On a three dimensional Compton scattering tomography system with fixed source

TL;DR

This paper introduces a novel 3D Compton scattering tomography system with a fixed source and a movable detector on a spherical surface, modeling data via a toric Radon transform and demonstrating its invertibility with numerical reconstructions.

Contribution

The paper presents a new 3D Compton tomography modality with a fixed source and a spherical detector path, along with a mathematical analysis of the associated Radon transform and reconstruction algorithm.

Findings

Proved the invertibility of the toric Radon transform.

Developed a numerical reconstruction method using Tikhonov regularization.

Provided a public implementation of the spherical harmonic expansion algorithm.

Abstract

Compton scatter tomography is an emerging technique with attractive applications in several fields in imaging such as non-destructive testing and medical scanning. In this paper, we introduce a novel modality in three dimensions with a fixed source and a single detector that moves on a spherical surface. We also study the Radon transform modeling the data that consists in integrals on toric surfaces. Using spherical harmonics we arrive to a generalized Abel s type equation connecting the coefficients of the expansion of the data with those of the function. We show the uniqueness of its solution and so the invertibility of the toric Radon transform. We illustrate this through numerical reconstructions in three dimensions using Tikhonov regularization. A preliminary version of the algorithm for discrete spherical harmonic expansion is available in a public code repository.