EEE5067-WAVELET THEORY AND MULTIRESOLUTION SIGNAL ANALYSIS

COURSE OBJECTIVES:

Wavelets have made quite a splash recently in the signal and image processing communities, especially with regard to applications like compression, modeling, restoration, decomposition, and reconstruction of signals. Until recently, they have been studied under different names by the different communities; mathematics, quantum physics, and signal processing. This course will present the recent body of knowledge that has contributed a unified understanding of this field and will describe the relationship between wavelets, multirate filters banks, and multiresolution analysis studied in the mathematics, signal processing, image processing, and biomedical signal analysis, respectively. The treatment will have a signal processing flavor with sufficient mathematical rigor. Applications of wavelets to signal compression, modelling image analysis and restoration, and biomedical signal analysis will be addressed. Subject covered: Continuous and discrete wavelet transform, multiresolution analysis, filter banks, orthonormal wavelets, biorthogonal wavelets, wavelet packet, and wavelet applications.

SYLLABUS:

WEEK 1--	Fourier Transform; The Time Domain Signal (Digital and Analog Signals), The Frequency Domain Signal HAFTA(Digital and Analog Signals), Sampling Theorem, Subsampling and Upsampling of a Discrete Signal, The Heisenberg Uncertainty Principle, Time-Frequency Plane
WEEK 2--	Basic Definition of Wavelet; Basics of Wavelet Analysis, The Haar Wavelet, Shifting and Scaling the Haar Wavelet, General Description of the Continuous Wavelet Transform, General Description of the Discrete Wavelet Transform
WEEK 3--	Filters; Definitions of Digital Filters, Inverse and Bandpass Filters, QMF and PRF Filters
WEEK 4--	Filter Bank; Analysis and Synthesis, Perfect Reconstruction, Losless Filter Bank, Polyphase Matrix, Note on Orthogonality
WEEK 5--	Multiresolution Analysis; Definition of a Multiresolution Analysis, Bases and Frames, The scaling function or Father function, Solution of Dilation Equations, Properties of the Scaling Function, Mother Wavelet, Existence of the Wavelets
WEEK 6--	Wavelet Transform and Filter Banks; Wavelet Expansion and Filtering, Filter Bank Interpretation, Fast Wavelet Transform, Wavelets and Linear Algebra
WEEK 7--	Approximation Properties and Wavelet Design; Approximation, Design Properties:Overwiew, Some Well-known Wavelets: Shannon or Sinc Wavelet, Meyer Wavelets, Daubechies Wavelet, Coiflets Wavelets, and Spline Wavelets, Battle-Lemarie Wavelet, Redundant Discrete Wavelet Transform
WEEK 8--	Wavelet Packets; Tree Structured Filter Banks and Wavelet Packets, Multichannel Filter Banks, Lapped Orthogonal Transform
WEEK 9--	Orthonormal Wavelets; Orthonormal Wavelet Spaces, Regularity of Orthonormal Wavelet Bases, Connecting with Subband Structures
WEEK 10--	Biorthogonal Wavelets; Dual Spaces, Compactly Supported Spline Wavelets, The Duality Principle
WEEK 11--	Signal Processing Applications; Music and Audio Signals, NMR Spectroscopy, ECG and EEG Signals
WEEK 12--	Image Processing Applications; Image Compression, Edge Detection, Texture Analysis, Computer Graphics
WEEK 13--	Project Evaluation
WEEK 14--	Midterm Exam

TEXTBOOK: Charles K. Chui, "Wavelets:A Mathematical Tool for Signal Analysis", Siam Publisher, Philadelphia, (Revision Edition) 1999.

SUPPLEMENTARY TEXTBOOKS:

1. G. Strang, T.Q. Nguyen, "Wavelets and Filter Banks", Wellesley-Cambridge Press, Wellesley, MA, 1998.
2. G. Kaiser, "A Friendly Guide to Wavelets", Birkhauser, 1994.
3. M. Vertelli and J. Kovacevic, "Wavelets and Subband Coding", Prentice Hall, Englewood Cliffs, NJ, 1995.

LECTURE NOTES: