Leonid Kunyansky
Professor, Applied Mathematics - GIDP
Professor, BIO5 Institute
Professor, Mathematics
Primary Department
Department Affiliations
(520) 621-4509
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

Kunyansky, L., & Holman, B. R. (2015). Gradual time reversal in thermo- and photo- acoustic tomography within a resonant cavity. Inverse Problems, 31, 035008.
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.

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.

Cox, B. T., Holman, B., & Kunyansky, L. (2013). Photoacoustic tomography in a reflecting cavity. Progress in Biomedical Optics and Imaging - Proceedings of SPIE, 8581.

Abstract:

Almost all known photoacoustic image reconstruction algorithms are based on the assumption that the acoustic waves leave the object (the imaged region) after a finite time. This assumption is fulfilled if the measurements are made in free space and reflections from the detectors are negligible. However, when the object is surrounded by acoustically hard detectors arrays (and/or by additional acoustic mirrors), the acoustic waves will bounce around in such a reverberant cavity many times (in the absence of absorption, forever). This paper proposes fast reconstruction algorithms for the measurements made from the walls of a rectangular reverberant cavity. The algorithms are tested using numerical simulations. © 2013 Copyright SPIE.

Bruno, O. P., & Kunyansky, L. A. (2001). Surface scattering in three dimensions: An accelerated high-order solver. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 457(2016), 2921-2934.

Abstract:

We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order, accurate solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of high-order equivalent source approximations, which allow for fast evaluation of non-adjacent interactions by means of the three-dimensional fast Fourier transform (FFT). Our acceleration scheme has dramatically lower memory requirements and yields much higher accuracy than existing FFT-accelerated techniques. The present algorithm computes one matrix-vector multiply in O(N6/5 log N) to O(N4/3 log N) operations (depending on the geometric characteristics of the scattering surface), it exhibits super-algebraic convergence, and it does not suffer from accuracy breakdowns of any kind. We demonstrate the efficiency of our method through a variety of examples. In particular, we show that the present algorithm can evaluate accurately, on a personal computer, scattering from bodies of acoustical sizes (ka) of several hundreds.