Range characterization of the weighted divergent beam and cone integral transforms

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.