I am interested in the mathematical theory that underlies signal/image processing, communication theory,
compression (dimension reduction) and time-frequency analysis. These all fall under the umbrella of
information theory, statistical data analysis, and applied harmonic analysis.
Application-wise my passion is in audio, speech and in the general nature of communication and perception.
I am also interested in biomedical applications.
Implicit in all of these fields is the concept of optimization, which is a cornerstone of applied mathematics.
UC Davis
Compressed Sensing and Sparse Representations
(Partially supported by the National Science Foundation,
VIGRE
Grant No. DMS-0135345)
I am currently working with Professor
Thomas Strohmer
studying ways to apply sparse representations to the
pseudodifferential operators
that model
time-varying communications channels. This research uses the general results from compressed sensing.
Compressed sensing is a relatively new area of research which is gathering a lot of attention.
It has been pioneered primarily by
Donoho,
Candes,
and Tao.
The theory relies on the fact that a compressible signal can be well-approximated by a sparse signal.
This amounts to a reduction in the dimensionality of the signal space. Much of the recent progress has come
from using the tools of random matrix theory, functional analysis, convex geometry and probability theory.
UC Davis' Professor Roman Vershynin and his collaborators
have also made significant contributions to the field. The compressed sensing group at Rice University maintains
an excellent
resource page. Terrence Tao also has a good
outline
on the history of the field.
There is strong evidence which shows that many of the brain's sensory signals can be modeled
with sparse representations and over-complete bases. I am interested in this, and other biomedical
applications of signal processing. I gave a brief presentation on this topic in March 2007
for the course Bio-Math 227.
PSO is a type of optimization based on biological organisms which exhibit
swarming, flocking and herding behavior such as birds, fish, bees, ants, and even humans.
It is a population-based tool in the sense that information is distributed amongst the individual
particles. The particles must communicate with each other and constantly evaluate their relative
position so as to be able to home in on a target (i.e., a global minimum or maximum).
San Francisco State University
Quantization Error and Coding Gain
At SFSU I did research with Professor
Shidong Li
on minimizing the quantization error arising in biorthogonal wavelet filter banks. Quantization error
minimization is also known as coding gain and deals with the optimal allocation of bits. Subband coding
is a technique that splits a signal into multiple frequency bands. More bits are allocated to the bands
which have greater signal energy. This gives finer resolution in the bands where the signal has the most
presence, and this reduces the overall quantization error. Similar biorthogonal wavelets are used
ubiquitously in applications such as JPEG-2000 for the compression of images.
We used the technique of Pseudoframes for Subspaces (PFFS) to design filter banks that had a higher
coding gain (for a certain statistical class of signals) than similar filters that are currently being used.
I presented our results in a paper entitled
Biorthogonal Wavelets of Maximum Coding Gain through Pseudoframes for Subspaces
at the SPIE Conference on Optics & Photonics: Session on Mathematics of Data/Image Pattern Recognition,
Compression, and Encryption with Applications in San Diego, August 2006.
SUNY Buffalo
Bandwidth-efficient Pulse Shaping
At the University of Buffalo I worked with Professor
Ozan Tonguz
on an industry-related project that dealt with finding bandwidth-efficient pulse shapes for wireless
digital communications. We worked with a engineering team at a company in Buffalo to design a
high-speed digital radio (such as the kind that transmits credit card information to the bank when
one uses a gas station pump).
From time-frequency analysis we know that the bandwidth of a signal must necessarily increase as the
data rate increases. The challenge was to keep the frequency signature of the signal within the
spectral bands established by the FCC while increasing the transmission capacity of the radio. We managed
to do this by using a clever combination of Partial-Response Continuous Phase Modulation (PR-CPM), Inter-Symbol
Interference (ISI) and a Viterbi decoding algorithm.
Other Projects
Course on Fourier array imaging and synthetic aperture radar (SAR)