Math 414-501 Projects — Spring 2018

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 a written version of project, a brief oral presentations will be made to the class.

There are two types of projects that will be allowed. The first is using wavelets to analyze a physical problem. (See the list below for possible topics.) These projects generally will involve MATLAB or MAPLE or similar software. Writing large programs that duplicate the function of MATLAB, say, is neither necessary or desirable.

The second type is a report on the mathematical content of some paper involving a wavelet topic not covered in the course. If you choose to do this type of project, you should talk to me about selecting a paper. Examples of topics include wavelet packets, bi-orthogonal and semi-orthogonal wavelets, approximation power of wavelets, continuous wavelets, and frames.

Due date: Tuesday, May 8, 2018, by noon. (I need time to grade the projects.)

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. 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. 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 suggestions 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.

  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. Image compression, noise removal, and singularity detection. (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.)

  5. Compressive sensing. This is a probabilistic method of recovering a signal or image from small amounts of data. Wavelets are used in the method. How does it work?

  6. Biorthogonal wavelets. What are they? How are they used?
Updated: 3/28/2018 (fjn)