Range characterization of the weighted divergent beam and cone integral transforms

TL;DR

This paper establishes mathematical range conditions for weighted integral transforms over conical surfaces, relevant to imaging techniques like Compton camera imaging, using factorization into beam and spherical transforms.

Contribution

It provides the first comprehensive range characterizations for weighted conical Radon and Compton transforms in two different geometries, generalizing previous results.

Findings

Range conditions for the conical Radon transform in convex geometries.

Range conditions for the Compton transform with planar detector geometry.

Complete range descriptions combining beam and spherical transform analyses.

Abstract

We establish range characterizations, or data consistency conditions, for an integral transform that maps a function to its weighted integrals over conical surfaces in $\mathbb{R}^n$. We consider two different geometries for the cone vertices, which lead to mathematically distinct range conditions. We use the term \emph{conical Radon transform} when the vertex set is a bounded convex subset of $\mathbb{R}^n$ including support of the unknown function. The second geometry is motivated by Compton camera imaging: the vertex set represents planar detector locations and is disjoint from the support of the radiation density. We refer to the corresponding transform as the \emph{Compton transform}. Our approach is based on a factorization into the $k$-weighted divergent beam transform and the spherical section transform. In the bounded convex vertex geometry, the range of the divergent beam component is described by a higher-order transport boundary-value problem, as studied by Derevtsov, Volkov, and Schuster \cite{Derevtsov2021}. In the planar detector geometry, we derive range conditions for the $k$-weighted divergent beam transform that generalize the planar cone-beam consistency conditions of Clackdoyle and Desbat \cite{ClackdoyleDesbat2013}. Combining these results with the range characterization of the spherical section transform yields complete range descriptions for both the $k$-weighted conical Radon transform and the $k$-weighted Compton transform.