The course grade is based primarily on the class project (presentation + final report). Approximate breakdown: Project: 90%, Homework: 10%.
READINGS refer to the Text: Strang and Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, 1997.
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 lecture 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.
| SES # | TOPICS | READINGS | LECTURE NOTES | ASSIGNMENTS |
|---|---|---|---|---|
| 1 | Discrete-time Filters: Convolution; Fourier Transform; Lowpass and Highpass Filters | Sec 1.1-1.4, 2.1 | ||
| 2 | Sampling Rate Change Operations: Upsampling and Downsampling; Fractional Sampling; Interpolation | Sec 3.1-3.3 | ||
| 3 | Filter Banks: Time Domain (Haar example) and Frequency Domain; Conditions for Alias Cancellation and no Distortion | Sec 4.1 | HW 1 out (PDF) | |
| 4 | Filter Banks (contd.): Perfect Reconstruction; Halfband Filters and Possible Factorizations | Sec 4.1 | ||
| 5 | Modulation and Polyphase Representations: Noble Identities; Block Toeplitz Matrices and Block z-transforms; Polyphase Examples | Sec 3.4, 4.1-4.4 | ||
| 6 | MATLAB® Wavelet Toolbox |
HW 1 due HW 2 out (PDF) |
||
| 7 | Orthogonal Filter Banks: Paraunitary Matrices; Orthogonality Condition (Condition O) in the Time Domain, Modulation Domain and Polyphase Domain | Sec 5.1-5.2 | ||
| 8 | Maxflat Filters: Daubechies and Meyer Formulas. Spectral Factorization | Sec 5.3-5.5 | ||
| 9 | Multiresolution Analysis (MRA): Requirements for MRA; Nested Spaces and Complementary Spaces; Scaling Functions and Wavelets | Sec 1.5, 6.1 | ||
| 10 | Refinement Equation: Interative and Recursive Solution Techniques; Infinite Product Formula; Filter Bank Approach for Computing Scaling Functions and Wavelets | Sec 6.2-6.4 |
HW 2 due HW 3 out (PDF) |
|
| 11 | Project Brief | |||
| 12 | Orthogonal Wavelet Bases: Connection to Orthogonal Filters; Orthogonality in the Frequency Domain. Biorthogonal Wavelet Bases | Sec 6.2, 6.4, 6.5 | ||
| 13 | Mallat Pyramid Algorithm | Sec 1.6, 6.2 | ||
| 14 | Accuracy of Wavelet Approximations (Condition A); Vanishing Moments; Polynomial Cancellation in Filter Banks | Sec 7.1 | ||
| 15 | Smoothness of Wavelet Bases: Convergence of the Cascade Algorithm (Condition E); Splines. Bases vs. Frames | Sec 7.2-7.4 | HW 3 due | |
| 16 | Signal and Image Processing: Finite Length Signals; Boundary Filters and Boundary Wavelets; Wavelet Compression Algorithms | Sec 8.1-8.3, 8.5, 10.1, 11.1-11.5 | ||
| 17 | Guest Lecture. Physical Wavelets and their Sources: Real Physics in Complex Spacetime | |||
| 18 | Lifting: Ladder Structure for Filter Banks; Factorization of Polyphase Matrix into Lifting Steps; Lifting Form of Refinement Equation | Sec 6.5 | ||
| 19 | Wavelets and Subdivision: Nonuniform Grids; Multiresolution for Triangular Meshes; Representation and Compression of Surfaces | |||
| 20 | Numerical Solution of PDEs: Galerkin Approximation; Wavelet Integrals (Projection Coefficients, Moments and Connection Coefficients); Convergence. Subdivision Wavelets for Integral Equations. Compression and Convergence Estimates | Sec 11.6 | ||
| 21 | M-band Wavelets: DFT Filter Banks and Cosine Modulated Filter Banks. Multiwavelets | Sec 7.5, 9.1-9.4 | ||
| 22 | Project Presentations | |||
| 23 | Project Presentations | |||
| 24 | Project Presentations | |||
| 25 | Project Presentations |
