Quantitative analysis of finite-difference approximations of free-discontinuity problems

Motivated by applications to image reconstruction, in this paper we analyse a finite-difference discretisation of the Ambrosio–Tortorelli functional. Denoted by " the elliptic-approximation parameter and by i the discretisation step-size, we fully describe the relative impact of " and i in terms of limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when " and i are of the same order, the underlying lattice structure affects the -limit which turns out to be an anisotropic free-discontinuity functional.