Differential geodesic mathematical morphology

Following a PDE-based formulation of low-level vision, recent works have attempted to cast classical mathematical morphology into the axiomatic framework of scale-space theory. This effort has led to derive continuous elementary morphological operators and revealed deep connections with the theory of reactive (hyperbolic) PDEs. Until now, researchers have focused their attention on Euclidean morphology. This article aims at setting up the foundations of differential geodesic mathematical morphology. Specifically, we define multiscale geodesic erosions and dilations, and derive their generating PDEs for arbitrary n-dimensional structuring sets or functions. Geodesic reconstruction then corresponds to steady-states of these equations for particular initial conditions. Geodesic morphological operators are further embedded into a general class of one-parameter operator semigroups, called geodesic scale-space operators. Within this framework, regularized geodesic operators are defined in a natural fashion by augmenting the basic PDEs with a diffusive (parabolic), scale-space-admissible component. Finally, efficient numerical implementations based on monotonic conservative schemes are presented in details. These developments provide the theoretical basis for PDE-based formulations of watershed segmentation and geodesic skeleton computation.