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. (2015). Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform. Inverse Problems, 31(9).
Kuchment, P., & Kunyansky, L. (2008). A survey in mathematics for industry: Mathematics of thermoacoustic tomography. European Journal of Applied Mathematics, 19(2), 191-224.

Abstract:

The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography. © 2008 Cambridge University Press.

Kunyansky, L. A. (1992). Generalized and attenuated radon transforms: Restorative approach to the numerical inversion. Inverse Problems, 8(5), 809-819.

Abstract:

The problem of the function reconstruction on its line integrals with known weight function is considered. The approach studied consists of treating the attenuated projections by the radon transform inversion formula and considering the result of the inversion as a distorted image. A helpful formula describing the distortion is obtained. The norm of the distortion operator is estimated and several iterative restoration algorithms based on the integral transfers are investigated. The results of the numerical inversion of the attenuated radon transform are presented to demonstrate the features of the algorithms.

Agranovsky, M., Kuchment, P., Kunyansky, L., & Wang, L. (2012). On Reconstruction Formulas and Algorithms for the Thermoacoustic Tomography. PHOTOACCOUSTIC IMAGING AND SPECTROSCOPY, 144, 89-101.
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.