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.
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. Here are a few
possibilities.
- Wavelet analysis of EEG, EKG, and so on. 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).
- Turbulence in fluids and gases. One of the applications of
wavelet analysis is to study the fine-scale, fractal geometry involved
in turbulent flow. Another is the low scale appearance of regularities
- coherent structures - in turbulent flow. A third arises in trying to
solve the Navier-Stokes equation via wavelet methods. Any of these
would make a very good project.
- 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.
- 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.)
- Medical imaging problems. There are a numebr 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?
- 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.
- 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.
- 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.
- 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.