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

Abstract

The spherical means Radon transform $\mathcal{M}f(x,r)$ is defined by the integral of a function $f$ in $\mathbb{R}^{n}$ 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 $\mathcal{M}f(x,r)$ where $x$ belongs to a hypersurface $Γ\subset\mathbb{R}^{n}$ and $r \in(0,\infty)$ has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When $Γ$ coincides with the boundary $\partialΩ$ of a bounded (convex) domain $Ω\subset\mathbb{R}^{n}$, 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 $Γ=\partialΩ$, such formulas are only known for the case when $Γ$ is an ellipsoid (or one of its partial cases). This gives rise to the natural 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.