wavelets [33] such as spherical wavelets [26] Lifting is also closely related to several other techniques [13, 22, 37, 34, 20, 4, 15, 7, 3, 19, 28]. Rather than giving the general structure of lifting at this point, we show how to rewrite the Haar and S transforms using lifting. We rewrite (3.1) in two steps which need to be executed sequentially. First compute the difference and then use the difference in the second step to compute the ....

T. A. C. M. Kalker and I. Shah. Ladder Structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. In Proceedings of the SPIE Conference on Visual Communications and Image Processing (Boston), pages 12--20, 1992.

....on which it is based has been around for at least several years. This realization strategy is based on a ladder network implementation of a QMFB s polyphase matrices. Such ladder networks were first proposed by Bruekers and Van Den Enden [2] and were also studied in some detail by Kalker and Shah [5] [6] Until more recently, many of the benefits of the lifting realization were not fully appreciated. In the context of wavelet transforms, Sweldens first coined the term lifting , and detailed the many advantages of this realization strategy [8] 9] In his work with Daubechies [4] he also ....

A. A. C. Kalker and I. A. Shah. Ladder structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. In Proceedings of SPIE: Visual Communications and Image Processing, volume 1818, pages 12--20, Boston, MA, November 1992.

....R[z; z Gamma1 ] The proof relies on the 2000 year old Euclidean algorithm. In the filter bank literature subband transform built using elementary matrices are known as ladder structures and were introduced in [5] Later several constructions concerning factoring into ladder steps were given [28, 41, 48, 32, 33]. Vetterli and Herley [56] also use the Euclidean algorithm and the connection to diophantine equations to find all high pass filters that, together with a given low pass filter, make a finite filter wavelet transform. Van Dyck et al. use ladder structures to design a wavelet video coder [20] In ....

....details, we refer to [37] Acknowledgments. The authors would like to thank Peter Schroder and Boon Lock Yeo for many stimulating discussions and for their help in computing the factorizations in the example section, Jelena Kovacevi c and Martin Vetterli for drawing their attention to reference [28], Paul Van Dooren for pointing out the connection between the M band case and the Smith normal form, and Geert Uytterhoeven, Avraham Melkman, Mark Maslen, and Paul Abbott for pointing out typos and oversights in an earlier version. Ingrid Daubechies would like to thank NSF (grant DMS 9401785) ....

T. A. C. M. Kalker and I. Shah. Ladder Structures for multidimensional linear phase perfect reconstruction filter banks and wavelets. In Proceedings of the SPIE Conference on Visual Communications and Image Processing (Boston), pages 12--20, 1992.

....fG 0 ; H 0 g in the low pass channel. The filters in the high pass channel are up to an even delay given by fz Gamma1 G 1 (z) zH 1 (z)g = fz Gamma1 H 0 ( Gammaz) zG 0 ( Gammaz)g. With this restriction, the perfect reconstruction (PR) property of the filter bank is expressed by the formula [6] (G 0 (z)G 1 ( Gammaz) e = 1; 1) where we use the notation f e (f o ) to denote the even (odd) part of a polynomial f . The question is how to design good filters fG 0 ; G 1 g The goodness of filters here is measured against two criteria. First of all, the filters are required to have a good ....

....in terms of polynomials. The low pass character of a filter G can be stated by requiring that PG (x) satisfies PG (x) ae 0 Gamma1 x 0 1 0 x 1 (2) Let f and g be the polynomials corresponding to G 0 and G 1 . The perfect reconstruction condition is easily seen to translate to [6] (f(x)g( Gammax) e = 1. The condition on the number of zeros also carries over: it can be checked that if a zero phase filter G has n zeros at e j , then n is even, and the corresponding PG (x) has a zero of order n=2 at x = Gamma1. Section II presents the analytical construction of pairs ....

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A. Kalker and I. Shah. Ladder Structures for Multidimensional Linear Phase Perfect Reconstruction Filter Banks and Wavelets. In Proceedings of the SPIE Conference on Visual Communications and Image Processing (Boston), pages 12--20, November 1992.

....allow a natural formulation of LP conditions. 1. INTRODUCTION Structures for perfect reconstruction filter banks (PRFB) are well known. A classical example is the lattice structure [8] In this paper we discuss the advantages of a relatively new structure, the so called ladder structure [1] [5]. After a short introduction to ladder structures, and the way they are used to build PRFBs, we consider two topics. First we show that that by posing some mild conditions on ladders we can enforce the filters of the resulting filter banks to be linear phase. The method consists of extending so ....

A. Kalker, I. Shah, Ladder Structures for Multidimensional Linear Phase Perfect Reconstruction Filter Banks and Wavelets, Proc. of the SPIE Visual Communications, Boston, Nov. 1992, pp.12-20.

....not only being insensitive to coefficient quantization but, under fairly general conditions, also to quantization of some of the internal intermediate results. The approximation properties of these structures under quantization are not clear as yet. Similar structures have also appeared in [2] In [5] it is shown that the class of 2 band PRFB s which are realized by a ladder structure fall in the class of 2 band bi orthogonal (BO) filter banks. However,the extension to the general problem of realizability by ladder structures of arbitrary (i.e. m band, m 2) multidimensional (M D) BO filter ....

A. Kalker, I. Shah, Ladder Structures for Multidimensional Linear Phase Perfect Reconstruction Filter Banks and Wavelets, SPIE Visual Communications, Boston, Nov. 1992.

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