A backward-forward regularization of the Perona-Malik equation
It is shown that the Perona-Malik equation \[ u_t = \nabla \cdot \left( \frac{1}{1+ |\nabla u |^2} \nabla u \right ) \] admits a natural regularization by forward-backward diffusion possessing better analytical properties that the Perona-Malik equation itself. This novel regularization is even milder than that previously proposed see [the author and \textit{J. V. Lambers}, ``Two new nonlinear nonlocal diffusions for noise reduction'', J. Math. Imaging Vis. 33, No. 1, 25--37 (2009; \url{doi:10.1007/s10851-008-0108-z})] but has the advantage of having a variational structure. The regularization leads \[ u_t = \nabla \cdot \left( \left [ \frac{1}{1+ |\nabla u |^2} + \delta \right ] \nabla u \right ). \] Well-posedness of the regularizing problem along with a complete understanding of its long time behavior are obtained by resorting to weak Young measure valued solutions. Solutions are unique (to some extent) but can exhibit ``micro-oscillations'' (in the sense of minimizing sequences and in the spirit of the material science) between ``preferred'' gradient states. In the limit of vanishing regularization, the preferred gradients have size zero or infinity, thus explaining the well-known phenomenon of staircasing. The theoretical results do completely confirm numerical observations concerning the generic behavior of solutions.
scientific article; zbMATH DE number 6012785
scientific article