Centroidal Voronoi diagrams for isotropic surface remeshing

Abstract

This paper proposes a new method for isotropic remeshing of triangulated surface meshes. Given a triangulated surface mesh to be resampled and a user-specified density function defined over it, we first distribute the desired number of samples by generalizing error diffusion, commonly used in image halftoning, to work directly on mesh triangles and feature edges. We then use the resulting sampling as an initial configuration for building a weighted centroidal Voronoi diagram in a conformal parameter space, where the specified density function is used for weighting. We finally create the mesh by lifting the corresponding constrained Delaunay triangulation from parameter space. A precise control over the sampling is obtained through a flexible design of the density function, the latter being possibly low-pass filtered to obtain a smoother gradation. We demonstrate the versatility of our approach through various remeshing examples.

Centroidal Voronoi tessellation in universal covering space of manifold surfaces

“…Surazhsky et al [2003] computed the constrained CVT by projecting the 1-ring neighbors of a site in the dual triangle mesh onto the tangential plane, and then finding the centroid of its Voronoi cell in the plane. Contrast to this local parametrization approach, Alliez et al [2005] used a global parametrization by cutting the surface into a disk-like topology, and computing a 2D CVT in the Euclidean parametrization domain. Valette et al [2008] directly computed an approximation of the constrained CVT as clusters of triangles.…”

“…Another alternative is to use the 3D Euclidean distance as an approximation Rong et al, 2011;Yan et al, 2009], but this may lead to disconnected Voronoi cells if two regions are very close in 3D Euclidean space but are far away along the surface. A better approach is to compute the CVT in a 2D parametrization domain of the surface [Alliez et al, 2005]. By assigning appropriate density values, the computed CVT is very close to the constrained CVT computed using the geodesic distance.…”

“…This makes the sites unable to cross the boundaries in the parametrization domain, and leads to visible artifacts along the cutting edges. In [Alliez et al, 2005], a great deal of special care and delicate strategies, such as minimizing the total cutting edge length and matching the cut graph with the feature skeleton, are required. If the cut graph does not coincide much with a set of feature edges, the remeshing results become unacceptable as indicated in [Alliez et al, 2005].…”

“…For closed genus-1 surfaces, their universal covering space can be embedded in 2D Euclidean space R 2 , so the computation of the CVT in the universal covering space is similar as in [Alliez et al, 2005] except that sites can move freely across the cutting boundaries. The universal covering space of closed genus-0 surfaces can be embedded in 2D spherical space S 2 .…”

“…Compared with previous methods using parametrization in 2D Euclidean space such as [Alliez et al, 2005], the main advantage of using the CVT in universal covering space is that the sites can move freely anywhere on the surface.…”