## Math 371: Wavelet and Fourier Analysis

Spring 2006

Final Project

The second of two projects for this course will be due at the end of the semester. However, you need to choose a project now. This project should involve a little more research than the first one, and it is likely that you will need to get some journal articles or inter-library loan books. You should allow the necessary time for this.

The written part of the project is due on Thursday, May 4, during the Reading Period before final exams, or earlier. It should be a research paper, about 5 to 10 pages long. It can include historical and background information, but should also include at least some mathematics.

The project will also include a presentation. There are two options about how this will be scheduled. (The class can vote on this.) (1) The presentations could be scheduled during the last week of classes. In that case, the second test will be given during the scheduled final exam period, 7:00 PM on Sunday, May 7. (2) We could have the second test on the last day of class, with presentations during the final exam period. In either case, the length of the exam will be the same, about the same as the length of the midterm.

The final project and presentation count for 15% of the grade. (The first project counted for 10%.)

The list below suggests some possible topics. You are not necessarily required to choose a topic from this list. If you have other ideas, you are welcome to discuss them with me. The topic should be in the general area of applications of wavelets. A project that compares the wavelet and the Fourier approach to some application could also be acceptable. It is not required that everyone in the class choose a different topic; however, if two people do work on the same topic, they will have to coordinate their presentations to some extent to avoid having too much overlap.

Possible topics:

- The use of wavelets in the JPEG2000 image compression standard.
- Video compression. (As far as I know, wavelets are not used in any current practical implementation, but wavelets have been considered for this application. The paper might be mostly about Fourier and other techniques.)
- The FBI fingerprint compression project. There is a fair amount of information on this in our textbook and in the handout on applications of wavelets. If you want to do this topic, you should go beyond what is already given in these two sources.
- More about noise reduction and/or compression of audio signals. Although this is covered extensively in the textbook, there is more that can be said about it and there are references to additional material in the text.
- More about wavelets and image processing. Again, there is significant coverage of this in the textbook, but there is a lot more to say about it. The "image recognition" problem could be particularly interesting.
- Wavelets, fractals, and turbulence.
- Wavelets and sensory processing (vision and/or hearing), perhaps including historical background information on "pyramid algorithms" in vision research.
- Medical application of wavelets.
- Scientific applications of wavelets

David Eck, 23 March 2006