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Topics in Applied Mathematics

3-0-9 units · Graduate

Topics vary from year to year.

Spring 2003 · Wavelets, Filter Banks and Applications

Topics

  • Discrete-time Filters: Convolution; Fourier Transform; Lowpass and Highpass Filters
  • Sampling Rate Change Operations: Upsampling and Downsampling; Fractional Sampling; Interpolation
  • Filter Banks: Time Domain (Haar example) and Frequency Domain; Conditions for Alias Cancellation and no Distortion
  • Filter Banks (contd.): Perfect Reconstruction; Halfband Filters and Possible Factorizations
  • Modulation and Polyphase Representations: Noble Identities; Block Toeplitz Matrices and Block z-transforms; Polyphase Examples
  • MATLAB ® Wavelet Toolbox
  • Orthogonal Filter Banks: Paraunitary Matrices; Orthogonality Condition (Condition O) in the Time Domain, Modulation Domain and Polyphase Domain
  • Maxflat Filters: Daubechies and Meyer Formulas. Spectral Factorization
  • Multiresolution Analysis (MRA): Requirements for MRA; Nested Spaces and Complementary Spaces; Scaling Functions and Wavelets
  • Refinement Equation: Interative and Recursive Solution Techniques; Infinite Product Formula; Filter Bank Approach for Computing Scaling Functions and Wavelets
  • Project Brief
  • Orthogonal Wavelet Bases: Connection to Orthogonal Filters; Orthogonality in the Frequency Domain. Biorthogonal Wavelet Bases
  • Mallat Pyramid Algorithm
  • Accuracy of Wavelet Approximations (Condition A); Vanishing Moments; Polynomial Cancellation in Filter Banks
  • Smoothness of Wavelet Bases: Convergence of the Cascade Algorithm (Condition E); Splines. Bases vs. Frames
  • Signal and Image Processing: Finite Length Signals; Boundary Filters and Boundary Wavelets; Wavelet Compression Algorithms
  • Guest Lecture. Physical Wavelets and their Sources: Real Physics in Complex Spacetime
  • Lifting: Ladder Structure for Filter Banks; Factorization of Polyphase Matrix into Lifting Steps; Lifting Form of Refinement Equation
  • Wavelets and Subdivision: Nonuniform Grids; Multiresolution for Triangular Meshes; Representation and Compression of Surfaces
  • Numerical Solution of PDEs: Galerkin Approximation; Wavelet Integrals (Projection Coefficients, Moments and Connection Coefficients); Convergence. Subdivision Wavelets for Integral Equations. Compression and Convergence Estimates
  • M-band Wavelets: DFT Filter Banks and Cosine Modulated Filter Banks. Multiwavelets
  • Project Presentations

Readings

Prerequisites & course information

Prereq: Permission of instructor Acad Year 2027-2028: G (Fall) Acad Year 2028-2029: Not offered 3-0-9 units Can be repeated for credit.

Sources