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.