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We study in this note the stability and inclusion of the jump set ofminimizers of convex denoising functionals, such as the celebrated"Rudin-Osher-Fatemi" functional, for scalar or vectorial signals. We show thatunder mild regularity assumptions on the data fidelity term and theregularizer, the jump set of the minimizer is essentially a subset of theoriginal jump set. Moreover, we give an estimate on the magnitude of jumps interms 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 generalcases. We also consider the case where the original datum has unboundedvariation, and define a notion of its jump set which, again, must contain thejump set of the solution.

Inclusion and estimates for the jumps of minimizers in variational denoising