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University of St. Thomas Center for Applied Mathematics

MINICOURSE OVERVIEW

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Flower Transform

Wavelets and Applications: A Multi-Disciplinary Undergraduate Course with an Emphasis on Scientific Computing

The purpose of this minicourse is to give participants an elementary introduction to wavelets and applications and an outline of the Wavelets and Applications course offered at the University of St. Thomas so that they might offer a similar course at their home institution. The minicourse is sponsored by the Mathematical Association of America.

Registration is available online or on site at the Joint Mathematical Meetings in New Orleans, LA. As of December 27, 2006, there are still 2 open spots in the workshop.

MAA

The workshop will be conducted at the Joint Mathematical Meetings in New Orleans, LA. There are two 2-hour sessions. Both sessions will be conducted in the Nottoway Room located on the 4th Floor of the Sheraton. The first session will be held on Saturday, January 6, 2007 from 10:30am to 12:30pm and the second session will run on Monday, January 8, 2007 from 1:00pm to 3:00pm.

The minicourse is offered by University of St. Thomas Mathematics Professor Patrick J. Van Fleet. He serves as the Director of the Center for Applied Mathematics at the University of St. Thomas, has written research papers in the area of (multi)wavelets, and spent the past several years developing the Wavelets and Applications course that will be discussed during the workshop. Support for the development of the course materials has been provided by the National Science Foundation.

Who Should Attend?

This minicourse is designed for anyone interested in learning about wavelets at a very elementary level or those interested in offering a course (see description below) in wavelets at their home institution. Since the course offered at the University of St. Thomas requires only a general computer science course, Calculus II, and a sophomore linear algebra course as prerequisites, the workshop is entirely self-contained. It is desirable that participants have some knowledge of Fourier series and Fourier transforms, but not necessary. Participants should be comfortable working with either Mathematica or Matlab.

JMM

What to Bring?

The Mathematical Association of America will provide computers equipped with Mathematica and a connection to the internet. All minicourse materials will be available on this website. A CD with the course materials will be distributed during the first session. A draft of the text Discrete Wavelet Transformations - An Elementary Approach with Applications by Patrick J. Van Fleet will also be included on the CD. Participants do not need to bring laptop computers unless they want to work on the material at other times during their trip to New Orleans. We will be processing digital images during the minicourse. Several test images will be provided but participants can bring their own images (or make them accessible via the web or email) if they so desire.

Minicourse Structure

NSF

Each session will be divided into two (somewhat) equal parts. One part will be traditional lecture and question/answer and the second part will be computer work. All computer work will be performed using Mathematica. Previous experience with Mathematica is helpful but not at all required. The computer labs are set up so that minimal coding is required. The lectures (in pdf format) and Mathematica notebooks will be posted to this web site a few days before the minicourse.

Wavelets and Applications Course

The course was first offered at the University of St. Thomas during the Spring 1998 semester as Math 316 Applied Mathematics and Modeling II. The course is part of a two-course sequence that is required of mathematics majors in the applied track. The purpose of the sequence is to expose students to current topics that have applications that are of interest to professionals in business or industry. As the development of the course progressed and I learned more about the type of students who take the course, it became clear that the topic of wavelets served many student needs.

Strong Computer Skills: Strong programming skills (or at least the ability to master sophisticated software packages)is necessary for many professionals who hold positions desired by mathematics majors. As more and more occupations require computer mastery, the typical required computer science course (or at least the one at the University of St. Thomas) is required to serve a wider variety of students. As a consequence, students typically do not see as much scientific programming as they did in past years. Since scientific programming is not only necessary in several occupations but also can be used as an effective learning tool, I decided the course should make heavy use of the computer and that my students would do a large amount of programming (the course in its current version allows the instructor to widely vary the amount of programming performed by the students). Since the Mathematics Department at UST requires the students to learn Mathematica as they take their calculus sequence, all software was developed using this package. We will soon begin the process of porting the software to Matlab and then Maple.
In Wavelets and Applications, students write modules for almost every computational aspect of the course. They develop routines for image encoding/decoding, quantitative and qualitative measures useful in applications, and discrete wavelet transforms (1D and 2D). A Mathematica package template is made available to them so that they can add these routines to a package. The package has help and integrates into the Help Browser. Students then use these routines in several applications. They download or create their own images/audio files and learn about digital images/audios from a very basic level. An emphasis is placed on creating code that is readable but also runs in an expeditious manner.
Several class periods are dedicated to programming or lab work and students invariably open their package to work along with me while I lecture. In this way, they come to see scientific programming as an effective learning tool.
Problem Solving: "What can I do with a major in mathematics?" "How can I really use mathematics in an application?" Most of us entertain these questions on a regular basis. The best way I have found to answer these questions is to get students immediately immersed in applications. After attending a workshop conducted by Gilbert Strang of MIT and Truong Nguyen of University of California San Diego, I became convinced that the best way to approach the course was to develop the idea of filtering. This approach requires an introduction to Fourier series, convolution, and filters. Once the students are comfortable with these ideas, it is straight-forward to develop the Discrete Haar Wavelet Transform and dive into applications such as naive image compression and edge detection. Some deficiencies of the HWT are made apparent by the applications, so a little more work gives us the Daubechies filters that we can insert into our transform matrices. Understand that the development of these discrete wavelet transforms is totally ad hoc - the advantage is that the students can immediately delve into applications but the downside is there is little introduction to the classical theory.
It is interesting to watch the reactions of the students as they develop the transformations and study the applications. The engineering and physics majors (the course is multi-disciplinary in nature and popular as a minor elective) totally embrace the idea of the ad hoc development and attack the applications much like they would problems in their own courses. The mathematics students are quite ambivalent about the process. They learn that they don't necessarily like the answers to the questions asked at the beginning of this section! In so many of their classes, mathematics majors are given explicit instructions for solving problems. The approach in this class is thus new to them. Most come to appreciate the development of the material in this manner. About halfway through the course, I show the students results of an image compression application that compares the filters they've developed to the so-called biorthogonal filters (these filters are used by the FBI to compress fingerprints as well in the JPG2000 compression algorithm). The bi-orthogonal filters outperform the filters they have developed and the students are naturally interested to learn how these filters were derived. At this point (and 6 weeks into the class!), I am ready to tell them that in order to develop highly sophisticated filters, we must model the wavelet transform in a more classical and general mathematical setting. Having seen what can be done with wavelets, the students are very comfortable and motivated to move into the more general mathematical setting. It is very illuminating to watch students (especially the non-majors) take to general theory once they have an idea why it is useful.
Solidify Ideas from Calculus and Linear Algebra and Motivation for Higher Level Courses: The ad hoc development has lead me to realize that the following statement is generally true: Many sudents, upon completion of the courses, possess the solid understanding of calculus and linear algebra necessary to excel in subsequent upper level courses. In this regard, a course such as Wavelets and Applications serves students well. The mathematics required to get them "up and running" in the course is minimal and the ad hoc design of the discrete transforms is an implicit review of calculus and linear algebra. Writing the computer modules also strengthens students matrix arithmetic skills.
In order to move through the classical theory, it is necessary to take some ideas for granted. In particular, pointwise convergence and precise manipulation of Fourier series are some of the first casualities in this approach. A multiresolution analysis is a set of nested subspaces in L² (R) and it is unrealistic to expect students at this level to understand the ideas of measurable sets. In this regard, the course is unpopular with mathematicians who insist on a high level of rigor in their courses. The advantage of this approach is that the students can see why they might want to learn more about the theoretical aspects glossed over in the course. In this way, the course provides motivation for taking higher-level mathematics courses.
Learn About a Current Topic: Many students have indicated to me that much of what they see in mathematics texts is several hundred years old. While I point out that the mathematics they learn is timeless, I am sympathetic to their desire to see mathematics applied to a current problem. Wavelets are a natural topic for illustrating to students how mathematics is used in today's applications. We actually step through a naive version of the JPG2000 compression algorithm and look at some popular methods for de-noising signals via wavelets. While the students do not leave the course ready to conduct theoretical research in the area of wavelets, they do leave with a solid appreciation of how mathematics is used in applications such as image compression and signal de-noising. They also understand the general problems still facing researchers and appreciate the amount of knowledge required to tackle such problems. In some cases, the course has motivated students to continue on in mathematics when they previously might not have done so.

The Course and the Minicourse

What I have learned developing the undergraduate class Wavelets and Applications is that the topic provides a wonderful arena for solidifying ideas learned in calculus and linear algebra, introducing important ideas and uses of complex variables, demonstrating real-world applications of mathematics, and developing scientific programming skills that students need if they are to succeed in the high-tech workforce. I have found that it is entirely tractable to offer this course to students who have completed Calculus I, II, a computer programming course, and sophomore linear algebra. I view the course as a post-sophomore capstone course that strengthens student knowledge in the prerequisite courses and provides some rationale and motivation for the mathematics they will see in courses such as real analysis. The course is entirely self-contained, makes heavy use of technology, and concludes with group projects that emphasize the communication skills paramount for success in any occupation.

In this minicourse, we will step through the learning process the students encounter. We will learn about digital image basics and the ad hoc development of the discrete transform and from there, move into the classical presentation of wavelet theory. We will spend a significant amount of time on applications - we will move from downloading our own images from the internet to denoising or compressing them.

The course serves many needs and it is popular among students. It is a natural class for those in engineering or physics who seek a minor in mathematics, it provides much-needed rationale for why we learn mathematics to prospective teachers, and it gives a gentle introduction to many ideas students will see in higher level mathematics courses. Hopefully, you will leave the workshop with the idea of offering the course at your institution!