The Perona-Malik equation (PME) is a forward-backward nonlinear
diffusion which was proposed in the context of image processing as an
image enhancement tool capable of preserving sharp features such as
edges. To this day its mathematical nature has not been fully
understood in spite of many an attempt.  After a brief historical
overview of the mathematical results available for the equation and
its many regularizations/relaxations, the talk will introduce and
analyze a novel, rather natural, regularization which will shed light
on the nature of PME. The regularization is quite mild in that PME is
regularized by a family of forward-backward equations, the solutions
of which are, however, better behaved.