The conical Radon transform with vertices on triple lines

TL;DR

This paper derives an explicit inversion formula for a restricted conical Radon transform relevant to imaging techniques like Compton camera imaging, using triple line sensors to simplify the over-determined problem.

Contribution

It introduces a novel restricted conical Radon transform model with triple line sensors and provides an analytic inversion formula for this specific case.

Findings

Derived an explicit inversion formula for the restricted conical Radon transform.

Defined and inverted a new ray transform adapted to triple line sensors.

Enhanced understanding of cone-based integral transforms in imaging applications.

Abstract

We study the inversion of the conical Radon which integrates a function in three-dimensional space from integrals over circular cones. The conical Radon recently got significant attention due to its relevance in various imaging applications such as Compton camera imaging and single scattering optical tomography. The unrestricted conical Radon transform is over-determined because the manifold of all cones depends on six variables: the center position, the axis orientation and the opening angle of the cone. In this work, we consider a particular restricted transform using triple line sensors where integrals over a three-dimensional set of cones are collected, determined by a one-dimensional vertex set, a one-dimensional set of central axes, and the one-dimensional set of opening angle. As the main result in this paper, we derive an analytic inversion formula for the restricted conical Radon transform. Along that way we define a certain ray transform adapted to the triple line sensor for which we establish an analytic inversion formula.