Discrete wavelets with quaternion and Clifford coefficients
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 quaternionic case, and of length 10 in the \(\text{Cl}(1,1)\) and \(\text{Cl}(2,0)\) cases. Matrix-valued scaling and wavelet functions are obtained via the cascade algorithm just as in the scalar-valued case, using the algebra property that function values can be multiplied. Solutions are plotted by plotting various of their coordinate plane projections. For example, in the \(\text{Cl}(1,1), \text{Cl}(2,0)\) and quaternion cases the functions are five-dimensional objects (one time variable and four space variables), so there are \(\binom{5}{2}=10\) planar projections and \(\binom{5}{3}=10\) 3-D curve projections, and each of these is plotted for length 10, 14 and 18 filters.
scientific article; zbMATH DE number 6928493
scientific article