Medial Spheres for Shape Representation - CIM - McGill University

three categories: Voronoi methods, spatial subdivision methods, and tracing methods. We
now describe each type of method in turn.
2.2.1 Voronoi Methods
Given a set of points sampled on a surface, the Voronoi diagram of the sampled points
is a powerful tool for computing the medial surface of the solid bounded by this surface.
Voronoi methods use a subset of the Voronoi faces, edges, and vertices to approximate the
medial surface and a subset of the Voronoi edges and vertices to approximate the medial
axis. As an example, Figure 2–4 shows the Voronoi diagram of a 2D point set and Voronoi
edges that approximate the medial axis are highlighted. For 3D point inputs, the modern
CGAL library [1] includes code that computes the Voronoi diagram of a set of points in
a manner that is both highly robust and efficient. Issues arising in the implementation of
such algorithms are to determine the appropriate sampling rate for the surface points and to
determine the right subset of the Voronoi diagram that provides a desirable approximation
to the medial surface. Voronoi-based methods can provide theoretical bounds on the qual-
ity of a medial surface approximation in terms of homotopy equivalence and geometric
proximity to the true medial surface.
The difficulty with using Voronoi diagrams of 3D points to approximate the medial
surface is that not all Voronoi vertices lie near the medial surface. Specifically, among
the sample points considered, there may exist four nearly co-planar nearby points that de-
termine a small sphere empty of other points, i.e., a Voronoi vertex. This vertex can be
arbitrarily far from the medial surface. Amenta et al. [7] defines a subset of the Voronoi
vertices, called the poles, and shows that the set of poles converges to the medial surface
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