  The study of the regularity of the minimizer of the weighted anisotropic
total variation with a general fidelity term is at the heart of this paper. We
generalized some recent results on the inclusion of the discontinuities of the
minimizer of the image denoising problem. In particular, we proved that for
well-chosen weights and anisotropies, it is actually possible to create
discontinuities that were not contained in the original image. We also observed
a reduced jump property at the discontinuities of the minimizer. To prove these
results we used some regularity theorems for minimal surfaces that we had to
adapt to our setting. We also illustrated our theoretical results with several
numerical simulations.
