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

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. (2002). Differential operators on graphs and photonic crystals. Advances in Computational Mathematics, 16(2-3), 263-290.

Abstract:

Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: -Δu = λεu, where ε(x) (the dielectric constant) is a periodic function that assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neumann type on graphs. The paper contains a technique of approximating these pseudo-differential spectral problems by much simpler differential ones that can sometimes be resolved analytically. Numerical experiments show amazing agreement between the spectra of the pseudo-differential and differential problems.

Kuchment, P., & Kunyansky, L. A. (1999). Spectral properties of high contrast band-gap materials and operators on graphs. Experimental Mathematics, 8(1), 1-28.

Abstract:

The theory of classical waves in periodic high contrast photonic and acoustic media leads to the spectral problem -Δu = λεu, where the dielectric constant ε(x) is a periodic function which assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. Existence and locations of spectral gaps are of primary interest. The high contrast asymptotics naturally leads to pseudodifferential operators of the Dirichlet-to-Neumann type on graphs and on more general structures. Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of the spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.

Kunyansky, L. (2011). Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra. Inverse Problems, 27(2).

Abstract:

We present explicit filtration/backprojection-type formulae for the inversion of the spherical (circular) mean transform with the centers lying on the boundary of some polyhedra (or polygons, in 2D). The formulae are derived using the double-layer potentials for the wave equation, for domains with certain symmetries. The formulae are valid for a rectangle and certain triangles in 2D, and for a cuboid, certain right prisms and a certain pyramid in 3D. All the present inversion formulae yield exact reconstruction within the domain surrounded by the acquisition surface even in the presence of exterior sources. © 2011 IOP Publishing Ltd.

Kunyansky, L., & Kunyansky, L. (2012). Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems and Imaging, 6(1), 111-131.

Abstract:

We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors placed on a circle (in 2D) or a sphere (in 3D) surrounding the object of interest. The third inversion algorithm works with the data measured by the integrating line detectors arranged in a cylindrical assembly rotating around the object. The number of operations required by these techniques is equal to O(n 3 log n) and O(n 3 log 2n) for the 3D techniques (assuming the reconstruction grid with n 3 nodes) and to O(n 2 log n) for the 2D problem with n × n discretizetion grid. Numerical simulations show that on large computational grids our methods are at least two orders of magnitude faster than the finite-difference time reversal techniques. The results of reconstructions from real measurements done by the integrating line detectors are also presented, to demonstrate the practicality of our algorithms. © 2012 American Institute of Mathematical Sciences.

Kunyansky, L. A. (2001). A new SPECT reconstruction algorithm based on the Novikov explicit inversion formula. Inverse Problems, 17(2), 293-306.

Abstract:

We present a new reconstruction algorithm for single-photon emission computed tomography. The algorithm is based on the Novikov explicit inversion formula for the attenuated Radon transform with non-uniform attenuation. Our reconstruction technique can be viewed as a generalization of both the filtered backprojection algorithm and the Tretiak-Metz algorithm. We test the performance of the present algorithm in a variety of numerical experiments. Our numerical examples show that the algorithm is capable of accurate image reconstruction even in the case of strongly non-uniform attenuation coefficient, similar to that occurring in a human thorax.