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

Motivated by applications to image reconstruction, in this paper we analyse a \emph{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 $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-limit which turns out to be an anisotropic free-discontinuity functional.