Math 414-501 Projects — Spring 2022

Requirements: Projects can be team efforts, with two or three students per project. Projects done alone are also acceptable; a team effort is however preferable. The project should be focused; avoid broad topics that cannot be completed by the end of the semester. Projects must involve wavelets. They must also have mathematical content. "Show-and-tell" or "golly-gee-whiz" projects are not acceptable. In addition to submitting the written version of project, students Will be required to give the class brief oral presentations concerning their projects.

There are two types of projects that will be allowed. The first is using wavelets to analyze a physical problem. The second type is a report on the mathematical content involving of a wavelet topic not covered in the course.

Due date: The project, in written form, is due in my office by 4 pm on Tuesday, May 10, 2022.

Style: The project report must be typed and written in good English prose; use 12 point font, 1.5 line spacing, and reasonable page margins. The length and content will generally vary, but will probably come to about ten to twelve pages, not including diagrams, programs, or references; it should not exceed twenty-five pages. Multimedia context should be delivered on a CD or DVD.

1. Introduction. This should briefly describe the project and summarize the rest of the paper.
2. Mathematical background. Discuss the wavelet(s) to be used and other mathematical topics involved – fractals, noise, etc. (Ask me if you're not sure.)
3. The application. Use wavelets to deal with the mathematical or physical problem you want to look at.
4. Conclusions. What information did your analysis yield? Include relevant charts, pictures, and other related things.
5. References. References, including web sites, must be properly cited. Be aware that plagiarism is a legal as well as moral offense.

Suggestions for Topics: The best projects are based on topics that a student wants to investigate. However, that isn't always possible, so here are several general areas for topics. These are not the only topics you may use. You are free to choose any topic closely related to wavelets, subject to my approval. Also, there are many different wavelets and wavelet transforms. Use the Haar wavelet only in conjunction with some other wavelet.

1. Wavelet analysis, fractals, and heart rates. Recent studies have shown that a healthy person's heart rate is fractal (cf. SIAM News). Also, wavelets have been used to determine fractal dimension and other quantities associated with fractals. One project would be to use wavelets to determine the fractal properties of a set of heart-rate data. See PhysioNet for data. Other places on the web also have it. Fractals — dynamical systems and other applications

2. Singularity detection. See chapter 6 in the text.
   • Failure of Mechanical Devices. The sound of a machine changes when it starts to fail. (Think high-pitched squeal from under the hood of your car.) Wavelets can be used to detect the discontinuity in the monitored sound of a machine, and then warn of failure.
   • Chirps. A chirp is a fast change in frequency. Wavelets and wavelet packets can be used to analyze such rapid frequency changes.
   • Restoring damaged or noisy recordings.

3. Stock market trends and applications to economics.

4. Images and audio files.
   • Compression and noise removal. (Compressing sound files is not recommended. The standard MPEG compression makes use of what the ear can hear. It is much more sophisticated than simple file compression.)
   • Detection of altered images and audio files. Detection of splices in a video clip.
   • Hiding an image (steganography).

5. Mathematical topics: biorthogonal wavelets — what are they? how are they used? Wavelet packets, continuous wavelet transforms, aliasing in wavelets.