12:30pm
Fine Hall PL (Professor's Lounge)
Title: Wavelets, Multiwavelets and Applications
Abstract: A refinable function generates multiresolution analysis (MRA). Standard (scalar) MRA assumes that there is only one scaling function. We make a step forward and allow several of them. New object is called a multi-scaling function. It has new features determined by the matrix structure of the dilation equation. For example, a multi-scaling function can combine orthogonality, symmetry and high approximation order while a scalar scaling function cannot.
We show how approximation, regularity and symmetry of multi-scaling functions can be studied using an original technique, based on two-scale factorization of matrix trigonometric polynomials. Using this technique we suggest an efficient way to construct biorthogonal multiwavelet bases with approximation and symmetry. As an example, we present a biorthogonal pair of smooth bases built from piecewise polynomial finite elements.
We argue why multiwavelets are promising in practice, describe a few possible applications and point out suddleties of their implementation. In particular, we apply multiwavelet filter banks for noise removal.