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This course is a systematic introduction to mathematical basic building
blocks (e.g., wavelets, local Fourier basis, etc.), which are useful
for diverse fields such as signal and image processing, numerical
analysis, and statistics. The course will emphasize the connection
between the continuum (i.e., analog) world and the discrete world, and
use approximation and compression of functions and data as main
examples.
Prerequisite:
(MAT 125B or MAT 201C) and (MAT 128B or MAT 167) and (MAT 129 or equivalent), or consent of instructor.
Topics:
Overture and Motivations
Basics
Fourier Transforms, Sampling Theorems, Fourier Series
Discrete Fourier Transform
Discrete Cosine/Sine Transform
Then we will cover the following topics as much as we can (which
depends on our progress and your interests):
Dealing with
stochastic processes/a collection of signals-in a traditional way
Heisenberg's uncertainty principle and Gabor functions
Various measures of concentration
Prolate Spheroidal Wave Functions and Their Applications (if
time allows)
Frame Theory
Importance of redundancy in representation
Windowed Fourier frames
Wavelet frames
Tools using Time/Space Domain Partitioning
Local Cosine/Sine Transform
Fast Laplace/Poisson Solvers
How to deal with boundary: Polyharmonic Local Sine Transform
Application to Image Approximation and Compression
Tools using Frequency Domain Partitioning
Shannon-Littlewood-Paley Wavelets
Haar Wavelets
Multiresolution Analysis and Wavelet Bases
Discrete Wavelet Transforms
Walsh Transform
Discrete Wavelet Packet Transforms
Application to Image Approximation and Compression
Compressed Sensing (if time allows)
Harmonic Analysis on Graphs and Networks (if time allows)
Textbooks:
The following textbooks are used as references, and good books to keep
on your desk, but not required.
W. L. Briggs and V. E. Henson: The DFT: An Owner's Maunal for the Discrete Fourier Transform, SIAM, 1995.
S. Jaffard, Y. Meyer, R. D. Ryan: Wavelets: Tools for Science and
Technology, SIAM, 2001.
S. Mallat: A Wavelet Tour of Signal Processing, 3rd Edition,
Academic Press, 2009.
I will also hand out in class (or post via webpages) many notes and copies of
original papers.
Class Web Page:
I will maintain the Web pages for this course (one of which you are
looking at now). In particular, please read the comments, handouts, and reference page often.
After each class, I will put relevant comments and references as well as
most of my handouts in class in this page that should
serve as a guide to further understanding of the class material.
You can access the Applied & Computational Harmonic Analysis home page at
https://www.math.ucdavis.edu/~saito/courses/ACHA/ from which you can
access to
this syllabus page, the
homework page.
Grading Scheme:
50% Homework
50% Final Report
Homework:
I will assign homework every other week for you to
solve, including both analytical and programming exercises. More
detailed, i.e., actual problems, due dates, etc. will be announced at our
homework page. LATE HOMEWORK WILL NOT BE ACCEPTED. A subset of
these problems will be graded.
Final Report:
The other half of your grade will be determined by your final report.
Here, you need to write a report on one of the following topics:
Describe how some of the methods you learned in this course will be
used in your
research.
Find out a practical application yourself (not copying from
papers/books) using the methods you learned in this course; describe
how to use them; describe the importance of that application; what
impact would you expect if you are successful?