Announcements:
Project Progress Report due. A progress report on your project is due Monday November 5. This should contain a detailed account of what you have accomplished so far on the project. I will be happy to meet with you to discuss this in person if you want.
Project Rough Draft. A rough draft of your project will be due on Monday November 26. This should be as complete as possible, but need not be in its final form. I will return it with comments the following week.
Deadlines. Please be aware of all relevant deadlines
Class Notes and Homework
08-27-2007 Orthonormal Bases in Hilbert Spaces. The 10 exercises in the notes are due Monday September 10.
09-10-2007 Review of Fourier Analysis. For homework do problems 3.4-3.7, 3.15 and 3.31 in An Introduction to Wavelet Analysis. These will be due on September 17. Also you should make sure to do the verifications in Exercises 3.44-3.46 in An Introduction... These will not be collected. These notes correspond to Chapter 3 of An Introduction to Wavelet Analysis and Sections 1.1, 1.2, 1.4, 1.5, 2.1-2.3 of Foundations of Time-Frequency Analysis.
09-17-2007 Wavelet Orthonormal Bases for L^2(R). These notes correspond roughly to Chapter 3 of An Introduction to Wavelet Analysis. The direct characterization of orthonormal wavelet bases given in the notes does not appear in the book but is taken from Hernandez and Weiss, A First Course in Wavelets. There is no homework assignment for this week.
09-24-2007 Multiresolution Analysis on L^2(R). These notes follow basically Sections 7.1-7.5 of An Introduction to Wavelet Analysis. The verification of several details in the proofs of the theorems are left as assigned exercises in the attached notes. All of this stuff is fairly routine and indeed appears already in the book, but is designed to force you to follow the mathematical argument more carefully. These exercises will be due Monday Oct. 1.
10-01-2007 Quadrature Mirror Filter (QMF) Conditions and the Discrete Wavelet Transform (DWT). These notes follow basically Sections 8.1-8.4 of An Introduction to Wavelet Analysis. I will in this class show you some MATLAB demos of wavelet decompositions of one and two-dimensional signals (images). This should give you a good intuition of what the wavelet coefficients are telling you about the function being analyzed.
10-09-2007 Daubechies Wavelets. These notes follow basically Chapter 9 of An Introduction to Wavelet Analysis. I will go over the construction of compactly supported wavelets of arbitrary smoothness from the point of view of filter design. There will be a fair amount of MATLAB demonstrations illustrating the construction and also applications of the wavelets to images.
10-15-2007 Wavelet Packets and Best Bases. These notes follow basically Chapter 11 of An Introduction to Wavelet Analysis. Wavelet packets are an introduction to wavelet-like bases corresponding to more general tilings of the time-frequency plane, and also the notion of a "library" of orthonormal bases from which a basis best-suited to the application at hand can be chosen. It turns out that the algorithm for computing decompositions in these bases is very fast. The exercises at the end of these notes are due in two weeks, so on October 29.
10-22-2007 Compression of Images with Wavelets. These notes follow basically Chapters 12 and 13 of An Introduction to Wavelet Analysis. Here I will give the basic idea behind two successful applications of wavelet ideas in applications. Most applications of wavelets will be at least conceptually related to one of these algorithms.
10-29-2007 Local Trigonometric Bases. These notes follow some sections in Chapter 1 of Hernandez and Weiss, A First Course in Wavelets, a copy of which has been handed out to you. In this construction, an orthonormal basis is found whose elements are very well-localized in time and frequency and such that the time localization of the elements are on an arbitrary partition of the real line. Homework Exercises related to this lecture are here and are due on November 12.
11-05-2007 Nonorthogonal Bases and Frames in Hilbert Space. These notes are a primer on nonorthogonal bases and frames in Hilbert Spaces, focusing on the notion of a Riesz basis (which is one step away from an orthonormal basis) and the notion of a frame, which can be thought of as an overcomplete system. These notions will be important in our discussion of Gabor systems. Much of the discussion of frames was taken from this paper (link here). Also some of the frame discussion appears in Section 5.1 of Grochenig, Foundations of Time-Frequency Analysis.
11-12-2007 Gabor Systems: Existence and Basic Properties. This lecture covers portions of Chapter 6 in Grochenig, Foundations of Time-Frequency Analysis.
11-19-2007 Gabor Systems: Duality and Density. This lecture covers portions of Chapter 7 in Grochenig, Foundations of Time-Frequency Analysis.
11-26-2007 The Zak Transform and the Balian-Low Theorem. This lecture covers portions of Chapter 8 in Grochenig, Foundations of Time-Frequency Analysis.
Solutions to Homework sets
About the project
What your project should be like.
Your semester project should fall into one of two categories:
(1) Development of some mathematical aspect of wavelet theory not covered already in class;
(2) Exposition of some application of wavelet theory also not covered already in class.
Projects of both kinds can overlap with class material but must extend it in some significant way.
Details can be worked out with me.
What I am expecting.
Your project should consist of two parts:
(1) a write up of approximately 15 pages, and
(2) a 20-minute presentation of your project for the class
(3) It is recommended that you use either the overhead projector or the beamer for the presentation. The clasroom computer is available for this or you can hook up your own laptop to the projector.
Some project ideas.
There are some ideas for possible projects in the appendix to the textbook entitled: Excursions in Wavelet Theory.
Some of the more appropriate ones are listed below:
Another possibility is the following.
Chapter 13 of the book describes the BCR algorithm in the
context of two examples of integral operators:
Sturm-Liouville Boundary Value Problems, and The Radon Transform.
Implementation and experimentation with the BCR algorithm in relation to some of
these operators would be very interesting. See me for details.
About the presentations.
To contact me, send mail to: dwalnut@gmu.edu.