Compton scatter tomography in annular domains

Abstract

Compton scatter tomography (CST) is an imaging process which reconstructs the electric charge density in a two-dimensional slice of an object. We describe a recent CST modality, introduced in 2010, designed to image an object from the data formed by the integrals of its electric charge density on a two parameter set of circular arcs in the plane, subtended by a rotating diameter of a fixed circle. Through a new approach based on a change of radial variables, introduced recently by Webber and Holman (2019 Inverse Problems Imaging 13 231–61) in a three-dimensional function reconstruction problem, the CST Radon problems (interior and exterior to the fixed circle) can be mapped to the classical Radon transform. This relation provides not only an elegant solution to the CST reconstruction problems but also provides a link from the CST Radon problems on annular domains (interior and exterior to the fixed circle) to the well-known exterior classical Radon problem, for which results on function reconstruction have been fully worked out by Quinto. Such CST Radon problems on annular domains may arise, in practice, from restrictions near grazing or near grazing back scattering Compton scattering, as missing data tomographic problems, for which the proposed connection offers a concrete solution.