Math 671: Numerical Analysis - Wavelets - Fall 1996

Instructor: Robert Krasny, 2842 East Hall, 763-3505, krasny@umich.edu

Time and Location: MW, 8:45-10am, 3439 Mason Hall

Wavelets provide a new way of representing signals using translation and scaling. In some cases, wavelet expansions have better properties than classical orthogonal bases. For example, the November 1995 issue of the AMS Notices has an article explaining why the FBI switched from a discrete cosine transform to a wavelet algorithm for compressing fingerprint data.

This course will present the basic theory and computer implementation of algorithms for wavelet analysis. We start with techniques from Fourier analysis and progress to wavelets. The topics include: discrete Fourier transform, fast Fourier transform, local Fourier transform, quadrature filters, orthogonal and biorthogonal wavelets, discrete wavelet transform, multiresolution analysis, wavelet packets, adapted bases, wavelet representation of distributions, multidimensional wavelets, applications to differential and integral equations, data compression, signal processing.

The aim of the course is to bring students to the point where they know enough about wavelets to read the current literature and to consider using wavelets in their own research.

Text:
Adapted Wavelet Analysis from Theory to Software by Victor Wickerhauser, A. K. Peters Publishing Co.

References:
Ten Lectures on Wavelets, by Ingrid Daubechies, SIAM
Wavelets and Other Orthogonal Systems with Applications, by Gilbert Walter, CRC Press
Wavelets and Filter Banks, by Gilbert Strang and Truong Nguyen, Wellesley-Cambridge Press

Prerequisites:
Students should have a good background in linear algebra. Some familiarity with Fourier series, Fourier transform and Hilbert space will be helpful, also basic programming skills (e.g. input/output, arrays, if-then-else, loops, plotting). I'll use a C pseudocode to present the algorithms, but assignments may be done in any language (e.g. Fortran, C, Matlab, ......). I plan to review the math and programming prerequisites as needed, depending on the level of the students.

Course Requirements:
Homework will be assigned, including programming exercises.