WAVELETS, FILTER BANKS AND APPLICATIONS

NEWS

COURSE INFORMATION

Schedule for Spring 2004:    Monday and Wednesday 1:30-3:00 in Room 1-390.

Text:           WAVELETS AND FILTER BANKS by Strang and Nguyen, Wellesley-Cambridge Press, 2nd Ed. (1997.)

Syllabus:      Please see the class schedule. Also, see the course announcement for general course overview.

ANNOUNCEMENTS

• Posted 04/20/2004: Solutions to Problem Set 3 have been posted. Look under Solutions.
• Posted 04/05/2004: Solutions for Problem Set 2 have been posted. Look under Solutions.
• Posted 03/10/2004: Problem Set 2 is due today in class and Problem Set 3 is out (Look under Problem Sets)
• Posted 03/03/2004: Solutions to Problem Set 1 have been posted. Look under Solutions.

LECTURE NOTES

• Lecture notes are posted here.

SUPPLEMENTARY NOTES

• Part I: Review of discrete-time signal processing; Half-band condition; Polyphase representation.
• Part II: Basis functions and filters; Multiresolution analysis.
• Part III: The Lifting Scheme; Second Generation Wavelets; Wavelet-Galerkin Methods [To Be Completed].

PROBLEM SETS

• Problem Set 1. Out 11^th February 2004, Due 23^th February 2004.
• Problem Set 2. Out 23^rd February 2004, Due 10^th March 2004.
• Problem Set 3. Out 10^th March 2004, Due 7^th April 2004.

SOLUTIONS

• Solutions to Problem Set 1. Posted 3^rd March 2004.
• Solutions to Problem Set 2. Posted 5^th April 2004
• Solutions to Problem Set 3. Posted 20^th April 2004.

INTERESTING LINKS

• Wavelet digest homepage.
• References

SOFTWARE

Matlab routines

• Function to generate half-band product filters
• Polyphase implementation of a filter
• Computation of Daubechies' filter coefficients (cepstrum method.)
• Computation of minimum phase spectral factors (cepstrum method.)
• Computation of scaling function and wavelet by recursion.

Matlab examples

• Example 1: Basic filters, upsampling and downsampling
• Example 2: Product filter examples (Needs prodfilt.m)
• Example 3: 1-D signal analysis.
• Example 4: 2D image analysis.
• Example 5: Polyphase filter implementation. (Needs polyfilt.m)
• Example 6: Generation of orthogonal scaling functions and wavelets. (Needs phivals.m)
• Example 7: Generation of biorthogonal scaling functions and wavelets. (Needs biphivals.m)
• Example 8: Mallat pyramid decomposition for functions in L^2.
• Example 9: Approximation power of wavelet bases.
• Example 10: Polynomial cancellation in filter banks.
• Example 11: Smoothness of wavelet bases.
• Example 12: Treatment of boundaries.

Java Applets (by Pavel Grinfeld)

• Inner Products of Functions
• Sums of a Trigonometric Series
• Fourier Serier
• Gibbs Phenonemon
• Aliasing

Matlab Minicourses

Matlab Wavelet Toolbox

To launch the graphical user interface for the Matlab wavelet toolbox, type "wavemenu" at the Matlab prompt. For starters, try selecting "Wavelet 1-D", then go to "File | Load Signal". To locate the sample signals on Athena, type the following path into the "Filter" field and hit the return key.

/mit/matlab_v6.5.1/distrib/toolbox/wavelet/wavedemo/*.mat

You will then be able to choose a signal from the list of files. A good one to start with is noisdopp.mat. Once the signal has loaded, choose a wavelet and the number of decomposition levels. The default wavelet filter is "haar". To use the Daubechies 4-tap filter, for example, you would choose "db2". The 2 means that the frequency response has 2 zeros at pi. Similarly, to use the linear phase 9/7 filters, you would choose "bior4.4". In this case, 4.4 means that the synthesis lowpass filter has 4 zeros at pi and the analysis lowpass filter has 4 zeros at pi (in that order.) Next, click "Analyze" to perform the wavelet analysis. You will then see graphs of the original signal (s) and the various subbands of the signal (an,dn,...,d1) where n is the number of levels. Note that the subband signals are of full length in this display. In other words, what is being displayed is not the actual coefficients produced in the output of the analysis bank, but the reconstructed version of each channel that is seen in the synthesis filter bank immediately prior to recombining the channels. Note that you can look at the statistics of the signal and even compress or denoise it.

The "Wavelet 2-D" menu item should be straightforward to use once you have seen the 1-D version. A good image to start with is woman.mat, which is the famous Barbara image.

Also interesting to beginners is the "Wavelet Display" menu item. This allows you to view the scaling function(s) and wavelet(s) and their associated filters.

Documentation

SUGGESTIONS