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.
Kunyansky, L. A. (2008). Thermoacoustic tomography with detectors on an open curve: An efficient reconstruction algorithm. Inverse Problems, 24(5).
Abstract:
Practical applications of thermoacoustic tomography require numerical inversion of the spherical mean Radon transform with the centers of integration spheres occupying an open surface. A solution of this problem is needed (both in 2D and in 3D) because frequently the region of interest cannot be completely surrounded by the detectors, as happens, for example, in breast imaging. We present an efficient numerical algorithm for solving this problem in 2D (similar methods are applicable in the 3D case). Our method is based on the numerical approximation of plane waves by certain single-layer potentials related to the acquisition geometry. After the densities of these potentials have been precomputed, each subsequent image reconstruction has the complexity of the regular filtration backprojection algorithm for the classical Radon transform. The performance of the method is demonstrated in several numerical examples: one can see that the algorithm produces very accurate reconstructions if the data are accurate and sufficiently well sampled; on the other hand, it is sufficiently stable with respect to noise in the data. © 2008 IOP Publishing Ltd.
