Discrete wavelets with quaternion and Clifford coefficients
type: Item
instance of: scholarly article
title: Discrete wavelets with quaternion and Clifford coefficients
zbMATH Open document ID: 1407.42031
author: Peng Zhang
published in: Advances in Applied Clifford Algebras
publication date: 2018-08-29T00:00:00Z
review text: Clifford-valued analogues of compactly supported scaling filters and wavelets are developed. Ginzburg's equations, e.g., [\textit{P. Ginzberg} and \textit{A. T. Walden}, IEEE Trans. Signal Process. 61, No. 6, 1357--1367 (2013; Zbl 1393.94242)] are direct analogues of scaling equations but with real coefficients \(s_k\) replaced by matrices \({\mathbf{S}}_k\), \(4\times 4\) in the quaternionic case. In Ginzburg's work, these matrices are chosen to provide an isomorphism with a basis of quaternions, multiplied by scalar scaling coefficients. One uses the notation \(\text{Cl}(p,q)\) to denote a Clifford structure with basis \(e_1,\dots, e_{p+q}\) such that \(e_1^2=\cdots=e_p^2=1=-e_{p+1}^2=\cdots=-e_{p+q}^2\). The basis scaling equations are \(\sum_{k=0}^{L-1} {\mathbf{S}}_k=\sqrt{2}\, {\mathbf{I}}_n\) (balancing) and \(\sum_{k=0}^{L-1-2m} {\mathbf{S}}_k\,{\mathbf{S}}_{k+2m}^T=\delta_{m,0} {\mathbf{I}}_n\) (orthogonality) with other conditions for vanishing moments. The current work starts observing that the plane determined by two vectors in \(\mathbb{R}^3\) can be rotated into a coordinate plane and, consequently, in Ginzburg's equations one can assume that the coefficient of \(\mathbf{j}\) in one quaternion and those of \(\mathbf{j}\) and \(\mathbf{k}\) in a second quaternion, in their real matrix representations, are zero. This makes it much simpler to solve the Ginzburg equations, after which one can rotate the solution back to the desired quaternionic coordinates. In the present work, the author alternately uses Gröbner bases or a genetic algorithm search to find solutions of the Ginzburg scaling equations, for certain values of \(L\) (the filter length). Scaling solutions with matrices corresponding to \(\text{Cl}(1,0), \text{Cl}(1,1)\) and \(\text{Cl}(2,0)\) structures can be extended to larger Clifford algebras. Several tables (Tabs. 1--15) list scaling coefficients of filters of length 6, 10, and 14 in the \(\text{Cl}(1,0)\) case, lengths 10, 14 and 18 in the…
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 65T60
zbMATH DE Number: 6928493
zbMATH Keywords: quaternion wavelets
zbMATH Keywords: Clifford wavelets
zbMATH Keywords: wavelet functions
reviewed by: Joseph Lakey
describes a project that uses: Quaternion
describes a project that uses: Clifford Multivector Toolbox
MaRDI profile type: Publication
full work available at URL: s00006-018-0876-5
OpenAlex ID: W2806387139
cites work: The theory and use of the quaternion wavelet transform
cites work: Orthonormal bases of compactly supported wavelets
cites work: Matrix representation of quaternions
cites work: Wavelet analysis of matrix–valued time–series
cites work: Q4382669
cites work: Matrix-Valued and Quaternion Wavelets
cites work: Q4449255
cites work: Multiresolution Approximations and Wavelet Orthonormal Bases of L 2 (R)
cites work: Applications of complex valued wavelet transforms to subband decomposition
cites work: Clifford algebras and spinors.
cites work: Clifford multivector toolbox (for MATLAB)
cites work: Q3143213
cites work: Orthogonal complex filter banks and wavelets: Some properties and design
DOI: 10.1007/S00006-018-0876-5
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description: scientific article; zbMATH DE number 6928493