Applied & Computational Harmonic Analysis Homework Page (Winter 2014)
Course: MAT 271
CRN: 84077
Title: Applied & Computational Harmonic Analysis
Class: TR 1:40pm-3:00pm, Physics 140
Instructor: Naoki Saito
Office: 2142 MSB
Phone: 754-2121
Email: saito@math.ucdavis.edu
Office Hours: TR 3:10pm-4:00pm, or by appointment
Homework #1 Due Thursday January 23, 2014
PDF file
Please take a look at my comments on HW1.
Homework #2 Due Thursday February 13, 2014
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 Thursday February 27, 2014
PDF file
Please read my comments to HW3, which contain useful and important information.
Final Project Assignment Due 5pm Monday, March 17, 2014
- 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,
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;
- Dave Donoho's WaveLab software downloadable from http://www-stat.stanford.edu/~wavelab; or
- Gabriel Peyré's The Numerical Tour toolbox downloadable from https://www.ceremade.dauphine.fr/~peyre/numerical-tour.
- 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.
Please email
me if you have any comments or questions!
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