The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes

TL;DR

This paper studies the inversion of a Radon transform over cones with vertices on a sphere, relevant to medical imaging, providing theoretical invertibility, a new inversion method, and numerical validation.

Contribution

It introduces a novel inversion approach for cone Radon transforms with vertices on a sphere and orthogonal axes, including a uniqueness result for related Abel equations.

Findings

Proved invertibility of the cone Radon transform under specified conditions.

Developed a series expansion-based inversion method.

Provided numerical implementation and results demonstrating effectiveness.

Abstract

Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such emission tomography using Compton cameras. In this paper we investigate the case where the vertices of the cones of integration are restricted to a sphere in $n$-dimensional space and symmetry axes are orthogonal to the sphere. We show invertibility of the considered transform and develop an inversion method based on series expansion and reduction to a system of one-dimensional integral equations of generalized Abel type. Because the arising kernels do not satisfy standard assumptions, we also develop a uniqueness result for generalized Abel equations where the kernel has zeros on the diagonal. Finally, we demonstrate how to numerically implement our inversion method and present numerical results.