tion of binary image operators, known as morphological sampling, has been proposed by Heijmans and Toet [4] We present new results on the problem of morphologically sampling binary images and image operators, thus enhancing and Heijmans [5,6]. This theory considers continuousthe morphological sampling theory of Heijmans and Toet. space binary images to be closed subsets of β«ήβ¬ d , is These include sampling composite operators and approximatconsistent with the underlying topological structure of ing continuous translation invariant erosions, openings, and mathematical morphology (determined by the so-called closings by discrete erosions, openings, and closings, respechit-or-miss topology), and satisfies useful morphological tively. The results presented in this paper have been already properties. In this paper, we present a number of new used in a theory of morphologically sampling random binary results that enhance the theory of morphological samimages modeled as random closed sets. Β© 1997 Academic Press pling. In particular, and under suitable conditions, we show that morphological sampling is compatible with composition of a useful class of binary image operators, * This work was supported by the Office of Naval Research, Mathematifor the covering discretization and provide a proof of one cal, Computer, and Information Sciences Division, under ONR Grant of the (unproved) results of Lemma 7.2 in [5] (see also N00014-90-1345.