Prof. Minh N. Do (115 CSL, 244-4782, minhdo@).

Efficient or sparse signal representation provides the foundation for many signal processing tasks, including sensing, reconstruction, denoising, compression, and feature extraction. This course aims to establish the theory necessary to understand and to effectively use local Fourier and wavelet bases, and related constructions. A particular emphasis will be put on representations that are amenable to fast algorithms, since these are the ones that are likely to have an impact in applications. The course also includes machine problems and independent research projects.

- M. Vetterli, J. Kovacevic, and V. K. Goyal, "Signal Processing" (Vol. 1: Foundations; Vol. 2: Fourier and Wavelet Representations). Available online at http://www.fourierandwavelets.org
- Research papers.

- S. Mallat, "A Wavelet Tour of Signal Processing," Academic Press, Second Edition, 3rd ed., 2008.
- G. Strang and T. Q. Nguyen, "Wavelets and Filter Banks," Wellesley-Cambridge Press, Revised Edition, 1998.
- I. Daubechies, "Ten Lectures on Wavelets," SIAM, 1992.
- P. P. Vaidyanathan, "Multirate Systems and Filter Banks," Prentice Hall, 1993.
- M. Vetterli and J. Kovacevic, "Wavelets and Subband Coding," Prentice Hall, 1995; downloadable from http://www.waveletsandsubbandcoding.org

1.

a. Vector and Hilbert spaces

b. Approximations, projections, and decompositions

c. Bases and frames

d. Computational aspects

2.

a. Discrete-time signals and systems

b. Discrete-time Fourier transform and z-transform

c. Multirate signal processing

3.

a. Orthogonal and biorthogonal two-channel filter banks

b. Design of two-channel filter banks

c. Tree-structured filter banks

d. Discrete wavelet transform

e. Non-linear approximation in the wavelet domain

4.

a. N-channel filter banks

b. Cosine-modulated local Fourier bases

c. Lapped transforms

d. Time-frequency analysis

5.

a. Multidimensional filter banks

b. Directional filter banks

c. Multiscale geometric representations

6.

a. Sparse signal processing

b. Speech, audio, image, and video compression

c. Signal denoising

d. Feature extraction

e. Compressed sensing

- Homework: 25%
- First midterm: 25%
- Second midterm: 25%
- Final project: 25%

- Syllabus
- Projects
- Introduction
- Summary of Perfect Reconstruction Filter Banks
- Wavelet Denoising
- Summary of Multiresolution Analysis
- Multidimensional Filter Banks and Multiscale Geometric Representations

- Homework 1. Solutions 1
- Homework 2. Solutions 2
- Homework 3. Solutions 3
- Homework 4. Solutions 4
- Homework 5. Solutions 5
- Homework 6. Solutions 6
- Homework 7. Solutions 7

- A previous Midterm 1 (Fall 2007). Solutions
- Midterm 1 (Fall 2012). Solutions
- A previous Midterm 2 (Fall 2007). Solutions
- Midterm 2 (Fall 2012). Solutions

- Wavelets,
by G. Strang,
*American Scientist***8**(April 1994) 250-255. - Wavelets and signal processing, by O.
Rioul and M. Vetterli,
*IEEE Signal Processing Magazine*, vol. 8, no. 4, Oct. 1991, pp. 14-38. - A
theory for multiresolution signal decomposition: the wavelet
representation, by S. Mallat,
*IEEE Transaction on Pattern Analysis and Machine Intelligence*, vol. 11, p. 674-693, July 1989. - Wavelets and filter banks: theory and design,
by M. Vetterli and C. Herley,
*IEEE Transactions on Signal Processing*, vol. 40, Sep. 1992, pp. 2207-2232. - Wavelets,
approximation and compression, by M. Vetterli,
*IEEE Signal Processing Magazine*, vol. 18, no. 5, Sep. 2001, pp. 59-73. - Theoretical
foundations of transform coding, by V. K. Goyal,
*IEEE Signal Processing Mag.*, vol. 18, no. 5, pp. 9-21, Sept. 2001.

- Amara's Wavelet Page: An extensive collection of wavelet resources on the Web.
- Wavelet Tutorial: An excellent wavelet tutorial for engineers.
- The Wavelet Digest: Latest news on wavelets.
- Compressed sensing: Collection of online papers and software in compressed sensing.