A backward–forward regularization of the Perona–Malik equation

It is shown that the Perona–Malik equation (PME) admits a natural regularization by forward–backward diffusions possessing better analytical properties than PME itself. Well-posedness of the regularizing problem along with a complete understanding of its long time behavior can be obtained by resorting to weak Young measure valued solutions in the spirit of Kinderlehrer and Pedregal (1992) [1], and Demoulini (1996) [2]. Solutions are unique (to an extent to be specified) but can exhibit “micro-oscillations” (in the sense of minimizing sequences and in the spirit of material science) between “preferred” gradient states. In the limit of vanishing regularization, the preferred gradients have size 0 or ∞ thus explaining the well-known phenomenon of staircasing. The theoretical results do completely confirm and/or predict numerical observations concerning the generic behavior of solutions.