Mathematics Colloquium
Thursday, November 14, 1996, 4:00pm
102 Bradley Hall
Professor Yiping Mao
University of South Carolina
speaks on
Compact supported wavelet bases with group action
invariance in higher dimension
Abstract. Wavelets are functions generated from one basic function by
dilations and translations. They are used as analyzing tools, by both pure
mathematicians (in harmonic analysis) and electrical engineers (in signal
analysis). A particularly interesting development is the discovery of
orthonormal (and biorthonormal) wavelets bases of $L^2(\Bbb{R}^n)$. The
so-called subband coding schemes of perfect reconstruction (PR) is a
general and efficient way to construct (bi)orthonormal bases of wavelets.
In this talk, we will discuss a class of subband coding schemes allowing
perfect reconstruction for signals sampled on the general lattices
$\Gamma$ of $\Bbb{R}^n$ and describe the construction of compactly
supported biorthonormal (or orthonormal) wavelet bases of
$L^2(\Bbb{R}^n)$. The most interesting feature in our construction is that
the sets of generating functions $\{\psi_i\}_{i=0}^{k}$ for the synthesis
and $\{\tilde{\psi}_i\}_{i=0}^{k}$ for the analysis, as well as the scaling
functions $\phi$ and $\tilde{\phi}$, are globally invariant under the action
of a finite group generated by an operator $P: \Gamma^*\setminus
\Bbb{R}^n \rightarrow \Gamma^*\setminus \Bbb{R}^n$. We classify all
possible such group actions and present a algorithm to solve Bezout
equation which is the key part of seeking PR subband coding schemes.
Tea. High tea will be served at 3:30pm in the Lounge.
Emmy's. Certain refreshments will be available at Emmy's after the talk.
Host. Carolyn Gordon is the hostess. Anybody who is interested in having
dinner with the speaker should contact Carolyn at 646-047.