Applied & Computational Harmonic Analysis Homework Page (Winter 2010)

Course: MAT 271
CRN: 63579
Title: Applied & Computational Harmonic Analysis
Class: MW 5:10pm-6:30pm, 2112 Math. Sci. Bldg.

Instructor: Naoki Saito
Office: 2142 MSB
Phone: 754-2121
Email: saito@math.ucdavis.edu
Office Hours: MW 3:10pm-4:00pm, or by appointment

Homework #1 Due Wednesday January 20 PDF file
Please take a look at my comments on HW1.

Homework #2 Due Monday February 8 PDF file
My comments to HW2 is here. Also, please take a look at my MATLAB codes and run them!

prob2.m

prob3.m

prob4.m

Homework #3 Due Monday February 22 PDF file Please read my comments to HW3, which contain useful and important information.

Final Project Assignment Due noon Monday March 22, 2010

Please use the word processor to write your report.
You should not write more than 10 pages.
The best possible scenario is to write a report on a project you yourself come up with. Examples are:
- Description of your Ph.D. research projects where you plan to use the tools you learned (or will learn) in this course.
- A survey of some particular area of applications of the ideas learned (or to be learned) in this course, e.g., image compression, denoising, interpolation, fast algorithms, numerical analysis, statistics, etc.
If you have difficulty specifying your project and final report, I would suggest that you do one of the following possible projects. Regardless of which project you choose, I would recommend to use either Wavelet Toolbox officially supplied by MATLAB or Dave Donoho's WaveLab software downloadable from http://www-stat.stanford.edu/~wavelab.
- Project A: Discriminant Signal Analysis
  - Record your voice (some words) via microphone.
  - Get the digital waveforms of your voice.
  - If you are male, then ask a female student (not necessarily in this class) to get her voice recording of the same words as you recorded. If you are female, get voices of a male student.
  - Apply the Windowed Fourier Transform to these voice recordings to compute the spectrograms.
  - Interpret the results. Can you identify male voices and female voices by looking at the spectrograms?
  - Repeat the same experiments using the Continuous Wavelet Transform and the scalograms. Which are easier to interpret the voices, spectrograms or scalograms?
  - Describe your further thoughts for recognizing male/female voices.
  - Instead of male/female voice signals, you can use any two classes of signals that show different time-frequency characteristics. Examples include: comparison of historical records of some stock prices of two companies or market indices of two countries, or seismograms (seismic signals) caused by earthquakes vs those by nuclear explosions, to name a few.
- Project B: Image Compression Experiments
  - Get your favorite digital photos including those taken by digital cameras by you. At least use a couple of different images with different characters, for example, one image should be edge dominant (i.e., close to piecewise smooth functions, e.g., cartoon images), the other should be texture rich (i.e., photos of the ocean, photos of zebras, etc.)
  - Implement JPEG like algorithm (i.e., split an input image into a set of blocks of 8 x 8 pixels; apply 2D DCT to each block; then keep a certain number of the largest coefficients). I am not asking you to implement quantization part (of course, you can do that if you are interested.) The program should have an option to: 1) sort all the coefficients of all the blocks in the decreasing order and keep top k largest coefficients in magnitude; or 2) sort all the coefficients within each block and keep top few coefficients per block.
  - Approximate the original image using that program; plot the curves of the relative L^2 error between the original and approximation vs the number of terms retained.
  - Apply several different discrete wavelet transforms to the entire image (not splitting into blocks). Sort the wavelet coefficients in decreasing order, keep only top k coefficients, and then reconstruct the approximation from the top k coefficients.
  - Plot the curves of the relative L^2 error between the original and approximation vs the number of terms retained.
  - Interpret the results and discuss the pros and cons of JPEG/DCT and wavelets and their suitability to the type/class of images.
  - Describe your thoughts about the future of image compression.

Please email me if you have any comments or questions!
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