Error estimates for finite differences approximations of the total variation

Corentin Caillaud (1) , Antonin Chambolle (1, 2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We present a convergence rate analysis of the Rudin-Osher-Fatemi (ROF) denoising problem for two different discretizations of the total variation. The first discretization is the well-known isotropic total variation that suffers from a blurring effect in a special diagonal direction. We prove that in the setting corresponding to this direction, the discrete ROF energy converges to the continuous one in O(h^2/3). The second total variation is based on Raviart-Thomas fields and achieves a O(h) convergence rate for the same quantity under some standard hypotheses.

Mots clés: total variation; finite differences; finite elements; error bounds; convergence rates; image denoising

Domaines: Analyse numérique [math.NA]

Corentin Caillaud, Antonin Chambolle. Error estimates for finite differences approximations of the total variation. IMA Journal of Numerical Analysis, 2023, 43 (2), pp.692--736. ⟨10.1093/imanum/drac001⟩. ⟨hal-02539136v2⟩