Math 414 Projects

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. The project should be focused; avoid broad topics that cannot be completed by the end of the semester.

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.

Topics: You are free to choose any topic closely related to wavelets, subject to my approval. Here are a few possibilities.

  1. Wavelet analysis of EEG, EKG, and other medical data. Medical data 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 and, for cardiac dynamics, Center for Polymer Studies. (Click on Research Projects, and then on Cardiac Dynamics).

  2. 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.

  3. 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. (The latest version of JPEG compression contains algorithms using wavelets.)

  4. Wavelet analysis of meteorological and geophysical data. Doing a wavelet analysis of meteorological data taken from satellites is a problem. Unlike signals or images, the samples taken are not from sites on a uniform grid, nor can they be adjusted to be on one. The largest available equally spaced grid on the sphere are the twenty vertices of the dodecahedron. Data is thus collected at scattered sites, rather than grid points. Various versions of spherical wavelets have been proposed to deal with this problem, but none have been entirely successful. Numerical comparisons of existing algorithms might produce some interesting results.

  5. Fuel burnout in a rocket. Wavelets are very good at detecting discontinuities of functions and their derivatives. One application is the detection of the precise moment at which a rocket's fuel burns out; fuel-burnout produces a discontinuity in the force on the rocket, resulting in a discontinuity in the second derivative of the trajectory. Analyze what happens with noisy data. In particular, how do we model the noise, since only simulations are available.

  6. Multiwavelets. A disadvantage of the usual wavelets is there is a tradeoff between how smooth a wavelet is and how what is ``length'' is. Multiwavelets use several functions for wavelets, instead of just one. Compare these with the usual wavelets and with FFT based techniques for signal processing tasks.

  7. Fractals. Wavelets have been used to study fractals, to compute fractal dimensions, to analyze turbulence in fluids, and to study chaotic behavior in systems. A possible project is to take a chaotic signal or fractal and analyze it with different wavelets.