Leonid Kunyansky

Leonid Kunyansky

Professor, Mathematics
Professor, Applied Mathematics - GIDP
Professor, BIO5 Institute
Primary Department
Department Affiliations
Contact
(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. 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.

Ambartsoumian, G., & Kunyansky, L. (2014). Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Problems and Imaging, 8(2), 339-359.

Exterior inverse problem for the circular means transform (CMT) arises in the intravascular photoacoustic imaging (IVPA), in the intravascular ultrasound imaging (IVUS), as well as in radar and sonar. The reduction of the IPVA to the CMT is quite straightforward. As shown in the paper, in IVUS the circular means can be recovered from measurements by solving a certain Volterra integral equation. Thus, a tomography reconstruction in both modalities requires solving the exterior problem for the CMT.

Nguyen, L., & Kunyansky, L. (2016). A dissipative time reversal technique for photo-acoustic tomography in а cavity. SIAM J. Imaging Science, 9(2), 748–769.
Kunyansky, L. (2012). A mathematical model and inversion procedure for magneto-acousto-electric tomography. Inverse Problems, 28(3).

Abstract:

Magneto-acousto-electric tomography (MAET), also known as the Lorentz force or Hall effect tomography, is a novel hybrid modality designed to be a high-resolution alternative to the unstable electrical impedance tomography. In this paper, we analyze the existing mathematical models of this method, and propose a general procedure for solving the inverse problem associated with the MAET. It consists in applying to the data one of the algorithms of thermo-acoustic tomography, followed by solving the Neumann problem for the Laplace equation and the Poisson equation. For the particular case when the region of interest is a cube, we present an explicit series solution resulting in a fast reconstruction algorithm. As we show, both analytically and numerically, the MAET is a stable technique yielding high-resolution images even in the presence of significant noise in the data. © 2012 IOP Publishing Ltd.

Kunyansky, L. A. (2007). A series solution and a fast algorithm for the inversion of the spherical mean Radon transform. Inverse Problems, 23(6), S11-S20.

Abstract:

An explicit series solution is proposed for the inversion of the spherical mean Radon transform. Such an inversion is required in problems of thermo- and photo-acoustic tomography. Closed-form inversion formulae are currently known only for the case when the centres of the integration spheres lie on a sphere surrounding the support of the unknown function, or on certain unbounded surfaces. Our approach results in an explicit series solution for any closed measuring surface surrounding a region for which the eigenfunctions of the Dirichlet Laplacian are explicitly known - such as, for example, cube, finite cylinder, half-sphere etc. In addition, we present a fast reconstruction algorithm applicable in the case when the detectors (the centres of the integration spheres) lie on a surface of a cube. This algorithm reconstructs 3D images thousands times faster than backprojection-type methods. © 2007 IOP Publishing Ltd.