Computational Harmonic Analysis Syllabus Page (Winter,
2002)
Course: MAT 280
CRN: 61380
Title: Computational Harmonic Analysis
Class: MW 5:40pm-7:00pm, 693 Kerr Hall
Instructor: Naoki Saito
Office: 675 Kerr
Phone: 754-2121
Email:saito@math.ucdavis.edu
Office Hours: TTh 2:00pm-3:00pm or by appointment via email
Course Objective:
This course is a systematic introduction to mathematical basic building
blocks (e.g., wavelets, local Fourier basis, prolate spheroidal wave functions),
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.
Prerequisite:
MAT 121, 127, 128, 167, 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
- The Uncertainty Principle and Bandlimited Signals
- Heisenberg's uncertainty principle and Gabor functions
- Various measures of concentration
- Prolate Spheroidal Wave Functions and Their Applications
- New Tools I: Splitting Time/Space Axis
- Local Cosine/Sine Transform
- Local Fourier Transform
- New Tools II: Splitting Frequency Axis
- Multiresolution Analysis and Wavelet Bases
- Discrete Wavelet Transforms
- Discrete Wavelet Packet Transforms
- New Tools III: Tools on Nonuniform Lattices
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.
- Y. Meyer: Oscillatory Patterns in Image Processing and Nonlinear
Evolution Equations, AMS, 2001.
- S. Mallat: A Wavelet Tour of Signal Processing, 2nd Edition, Academic
Press, 1999.
I will also hand out many notes and copies of original papers in
class.
Class Web Page:
Class Mailing List:
The class mailing list was created.
You can 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, everyone will receive it.
So, please use this wisely and politely. Its name is:
mat280-cha-w02@ucdavis.edu.
Grading Scheme:
- 30% Attendance
- 20% Quiz/Homework
- 50% Final Report
Homework:
I will occasionally (i.e., not every week) assign quiz/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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