Binary random fields, random closed sets, and morphological sampling

We theoretically formulate the problem of processing continuous-space binary random fields by means of mathematical morphology. This may allow us to employ mathematical morphology to develop new statistical techniques for the analysis of binary random images. Since morphological transformations of continuous-space binary random fields are not measurable in general, we are naturally forced to employ intermediate steps that require generation of an equivalent random closed set. The relationship between continuous-space binary random fields and random closed sets is thoroughly investigated. As a byproduct of this investigation, a number of useful new results, regarding separability of random closed sets, are presented. Our plan, however, suffers from a few technical problems that are prominent in the continuous case. As an alternative, we suggest morphological discretization of binary random fields, random closed sets, and morphological operators, thereby effectively implementing our problem in the discrete domain.