A Hierarchical Approach for Regular Centroidal Voronoi Tessellations

Abstract

International audienceIn this paper we consider Centroidal Voronoi Tessellations (CVTs) and study their regularity. CVTs are geometric structures that enable regular tessellations of geometric objects and are widely used in shape modeling and analysis. While several efficient iterative schemes, with defined local convergence properties, have been proposed to compute CVTs, little attention has been paid to the evaluation of the resulting cell decompositions. In this paper, we propose a regularity criterion that allows us to evaluate and compare CVTs independently of their sizes and of their cell numbers. This criterion allows us to compare CVTs on a common basis. It builds on earlier theoretical work showing that second moments of cells converge to a lower bound when optimising CVTs. In addition to proposing a regularity criterion, this paper also considers computational strategies to determine regular CVTs. We introduce a hierarchical framework that propagates regularity over decomposition levels and hence provides CVTs with provably better regularities than existing methods. We illustrate these principles with a wide range of experiments on synthetic and real models

A Constrained Resampling Strategy for Mesh Improvement

“…Mesh Improvement and Quality Remeshing: Smoothing methods represented by the Centroidal Voronoi Tessellation (CVT) [DFG99] and its variants [DGJ03,WHWB16,JFL14,DW03] work by moving vertices to optimize an energy function. Other optimization‐based smoothing techniques for various quality objectives including angle bounds, edge length and triangle areas compute locally optimal moves [ABE99, Ren16].…”

On Volumetric Shape Reconstruction from Implicit Forms

“…However, degenerate tetrahedra, typically slivers, can still appear, although their number can be reduced by global [18] or local [19] optimization techniques. Centroidal Voronoi Tessellations (CVTs) are a special type of Voronoi tessellations with regular Voronoi cells [20]. Such tessellations are known to be optimal quantizers [21] and their cells, mostly truncated octahedra, are more isotropic than cubes or tetrahedra.…”

“…Consequently, CVTs have been used to discretize 2D and 3D shapes in many scientific domains [22]. While methods have been proposed to clip a CVT to a surface mesh [20] [23] [24], to the best of our knowledge, none is able yet to handle implicit forms. We introduce therefore in the next section a new clipping method for CVTs.…”

“…This is the case with CVTs, for which the ideal cell shape is known to be a truncated octahedron as mentioned in [21]. In this work, the dimensionless second moment of a polytope is introduced and can be used as a measure of the regularity of a CVT cell [20]. Metrics have also been proposed for other types of cells, such as hexahedra [26], as well as algebraically for general cells [28].…”

“…To this purpose, CVT algorithms alternatively re-estimate site locations and their associated cells, in an iterative manner. Approaches exist that compute CVTs given explicit forms for shapes, usually meshes as in [20] [23] [24]. We consider here the case of shapes defined by implicit forms in 3D and also, more specifically, implicit forms obtained from point clouds, a frequent case when modeling shapes with visual observations.…”