Extension of Mathematical Morphology in Riemannian Spaces

Mathematical morphology remains an efficient image analysis tool due to its morphological scale-spaces capability. It can be formulated using partial differential equations that are in fact a particular case of the first order Hamilton-Jacobi equations for which viscosity solutions exist and are given by Hopf-Lax-Oleinik (HLO) formulas. In this study, we propose an extension of HLO formulas in Riemannian manifolds by considering a general Cauchy problem, and prove the existence of a unique viscosity solution. Some properties are derived, and example on the hyperbolic ball is also provided.