Discrete Wavelets

Final Projects

Helmut  Knaust - Department of Mathematical Sciences, University of Texas at El Paso, hknaust@utep.edu
04/18/2022

Final Project presentations will take place during final exam time on Tuesday, May 10 at 13:00 – 15:45.

Each student group will give a 20 minute presentation.  The presentation needs to include a Mathematica notebook. You will be graded on the mathematical correctness of the presentation,  the quality and completeness of  the notebook and other material you prepare, and the quality of your oral presentation. All group members must participate equally in the presentation. Give references to sources you use other than the ones listed below.

All electronic material presented has to be turned in to the instructor by Thursday, May 12.

Final project selection will take place on Thursday, April 28, during class time.

Project 1
Lossless JPEG2000 with Lifting

There is a lossless version of the JPEG2000 algorithm designed for archiving. 

Implement  the grayscale algorithm as described in our textbook, pp. 461-471 and Example 12.4. What is the entropy of the transformed image? Check that the original image seen by the sender will be identical to the one reconstructed by the receiver.

Project 2
Cascade Algorithm

The Cascade Algorithm can be used to construct approximations to the scaling and wavelet functions using the coefficients of the discrete wavelet algorithm.

1. Implement the continuous algorithm described in Ruch & van Fleet's book for the scaling functions and the wavelet functions for D4 and D8.
2. Implement the algorithm “Construction 3” described in Gilbert Strang: Wavelets and Dilation Equations. SIAM Review, Vol. 31, No. 4. (1989), pp. 614-627, on pages 0-3 and 0-4. Again, do this for D4 and D8. 

Below are pictures of the scaling function (blue) and wavelet function (red) for D6.

Project 3
JPEG Algorithm

Explain and implement the JPEG algorithm for grayscale images (see Section 12.1 in the textbook). Research whether there are other normalization matrices - what is the rationale for the choice of the coefficients?