
Teaching
Research
Vita
IMA Activities


Research Interests
Originally I worked
in algebra at Moscow State University, Moscow, Russia, specializing in
regular matrix rings and algebras. After completing my first Ph.D. I
decided to come to the USA and switch my specialty to applied
mathematics. The first applied project I worked on was a paint mix
color optimization problem at Pittsburgh Paint and Glass (PPG
Industries).
The decision to turn from
pure mathematics to applied mathematics proved to be the right choice
since I found that I enjoyed working on applied problems as much as I
had previously enjoyed my work in algebra. It gave me the opportunity
to learn something about the new area of wavelets and their
applications in diverse fields. It also became necessary to extensively
use computers since often the resulting equations could not be solved
analytically nor adequately tested for industrial use.
Meetings with scientists from GE
Medical Systems led me to pursue the field of medical imaging,
particularly computerized tomography (CT). That, and the
influence of my advisors, helped to define several major projects
revolving around medical imaging (or, more generally, image processing)
and applied wavelets.
In CT, an image must be
reconstructed from its sampled projections in the form of the Radon
transform. The usual requirements on any reconstruction procedure
are its computational effectiveness as well as quality of recovered
image. Standard techniques involve twodimensional wavelets and wavelet
transforms. In my dissertation and recent works it has been shown that
onedimensional wavelet approximations in the Radon transform domain
provide one with algorithms that meet both of the above
requirements. When the raisedcosine spectrum sampling function
(which happens to have a closed form – quite a rare property of a
scaling function!) is used, all integrations are avoided. A
positive summability method can be applied to further improve the
quality of the reconstructed picture, especially along its
discontinuities.
A recent surge of interest –
mine as well as generally  in prolate spheroidal wave functions
(PSWFs) has prompted the exploration of their possible use in CT.
These bandlimited functions are unique in that they are maximally
concentrated on a finite time interval. Computationally they behave as
if they were finite in both time and frequency domain – a distinct
impossibility theoretically, but a practical necessity for anyone
working in signal or image processing. It was no surprise when
our PSWFbased CT reconstruction algorithm produced superb
results.
Another interesting venue
that opens up is related to the absence of a closed form for
PSWFs. We have recently shown that instead of the traditional
LegendreBessel approximation approach to PSWFs, one can nicely and
rather easily approximate them by series in the wellknown sinc
function with a minimum of computation.
Our most recent project concerns
the subject of functional magnetic resonance imaging (fMRI). In this
instance the data consists of sampled values of the Fourier transform
of the magnitude of the image and the goal is to estimate the amount of
the activity in a region of the brain. We are currently looking into
alternative ways of estimating this quantity so that only a portion of
the data is used thereby enabling one to increase the speed of
calculation.
This year's IMA's inspirational program on Imaging piqued
my interest in the field of RADAR imaging. We have reason to believe
that there might be a good use for wavelets based on PSWF in this area
as well. I plan to continue both of these investigations and perhaps
expand into other areas of imaging science.

