When and Where Semester: Fall 2009 Meeting time: Wed 4-5:30, Fri 3-4:30 Meeting place: PGH 348 Office Hours: Mon 3-4, Wed 3-4 (or by appointment) Instructor Name: Demetrio Labate E-mail address: dlabate@math.uh.edu URL: http://www.math.uh.edu/~dlabate	An FBI-digitized left thumb fingerprint. The image on the left is the original; the one on the right is reconstructed using wavelets with a 26:1 compression.

Wavelet theory stands at the crossroad of harmonic analysis, signal processing and scientific computing. Its goal is to provide a coherent set of mathematical methods that are adapted to the study of a variety of nonstationary signals and are also suitable for efficient algorithmic implementation. The past decade has witnessed an explosion of research on wavelets, including thousands of papers and a great number of successful applications. These applications include, to name a few, the new FBI fingerprint database and JPEG2000, the new standard for image compression (replacing the old JPEG).

This course will provide an introduction to the theory of wavelets and its applications in mathematics and signal processing. Some of the topics I will address are:

Orthonormal bases and frames: A basic problem in mathematics and engineering is to represent a function or a signal as superposition of elementary components. I will introduce the theory of frames and show that it provides the general framework to address this problem. Orthonormal bases are a special example of frames.

Wavelet bases: The first wavelet basis, the Haar basis, was discovered in 1909 before wavelet theory was born. Unfortunately, the elements of this basis are not continuous. The success of the wavelet theory is due to the ability to construct a variety of wavelet bases with very nice mathematical properties such as smoothness, compact support, vanish moments, etc. I will present several examples of wavelet bases and describe what kind of features are desirable in such a basis.

Multiresolution Analysis: Multiresolution analysis is a general method for constructing wavelet bases. I will describe how to use this approach to construct the Shannon wavelets, the Daubechies wavelets and the spline wavelets.

Wavelets and approximation theory: One striking feature of wavelets is their ability to represent function with discontinuities. In fact wavelets have optimal approximation properties for several classes of functions and signals. I will introduce linear and nonlinear approximations, examine the approximation properties of wavelets and compare them to Fourier methods.

Wavelets and signal processing: Wavelets appear today in a variety of advanced signal processing applications, including analysis and diagnostics, quantization and compression, transmission and storage, noise reduction and removal. I will describe the connection between wavelet theory and filter banks theory in signal processing. I will present applications of wavelets to data/image compression and denoising. Some of this applications will be further explored by the students as individual or group projects.

Prerequisite

It is strongly encouraged that students have a background in Linear Algebra and Real Analysis.

Links


Review on : Lebesgue Integration (by Prof. C.Heil)

Review on : Banach and Hilbert spaces (by Prof. C.Heil)

HOMEWORK

Homework 1. Due Wed Sep 16. Solution: Solution HW1
Homework 2. Due Wed Oct 7. Solution: Solution HW2
Homework 3. Due Wed Oct 28.
Homework 4. NOTICE: you need the file mysigex.m. Due Dec 1.

Text

I will select most material from: A Mathematical Introduction to Wavelets , by P. Wojtaszczyk, 1997 and from A First Course on Wavelets , by E. Hernandez and G. Weiss, 1996. Additional notes and papers will be provided by the instructor.