Semi-Continuity of Skeletons in Two-Manifold and Discrete Voronoi Approximation
Published in IEEE Transactions on Pattern Analysis and Machine Intelligence, 2015
Abstract: 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 two-manifold $\mathcal{M}$, based on a geodesic metric. We present a formal definition of the skeleton $S(Ω)$ for a shape $Ω$ in $\mathcal{M}$ and show several properties that make $S(Ω)$ distinct from its Euclidean counterpart in $\mathbb{R}^{2}$. We further prove that for a shape sequence ${Ω_i}$ that converge to a shape $Ω$ in $\mathcal{M}$ , the mapping $Ω→\bar{S}(Ω)$ 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 $Ω$ in $\mathcal{M}$ , 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.
Recommended citation: Yong-Jin Liu (2015) Semi-continuity of skeletons in 2-manifold and discrete Voronoi approximation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 37, No. 9, pp. 1938-1944, 2015.