Arcs and Curves in Digital Pictures

Characterizations of digital “simple arcs” and “simple closed curves” are given. In particular, it is shown that the following are equivalent for sets S having more than four points: (1) S is a simple curve; (2) S is connected and each point of S has exactly two neighbors in S ; (3) S is connected, has exactly one hole, and has no deletable points. It follows that if a “shrinking” algorithm is applied to a connected S that has exactly one hole, it shrinks to a simple curve.