Speaker:      Ivan Selesnick
Title:        Groebner Bases and Wavelet Design
Affiliation:  Electrical Engineering
              Polytechnic University
              Brooklyn, New York USA
URL:          http://taco.poly.edu/selesi/
Email:        selesi@taco.poly.edu

Abstract

The most convenient approach to the construction of orthonormal wavelet bases
is based on the spectral factorization of an appropriately designed
autocorrelation sequence. However, that approach does not extend to many
generalizations of the standard type of wavelet basis. For example, the design
of certain multiwavelet bases and wavelet frames demand the solution to
nonlinear design equations. In this talk we describe some problems in
specialized wavelet design, and describe our experiences using Groebner bases
to solve the nonlinear design equations that arise. We focus on the design of
wavelet bases possessing properties that make them useful for signal processing
applications.

Using Groebner bases in practice can be a challenge because of the high
computational and memory requirements. In one example (albeit atypical), the
calculation of a Groebner basis (GB) took us 4 weeks on a 200MHz Sun Ultra 2.
The size of the resulting ASCII file, into which the GB itself was written, was
18.9 Mbytes. However, removing repeated roots by polynomial factorization, and
recomputing the GB, the GB file was reduced to only 370 Kbytes.

Despite their high computational and memory requirements, our experiences and
results support the utility of GBs in solving certain nonlinear design
problems. However, the effective use of GBs requires a knowledge of how apply
operations like GB factorization, the FGLM algorithm, etc.