The Radon Transform over Cones with Vertices on the Sphere and Orthogonal Axes

Source abstract

Recovering a function from its integrals over circular cones recently gained significance because of its relevance to novel medical imaging technologies such as emission tomography using Compton cameras. In this paper, we investigate the case where the vertices of the cones of integration are restricted to a sphere in $n$-dimensional space and where the symmetry axes are orthogonal to the sphere. We show invertibility of the considered transform and develop an inversion method based on series expansion and reduction to a system of one-dimensional integral equations of generalized Abel type. Since the arising kernels do not satisfy standard assumptions, we also develop a uniqueness result for generalized Abel integral equations where the kernel has zeros on the diagonal. Finally, we demonstrate how to efficiently implement our inversion method and present numerical results.