Discontinuities of the minimizers of the weighted or anisotropic total variation for image reconstruction

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.