Morphology on Convolution Lattices with Applications to the Slope Transform and Random Set Theory

This paper develops an abstract theory for mathematical morphology on complete lattices. It differs from previous work in this direction in the sense that it does not merely assume the existence of a binary operation on the underlying lattice. Rather, the starting point is the recognition of the fact that, in general, objects are only known through information resulting from a given collection of measurements, called evaluations. Such an abstract approach leads in a natural way to the concept of convolution lattice, where ‘convolution’ has to be understood in the sense of an abstract Minkowski addition. The paper contains various examples. Two applications are treated in great detail, the morphological slope transform and random set theory.