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 δ, the
discretized functionals Γ-converge to the Mumford-Shah functional only
if δ≪ε, ε being the elliptic approximation
parameter of the Ambrosio-Tortorelli functional. Discretizing with respect to
stationary, ergodic and isotropic random lattices we prove this
Γ-convergence result also for δ∼ε, a regime at which
the discretization with respect to a periodic lattice converges instead to an
anisotropic version of the Mumford-Shah functional.Comment: 36 pages, 6 figures. Added some numerical example