Wavelets and Applications: A Multidisciplinary Undergraduate Course with an Emphasis on Scientific Computing
Patrick Van Fleet, University of St. Thomas; Catherine Beneteau,
University of South Florida; Caroline Haddad, SUNY Geneseo; and David
Ruch, Metropolitan State College of Denver
June 4-7, 2008
University of St. Thomas
St. Paul, MN
Primary funding provided by the National Science Foundation Division of
Undergraduate Education (DUE-0442684).
Registration Fee: $325 by April 28, $450 after
Wavelet theory draws from mathematically rich areas such as harmonica analysis, analysis, approximation theory, and functional analysis. While the classical development of the theory is beautiful mathematics, it is perhaps not the easiest way to learn about the development of the Discrete Wavelet Transformations (DWTs) that are used in so many of today’s digital-image and signal-processing applications. We will motivate the development of the DWT in an ad hoc manner and provide rationale for why the topic makes for an excellent course on mathematical applications at an early point in an undergraduate’s career. We will construct the discrete Haar wavelet transformation (it is simple averaging and differencing) and learn how it can be used to perform image edge detection and naïve image compression. These applications will lead us to discover that the wavelet transformation we desire has features not available in the Haar transform. We will continue the development of more advanced DWTs in an ad hoc manner and use them to perform signal and image denoising. At this point, participants (and students in the class!) will have a strong understanding of the DWT and its use in applications. In order to develop the sophisticated DWTs used in applications such as the JPEG2000 image compression standard or the FBI wavelet-scalar quantization standard for archiving fingerprints, we must now model the transform development in a more structured way. The workshop will conclude with a discussion of this mathematical model (i.e., developing the filters for the DWT using Fourier methods) and the use of the DWT in the JPEG2000 image compression standard.
Participants will learn how an introductory topics course (requiring only calculus and linear algebra) on DWTs and their applications can be of great benefit to their students. Taking an “applications first” approach (1) shows students that real-world problems are typically solved by using different areas of mathematics; (2) solidifies ideas from sophomore calculus and linear algebra; (3) establishes the computer as an effective learning tool; (4) provides strong motivation for taking upper-level classes such as real analysis; and (5) allows students to learn about a current topic and its uses in real-world applications. Students are excited to see current applications, and they develop a new appreciation for linear algebra when they see how it can be used to process digital images and audio clips!
Software for the course is available in Mathematica or MATLAB. Participants are encouraged to bring their own digital images or digital audio files for use in the applications portion of the workshop. Workshop participants will receive software and lecture materials that can be used to offer the course at their home institution. The workshop will follow the presentation given in the book Discrete Wavelet Transformations: An Elementary Approach with Applications (Wiley, 2008). The workshop features two instructors who were participants at past offerings of this PREP workshop and offers an opportunity for participants to further develop materials for the course at a workshop at a subsequent MathFest meeting.More information regarding the workshop can be found online at http://www.stthomas.edu/wavelets.
Questions about PREP? Contact Olga Dixon at 202-319-8498 or email@example.com.