Leonid Kunyansky

Leonid Kunyansky

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

Work Summary

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

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

Kuchment, P., & Kunyansky, L. (2011). 2D and 3D reconstructions in acousto-electric tomography. Inverse Problems, 27(5).

Abstract:

We propose and test stable algorithms for the reconstruction of the internal conductivity of a biological object using acousto-electric measurements. Namely, the conventional impedance tomography scheme is supplemented by scanning the object with acoustic waves that slightly perturb the conductivity and cause the change in the electric potential measured on the boundary of the object. These perturbations of the potential are then used as the data for the reconstruction of the conductivity. The present method does not rely on 'perfectly focused' acoustic beams. Instead, more realistic propagating spherical fronts are utilized, and then the measurements that would correspond to perfect focusing are synthesized. In other words, we use synthetic focusing. Numerical experiments with simulated data show that our techniques produce high-quality images, both in 2D and 3D, and that they remain accurate in the presence of high-level noise in the data. Local uniqueness and stability for the problem also hold. © 2011 IOP Publishing Ltd.

Terzioglu, F., Kuchment, P., & Kunyansky, L. (2017). Compton Camera Imaging And The Cone Transform. A Brief Overview. Inverse Problems.
Bruno, O. P., & Kunyansky, L. A. (2000). Fast, high-order solution of surface scattering problems. IEEE Antennas and Propagation Society, AP-S International Symposium (Digest), 4, 1860-1863.

Abstract:

A fast, high-order algorithm for the solution of problems of acoustic scattering from smooth surfaces in three dimensions is presented. Numerical experiments indicate that this algorithm performs exceptionally well.

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.