Salient and Reentrant Points of Discrete Sets

The border-salient and reentrant points of a discrete set are special points of the border of the set. When they are given with multiplicity they completely characterize the set, and without multiplicity they characterize the set if all its 8-components are 4-connected. The inner-salient and reentrant are defined similarly to the border ones, but we show that, in general, they do not characterize the set, even if this set is 4-simply connected. We also show that the genus of a set can be easily computed from the number of salient and reentrant points. A discrete set is a finite subset of the integer plane Z. Intuitively a discrete set can be described by its border, but this border can also be characterized by the points where there is a change of direction. In this paper these points are said to be salient and reentrant points. In fact we define two types of salient and reentrant points: the first ones are on the border of the set (the border points), and the second ones are in the inner of the set (the inner points). Section 1 of this paper presents some preliminary definitions and properties. The section 2 is devoted to the border salient/reentrant points: we first prove that the genus of a set can be computed from the number of salients points and the number of reentrant points, it is in fact a reformulation of the well known result that the genus of a set can be locally computed. Then we prove that any discrete set is completely characterized by its set of border salient/reentrant points given with multiplicities. If the multiplicities are not given then the set must satisfy some connectivity constraints to be characterized. 1 Partially supported by the Project EPML-9 of French CNRS. Preprint submitted to Elsevier Science 27 April 2004 Section 3 presents the inner salient/reentrant points. We show that in general discrete sets are not characterized by their sets of inner salient/reentrant points even if the points are given with multiplicities and the sets are connected, nevertheless we show that there is a characterization theorem if the sets are supposed to satisfy convexity constraints.