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 [1] 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 E. T. 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.

Analytical reconstruction formula with efficient implementation for a modality of Compton scattering tomography with translational geometry

“…Thus, first order Compton scattering is the only source of attenuation for radiation and data acquisition is performed with a pair source detector, assumed to be point-like. These conditions are common in the literature[18,35,36,3,31,29,27] and have been already discussed in[19,33,35,27].…”

An analytical reconstruction formula with efficient implementation for a modality of Compton Scattering Tomography with translational geometry

“…Thus, first order Compton scattering is the only source of attenuation for radiation and data acquisition is performed with a pair source detector, assumed to be point-like. These conditions are common in the literature[16,26,29,30,31,20,24] and have been already discussed in[15,19,26,24].…”

A joint reconstruction and lambda tomography regularization technique for energy-resolved X-ray imaging

“…The scanner depicted in figure 1 consists of a row of fixed, switched, monochromatic fan beam sources (S), a row of detectors (D A ) to measure the transmitted photons, and a second (slightly out of plane) row of detectors (D C ) to measure Compton scatter. The detectors are assumed to energy-resolved, a common assumption in CST [35,51,41,43,55], and the sources are fan-beam (in the plane) with opening angle π (so there is no restriction due to cropped fan-beams).…”

“…When the attenuation of the incoming and scattered rays is ignored, the Compton scattered intensity in two-dimensions can be modelled as integrals of n e over toric sections [35,51,55] (1.1)…”

“…A toric section T = C 1 ∪ C 2 is the union of two intersecting circles of the same radii (as displayed in figure 2), and dt is the arc measure on T . The recovery of n e is equivalent to inversion of the toric section Radon transform [34,35,51,55]. See figure 2.…”