Anisotropic polygonal remeshing

Abstract

In this paper, we propose a novel polygonal remeshing technique that exploits a key aspect of surfaces: the intrinsic anisotropy of natural or man-made geometry. In particular, we use curvature directions to drive the remeshing process, mimicking the lines that artists themselves would use when creating 3D models from scratch. After extracting and smoothing the curvature tensor field of an input genus-0 surface patch, lines of minimum and maximum curvatures are used to determine appropriate edges for the remeshed version in anisotropic regions, while spherical regions are simply pointsampled since there is no natural direction of symmetry locally. As a result our technique generates polygon meshes mainly composed of quads in anisotropic regions, and of triangles in spherical regions. Our approach provides the flexibility to produce meshes ranging from isotropic to anisotropic, from coarse to dense, and from uniform to curvature adapted.

Variational shape approximation

“…This is a consequence of the natural anisotropic nature of most surfaces: as brought to light in recent graphics work [Interrante et al 1996;Girshick et al 2000;Rössl and Kobbelt 2000;Hertzmann and Zorin 2000], the main traits of an originally oversampled mesh can be extracted from a close inspection of the curvature tensor field. Aligning either strokes (as done by caricaturists) or mesh edges (as done in anisotropic remeshers [Botsch and Kobbelt 2001;Alliez et al 2003]) along these curvature lines results in a particularly effective way to describe the geometry of a surface by respecting local symmetries and key features that govern lighting effects. Although such a strategy increases the mesh efficiency by matching the conditions of optimality for the L 2 metric in the limit (see Section 2.1), there is no theoretical guarantee of its efficiency of approximation at coarse scales; additionally, local approximations of differential curvatures, known to be arguable on discrete meshes, render these methods more prone to suboptimal results.…”

Constructing streak surfaces for 3D unsteady vector fields

“…This forms an intermediate triangle. The triangle is then sub-divided into three quads -a method inspired by Alliez et al [Alliez et al 2003]. In the top row, a new T-junction is created.…”

“…The updated topology forms an intermediate triangle in the mesh. In order to maintain a quad-based topology we decompose the triangle into three quads using a method adapted from Alliez et al [Alliez et al 2003]. …”

“…Some benefits of quad-based meshes are demonstrated by Alliez et al [Alliez et al 2003] and Tong et al [Tong et al 2006].…”

Estimating curvatures and their derivatives on triangle meshes

“…The principal curvatures and principal directions have been widely used in computer graphics, appearing in applications such as remeshing [Alliez et al 2003], smoothing [Desbrun et al 1999], segmentation [Trucco and Fisher 1995], visualization [Interrante et al 1995], and nonphotorealistic rendering [Hertzmann and Zorin 2000;Praun et al 2001;DeCarlo et al 2003]. We may classify existing methods for estimating principal curvatures and directions (as opposed to methods that estimate only the mean curvature H = (κ 1 + κ 2 )/2 or Gaussian curvature K = κ 1 κ 2 ) into three general categories:…”