Partial differential equations and mathematical morphology (Q1277403)

The paper is devoted to the behaviour of a class of iterated filters arising in image processing. In particular, the author considers the so-called morphological filters (i.e. monotone and commuting with translations and with strictly increasing contrast changes) defined by \[ IS_su(x):=\inf_{B\in{\mathbb B}}\sup_{y\in B}u(x+s^{1/N}y), \qquad SI_su(x):=\sup_{B\in{\mathbb B}}\inf_{y\in B}u(x+s^{1/N}y), \] where \(N\) is the space dimension, and \({\mathbb B}\) is the so-called set of structuring elements of the filter. Under suitable assumptions on \({\mathbb B}\), one of the main results of the paper is the convergence of the iterates of the alternate operator \(T_h=SI_s IS_s\) (with \(h=s^{2/(N+1)}\)), namely \(u(x,t)=T_h^{[t/h]}(x)\) uniformly converge on compact subsets of \({\mathbb R}^N\times (0,\infty)\) to the solution, in the viscosity sense, of the fully nonlinear PDE \[ u_t=c| Du| \Biggl(\Pi_{i=1}^{N-1}\lambda_i\Biggr)^{1/(N+1)} H(\lambda_1,\ldots,\lambda_{N-1}) \] where \(\lambda_1,\ldots,\lambda_{N-1}\) are the principal curvatures of the level set of \(\{y: u(t,y)=u(t,x)\}\) containing \(x\) and \(H(t_1,\ldots,t_{N-1})=1\) of all \(t_i\)'s are positive, \(H(t_1,\ldots,t_{N-1})=-1\) of all \(t_i\)'s are negative and \(H(t_1,\ldots,t_{N-1})=0\) in all the other cases. The author also proves convergence of the iterated median filters to the classical mean curvature motion.