Mathematicians from harmonic analysis, functional analysis, and approximation theory have helped to develop wavelet theory and to construct algorithms for use in applications. In this workshop, we will discuss the Haar wavelet transform: simple averaging and differencing of pairs of numbers. This elementary transform lays the groundwork for the software development we will undertake and allows us to connect Fourier series and convolution to wavelet filter development.
Through the development of an undergraduate course on wavelets, the workshop organizer realized that the way in which wavelet theory came into being is an effective way to present the material to undergraduates. Constructing discrete wavelet transforms in an ad hoc manner (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.
This course takes an "applications first" approach: the discrete wavelet transforms are developed in an ad hoc manner. We will delve into applications such as image boundary detection, naive compression, and signal de-noising. We will write all the software we need to perform these tasks. In order to build so-called biorthogonal wavelet filters, we need to look at our construction in a more general mathematical context. Once we have learned this construction, we will return again to applications. The biorthogonal wavelet transform is now used in the new JPEG2000 format and in the FBI Fingerprint Standard. We will end the workshop by studying one of these applications in detail.
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.
More information regarding the workshop can be found online at http://cam.mathlab.stthomas.edu/wavelets.