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

TL;DR

This paper investigates how finite-difference discretizations of the Ambrosio-Tortorelli functional behave under different scaling regimes, revealing the influence of lattice structure on the resulting free-discontinuity models.

Contribution

It provides a comprehensive analysis of the impact of discretization and approximation parameters on the Gamma-limit of the functional, especially highlighting anisotropic effects.

Findings

Gamma-limit depends on the relative scaling of b5 and elta

Lattice structure influences the anisotropy of the limit functional

Different regimes lead to distinct free-discontinuity models

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 $\delta$ the discretisation step-size, we fully describe the relative impact of $\varepsilon$ and $\delta$ in terms of $\Gamma$-limits for the corresponding discrete functionals, in the three possible scaling regimes. We show, in particular, that when $\varepsilon$ 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.