Inverting Spherical Radon Transform by a Closed-form Formula: A Microlocal Analytic Point of View

TL;DR

This paper develops a microlocal analytic framework for inverting the spherical Radon transform restricted to spheres centered on a hypersurface, deriving a closed-form inversion formula through oscillatory integral analysis.

Contribution

It introduces a microlocal analytic approach to invert the spherical Radon transform with a new closed-form formula based on oscillatory integrals.

Findings

Derived microlocal properties of the inversion formula

Established conditions for the inversion formula's effectiveness

Provided insights into the transform's behavior on observation surfaces

Abstract

Let $\mR$ be the restriction of the spherical Radon transform to the set of spheres centered on a hypersurface $\mS$. We study the inversion of $\mR$ by a closed-form formula. We approach the problem by studying an oscillatory integral, which depends on the observation surface $\mS$ as a parameter. We then derive various microlocal analytic properties of the associated closed-form inversion formula.