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 $\Gamma \subset {\mathbb{R}}^n$ 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.

11 pages, 1 figure

References in corpus (4)

• Universal inversion formulas for recovering a function from spherical means
• Inversion of circular means and the wave equation on convex planar domains
• A uniform reconstruction formula in integral geometry
• Exact Reconstruction Formula for the Spherical Mean Radon Transform on Ellipsoids