Semi-Continuity of Skeletons in 2-Manifold and Discrete Voronoi Approximation

The skeleton of a 2D shape is an important geometric structure in pattern analysis and computer vision. In this paper we study the skeleton of a 2D shape in a 2-manifoldM, based on a geodesic metric. We present a formal definition of the skeleton S(Ω) for a shape Ω inM and show several properties that make S(Ω) distinct from its Euclidean counterpart in R2. We further prove that for a shape sequence {Ωi} that converge to a shape Ω inM, the mapping Ω→ S(Ω) is lower semi-continuous. A direct application of this result is that we can use a set P of sample points to approximate the boundary of a 2D shape Ω inM, and the Voronoi diagram of P inside Ω ⊂M gives a good approximation to the skeleton S(Ω). Examples of skeleton computation in topography and brain morphometry are illustrated.