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
 
 

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.

Multiresolution representation of a complex shape