MATH 772-001 - Wavelet Theory - Fall 2011



Project Deadlines:

(1)   Tuesday September 13. Paper that you will be basing your project on must be selected this week.You should each make an appointment to see me this week to discuss the details of what you will do for your project.

(2)   Tuesday September 20. Project proposal due.  This will consist of a detailed summary of our discussion of the previous week and will outline exactly what you intend to do for your project.  This proposal should be only about one page long.

(3)   Tuesday October 25. Project progress report due.  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, and I will ask for a meeting if I think it is warranted. This report should contain a draft of what you have already done (rough is fine), and an outline and time-table for what you intend to do.

(4)   Tuesday November 29. Final draft of written portion of project due. I will get back to each of you by Thursday, December 1 with any suggestions or additions that I think you need to make.  This draft should look very much like the final write-up that you turn in, and in fact should be essentially complete. Your final grade will be based in part on this draft.  You should also have an outline of your final presentation ready at this time and discuss it with me.

(5)   Week of December 6. Project presentations.

(6)   Tuesday December 13. Final project write-ups due. This should look essentially like the document you turned in on Nov. 29 with small changes made.


Deadlines.  Please be aware of all relevant deadlines

Course syllabus pdf

Class Notes and Homework


08-30-2011   Orthonormal Bases in Hilbert Spaces.  The 8 exercises in the notes are due Thursday September 08.


09-01-2011   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 Thursday, September 15.  Also you should make sure to do the verifications in Exercises 3.44-3.46 in Introduction.  These will not be collected.  These notes correspond to Chapter 3 of Introduction and Sections 1.1, 1.2, 1.4, 1.5, 2.1-2.3 of Foundations of Time-Frequency Analysis. 


09-06-2011   Wavelet Orthonormal Bases for L^2(R).   These notes correspond roughly to Chapter 3 of Introduction.  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.


09-13-2011   Multiresolution Analysis on L^2(R).  These notes follow basically Sections 7.1-7.5 of Introduction.  The verification of several details in the proofs of the theorems is 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 Thursday September 23.


09-20-2011   Quadrature Mirror Filter (QMF) Conditions and the Discrete Wavelet Transform (DWT).  These notes follow basically Sections 8.1-8.4 of Introduction.  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.


09-27-2011   Daubechies Wavelets.  These notes follow basically Chapter 9 of Introduction.   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.


10-04-2011  Compression of Images with Wavelets.  These notes follow basically Chapters 12 and 13 of Introduction.   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.








11-03-2011   Frames and Riesz Bases 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 Foundations.


11-10-2011   Gabor Systems:  Existence and Basic Properties.  This lecture covers portions of Chapter 6 in Foundations.


11-17-2011   Gabor Systems:  Duality and Density.  This lecture covers portions of Chapter 7 in Foundations.


11-29-2011   The Zak Transform and the Balian-Low Theorem.  This lecture covers portions of Chapter 8 in Foundations.

Solutions to Homework sets


Math Department.

About the project

What your project should be like.

Your semester project should be based on a research or advanced expository paper published in or after the year 2000

on some theoretical or applied aspect of wavelet theory, Gabor systems, or more generally in time-frequency analysis.

The project can overlap with class material but must extend it in some significant way.

The paper you choose must be approved by me.


What I am expecting.

Your project should consist of two parts.

(1)    A write up of approximately 15 pages.  This will include background exposition material on the subject matter of the paper,

detailed proofs of theorems and lemmas in the paper, and implementation of any algorithms described in the paper.  Depending

on the depth or difficulty of the paper, all of the above mentioned goals can be negotiated.  The goal is that you gain an understanding

of an aspect of wavelet theory or time-frequency analysis that is of particular interest to you.  The write-up will be typed using some

kind of software capable of producing attractive mathematical output, such as TeX or MS Word.


(2)    A 30-minute presentation of your project for the class.  The presentations will be scheduled for the last week of class and should

be done using a projection system of some kind.  Do not go into detail about your entire project but set as your goal to present to

the class the basic ideas in your paper, and to convey why the results of your paper are interesting and important.


To contact me, send mail to: