Construction of Iso-Contours, Bisectors, and Voronoi Diagrams on Triangulated Surfaces

In the research of computer vision and machine perception, 3D objects are usually represented by 2-manifold triangular meshes {\cal M}. In this paper, we present practical and efficient algorithms to construct iso-contours, bisectors, and Voronoi diagrams of point sites on {\cal M}, based on an exact geodesic metric. Compared to euclidean metric spaces, the Voronoi diagrams on {\cal M} exhibit many special properties that fail all of the existing euclidean Voronoi algorithms. To provide practical algorithms for constructing geodesic-metric-based Voronoi diagrams on {\cal M}, this paper studies the analytic structure of iso-contours, bisectors, and Voronoi diagrams on {\cal M}. After a necessary preprocessing of model {\cal M}, practical algorithms are proposed for quickly obtaining full information about iso--contours, bisectors, and Voronoi diagrams on {\cal M}. The complexity of the construction algorithms is also analyzed. Finally, three interesting applications—surface sampling and reconstruction, 3D skeleton extraction, and point pattern analysis—are presented that show the potential power of the proposed algorithms in pattern analysis.