Wavelets have established themselves as an important tool in modern signal processing as well as in applied mathematics. The objective of this course is to establish the theory necessary to understand and use wavelets and related constructions. A particular emphasis will be put on constructions that are amenable to efficient algorithms, since ultimately these are the ones that are likely to have an impact. We thus study applications in signal processing, communications, and sensing where time-frequency transforms like wavelets play an important role. The course has computer and research projects involving independent study.

- M. Vetterli, J. Kovacevic, and V. K. Goyal, "The World of Fourier and
Wavelets: Theory, Algorithms and Applications," (manuscript); available for purchase from the IEEE store in Everitt Lab;
downloadable from http://www.fourierandwavelets.org.
**OR**

M. Vetterli and J. Kovacevic, "Wavelets and Subband Coding," Prentice Hall, 1995; downloadable from http://www.waveletsandsubbandcoding.org - Research papers.

- S. Mallat, "A Wavelet Tour of Signal Processing," Academic Press, Second Edition, 1999.
- 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.

1.

a. Why wavelets, filter banks, and multiresolution analysis?

b. Signal spaces and operators

c. Review of Fourier theory

d. Multirate signal processing

e. Time-frequency analysis

2.

a. Series expansions of discrete-time signals

b. Analysis and design of filter banks

c. Orthogonal and biorthogonal filter banks

d. Tree-structured filter banks

e. Discrete wavelet transform

3.

a. Multiresolution analysis

b. Iterated filter banks

c. Wavelets and filter banks

d. Wavelet series and its properties

e. Regularity and approximation properties

4.

a. Frame theory

b. Oversampled filter banks

c. Continuous wavelet and short-time Fourier transforms

5.

a. Sparse representation

b. Linear and non-linear approximation in various bases

c. Non-linear signal estimation

d. Multidimensional filter banks and wavelets

e. Multiscale geometric representation and processing

f. Compressed sensing

6.

a. Speech, audio, image, and video compression

b. Signal denoising

c. Feature extraction

d. Inverse problems

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

- 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.