Back-projection inversion of a conical Radon transform

Abstract

In 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 called here conical Radon transform (CRT). Concretely, we seek to recover a density function (Formula presented.) in (Formula presented.) from its integrals over such circular cone surfaces or its conical projections. Although the existence of the inverse CRT has been established, it is the aim of this work to use this result to extent the concept of back-projection, well known in Computed Tomography (CT) to this type of cone surfaces. We discuss in some details the features of back-projection in relation to the corresponding CRT adjoint operator as well as the filters that arise naturally from the exact solution of the inversion problem. This intuitive approach is attractive, lends itself to efficient computational algorithms and may provide hints and guide for more general back-projection methods on other classes of cone surfaces, for example, occurring in Compton camera imaging. Comprehensive numerical simulations results are presented and discussed to illustrate and validate this approach based on the concept of back-projection.