Spring 2007 MATH 2025
Syllabus and Course Materials
Wavelets Made Easy

Office Hours till the end of the Spring Semester:

Th 5/10, Fri 5/11, Mon 5/14, Tue 5/15 and Wed 5/16, 1 - 2 pm.

Instructor: Jacek Cygan
Office: 392 Lockett
Phone: 578-1565
e-mail: cygan@math.lsu.edu
Classes on: Monday, Wednesday, Friday 12:40 - 1:30, Room 134 Lockett

Introduction Prerequisites Texts Course Outline Home Work Practice Exercises Grading Scheme Schedule of Exams, Assignments, and Due Dates Final Grades

The theory of wavelets is a relatively recent mathematical theory.

Some Introductory Articles About Wavelets

It is the basic theory behind several modern applications in storage of electronic information, data compression, image reconstruction and electronic transmission of information.

For applications in science, Engineering, Medicine, or Finance see e.g. The Illustrated Wavelet Transform Handbook by Paul S. Addison and references there.

The applications also include the storage of fingerprints (see Fingerprints Go Digital by Christopher M. Brislawn in the Notices of the American Mathematical Society, Vol. 42, Number 11, November 1995, or The FBI Fingerprint Image Compression Standard); the new jpg-standard (see e.g. JPEG 2000); and computer graphics (see Wavelets for Computer Graphics)

The basic ideas of wavelet theory can be formulated using the language of linear algebra: Vector spaces, subspaces, linear maps, inner product, orthogonal projections, and basis.

Related concepts in analysis are: Vector spaces of functions, approximation of functions using basic functions (in our case wavelets), dilation and translation, change of basis.

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Prof. Olafsson's Fall 2004 lecture notes as linked in the Course Outline section below.

Textbook Wavelets Made Easy by Yves Nievergelt.

We also recommend the book The World According to Wavelets by Barbara Burke Hubbard for history and general information about wavelets.

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• We will start with Prof. Olafsson's lecture notes on Functions, Wavelets, and Signals. This will serve as an introduction to Chapter 1 of Nievergelt's book.
  
  
• Then we will discuss the simplest wavelet - the Haar wavelet in one dimension. The main topic is the Fast Haar Wavelet Transform. This is a special case of orthogonal projections and change of basis. This is Chapter 1 of Nievergelt's book.
  
  
• We will translate now these facts into the quite general language of linear algebra. This material is in Chapter 4 of Nievergelt's book, but we will follow Prof. Olafsson's notes. The important concepts are as mentioned above:
  
  
  • Vector spaces, subspaces, vector spaces of functions
    
    Notes for:

    vector spaces
    subspaces
    vector spaces of functions
    

  • linear maps, inner products and norms, basis and orthogonal basis, Gram-Schmidt orthogonalization
    
    Notes for:

    linear maps
    inner products and norms
    generating sets and bases
    Gram-Schmidt orthogonalization
    

  • orthogonal projections
    
    
  We will give several examples that will be used later.
  
  
  
• After that we will discuss the two dimensional Haar wavelet, general construction of wavelets, and some applications. This is in Nievergelt's book Chapter 2.
  
  
• The wavelet transform is only one example of integral transforms. The Fourier Transform is much older. It has become an indispensable tool in mathematics and applied sciences. We will discuss some aspects of the Fourier Transform, namely an introduction to the Discrete Fourier Transform and the Fast Fourier Transform. Before doing that, we will need to review the complex numbers, and introduce the complex exponential function. This material is again in Nievergelt's Chapter 5.
  
  
• If there is time we will look at Daubechies Wavelets from Chapter 3, and How to Make Wavelets in general, following R.S. Strichartz's paper in The American Mathematical Monthly, Vol. 100, No. 6 (June - July, 1993), pp. 539-556.
  
  
• A presentation showing several applications of the Wavelet Transform
• Alfred Haar's 1909 dissertation where, in Chapter III, he introduced "The Orthogonal Function System χ" later called "Haar's wavelets"

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To do well in a math class, it is essential to get sufficient practice working problems. Accordingly, we will have both graded and ungraded homework assignments.

The Practice Exercises below, as well as exercises in the webnotes and textbook, will not be collected, but will on occasion be discussed in class. They will also be indicative of problems you will encounter on exams. So please try to do them.

The 3 graded assignments - Assignment 1, 2, and 3 (about 6 problems each), due at the begining of the class January 31, March 5, and April 13 respectively, will be posted in the table Schedule of Exams, Assignments, and Due Dates.

There are 650 points distributed as follows:

Three Assignments, worth ~33 points each
Three in-class Quizzes, worth also ~33 points each
Three in-class Tests, worth 100 points each
Final Exam, worth 150 points

Wednesday, January 31	Assignment #1
Wednesday, February 7	Quiz #1
Friday, February 16	Test #1
Monday, March 5	Assignment #2
Friday, March 9	Quiz #2
Friday, March 16	Test #2
Friday, April 13	Assignment #3
Friday, April 20	Quiz #3
Friday, April 27	Test #3
Wednesday, May 9 7:30-9:30 AM	Final Exam*

* Suggested Review for the Final Exam: Assignment 1, Text pages 20 - 21 and p.7 of Practice Exercises for "Fast Haar Wavelet Transform"; Test 2 and Practice Exercises for "Orthogonal Projections" p.1; Assignment 3; FFT problems from Test 3; Practice Exercises for "How to Make Wavelets".

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