Inversion of the Spherical Mean Transform with Sources on a Hyperplane

TL;DR

This paper develops new inversion formulas for an integral operator averaging functions over half-spheres in the upper half-space, using Fourier and Radon analysis, with potential applications in practical reconstructions.

Contribution

It introduces novel inversion formulas for the spherical mean transform with sources on a hyperplane, generalizing previous approaches and linking to paraboloid integrals.

Findings

Derived new inversion formulas for the spherical mean transform.

Connected the transform to paraboloid integral operators.

Formulas are suitable for practical reconstruction applications.

Abstract

The object of this study is an integral operator $\mathcal{S}$ which averages functions in the Euclidean upper half-space $\mathbb{R}_{+}^{n}$ over the half-spheres centered on the topological boundary $\partial \mathbb{R}_{+}^{n}$. By generalizing Norton's approach to the inversion of arc means in the upper half-plane, we intertwine $\mathcal{S}$ with a convolution operator $\mathcal{P}$. The latter integrates functions in $\mathbb{R}^{n}$ over the translates of a paraboloid of revolution. Our main result is a set of inversion formulas for $\mathcal{P}$ and $\mathcal{S}$ derived using a combination of Fourier analysis and classical Radon theory. These formulas appear to be new and are suitable for practical reconstructions.