Mathematical Morphology for Real-Valued Images on Riemannian Manifolds

摘要

This paper introduces mathematical morphology for real-valued images whose support space is a Riemannian manifold. The starting point consists in replacing the Euclidean distance in the canonic quadratic structuring function by the Riemannian distance. Besides the definition of Riemannian dilation/erosion and Riemannian opening/closing, their properties are explored. We generalize also some theoretical results on Lasry-Lions regularization for Cartan-Hadamard manifolds. Theoretical connections with previous works on adaptive morphology and on manifold shape are considered. Various useful image manifolds are formalized, with an example using real-valued 3D surfaces.