<title>Binary random fields, random set theory, and the morphological analysis of shape</title>

Abstract

We discuss a number of issues related to the morphological analysis of random shape by means of discrete random set theory. Our purpose here is twofold. First, we would like to demonstrate that, in the discrete case, a number of problems associated with random set theory can be effectively solved. Furthermore, we would like to establish a direct relationship between discrete random sets and binary random fields. To accomplish this, we first introduce the cumulative distribution and capacity functionals of a discrete random set, and review their properties. Under a natural assumption, we show that there exists a one-to-one correspondence between the probability mass function of a discrete binary random field and the cumulative distribution functional of the corresponding discrete random set. The cumulative distribution and capacity functionals are then related to higher-order moments of a discrete binary random field. We show that there exists a direct relationship between the capacity functional of a discrete random set and the capacity functional of the discrete random set obtained by means of dilation, erosion, opening, or closing. These relationships allow us to derive an interesting result, regarding the statistical behavior of elementary morphological filters. Finally, we introduce moments for discrete random sets, and show that the class of opening-based discrete size distributions are higher-order moments of a discrete random set. This last observation allows us to argue that discrete size distributions are good statistical summaries for shape.