Morphological Counterpart of Ornstein–Uhlenbeck Semigroups and PDEs

Abstract

Morphological semigroups and corresponding Partial Dierential Equations are equivalent respectively to Hopf-Lax semigroups and the Cauchy problem of a family of rst-order HamiltonJacobi equations. They are related to Maslov idempotent analysis too. The OrnsteinUhlenbeck operator and OrnsteinUhlenbeck semigroup play the role of the Laplacian and the heat kernel semigroup if the Lebesgue measure is replaced by the standard Gaussian measure. In this paper we revisit some contributions on the idempotent analogue of the semigroups associated with the OrnsteinUhlenbeck semigroup, which are based on a Maslov measure, as well as the associated rst-order HamiltonJacobi equation. We study the relevance of the corresponding semigroups in the context of morphological erosions and dilations.