RESUMEN
Imaging processes built on the Compton scattering effect have been under continuing investigation since it was first suggested in the 50s. However, despite many innovative contributions, there are still formidable theoretical and technical challenges to overcome. In this paper, we review the state-of-the-art principles of the so-called scattered radiation emission imaging. Basically, it consists of using the cleverly collected scattered radiation from a radiating object to reconstruct its inner structure. Image formation is based on the mathematical concept of compounded conical projection. It entails a Radon transform defined on circular cone surfaces in order to express the scattered radiation flux density on a detecting pixel. We discuss in particular invertible cases of such conical Radon transforms which form a mathematical basis for image reconstruction methods. Numerical simulations performed in two and three space dimensions speak in favor of the viability of this imaging principle and its potential applications in various fields.
RESUMEN
Radon transforms defined on smooth curves are well known and extensively studied in the literature. In this paper, we consider a Radon transform defined on a discontinuous curve formed by a pair of half-lines forming the vertical letter V. If the classical two-dimensional Radon transform has served as a work horse for tomographic transmission and/or emission imaging, we show that this V-line Radon transform is the backbone of scattered radiation imaging in two dimensions. We establish its analytic inverse formula as well as a corresponding filtered back projection reconstruction procedure. These theoretical results allow the reconstruction of two-dimensional images from Compton scattered radiation collected on a one-dimensional collimated camera. We illustrate the working principles of this imaging modality by presenting numerical simulation results.