Centroidal Voronoi Tessellations for Line Segments and Graphs. technical report

Abstract

Figure 1: Starting from a mesh (A) and a template skeleton (B), our method fits the skeleton to the mesh (C) and outputs a segmentation (D). Our main contribution is an extension of Centroidal Voronoi Tesselation to line segments, using approx-imated Voronoi Diagrams of segments (E). Segment Voronoi cells (colors) are approximated by the union of sampled point’s Voronoi cells (thin lines, right half of D). Clipped 3D Voronoi cells are accurately computed, at a sub-facet precision (F). Centroidal Voronoi Tesselation (CVT) of points has many applications in geometry processing, including re-meshing and segmentation to name but a few. In this paper, we pro-pose a new extension of CVT, generalized to graphs. Given a graph and a 3D polygonal surface, our method optimizes the placement of the vertices of the graph in such a way that the graph segments best approximate the shape of the surface. We formulate the computation of CVT for graphs as a continuous variational problem, and present a simple approximated method to solve this problem. Our method is robust in the sense that it is independent of degeneracies in the input mesh, such as skinny triangles, T-junctions, small gaps or multiple connected components. We present some applications, to skeleton fitting and to shape segmentation