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

TL;DR

This paper proves that explicit universal backprojection formulas for reconstructing functions from spherical means are only valid for ellipsoidal domains, establishing the limitations of these formulas in imaging applications.

Contribution

The authors demonstrate that the universal backprojection inversion formulas cannot be extended beyond ellipsoids, identifying ellipsoids as the maximal class for such explicit formulas.

Findings

Universal backprojection formulas are only valid for ellipsoids.

Extension of these formulas to non-ellipsoidal domains is impossible.

Ellipsoids are the largest class of domains with explicit inversion formulas.

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 $\Gamma\subset\mathbb{R}^{n}$ and $r \in(0,\infty)$ has important applications in modern imaging modalities, such as photo- and thermo- acoustic tomography. When $\Gamma$ coincides with the boundary $\partial\Omega$ of a bounded (convex) domain $\Omega\subset\mathbb{R}^{n}$, a function supported within $\Omega$ can be uniquely recovered from its spherical means known on $\Gamma$. We are interested in explicit inversion formulas for such a reconstruction. If $\Gamma=\partial\Omega$, such formulas are only known for the case when $\Gamma$ 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 $\Gamma$? In this article we prove, for the so-called "universal backprojection inversion formulas", that their extension to non-ellipsoidal domains $\Omega$ is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.