Universal Inversion Formulas for Recovering a Function from Spherical Means

The problem of reconstruction of 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 of an arbitrarily shaped bounded convex domain with smooth boundary. Provided that the unknown function is supported inside that domain, the derived formulas recover the unknown function up to an explicitly computed 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.

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