Inversion formulas for the attenuated conical Radon transform: Plane and cylinder case

Abstract

Since the invention of Compton camera imaging, the conical Radon transform, which maps a given function defined on 3-dimensional Euclidean space to its surface integrals over cones, has been studied intensively. The problem of recovering such a function from its unrestricted conical Radon transform is overdetermined, since the set of all cones depends on the three dimensions of the vertex, the two dimensions of the central axis, and the one-dimensional opening angle. Therefore, various types of restricted conical Radon transforms have also been studied. However, most of these studies have neglected the attenuation of the medium. This article presents the study of attenuated conical Radon transforms with vertices on a plane (referred to as the plane case) or the cylinder (referred to as the cylinder case) in a 3-dimensional space. In all cases, the function is integrated over conical surfaces with an additional weight that decreases with the distance to the vertex of the cones. The main results provide explicit inversion formulas for the attenuated conical Radon transform in the plane and in the cylinder case.