[en] Recovering a function from integrals over conical surfaces has recently received 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 the first main result, we show the 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 increase the reconstruction quality from conical Radon data. (paper)