Finite random sets and morphology

In order to be able to optimally design morphological shape extraction algorithms operating on binary digital images, a probability theory is needed for finite random sets and probability relations that show how the probability changes as a finite random set is propagated through a morphological operation. In this paper, we develop such a theory for finite random sets. We then demonstrate how to apply this theory for calculating the probability that a set S perturbed by min or max noise N and dilated or eroded by a structuring element K is a subset, superset, or hits a given set R. In some cases we obtain exact results and in some cases we obtain bounds for the desired probability.