**
Project Ideas**

**Here is a list of 12 project ideas. Some of these ideas may already
have been worked on by me and my student collaborators as part of the NSF REU
(Research Experience for Undergraduates) held at
UWEC, known as
SUREPAM. If you are
interested in one of these ideas, please contact me at
walkerjs@uwec.edu (Jim Walker) so
that I can tell you if any work has been done already (even if it has, maybe we
can collaborate??). Also, if you do work that you intend to publish in
some form I would very much like to hear from you, and would be most happy to
assist you in any way that I can. As work develops with these and
other future projects, I hope to create an archive of final results (PDF files
or other formats), so let's keep in touch! **

**The projects are in three areas: (1) Music,
(2) Noise removal from audio, and (3)
Image Processing.**

**Music The application of mathematics to
musical theory is a very hot topic in both the math and music worlds. I
have published two papers in this area with students: **

*** James S. Walker and Jeremy F. Alm,
"Time-frequency analysis of musical
instruments'' , SIAM Review, 44**,** ****pp. 457-476, 2002**

*** James S.
Walker and Amanda J. Potts, "Time-frequency spectra of music'',
**
Advances in
Analysis: Proceedings of 4th International ISAAC Congress. World Scientific (2005),
487-493.

The most
important background on the subject can be found in my Primer and in the article that I have
written (currently under review) with a music professor, entitled
"Music: A time-frequency approach." (TFAM).
Based on that article, here are six problems that could be worked on:

1. Detailed analysis of
a passage of the song *Buenos Aires*
from the musical *Evita *using the methods discussed in
TFAM.

**2. Detailed analysis of another musical
passage. Some possibilities include: a passage from Stravinsky's Rite
of Spring, or a passage from Copland's Appalachian Spring, or a passage from Messiaen's Catalogue
d'Oiseaux (including comparison with bird song). But I am open to
other possibilities.**

**3. Use the inverse Gabor transform,
described in TFAM, to synthesize new musical passages.**

**4. Mathematical modeling and analysis
of the technique of percussion scalograms described in
TFAM.
Note: Significant progress was made on this
problem for drum based percusion in the 2007 SUREPAM program. See the
webpage:
http://people.uwec.edu/walkerjs/PicturesOfMusic/. Further work on
this topic would involve determining how to automatically determine percussion
scalograms for different musical instruments, based on their time-frequency
characteristics.**

**5. Using percussion scalograms,
investigate the algebraic structure of the "production rules" of rhythm and
tempo in music. For instance, do these rules constitute a mathematical
group?
Note: More details on the background of this
topic can be obtained from the preprint
Time-frequency analysis of
musical rhythm.**

**6. Develop and utilize (either in
Maple, or MatLab, or Visual Basic), the new method of
Reassigned Short-time Fourier Transforms and apply it to analyzing music
using the methods described in TFAM.**

**Noise Removal from Audio The
application of the method of Gabor transforms to audio noise removal is
described in detail in my Primer and in my article (currently under review) entitled
"Denoising Gabor Transforms"
(DGT) . Based on that article, here are three problems that could be
worked on:**

**7****. Generalizing the method described in
DGT to noise that has different standard deviations within different frequency
(one such noise, so-called "pink noise" occurs in real, environmental,
background noise).**

**8****. Generalizing the method described in
DGT to noise with changing standard deviation over time (non-stationary noise).
[This is a VERY difficult problem, but even partial results would be of great
interest.]**

**9****. Investigating "soft thresholding" (see the discussion of noise removal in TFAM) for generalizing the
noise removal technique in DGT.**

**Image Processing Here are three
projects in image processing. The first two relate to removing random
clutter from STM images:**

**Cluttered STM image
Processed STM image**

**Based on this initial work, here are two projects in STM clutter removal:**

**10****. Develop an improved algorithm, and
software, for both automatic and user-guided clutter removal from STM images.**

**11. Develop a model for the clutter and
prove that the algorithm from 9 is an effective
clutter removal method.**

**The third project deals with super-resolution image enhancement:**

**12. Develop an image enhancement
algorithm based on the Principles A and B of ASWDR (see my Wavelet Primer,
section 4.5): use those principles to project up to finer scale wavelet
values, followed by inverse wavelet transform, to produce a higher resolution
image.**