Inserted: 18 jun 2018
Last Updated: 7 oct 2020
Journal: Interfaces Free Bound.
Motivated by applications to image reconstruction, in this paper we analyse a 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.
Keywords: Gamma-convergence, finite-difference discretisation, Ambrosio-Tortorelli functional, elliptic approximation, free-discontinuity functionals