Applied & Computational Harmonic Analysis Homework Page (Winter 2016)

Homework #1 Due Wednesday January 20, 2016 PDF file
Please take a look at my comments on HW1.

Homework #2 Due Wednesday February 10, 2016 PDF file
My comments to HW2 is here. Also, please take a look at my MATLAB codes and run them!

prob3.m

prob4.m

prob5.m

Homework #3 Due Wednesday March 2, 2016 PDF file
Please read my comments to HW3, which contain useful and important information.

Final Project Assignment Due 5pm Saturday, March 19, 2016

Please use a word processing system to write your report (LaTeX strongly recommended).
You should not write more than 10 pages.
Submit your pdf file of your report to me via email.
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, I would suggest that you do one of the following possible projects. Regardless of which project you choose, I would recommend to use either:
1. Wavelet Toolbox officially supplied by MATLAB;
2. Dave Donoho's WaveLab software downloadable from http://www-stat.stanford.edu/~wavelab; or
3. Gabriel Peyré's The Numerical Tour toolbox downloadable from http://www.numerical-tours.com.
- 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 L² 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 L² 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.
- Project C: Synchrosqueezed Wavelet Transform Experiments
  - Get your favorite time series data such as music signals, stock market price curves, etc.
  - Read the following papers very carefully and learn what the synchrosqueezed wavelet transforms are:
    I. Daubechies, J. Lu, & H.-T. Wu: "Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool," Applied and Computational Harmonic Analysis, vol.30, no.2, pp.243-261, 2011.
    F. Auger, P. Flandrin, Y.-T. Lin, S. McLaughlin, S. Meignen, T. Oberlin, & H.-T. Wu: "Time-frequency reassignment and synchrosqueezing: An overview," IEEE Signal Processing Magazine, vol.30, no.6, pp.32-41, 2013.
  - Download the software from https://github.com/ebrevdo/synchrosqueezing.
  - Apply the synchrosqueezed wavelet transform to your favorite data you prepared above, and plot the time-frequency plane figure of your data.
  - Apply more conventional windowed Fourier transform and continuous wavelet transform to your data, and plot the time-frequency plane and time-scale plane figures generated by these transforms, respectively.
  - Interpret these time-frequency/time-scale plane figures, analyze your data on these time-frequency/time-scale planes, and discuss what you have found out about your data, and discuss pros and cons of each of these three transforms.
- Project D: Generalized Morse Wavelet Transform Experiments
  - Get your favorite time series data such as music signals, stock market price curves, etc.
  - Read the following papers very carefully and learn what the generalized Morse wavelet transforms are:
    S. C. Olheda and A. T. Walden: "Generalized Morse wavelets," IEEE Trans. Signal Process., vol.50, no.11, pp.2661-2670, 2002.
    J. M. Lilly and S. C. Olheda: "Higher-order properties of analytic wavelets," IEEE Trans. Signal Process., vol.57, no.1, pp.146-160, 2009.
    J. M. Lilly and S. C. Olheda: "On the analytic wavelet transform," IEEE Trans. Inform. Theory, vol.56, no.8, pp.4135-4156, 2010.
    J. M. Lilly and S. C. Olheda: "Generalized Morse wavelets as a superfamily of analytic wavelets," IEEE Trans. Signal Process., vol.60, no.11, pp.6036-6041, 2012.
  - Download the software from http://www.jmlilly.net/jmlsoft.html.
  - Apply the generalized Morse wavelet transform with various different parameters (β, γ) to your favorite data you prepared above, plot the time-frequency plane figure of your data.
  - Interpret these time-frequency plane figures, analyze your data on these time-frequency planes, and discuss what you have found out about your data, and discuss which parameter pair (β, γ) gives you the best result in terms of data interpretation.
- Project E: Coifman-Meyer Gaussian Bases Experiments
  - Get your favorite time series data such as music signals, stock market price curves, etc.
  - Read the following paper very carefully and learn what the Gaussian bases of Coifman and Meyer are:
    R. R. Coifman and Y. Meyer: "Gaussian Bases," Applied and Computational Harmonic Analysis, vol.2, no.3, pp.299-302, 1995.
  - Implement the Gaussian Bases using MATLAB.
  - Expand your favorite data you prepared above into the Gaussian Basis with a particular parameter ζ, plot the magnitude of the expansion coefficients in the nonincreasing order. Then do the same for different values of ζ and compare the results. In particular, examine the relationship between the speed of the decay of the magnitude of the coefficients and the parameter ζ.
  - Repeat the above experiments for a different signal and check your previous explanation and conclusion on the relationship between the coefficient decay speed and ζ still hold or not.