An algebraic structure of orthogonal wavelet space
type: Item
instance of: scholarly article
title: An algebraic structure of orthogonal wavelet space
zbMATH Open document ID: 0966.42027
author: Jun Tian
author: Raymond O. jun. Wells
published in: Applied and Computational Harmonic Analysis
publication date: 2001-08-12T00:00:00Z
review text: In this paper the authors study the space of compactly supported orthonormal wavelets over real numbers. The wavelet subspace \(WM(2,g)\) of rank 2 (corresponding to 2-band wavelets) and genus \(g\) (corresponding to scaling filters of length \(2g\)) is mapped one-to-one to the \((g-1)\) dimensional real torus. Because of the degenerate phenomenon of the paraunitary matrix, this parametrization is not onto. Instead there exists an onto mapping from the torus to the closure of the wavelet space \(\overline{WM(2,g)}:=WM(2,g)\cup WM(2,g-2)\cup\cdots\). By utilizing the factorization theory, the authors present a fast implementation of the discrete wavelet transform (DWT). In general the computational complexity of a rank \(m\) orthogonal DWT is \(O(m^2g)\). By starting with a given scaling filter, the authors construct additional \((m-1)\) wavelet filters so that DWT is implemented in \(O(mg)\) operations.
reviewed by: Radu Balan
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 94A12
zbMATH DE Number: 1471672
zbMATH Keywords: wavelet
zbMATH Keywords: parametrization
zbMATH Keywords: discrete wavelet transform
MaRDI profile type: Publication
OpenAlex ID: W1978782853
cites work: Q4003151
cites work: A general framework of compactly supported splines and wavelets
cites work: Ondelettes, analyses multirésolutions et filtres miroirs en quadrature. (Wavelets, multiscale analysis and quadrature mirror filters)
cites work: Biorthogonal bases of compactly supported wavelets
cites work: Q4849992
cites work: Orthonormal bases of compactly supported wavelets
cites work: Ten Lectures on Wavelets
cites work: Factoring wavelet transforms into lifting steps
cites work: De-noising by soft-thresholding
cites work: Rank M Wavelets with N Vanishing Moments
cites work: Q4027552
cites work: Q4327556
cites work: Pollen product factorization and construction of higher multiplicity wavelets
cites work: Tight frames of compactly supported affine wavelets
cites work: Necessary and sufficient conditions for constructing orthonormal wavelet bases
cites work: Multiresolution Approximations and Wavelet Orthonormal Bases of L 2 (R)
cites work: Q3135829
cites work: SU I (2, F [ z,1/z ]) for F A Subfield of C
cites work: Q4220597
cites work: On orthonormal wavelets and paraunitary filter banks
cites work: Theory of regular M-band wavelet bases
cites work: Q3143213
cites work: Q3138779
cites work: Wavelets and filter banks: theory and design
cites work: Q4847221
DOI: 10.1006/ACHA.2000.0300
description: scientific article; zbMATH DE number 1471672