Math 414-501 Projects — Spring 2009

Requirements: Projects should be team efforts, with three students per project. Once a team is formed, members should select a person who will serve as a contact. 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. Projects should be submitted to me for approval before being worked on.

Style: The project report must be typed and written in good English prose. The length and content will generally vary, but will probably come to about ten pages, not including diagrams, programs, or references; it should not exceed twenty-five pages. It should include a brief introductory section providing background material, setting out the purpose of the project, and summarizing what was accomplished. References, including web sites, must be properly cited. Be aware that plagiarism is a legal as well as moral offense.

Oral Presentation: At the end of the semester I will have the groups briefly present their projects to the whole class.

Topics: You are free to choose any topic closely related to wavelets, subject to my approval. Often students base projects on topics that come up in their courses. Here are a few suggestions.

  1. Wavelet analysis of EEG, EKG, and so on. This is one-dimensional medical data; it is typically analyzed by FFT methods. Would there be any benefit in using a wavelet analysis on an EEG or EKG signal? There are many types of wavelets, each type having its advantages and disadvantages. An interesting project would be to do a comparative study of the FFT techniques and analysis done with various types of wavelets. See PhysioNet for data.

  2. Image compression and transmission. The direct cosine transform is the basis of the technique currently used by JPEG. The reason is that errors in compressed, transmitted data show up as a ``blocking effect.'' Errors in transmission of wavelet coefficients tend to produce strange, global distortions. On the other hand, there are many different wavelets, including biorthogonal ones. One possible project in this direction would be to study the FBI's choice of a wavelet-based method to store and compress finger print data, and try to use it to do compression and lossy transmission without strange visual artifacts. (JPEG2000 algorithms use wavelets.)

  3. Medical imaging problems. There are a number of medical imaging techniques, including MRI, MRA, PET, and even ``old-fashioned'' X-rays. Current techniques are Fourier based. Would wavelets do a better job of compressing and reconstructing data? Would wavelets make it easier to find irregularities in the heart, tumors and so on?

  4. Image processing and reconstruction in astronomy. The questions here involve interpreting and correcting data from, say, the Hubble Space Telescope or dealing with information from faint stars.

  5. Noise detection in Mechanical Devices. Fourier analysis has long been recognized as a powerful tool for characterizing the dynamical behavior of mechanical systems, especially rotating machinery (e.g., motors, pumps, fans, compressors, and turbines). The traditional approach has been to apply the Fourier transform to signatures from various test signals, in the frequency domain, in order to identify defects and correlate them with sources. While Fourier analysis can effectively deal with stationary behavior, wavelet algorithms need to be developed to handle non-stationary and fast changing events -- such as shock waves.
  6. Broadcasting in digital. TV and radio stations will have to broadcast in a digital format, as well as analog. Would wavelets be a better tool for sending digital signals? One might compare Fourier methods with digital methods.

  7. Fractal behavior. Wavelets have been used to study fractals, to compute fractal dimensions, to analyze turbulence in fluids, and to study chaotic behavior in systems. There even has been recent interest in using wavelets in studying coast lines. A possible project here would be to track down some data and use wavelet methods to analyze it for fractal behavior.