Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform

TL;DR

This paper presents explicit inversion formulas for reconstructing a function from spherical means in corner-like domains, reducing the problem to the classical Radon transform, with applications in thermoacoustic and photoacoustic tomography.

Contribution

It provides explicit solutions for the inverse problem in 2D and 3D corner-like geometries, linking spherical mean data to Radon projections for accurate reconstruction.

Findings

Explicit inversion formulas for 3D octet boundaries.

Reconstruction in 2D with centers on intersecting rays.

Reduction of the inverse problem to Radon transform inversion.

Abstract

We consider an inverse problem arising in thermo-/photo- acoustic tomography that amounts to reconstructing a function $f$ from its circular or spherical means with the centers lying on a given measurement surface. (Equivalently, these means can be expressed through the solution $u(t,x)$ of the wave equation with the initial pressure equal to $f$.) An explicit solution of this inverse problem is obtained in 3D for the surface that is the boundary of an open octet, and in 2D for the case when the centers of integration circles lie on two rays starting at the origin and intersecting at the angle equal to $\pi/N$, $N=2,3,4,...$. Our formulas reconstruct the Radon projections of a function closely related to $f$, from the values of $u(t,x)$ on the measurement surface. Then, function $f$ can be found by inverting the Radon transform.