  Recovering a function from integrals over conical surfaces recently got
significant interest. It is relevant for emission tomography with Compton
cameras and other imaging applications. In this paper, we consider the weighted
conical Radon transform with vertices on the sphere. Opposed to previous works
on conical Radon transform, we allow a general weight depending on the distance
of the integration point from the vertex. As first main result, we show
uniqueness of inversion for that transform. To stably invert the weighted
conical Radon transform, we use general convex variational regularization. We
present numerical minimization schemes based on the Chambolle-Pock primal dual
algorithm. Within this framework, we compare various regularization terms,
including non-negativity constraints, $H^1$-regularization and total variation
regularization. Compared to standard quadratic Tikhonov regularization,
TV-regularization is demonstrated to significantly increase the reconstruction
quality from conical Radon data.
