Dominance and Incidence Structures with Applications to Stochastic Geometry and Mathematical Morphology

Mathematical morphology is based on set-theoretic notions such as inclusion and intersection. In this paper it is shown how such notions can be extended to complete lattices. This results in so-called dominance and incidence structures. Such structures can be regarded as the geometric counterpart of adjunctions, which play an important role in morphology. This paper discusses the theoretical foundations, some applications from mathematical morphology, and an application to the Buffon-Sylvester problem in stochastic geometry.