We study the discrete version of a family of ill-posed, nonlinear diffusion equations
of order $2n$. The fourth order $(n=2)$ version of these equations constitutes our main motivation, as
it appears prominently in image processing and computer vision literature. It was proposed by You
and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The
second order equation $(n=1)$ corresponds to another famous model from image processing, namely
Perona and Malik’s anisotropic diffusion, and was studied in earlier papers. The equations studied in
this paper are high order analogues of the Perona-Malik equation, and like the second order model,
their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a
recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the
discrete in space version of these high order equations in any space dimension, for a large class of
diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are
typically observed.