Universal inversion formulas for recovering a function from spherical means

The problem of reconstruction a function from spherical means is at the heart of several modern imaging modalities and other applications. In this paper we derive universal back-projection type reconstruction formulas for recovering a function in arbitrary dimension from averages over spheres centered on the boundary an arbitrarily shaped smooth convex domain. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed smoothing integral operator. For elliptical domains the integral operator is shown to vanish and hence we establish exact inversion formulas for recovering a function from spherical means centered on the boundary of elliptical domains in arbitrary dimension.Comment: [20 pages, 2 figures] Compared to the previous versions I corrected some typo

Inversion of circular means and the wave equation on convex planar domains

We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary \partial \Om of a smooth convex bounded domain \Om \subset \R^2. As a main result we establish back-projection type inversion formulas that recover any initial data with support in \Om modulo an explicitly computed smoothing integral operator \K_\Om. For circular and elliptical domains the operator \K_\Om is shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on \partial \Om. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography.Comment: [14 pages, 2 figures