Math 414-501 Projects — Spring 2015
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.
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 12, 2015, 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, single 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.
- Introduction. This should briefly describe the project and
summarize the rest of the paper. It should be about a page in length.
- Mathematical background. Discuss the wavelet(s) to be used and
other mathematical topics involved – fractals, noise, etc. (Ask
me if you're not sure.)
- The application. Use wavelets to deal with the
mathematical or physical problem you want to look at.
- Conclusions. What information did your analysis
yield? Include relevant charts, pictures, and other related things.
- References. References, including web sites, must be properly
cited. Be aware that plagiarism is a legal as well as moral offense.
Suggestions for Topics: Several suggestions for topics
are given below. 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.
- 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.
- 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).
- 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.
- Rocket burnout. When a missile burns up its fuel, the forces on
it change, because the thrust is 0. How well can this be detected in
the presence of noise?
- Failure of Mechanical Devices. The sound of a machine changes
when it starts to fail. (Think high-pitched squeal from under the hood
of your car.) Wavelets can be used to detect the discontinuity in the
monitored sound of a machine, and then warn of failure.
- Chirps. A chirp is a fast change in frequency. Wavelets and
wavelet packets can be used to analyze such rapid frequency changes.
- Restoring damaged or noisy recordings.
- 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.)
- Medical imaging problems. There are a number of medical imaging
techniques, including MRI, MRA, PET, and even ``old-fashioned''
X-rays. Data sets from these tend to be very large and possibly
noisy. Projects here involve compression and noise removal. In
addition, a wavelet analysis of an image can help in the detection of
tumors, say.
- Detection of altered images. Analyze a photograph that's been
altered or that there is a suspicion of its having been altered.
- Hiding an image (steganograpy). Wavelets break down an image into
smoothed piece and a piece containing details. Hide an image in the
details. The same applies to other types of signals -- radio
transmissions, for instance.
- Fingerprints. The FBI uses wavelets to compress and retrieve data
concerning fingerprints. How do they do it? What wavelets are used?
Retrieval is mathching a set of fingerprints with one in the data
base. How well does this work.
- Compressive sensing. This is a probabalistic method of
recovering a signal or image from small amounts of data. Wavelets
are used in the method. How does it work?
- Biorthogonal wavelets. What are they? How are they used?
Updated: 1/19/2015 (fjn)