Applied & Computational Harmonic Analysis
Syllabus Page (Spring 2004)
Course: MAT 280-001
CRN: 69827
Title: Applied & Computational Harmonic Analysis
Class: MW 4:10pm-5:30pm, 1070 Bainer Hall
Instructor: Naoki Saito
Office: 675 Kerr
Phone: 754-2121
Email:saito@math.ucdavis.edu
Office Hours: By appointment
Course Objective:
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 121, 127, 128AB, 167, 201, or their equivalents, or consent of the
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
- Karhunen-Lo\`eve Transform/Principal Component Analysis
- Independent Component Analysis (if time allows)
- The Uncertainty Principle and Bandlimited Signals
- 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
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, 2nd Edition,
Academic Press, 1999.
I will also hand out in class (or post via webpages) many notes and copies of
original papers.
Class Web Page:
Class Mailing List:
The class mailing list was created.
Important announcement will be communicated through this mailing list.
You can also submit your public comments, suggestions, and questions on HW,
and/or some useful information related to the class to this mailing
list. Once you send your email to this list, however, everyone will receive it.
So, please use this wisely and politely. Its name is: acha-s04@ucdavis.edu.
Grading Scheme:
- 40% Homework
- 60% Final Report
Homework:
I will occasionally (i.e., not every week) assign homework 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?
Please email me if you
have any comments or questions!
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