  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.
