MIT OpenCourseWare | Mathematics | 18.327 Wavelets, Filter Banks and Applications, Spring 2003

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TOPICS #	SLIDES	HANDOUTS


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)