Algorithms and Criteria for Volumetric Centroidal Voronoi Tessellations

Li Wang 1, 2
2 MORPHEO - Capture and Analysis of Shapes in Motion
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : This thesis addresses the problem of volumetric tessellations from three-dimensional shapes, i.e., given a three-dimensional shape that is usually represented by its boundary surface, how to optimally subdivide the interior of the surface into smaller shapes, called cells, according to several criteria concerning accuracy, uniformity and regularity. We consider centroidal Voronoi tessellation that is an effective approach for building uniform and regular volumetric tessellations.A centroidal Voronoi tessellation (CVT) of a shape can be viewed as an optimal subdivision with the cells whose centers of mass, called centroids, are optimally distributed inside the shape. CVTs have been widely used in computer vision and graphics because of their properties of uniformity and regularity that are immune to shape variations. However, the problems such as how to evaluate the regularity of a CVT and how to build a CVT from different types of shapes remain a challenge.One contribution of this thesis is that we propose regularity criteria based on the normalized second order moments of the cells. These regularity criteria allow evaluating volumetric tessellations and specially comparing the regularity of different CVTs without the assumption that their shape and number of sites should be the same. Meanwhile, we propose a hierarchical approach based on a subdivision scheme that preserves cell regularity and the local optimality of CVTs. Experimental results show that our approach performs more efficiently and builds more regular CVTs according to the regularity criteria than state-of-the-art methods.Another contribution is a novel CVT algorithm for implicit shapes and an extensive comparison of Marching Cubes, Delaunay refinement and our algorithm. The key of our algorithm is using convex hulls and the local improvement to build accurate boundary cells. We present a comparison of these three algorithms with different criteria including accuracy, regularity and complexity on a large number of variant data. The results show that Marching Cubes is the fastest one and our algorithm build more accurate and regular volumetric tessellations than the others.We also explore the applications such as a shape animation framework based on CVTs that generates plausible animations with real dynamics. And the source code of the whole work of this thesis is available online for the purpose of further research.
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Li Wang. Algorithms and Criteria for Volumetric Centroidal Voronoi Tessellations. General Mathematics [math.GM]. Université Grenoble Alpes, 2017. English. ⟨NNT : 2017GREAM002⟩. ⟨tel-01455701v3⟩

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