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 epsilon the elliptic-approximation parameter and by delta the discretisation step-size, we fully describe the relative impact of epsilon and delta in terms of Gamma-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when epsilon and delta are of the same order, the underlying lattice structure affects the Gamma-limit which turns out to be an anisotropic free-discontinuity functional.