James S. Walker    Music, Noise Removal from Audio, and Wavelets

Music  The application of mathematics to musical theory is a very hot topic in both the math and music worlds.  I have written 4 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. 
 
* James S. Walker, Xiaowen Cheng, and Jarod Hart, "Time-frequency analysis of musical rhythm'',  Accepted for publication in AMS Notices.

* James S. Walker and Karen Muir, "Analyzing Musical Rhythm'',  Submitted.

Some further background on the subject can be found in an 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.   Significant progress was made on this problem for drum based percusion in the 2007 SUREPAM program.  See the webpage:  http://www.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 paper 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 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.

Wavelet Image Processing  Wavelet analysis is a powerful technique for processing images (see, e.g., my introductory book A Primer on Wavelets and their Scientific Applications).  Recently I have developed a new wavelet-based method for removing random clutter from scanning tunneling microscope (STM) images:

                                           
                                                 Cluttered STM image                                    Processed STM image

Based on this initial work, here are three problems that we could work on:

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 problem 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.