Manifold Shape: from Differential Geometry to Mathematical Morphology

Much progress has been made in extending Euclidean mathematical morphology to more complex structures such as complete lattices or spaces with a non-commutative symmetry group. Such generalizations are important for practical situations such as translation and rotation invariant pattern recognition or shape description of patterns on spherical surfaces. Also in computer vision much use is made of spherical mappings to describe the world as seen by a human or machine observer. Stimulated by these developments the question is studied here of the shape description of patterns on arbitrary (smooth) surfaces based on mathematical morphology. The primary interest in this paper is to outline the mathematical structure of this description. Some concepts of differential geometry, in particular those of parallel transport and covariant differentiation, are used to replace the more restricted concept of invariance used so far in mathematical morphology. The corresponding morphological operators which leave the geometry on the surface invariant are then constructed.