A. A. M. L. Bruekens and A. W. M. van den Enden, \New networks for perfect inversion and perfect reconstruction," IEEE J. Selected Areas Commun., vol. 10, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....L = KM (drawn for M = 8) C. The Lifting Scheme or the Ladder Structure The lifting scheme, also known as the ladder structure, is a special type of lattice structure, a cascading FB construction using only elementary matrices identity matrices with one single non zero o diagonal element [7], 10] 11] Two channel lter banks implemented in ladder structures save roughly half of the computational complexity due to the exploitation of the inter subband relationship between coecients. It has been proven that any 2 2 polyphase matrix E(z) with unity determinant can be factored into ....

....u ij and their corresponding elementary matrices U ij form the upper triangular matrix U, i.e. U = Q i;j U ij . The diagonal scaling factors labelled i are contained in the diagonal matrix D and they hold the key to invertibility. This LU parameterization in FB design rst appears in [7] under the name ladder structure. Since the permutation matrix is a simple re routing, we usually ignore it in the parameterization. The diagonal scaling factors are strategically placed at the end of the structure so that they can be absorbed into the quantization stage whenever possible. This ....

A. A. M. L. Bruekens and A. W. M. van den Enden, \New networks for perfect inversion and perfect reconstruction," IEEE J. Selected Areas Commun., vol. 10, 1992.

....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a factorization result, similar to the one presented in this paper, was obtained earlier by Kalker and Shah in an unpublished manuscript [29] While our work goes into more detail concerning the non uniqueness, implementation, and ....

A. A. M. L. Bruekens and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. Selected Areas Commun., 10(1), 1992.

....Burke Hubbard [5] Wavelet analysis is known as a linear tool. However, it is starting to be recognized that nonlinear extensions are possible [8, 10, 11, 12, 13, 15, 18, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 40] The lifting scheme, recently introduced by Sweldens [45, 46, 47] see also [4] for a predecessor of this scheme, known as a ladder network ) has provided a useful tool for constructing nonlinear wavelet transforms. The enormous flexibility and freedom that the lifting scheme o#ers has challenged researchers to develop various nonlinear wavelet transforms [8, 10, 11, 12, ....

Bruekers, F. A. M. L., and van den Enden, A. W. M. New networks for perfect inversion and perfect reconstruction. IEEE Journal on Selected Areas in Communications 10 (1992), 130--137.

....connects to many earlier approaches. The basic idea behind lifting is a simple relationship between all filter banks that share the same lowpass or the same highpass filter, also observed by Vetterli and Herley in [58] Lifting also leads to a filter bank implementation known as ladder structures [2]. Moreover, it is well known that all one dimensional FIR filter banks fit into lifting [15, 31, 43, 53] In this paper we aim to provide a general recipe based on lifting for building filter banks and wavelets in any dimension, for any lattice and any number of primal and dual vanishing moments. ....

A. A. M. L. Bruekens and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. Selected Areas Commun., 10(1), 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 ....

A.A.M.L. Bruekers, A.W.M. van den Enden, New Networks for Perfect Inversion and Perfect Reconstruction, IEEE Journal on Selected Areas in Communications, Vol. 10, no. 1, January 1992.

....two methods that are widely used in the construction of perfect reconstructing filter banks. Both methods use a product of simple matrices to construct the matrices H p and G t p . The basic building blocks of the first method are known in signal processing as ladder structures or lifting steps [2, 4, 10]. They are given by the elementary transvections T 12 (a) 1 a 0 1 ; T 21 (b) 1 0 b 1 ; 3) with a; b 2 B, and by diagonal matrices in GL 2 (B) Figure 3 shows an example of such an implementation. # 2 2 2 z z Gamma1 # 2 p(z) q(z) q(z) p(z) Figure 3: Ladder ....

F. A. Brueckers and A. W. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. on Selected Areas in Communications, 10(1):130--137, 1992.

....the approximations cannot be done independently. This is a difficult problem, but it can be overcome by using structurally enforced perfect reconstruction. A well known example is the lattice structure (see [6] A more recently introduced structure is the so called ladder structure (see [7]) A ladder structure implements a filter transform over K as a cascade of ladder (or elementary) steps Phi : Z 2 Z 2 . Mathematically a ladder step is completely characterized by a triple (i; j; OE) where fi; jg = f1; 2g, and where OE : Z Z is a filter transform (see also Fig. 3) ....

....that ladders are particularly suitable for implementing perfect reconstructing filter banks. Each step in a ladder cascade is inverted by sign change. Therefore, if the analysis matrix A of a perfect reconstructing filter bank is implemented with a ladder, the synthesis matrix is immediately found [7]. If the matrix A is approximated in the implementation phase, the modification of the synthesis matrix immediately follows. Moreover, if a matrix A is realizable by a ladder, then its inverse is also realizable by a ladder (see Fig. 3) f f X X j Y i Y j X i X j i Fig. 3. Ladder ....

A. Bruekers and A. van den Enden, "New Networks for Perfect Inversion and Perfect Reconstruction," IEEE Journal on Selected Areas in Communications, vol. 10, pp. 130--137, January 1992.

....connects to many earlier approaches. The basic idea behind lifting is a simple relationship between all filter banks that share the same lowpass or the same highpass filter, also observed by Vetterli and Herley in [57] Lifting also leads to a filter bank implementation known as ladder structures [2]. Moreover, it is well known that all FIR filter banks fit into lifting [15, 29, 41, 52] In this paper we aim to provide a general recipe based on lifting for building filter banks and wavelets in any dimension, for any lattice and any number of primal and dual vanishing moments. To our ....

A. A. M. L. Bruekens and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. Selected Areas Commun., 10(1), 1992.

....filter banks and wavelets have been areas of active research for use in video and image communication systems. At the same time efficient structures for the implementation of such filters are of importance. In 1 D, the well known lattice structure and the recently introduced ladder structure [1] are attractive. However, their extensions to higher dimensions (m D) have been limited. In this paper we reintroduce the ladder structure, with the purpose of transforming the structure into m D using the McClellan transform. 1 Introduction Recently the ladder structure was introduced for the ....

....In this paper we reintroduce the ladder structure, with the purpose of transforming the structure into m D using the McClellan transform. 1 Introduction Recently the ladder structure was introduced for the design and efficient implementation of 1 D perfect reconstructing filter banks (PRFB) [1]. These structures have the advantage of not only being insensitive to coefficient quantization but, under fairly general conditions, also to quantization of intermediate results. In [6] it is shown that the class of 1 D 2 channel PRFB s which can be realized by a ladder structure coincides with ....

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A.A.M.L. Bruekers, A.W.M. van den Enden, New Networks for Perfect Inversion and Perfect Reconstruction, IEEE Journal on Selected Areas in Communications, vol. 10, no. 1, January 1992.

....filters. It is thus important to know the design space which can by implemented by a certain structure. This is referred to as the completeness problem. Recently the ladder structure was introduced for the design and efficient implementation of 1 D perfect reconstructing filter banks (PRFB) [1]. These structures have the advantage of 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 ....

....second theme, viz. ladder structures. We introduce the ladder as a structure for building perfect reconstructing filter banks, similar to lattice structures [13] 10] The basic definitions and properties of ladders are discussed. For a discussion of their implementational advantages we refer to [1]. Thirdly the completeness question of ladders with respect to BO systems is raised. In the 1 D case it is shown that the ladder structure is complete: i.e. every 1 D BO filter bank can be implemented with a ladder structure (to be referred to as realizable) In the M D case, completeness is no ....

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A.A.M.L. Bruekers, A.W.M. van den Enden, New Networks for Perfect Inversion and Perfect Reconstruction, IEEE Journal on Selected Areas in Communications, Vol. 10, no. 1, January 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 ....

A. A. M. L. Bruekens and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. Selected Areas Commun., 10(1), 1992.

....of a QMFB has become quite popular recently, the idea 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 ....

F. A. M. L. Bruekers and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE Journal on Selected Areas in Communications, 10(1):130--137, January 1992.

....common notation in this field, this is written as SL(n; R[z; z Gamma1 ] E(n; 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 ....

A. A. M. L. Bruekens and A. W. M. van den Enden. New networks for perfect inversion and perfect reconstruction. IEEE J. Selected Areas Commun., 10(1), 1992.

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F. A. Brueckers and A. W. van den Enden, "New networks for perfect inversion and perfect reconstruction," IEEE J. on Selected Areas in Communications 10(1), pp. 130--137, 1992.

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