Analytic inversion of a Radon transform on double circular arcs with applications in Compton Scattering Tomography

TL;DR

This paper introduces a new Radon transform for a novel Compton Scattering Tomography system with a single rotating detector, providing an analytic inversion formula and demonstrating practical feasibility through simulations.

Contribution

It presents the first analytic inversion formula for a Radon transform associated with a new CST modality using double circular arcs, and offers an efficient numerical implementation.

Findings

Analytic inversion formula established for the new Radon transform.

Numerical implementation based on Hilbert transform demonstrated.

Simulation results confirm the system's practical feasibility.

Abstract

In this work we introduce a new Radon transform which arises from a new modality of Compton Scattering Tomography (CST). This new system is made of a single detector rotating around a fixed source. Unlike some previous CST, no collimator is used at the detector. Such a system allows us to collect scattered photons coming from two opposite sides of the source-detector segment, hence the manifold of the associated Radon transform is a family of double circular arcs. As first main theoretical result, an analytic inversion formula is established for this new Radon transform. This is achieved through the formulation of the transform in terms of circular harmonic expansion satisfying the consistency conditions in Cormack's sense. Moreover, a fast and efficient numerical implementation via an alternative formulation based on Hilbert transform is carried out. Simulation results illustrate the theoretical feasibility of the new system. From a practical point of view, an uncollimated detector system considerably increases the amount of collected data, which is particularly significant in a scatter imaging system.