An inversion formula for the spherical mean transform with data on an ellipsoid in two and three dimensions

TL;DR

This paper extends existing inversion formulas for spherical mean transforms from the unit sphere to ellipsoids in 2D and 3D, enabling reconstruction of functions from spherical means centered on ellipsoids.

Contribution

It introduces modified methods to derive inversion formulas for spherical means centered on ellipsoids, expanding the applicability beyond the unit sphere.

Findings

Derived inversion formulas for ellipsoidal centers in 2D and 3D

Demonstrated the feasibility of reconstructing functions from ellipsoidal spherical means

Extended previous methods to more general geometric settings

Abstract

In the articles [1] and [2] of D. Finch, M. Haltmeier, S. Patch and D. Rakesh inversion formulas were found in any dimension $n\geq2$ for recovering a smooth function with compact support in the unit ball from spherical means centered on the unit sphere. The aim of this article is to show that the methods used in [1], [2] can be modified in order to get similar inversion formulas from spherical means centered on an ellipsoid in two and three dimensional spaces.