Discrete Wavelets

Final Projects

Helmut  Knaust - Department of Mathematical Sciences, University of Texas at El Paso, hknaust@utep.edu
04/16/2026

Final Project presentations will take place during the final exam period (Thursday, May 14 at 19:00).
Each student group will give a 15 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 to be presented has to be turned in to the instructor by Friday, May 15.

Final project selection will take place on 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 (p. 541f). 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
Fusing Grayscale and Color Images

In remote sensing, the transmission channel can be very "narrow". To conserve bandwidth, the sender may transmit a detailed grayscale image and a small color image of the same scene. It is then up to the sender to fuse the image to obtain a detailed color image. One such fusing procedure is described in Shutao Li, James T. Kwok, Yaonan Wang: Using the discrete wavelet frame transform to merge Landsat TM and SPOT panchromatic images.
The images can be downloaded here. You can use the standard wavelet transform method instead of the wavelet frame transform method.

+=?

Project 3
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 4
Zooming in on a Region of Interest in a Transformed Image

Problem

Given an image transformed by k iterations of a wavelet transform, write an algorithm that finds the inverse of a sub - matrix in the blur portion without inverting the entire matrix.  This is known in the medical field as "region of interest". 

This exploits the fact that wavelet transforms - unlike the Fourier transform - are localized in the time (space) domain.

Procedure

We wish to partially invert the Haar Wavelet Transform of an image:

This represents the transformed Image.

Below is the blurred portion of the image that corresponds to B above.

The red rectangle (the chair) is the region of interest in the transformed image and corresponds to the blue rectangle b in the image above in the blur.
Below are the 4 corresponding areas b, v, h, d assembled into one image.

The Inverse Haar Wavelet Transform is applied to this to obtain the following “Zoom” below.  

Project Outline:

1.  Given a matrix transformed by one iteration (k = 1) of the Haar WT, define and identify where in the transformed matrix you want to obtain the "region of interest". Decide what other portions of the transformed matrix will be needed to invert “the region of interest”.
2.  Write code to obtain the inverse.
3.  Test it! (This can only be done if the original matrix is known.)  
4.  Repeat Steps 1 - 3 for k = 2, and k=3.
5.  Write a module for arbitrary k.
6.  Modify the code replacing the Haar filter by the Daubechies-4 filter.  What problem occurs when you attempt to invert the region of interest of the given image transformed with the D4 filter? Any ideas how to fix this? (No coding required.)

Acknowledgement:

This project was jointly developed by Caroline Haddad (SUNY Geneseo), Dawit Haile (Virginia State University), and Helmut Knaust (University of Texas at El Paso)

Project 5
Image Compression for Color Images

1. Take a color picture of your choice, and extract the Red, Green and Blue channels. 
2. Apply the CDF 97 Filter three times to each of the color channels. 
3. Quantize the three results by using the JPEG 2000 quantization scheme. Use values for τ to get similar PSNRs in all three cases in the end.
4. Compute the entropy of each of the three quantized channels, and then compute the theoretical bit size needed to encode the quantized color image.
5. Apply the inverse transform three times to each of the channels.
6. Reassemble and display the quantized color image.
7. Compute the PSNR of the original color image and the quantized version.
8. Compute the overall compression factor obtained for the quantized color image. 
9. Repeat 1.-8. using  YCbCr channels instead of RGB channels.
10. Repeat 1.-8. using (Subscript[YCb, 1/2])(Subscript[Cr, 1/2]) channels.
11. Repeat the procedure for one more color image.
12. Compare the results obtained.