Connectivity in Digital Pictures

TLDR: It is shown that every simply-connected object in such a picture has elements which can be deleted without destroying its simple- connectedness, which makes it easy to prove that a well-known "shrinking" algorithm always works--that is, shrinks any simply- connected object down to a single element.

Abstract: Natural concepts of connectedness and simple-connectedness are defined for subsets of a digital picture. It is shown that every simply-connected object (with more than one element) in such a picture has elements which can be deleted without destroying its simple- connectedness. This makes it easy to prove that a well-known "shrinking" algorithm always works--that is, shrinks any simply-connected object down to a single element. It also becomes easy to show that the natural "edge-following" algorithm, in which one "keeps one's hand on the wall," follows completely around the edge of any simply-connected object; this result in turn can be used to show that a well-known "border-following" algorithm (in which one follows the border elements of the object rather than the "cracks" between the object and its comple- ment) always works. Various related questions are also treated, among them, that of whether there can exist a "parallel" shrinking algorithm.