18.327 | Spring 2003 | Graduate

Wavelets, Filter Banks and Applications

Lecture Notes

SLIDES open as color slides in a pdf document. HANDOUTS open as black-and-white slides in a pdf document. To view the color slides properly, you might need special fonts. If this happens, please refer to the handouts instead, which have all the fonts embedded in them and can be viewed or printed as-is.

Discrete-time Filters: Convolution; Fourier Transform; Lowpass and Highpass Filters Slides 1 (PDF) Handout 1 (PDF)
Sampling Rate Change Operations: Upsampling and Downsampling; Fractional Sampling; Interpolation Slides 2 (PDF) Handout 2 (PDF)
Filter Banks: Time Domain (Haar example) and Frequency Domain; Conditions for Alias Cancellation and no Distortion Slides 3 (PDF) Handout 3 (PDF)
Filter Banks (contd.): Perfect Reconstruction; Halfband Filters and Possible Factorizations Slides 4 (PDF) Handout 4 (PDF)
Modulation and Polyphase Representations: Noble Identities; Block Toeplitz Matrices and Block z-transforms; Polyphase Examples Slides 5 (PDF) Handout 5 (PDF)
MATLAB® Wavelet Toolbox Slides 6 (PDF) Handout 6 (PDF)
Orthogonal Filter Banks: Paraunitary Matrices; Orthogonality Condition (Condition O) in the Time Domain, Modulation Domain and Polyphase Domain Slides 7 (PDF) Handout 7 (PDF)
Maxflat Filters: Daubechies and Meyer Formulas. Spectral Factorization Slides 8 (PDF) Handout 8 (PDF)
Multiresolution Analysis (MRA): Requirements for MRA; Nested Spaces and Complementary Spaces; Scaling Functions and Wavelets Slides 9 (PDF) Handout 9 (PDF)
Refinement Equation: Iterative and Recursive Solution Techniques; Infinite Product Formula; Filter Bank Approach for Computing Scaling Functions and Wavelets Slides 10 (PDF) Handout 10 (PDF)
Orthogonal Wavelet Bases: Connection to Orthogonal Filters; Orthogonality in the Frequency Domain. Biorthogonal Wavelet Bases Slides 11 (PDF) Handout 11 (PDF)
Mallat Pyramid Algorithm Sec Slides 12 (PDF) Handout 12 (PDF)
Accuracy of Wavelet Approximations (Condition A); Vanishing Moments; Polynomial Cancellation in Filter Banks Slides 13 (PDF) Handout 13 (PDF)
Smoothness of Wavelet Bases: Convergence of the Cascade Algorithm (Condition E); Splines. Bases vs. Frames Slides 14 (PDF) Handout 14 (PDF)
Signal and Image Processing: Finite Length Signals; Boundary Filters and Boundary Wavelets; Wavelet Compression Algorithms Slides 15 (PDF) Handout 15 (PDF)
Lifting: Ladder Structure for Filter Banks; Factorization of Polyphase Matrix into Lifting Steps; Lifting Form of Refinement Equation Sec Slides 16 (PDF) Handout 16 (PDF)
Wavelets and Subdivision: Nonuniform Grids; Multiresolution for Triangular Meshes; Representation and Compression of Surfaces Slides 17 (PDF) Handout 17 (PDF)
Wavelets and Subdivision (contd.): Nonuniform Grids; Multiresolution for Triangular Meshes; Representation and Compression of Surfaces Slides 18 (PDF) Handout 18 (PDF)
Numerical Solution of PDEs: Galerkin Approximation; Wavelet Integrals (Projection Coefficients, Moments and Connection Coefficients); Convergence. Subdivision Wavelets for Integral Equations. Compression and Convergence Estimates Slides 19 (PDF) Handout 19 (PDF)
M-band Wavelets: DFT Filter Banks and Cosine Modulated Filter Banks. Multiwavelets Slides 20 (PDF) Handout 20 (PDF)