Discrete Wavelets

Final Projects

Helmut  Knaust - Department of Mathematical Sciences, University of Texas at El Paso, hknaust@utep.edu
11/5/2014

Final Project presentations will take place during final exam time on Monday, December 8, at 17:00-19:45 in Bell 130.

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. All electronic material to be presented has to be turned in to the instructor by Wednesday, December 10, at 17:00.

Final project selection will take place on Monday, November 10, during class time.

If all students in a group complete the course evaluation by the deadline, I will add 5 points to their final project grade.

Project A - Smoke Detection

The paper "CONTOUR BASED SMOKE DETECTION IN VIDEO USING WAVELETS", by B. Ugur Toreyin , Yigithan Dedeoglu and A. Enis Cetin discusses various techniques  - some of them using wavelets - to detect smoke in videos. Some sample videos are available at  http://signal.ee.bilkent.edu.tr/VisiFire/Demo/SampleClips.html. You  will need some software such as VirtualDub to extract the image frames from a video.

Project B - Lossless JPEG2000 with Lifting

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

1. Implement  the grayscale algorithm as described in our textbook, pp. 472-479.
2. Implement the color algorithm using Reversible Component Transformation (RCT) - a description can be found in a student paper.

Project C - 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?

[Alejandro, Hassan, Susana] Project D - 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, see Ruch & van Fleet's book. (A copy of the book is available for your perusal.)

Implement the Cascade Algorithm for the Haar, D6 and D12 scaling functions and wavelet functions.

[Agamyrat, Kylee, Suzana] Project E - 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.

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[Catherine, James] Project F - Matrix Completion

1. Explain the procedure outlined in Problem 7.19 in the textbook, then use Mathematica to present the solution.
2. Write commands similar to “D4” (see DW.m) for the Matrix Completion of the Daubechies-4 Filter developed in Problem 7.19.
3. Check that the matrix generated by your commands in Step 2 is orthogonal.
4. Take a grayscale picture of your choice, apply your transform once to the image, and check on artifacts at the edges of the blur portion of the transformed image. 
5. Can you implement a similar procedure for the D6 filter? The instructions in D6c may be helpful.

Project G - Face recognition

Wavelets can be used to identify a face from a databank of face images. Such an algorithm is described in C.Garcia, G.Zikos, G.Tziritas: Wavelet packet analysis for face recognition.

I will provide the group with sample images from the FERET database of face images.

[Ashley, Mario] Project H - 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 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 do you expect to happen?
7.  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 in Step 7.)

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)