Math 414-501 Projects — Spring 2010

Requirements: Projects should be team efforts, with two or three students per project. Projects done alone are not acceptable. Projects should involve using computers; MATLAB and the Wavelet Toolbox should be sufficient for almost any project, so writing large programs in, say, C or C++ is neither necessary or desirable. Other programs with the same capabilities may be used instead of MATLAB. The project should be focused; avoid broad topics that cannot be completed by the end of the semester.

Due date: Friday, May 7, 2010.

Style: The project report must be typed and written in good English prose; use 12 point font, single line spacing, and reasonable page margins. The length and content will generally vary, but will probably come to about twelve to fifteen 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. Inroduction. This should briefly describe the project and summarize the rest of the paper. It should be about a page in length.
  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. Discuss what you will use wavelets to analyze.
  4. Results of the analysis. 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.

Topics: Several suggestions for topics are given below. 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.

  2. More fractals: turbulence in fluids and gases, "strange attractors." One of the applications of wavelet analysis is to study the fine-scale, fractal geometry involved in turbulent flow. Another is to calculate fractal dimension of strange attractors arising in certain nonlinear ODEs – e.g., the Lorenz equations (meteorology) and Duffing's equation (mechancial sytems).

  3. Singularity detection and noise. Wavelets can be used to remove noise and to detect discontinuities in derivatives (cf. Chapter 6 in the text), even in the presence of noise. Here are a few sample applications.

  4. Signal compression. Wavelets can be used to greatly reduce the size of a sound file, with some loss, however. Pick a sound file and compress it using wavelets. Compare this with standard compression methods. Use several different wavelets.

  5. Image compression, noise removal, and singularity detection.