This course, Introduction to the Mathematics of Wavelets,
covers Wavelets (via Multiresolution Analysis) and background,
including but not limited to Hilbert Space, Fourier transforms
and (lightly!) Fourier series, and the Fast Fourier Transform.
See the syllabus for more information (link below, to a PDF document).
To get an early start, you may want to read the Introduction (link below),
try working the exercises there, and glance at the other documents as well.
If you have the text already, don't worry -- the author uses terminology
that wonn't take long to explain when school starts. And the Introduction
and the Lebesgue Facts might help too.
Math 5467 meets 3 times
weekly:
at 12:20 - 1:10pm MWF, VinH 20.
Office Hours: 1:30-2:30pm MWF or by appointment
-- or call 5-3855 to see if
I'm in and available, especially before class.
You can
ask questions by email too: jodeit@math.umn.edu
Please, send
me messages in TEXT mode only! Also, here is a Web link to
a list of
special codes for math symbols (most of them start with the \
character).
TEXT (ISBN 0-521-57894-9 paperback)
A Mathematical Introduction to Wavelets,
by P. Wojtaszczyk, Cambridge University Press, 1997.
The text has a lot more in it than we need, and in turn it needs
a great deal of background information, which will be placed on
the Web from time to time as PDF documents, with links below.
Announcements, hints, "news" will be here.
NEWS May 17: Grades are in. I wish you all well!
NEWS Apr 27: A new document, "Miscellany," is available.
It contains an elegant (not mine!) solution for SP2, a solution
for A9#1, and a note on "biorthogonality."
NEWS Apr 22: The Challenge Problem is on the Assignment Sheet.
An even newer version of "From a scaling function to a wavelet" is available.
NEWS Apr 18: Assignment 10 is ready, to be due Friday 4/18.
A newer version of "From a scaling function to a wavelet" is available.
NEWS Apr 11: A new note is available below, on getting a
wavelet, once we have our scaling function.
NEWS Apr 10: Assignment 9 is ready.
NEWS Apr 3: New versions of MRA to \phi and Low-pass to \phi;
links to old documents on Tempered Distributions
Solutions for Test 2.
NEWS Apr 2: Assignment 8 is ready.
NEWS Mar 27: Test 2 tomorrow, Friday March 28!
Assignment 7 is up; computer disaster delay...
NEWS Mar 26: Newer version of `From
the Low-pass filter to a "Scaling Function"';
Next Assignment planned for later today.
NEWS Mar 25: New version of `From
the Low-pass filter to a "Scaling Function"'
NEWS Mar 14: Special Problem 2, due March 31, has been added to the
Assignments
Test 2 will be March 28.
NEWS Mar 13: Here are some suggestions for projects. They are not
"definitive;" they can be used to "start." I probably won't be answering
email during the break, so go ahead and start your Project if you
wish -- when I check it, I may suggest some changes, hopefully minor.
(1) Explore creating wavelets that use a dilation factor different than
two. The Wavelet Digest may have information (this applies in all cases!)
(2) Investigate the effects of different kinds of compression using
wavelets, including using the highest X percent of absolute values and
using combined ranges of nbsp;j and nbsp;k. Show the results as
graphs, or pictures if you work in two dimensions, or as sounds.
(3) Look into the "fractal" nature of scaling functions and wavelets.
(4) Try to design a scaling function or wavelet that closely resembles
a signal that has some importance in your field -- I'm thinking of the
seismic reflection off dolomite as an example.
(5) Look up information on the role that Heisenberg's Uncertainty
Principle plays in signal analysis, and report on it with examples
done in detail.
(6) Find out about Gabor "wavelets" and others that don't use MRAs
as their starting point, but instead start with functions \psi that
satisfy \int |\hat\psi (\xi)|^2/|xi| d\xi =1, and whose "transform"
is given by < f, \psi_{ab} >, where \psi_{ab}(t)=2^{a/2}\psi(2^at-b),
and nbsp;a nbsp;b nbsp; take on all real values.
(7) Search the Wavelet Digest for a keyword you're interested in,
and then look into reporting on that in some detail, with examples.
NEWS Mar 12: The new Test dates will be March 28 and May 2.
I'll be asking you on Friday about your Project plans! Assignment 6
is on the Web. There should be a new Special Problem up by Friday.
There should be at most 4 Special problems.
NEWS Mar 6: Weighting schemes I, II and III and further details
will be put into a revised Syllabus. The proposals have been moved to
Old News, along with the project ideas from 2000.
The papergrader neeeds you to return Assignments 1 & 2, or Xerox copies
of the pages that show your score (add your name if necessary!)
because he forgot to write your scores down.
Suggested dates for Tests 2 and 3: March 28, May 2.
Please check your calendars.
Links, if any, to other Web pages pages related to this course
will go here.
The links below are to PDF files (you need
Adobe Acrobat Reader to view them).
Follow this link to download the latest Acrobat Reader
from Adobe at no charge.
This is the "Express" URL
Assignments 1--10, SP 1, 2, CP
The Gram-Schmidt "orthogonalization" process
From a "Scaling Function" to a "Wavelet:" Fair version 4/22
From the Low-pass filter to a "Scaling Function" 4/3
From a "Multiresolution Analysis" to a "Scaling Function" 4/3
The L^2 Fourier Trasform Exists!
The document below gives a mathematical introduction to the kinds of vector spaces that wavelets "live" in.
A Brief Introduction to Inner Product Spaces and Hilbert Spaces.