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  1. Journal of Fourier Analysis and Applications
  2. Journal of Fourier Analysis and Applications : Volume 4
  3. Journal of Fourier Analysis and Applications : Volume 4, Issue 3, May 1998
  4. Factoring wavelet transforms into lifting steps
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Journal of Fourier Analysis and Applications : Volume 23
Journal of Fourier Analysis and Applications : Volume 22
Journal of Fourier Analysis and Applications : Volume 21
Journal of Fourier Analysis and Applications : Volume 20
Journal of Fourier Analysis and Applications : Volume 19
Journal of Fourier Analysis and Applications : Volume 18
Journal of Fourier Analysis and Applications : Volume 17
Journal of Fourier Analysis and Applications : Volume 16
Journal of Fourier Analysis and Applications : Volume 15
Journal of Fourier Analysis and Applications : Volume 14
Journal of Fourier Analysis and Applications : Volume 13
Journal of Fourier Analysis and Applications : Volume 12
Journal of Fourier Analysis and Applications : Volume 11
Journal of Fourier Analysis and Applications : Volume 10
Journal of Fourier Analysis and Applications : Volume 9
Journal of Fourier Analysis and Applications : Volume 8
Journal of Fourier Analysis and Applications : Volume 7
Journal of Fourier Analysis and Applications : Volume 6
Journal of Fourier Analysis and Applications : Volume 5
Journal of Fourier Analysis and Applications : Volume 4
Journal of Fourier Analysis and Applications : Volume 4, Issue 6, November 1998
Journal of Fourier Analysis and Applications : Volume 4, Issue 4-5, July 1998
Journal of Fourier Analysis and Applications : Volume 4, Issue 3, May 1998
Factoring wavelet transforms into lifting steps
Sampling of Paley-Wiener functions on stratified groups
Mean oscillation of functions and the Paley-Wiener space
Single wavelets in n-dimensions
New type Paley-Wiener theorems for the modified multidimensional Mellin transform
Wavelets on fractals and besov spaces
Schrödinger equation and oscillatory Hilbert transforms of second degree
Gibbs' phenomenon for sampling series and what to do about it
Journal of Fourier Analysis and Applications : Volume 4, Issue 2, March 1998
Journal of Fourier Analysis and Applications : Volume 4, Issue 1, January 1998
Journal of Fourier Analysis and Applications : Volume 3

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Factoring wavelet transforms into lifting steps

Content Provider SpringerLink
Author Daubechies, Ingrid Sweldens, Wim
Copyright Year 1998
Abstract This article is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formulaSL(n;R[z, z−1])=E(n;R[z, z−1])); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e., non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers.
Starting Page 247
Ending Page 269
Page Count 23
File Format PDF
ISSN 10695869
Journal Journal of Fourier Analysis and Applications
Volume Number 4
Issue Number 3
e-ISSN 15315851
Language English
Publisher Birkhäuser-Verlag
Publisher Date 1998-01-01
Publisher Place Boston
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Wavelet lifting elementary matrix Euclidean algorithm Laurent polynomial Abstract Harmonic Analysis Approximations and Expansions Fourier Analysis Partial Differential Equations Applications of Mathematics Signal, Image and Speech Processing
Content Type Text
Resource Type Article
Subject Applied Mathematics Analysis
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