Minimally supported frequency composite dilation wavelets

Jeffrey Blanchard
Grinnell College
UNITED STATES

given at  strobl07 (19.06.07 16:50)
id:  554
length:  25min
status:  accepted
type:  talk
The system $\Psi = (\psi^1, \psi^2,\dots,\psi^L)^T\subset L^2({\mathbb{R}}^n)$ is an $(AB,\Gamma)$-\textit{Composite Dilation Wavelet} if
$\{D_a^jD_bT_k\psi^i : j\in{\bf{Z}}, b\in B, k\in\Gamma, i=1,\dots,L\}$ is an orthonormal basis for $L^2({\mathbb{R}}^n)$, where $A=\{a^j : j\in\mathbb{Z}\}$ is a group generated by an expanding matrix, $a$, $B$ is a subgroup of $GL_n({\mathbb{R}})$, and $\Gamma$ is a full rank lattice. Given a finite group $B$, we present admissibility conditions for arbitrary lattices and then for arbitrary expanding matrices. We show that these admissibility conditions are sufficient to generate minimally supported frequency, $(AB, \Gamma)$-composite dilation wavelets for $L^2({\mathbb{R}}^n)$. We then show that for any finite group $B$ whose fundamental region is bounded by hyperplanes through the origin, such as Coxeter groups or rotation groups, we can always find admissible lattices and expanding matrices. Given the existence of MSF, composite dilation wavelets for $L^2({\mathbb{R}}^n)$, we explore ideas to minimize the number of wavelet generators, $L$, for the system $\Psi$. We will present examples of singly generated composite dilation wavelets for $L^2({\mathbb{R}}^n)$ and examine how reducing the generators limits our freedom.