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.
- 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.
- 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.)
- 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?
- 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.
- 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.
- 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.
- 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.