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 two-dimensional wavelets and wavelet transforms. In my dissertation and recent works it has been shown that one-dimensional wavelet approximations in the Radon transform domain provide one with algorithms that meet both of the above requirements.  When the raised-cosine 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 band-limited 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 PSWF-based 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 Legendre-Bessel approximation approach to PSWFs, one can nicely and rather easily approximate them by series in the well-known 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.