Error estimates for finite differences approximations of the total variation

Abstract

Abstract We present a convergence rate analysis of the Rudin–Osher–Fatemi (ROF) denoising problem for two different discretizations of the total variation. The first is the standard discretization, which induces blurring in some particular diagonal directions. We prove that in a simplified setting corresponding to such a direction, the discrete ROF energy converges to the continuous one with the rate $h^{2/3}$. The second discretization is based on dual Raviart–Thomas fields and achieves an optimal $O(h)$ convergence rate for the same quantity, for discontinuous solutions with some standard hypotheses.