Incidence and lattice calculus with applications to stochastic geometry and image analysis

Incidence between subsets is a basic concept of stochastic geometry and mathematical morphology. In this note we discuss a formal generalisation of incidence (and the dual notion of dominance) in the setting of complete lattices. We discuss applications to mathematical morphology, random set theory and combinatorial geometrical probability. We also suggest possible applications to transmission microscopy, digital image discretization and robot motion planning. The generalised incidence structure turns out to be equivalent to the established idea of a lattice adjunction. Using this, many problems in stochastic geometry (Buffon-Sylvester problem, local knowledge, overprojection effects) can be reformulated as lattice calculations.