On the exactness of the universal backprojection formula for the spherical means Radon transform

Abstract

The spherical means Radon transform is defined by the integral of a function f in R n over the sphere S ( x , r ) of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data where x belongs to a hypersurface Γ ⊂ R n and r ∈ ( 0 , ∞ ) has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When Γ coincides with the boundary ∂ Ω of a bounded (convex) domain Ω ⊂ R n , a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If Γ = ∂ Ω , such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to a question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called ‘universal backprojection inversion formulas’, that their extension to non-ellipsoidal domains Ω is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.