Compton scattering tomography is an emerging scanning technique with attractive applications in several fields such as non-destructive testing and medical imaging. In this paper, we study a modality in three dimensions that employs a fixed source and a single detector moving on a spherical surface. We also study the Radon transform modeling the data that consists of 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 a regularized approach.

International audienceIn an effort to deal with many ionizing radiation imaging mechanisms involving the Compton effect,we study a Radon transform on circular cone surfaces having a fixed axis direction, which is calledhere conical Radon transform (CRT). Concretely we seek to recover a density function $f(x,y,z)$in $\mathbb{R}^{3}$ from its integrals over such circular cone surfaces or its conicalprojections. Although the existence of the inverse CRT has been established, it isthe aim of this work to use this result to extent the concept of back-projection well known inComputed Tomography (CT) to this type of cone surfaces. We discuss in some details the featuresof back-projection in relation to the corresponding conical Radon transform adjoint operator as well asthe filters that arise naturally from the exact solution of the inversion problem.This intuitive approach is attractive, lends itself to efficient computational algorithms and mayprovide hints and guide for more general back-projection methods on other classes of cone surfaces, for example occurring inCompton camera imaging. Comprehensive numerical simulations results are presented and discussedto illustrate and validate this approach based on the concept of back-projection