The cone-beam transform and spherical convolution operators

The cone-beam transform consists of integrating a function defined on the three-dimensional space along every ray that starts on a certain scanning set. Based on Grangeat's formula, Louis (2016 Inverse Problems 32 115005) states reconstruction formulas based on a new generalized Funk–Radon transform on the sphere. In this article, we give a singular value decomposition of this generalized Funk–Radon transform. We use this result to derive a singular value decomposition of the cone-beam transform with sources on the sphere thus generalizing a result of Kazantsev (2015 J. Inverse Ill-Posed Problems 23 173–85).