  We consider the one-dimensional Perona-Malik functional, that is the energy
associated to the celebrated forward-backward equation introduced by P. Perona
and J. Malik in the context of image processing, with the addition of a forcing
term. We discretize the functional by restricting its domain to a finite
dimensional space of piecewise constant functions, and by replacing the
derivative with a difference quotient.
  We investigate the asymptotic behavior of minima and minimizers as the
discretization scale vanishes. In particular, if the forcing term has bounded
variation, we show that any sequence of minimizers converges in the sense of
varifolds to the graph of the forcing term, but with tangent component which is
a combination of the horizontal and vertical directions.
  If the forcing term is more regular, we also prove that minimizers actually
develop a microstructure that looks like a piecewise constant function at a
suitable scale, which is intermediate between the macroscopic scale and the
scale of the discretization.
