Factoring wavelet transforms into lifting steps
type: Item
instance of: scholarly article
title: Factoring wavelet transforms into lifting steps
zbMATH Open document ID: 0913.42027
DOI: 10.1007/BF02476026
author: Ingrid Daubechies
author: Wim Sweldens
published in: The Journal of Fourier Analysis and Applications
publication date: 1999-05-20T00:00:00Z
full work available at URL: https://eudml.org/doc/59563
review text: Given a signal \((x_k)_{k \in Z}\), we may split it into even \(x_e = (x_{2k})\) and odd \(x_o = (x_{2k+1})\) components. A lifting step is defined to be a transformation of the form \(x \mapsto (x_o, x_e - P(x_o))\), where \(P(x_o)\) is a linear predictor for the even component based on the odd component. (For instance, one could take \(P(x_o)_{2k} = (x_{2k-1} + x_{2k+1})/2\)). A dual lifting step is a transformation of the form \((x_o,d) \mapsto (x_o + U(d), d)\), where \(U(d)\) is an update operator to the odd components based upon the detail \(d = x_e - P(x_o)\). (For instance, one can take \(U(d)_{2k} = (d_{2k-1} + d_{2k+1})/4)\). In this paper the authors show that any finite impulse response (FIR) filter bank or wavelet transform can be factored into a sequence of lifting steps and dual lifting steps, each of which is also given by an FIR filter. The method of proof relies upon the z-transform and then a factorization of a Laurent polynomial-valued matrix into elementary matrices by means of the Euclidean algorithm.
Mathematics Subject Classification ID: 42C40
zbMATH DE Number: 1220776
zbMATH Keywords: wavelet transforms
zbMATH Keywords: lifting
zbMATH Keywords: elementary matrix
zbMATH Keywords: finite impulse response filter bank
zbMATH Keywords: FIR filter
Wikidata QID: Q114830194
reviewed by: Terence Tao
MaRDI profile type: Publication
cites work: Families of multiresolution and wavelet spaces with optimal properties
cites work: Q5562621
cites work: Q3703780
cites work: Wavelet transforms that map integers to integers
cites work: Local decomposition of refinable spaces and wavelets
cites work: Q4003151
cites work: Q4321768
cites work: A Cardinal Spline Approach to Wavelets
cites work: A general framework of compactly supported splines and wavelets
cites work: Biorthogonal bases of compactly supported wavelets
cites work: Q4692841
cites work: Banded matrices with banded inverses. II: Locally finite decomposition of spline spaces
cites work: Q4880368
cites work: Orthonormal bases of compactly supported wavelets
cites work: Painless nonorthogonal expansions
cites work: Q3344703
cites work: Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape
cites work: Multiresolution Representation of Data: A General Framework
cites work: Q3043346
cites work: Wavelets and recursive filter banks
cites work: Q3999407
cites work: Multiresolution Approximations and Wavelet Orthonormal Bases of L 2 (R)
cites work: Fast algorithms for discrete and continuous wavelet transforms
cites work: Q4524850
cites work: Q3143213
cites work: The lifting scheme: A custom-design construction of biorthogonal wavelets
cites work: The Lifting Scheme: A Construction of Second Generation Wavelets
cites work: A family of polynomial spline wavelet transforms
cites work: Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property
cites work: Running FIR and IIR filtering using multirate filter banks
cites work: Wavelets and filter banks: theory and design
cites work: Q4847221
OpenAlex ID: W4251101141
description: scientific article; zbMATH DE number 1220776