Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra

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 the 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.Comment: 9 figure

A mathematical model and inversion procedure for Magneto-Acousto-Electric Tomography (MAET)

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 the present paper we analyze existing mathematical models of this method, and propose a general procedure for solving the inverse problem associated with 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, MAET is a stable technique yilelding high-resolution images even in the presence of significant noise in the data

Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography

The paper contains a simple approach to reconstruction in Thermoacoustic and Photoacoustic Tomography. The technique works for any geometry of point detectors placement and for variable sound speed satisfying a non-trapping condition. A uniqueness of reconstruction result is also obtained