Shape of an arbitrary finite point set in IR2

We study the shape of a finite point set in IR2, where the points are not bound to a regular grid like ℤ2. The shape of a connected point set in IR2 is captured by its boundary. For a finite point set the boundary is a directed graph that connects points identified as boundary points. We argue that to serve as a proper boundary definition the directed graph should regulate scale, be minimal, have an increasing interior and be consistent with the boundary definition of connected objects.We propose to use the directed variant of theα-shape as defined by Edelsbrunner et al (1983), which we call theα-graph. Theα-graph is based on a generalization of the convex hull.The computational aspects of theα-graph have been extensively studied, but little attention has been paid to the potential use of theα-graph as a shape descriptor or a boundary definition. In this paper we prove that theα-graph satisfies the aforementioned criteria. We also prove a relation between theα-graph and the opening scale space from mathematical morphology. In fact, theα-hull provides a generalization of this scale space.

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