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 input function. The second geometry is motivated by Compton camera imaging where the vertex set represents planar detector locations and is disjoint from the support of the input function representing the radiation density. We refer to the associated 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 [7]. 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 [5]. 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.