Explicit inversion of spherical Radon transforms in odd dimensions with partial radial data

TL;DR

This paper presents an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data, reducing the problem to solving ordinary differential equations and validating the approach through numerical simulations.

Contribution

It introduces a new explicit inversion method for odd-dimensional spherical Radon transforms with partial data, simplifying previous integral equation approaches.

Findings

Explicit inversion algorithm derived for odd dimensions

Reconstruction reduces to solving ordinary differential equations

Numerical simulations confirm theoretical validity

Abstract

We derive an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data. We prove that the reconstruction of the unknown function can be reduced to solving ordinary differential equations, thereby providing a more explicit approach in odd dimensions than solving Volterra integral equation of the first kind established in prior works. We also provide analytical solutions in some special cases. Finally, we present numerical simulations validating our theoretical results. Our work answers a question posed by Rubin in ``Inversion formulae for the spherical mean in odd dimensions and the Euler-Poisson-Darboux equation,'' Inverse Problems 24 (2008), no. 2, 025021, 10 pp.