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

Abstract

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