  We propose and analyze a finite-difference discretization of the
Ambrosio-Tortorelli functional. It is known that if the discretization is made
with respect to an underlying periodic lattice of spacing $\delta$, the
discretized functionals $\Gamma$-converge to the Mumford-Shah functional only
if $\delta\ll\varepsilon$, $\varepsilon$ being the elliptic approximation
parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to
stationary, ergodic and isotropic random lattices we prove this
$\Gamma$-convergence result also for $\delta\sim\varepsilon$, a regime at which
the discretization with respect to a periodic lattice converges instead to an
anisotropic version of the Mumford-Shah functional.
