Leonid Kunyansky

Professor, Mathematics

Professor, Applied Mathematics - GIDP

Professor, BIO5 Institute

Primary Department

Mathematics

Department Affiliations

Mathematics

BIO5 Institute

Contact

(520) 621-4509

leonk@math.arizona.edu

Work Summary

I develop mathematics of biomedical imaging. All modalities of tomography imaging rely heavily on mathematical algorithms for forming an image. I develop the theory and the algorithm enabling this technology.

Research Interest

Biomedical imaging, in general, and various modalities of tomography are now an important part of medical practice and biomedical research. I develop mathematics of biomedical imaging. All modalities of tomography imaging rely heavily on mathematical algorithms for forming an image. My work involves developing the theory and the algorithm enabling this technology. By developing these techniques further, I contribute to improving health and life in the 21st century. Keywords: Electromagnetic and acoustic scattering; wave propagation; photonic crystals; spectral properties of high contrast band-gap materials and operators on graphs; computerized tomography.

Publications

Asymptotic methods for thin high-contrast two-dimensional PBG materials

Axmann, W., Kuchment, P., & Kunyansky, L. (1999). Asymptotic methods for thin high-contrast two-dimensional PBG materials. Journal of Lightwave Technology, 17(11), 1996-2007.

Abstract:

This paper surveys recent analytic and numerical results on asymptotic models of spectra of electromagnetic (EM) waves in two-dimensional (2-D) thin high-contrast photonic bandgap (PBG) materials. These models lead to discovery of interesting phenomena, including extremely narrow bands that can be used for spontaneous emission enhancement, gaps in the long wave regions, and asymptotic periodicity of the spectrum. The asymptotic results provide unexpectedly good qualitative (and sometimes quantitative) description of spectral behavior for materials of finite contrast. In some cases, simple ordinary differential models can be derived that yield a good approximation of the spectra. In such situations, one can obtain approximate analytic formulas for the dispersion relations.

Photoacoustic tomography in a rectangular reflecting cavity

Kunyansky, L., Holman, B., & Cox, B. T. (2013). Photoacoustic tomography in a rectangular reflecting cavity. Inverse Problems, 29(12).

Abstract:

Almost all known image reconstruction algorithms for photoacoustic and thermoacoustic tomography assume that the acoustic waves leave the region of interest after a finite time. This assumption is reasonable if the reflections from the detectors and surrounding surfaces can be neglected or filtered out (for example, by time-gating). However, when the object is surrounded by acoustically hard detector arrays, and/or by additional acoustic mirrors, the acoustic waves will undergo multiple reflections. (In the absence of absorption, they would bounce around in such a reverberant cavity forever.) This disallows the use of the existing free-space reconstruction techniques. This paper proposes a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations. © 2013 IOP Publishing Ltd.

Leonid Kunyansky