Recovering Finite Parametric Distributions and Functions Using the Spherical Mean Transform

TL;DR

This paper presents a method to recover finite parametric distributions and functions from their spherical mean transform by converting the problem into a Prony's system, enabling parameter extraction under general position assumptions.

Contribution

The paper introduces a novel approach to reconstruct functions from spherical means using Prony's system, expanding the toolkit for inverse problems in integral geometry.

Findings

Successful conversion of the reconstruction problem into a Prony's system

Guarantee of regularity under general position assumptions

Effective parameter extraction for finite parametric distributions

Abstract

The aim of the article is to recover a certain type of finite parametric distributions and functions using their spherical mean transform which is given on a certain family of spheres whose centers belong to a finite set $\Gamma$. For this, we show how the problem of reconstruction can be converted to a Prony's type system of equations whose regularity is guaranteed by the assumption that the points in the set $\Gamma$ are in general position. By solving the corresponding Prony's system we can extract the set of parameters which define the corresponding function or distribution.