Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
Text
Strang, and Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, 1997.
Course Structure
The course will consist of lectures, homework assignments and a project on a topic related to the student's area of interest. We will aim for the right balance of theory and "applications". The course has no specific prerequisites, although a basic knowledge of Fourier transforms is recommended. We start with time-invariant filters and basic wavelets. The text gives an overall perspective of the field -- which has grown with amazing speed. The topics will include
- Analysis of Filter Banks and Wavelets
- Design Methods
- Applications
- Hands-on Experience with Software
These four key areas will be developed in detail.
Analysis
- Multirate Signal Processing: Filtering, Decimation, Polyphase, Perfect Reconstruction and Aliasing Removal.
- Matrix Analysis: Toeplitz Matrices and Fast Algorithms.
- Wavelet Transform: Pyramid and Cascade Algorithms, Daubechies Wavelets, Orthogonal and Biorthogonal Wavelets, Smoothness, Approximation, Boundary Filters and Wavelets, Time-Frequency and Time-Scale Analysis, Second-Generation Wavelets.
Design Methods
- Spectral Factorization, Cosine-Modulated Filter Banks, Lattice Structure, Ladder Structure (Lifting.)
Applications
- Audio and Image Compression, Quantization Effects, Digital Communication and Multicarrier Modulation, Transmultiplexers, Text-Image Compression: Lossy and Lossless, Medical Imaging and Scientific Visualization, Edge Detection and Feature Extraction, Seismic Signal Analysis, Geometric Modeling, Matrix Preconditioning, Multiscale Methods for Partial Differential Equations and Integral Equations.
Simulation Software
- MATLAB® Wavelet Toolbox, Software for Filter Design, Signal Analysis, Image Compression, PDEs, Wavelet Transforms on Complex Geometrical Shapes.
We encourage you to learn about wavelets and their applications.
Multiresolution representation of a complex shape. Courtesy of Igor
Guskov, University of Michigan. Used with permission.