The discontinuity set of solutions of the TV denoising problem and some extensions

The main purpose of this paper is to prove that the jump discontinuity set of the solution of the total variation based denoising problem is contained in the jump set of the datum to be denoised. We also prove some extensions of this result for the total variation minimization flow, for anisotropic norms, and for some more general convex functionals, which include the minimal surface equation case and its anisotropic extensions.