Inclusion and estimates for the jumps of minimizers in variational denoising

We study stability and inclusion of the jump set of minimizers of convex denoising functionals, such as the celebrated "Rudin-Osher-Fatemi" functional, for scalar or vectorial signals. We show that under mild regularity assumptions on the data fidelity term and the regularizer, the jump set of the minimizer is essentially a subset of the original jump set. Moreover, we give an estimate on the magnitude of jumps in terms of the data. This extends old results, in particular of the first author (with V. Caselles and M. Novaga) and of T. Valkonen, to much more general cases. We also consider the case where the original datum has unbounded variation, and define a notion of its jump set which, again, must contain the jump set of the solution.