Anisotropic quadrangulation

Abstract

Quadrangulation methods aim to approximate surfaces by semi-regular meshes with as few extraordinary vertices as possible. A number of techniques use the harmonic parameterization to keep quads close to squares, or fit parametrization gradients to align quads to features. Both types of techniques create near-isotropic quads; featurealigned quadrangulation algorithms reduce the remeshing error by aligning isotropic quads with principal curvature directions. A complementary approach is to allow for anisotropic elements, which are well-known to have significantly better approximation quality.In this work we present a simple and efficient technique to add curvature-dependent anisotropy to harmonic and feature-aligned parameterization and improve the approximation error of the quadrangulations. We use a metric derived from the shape operator which results in a more uniform error distribution, decreasing the error near features.

Anisotropic geodesics for live‐wire mesh segmentation

“…A number of tensor‐based anisotropic metrics have been proposed in the context of quad meshing and segmentation [CBK12, PSH*04, KMZ10]. However, when the eigenvectors of these tensors ( e 1 , e 2 ) are aligned with the principle curvature directions ( u 1 , u 2 ), the geodesics under these metrics either become unstable in feature‐less regions or may fail to snap to salient features.…”

“…Note that M x is discontinuous at an umbilical or saddle point, where curvature directions u 1 , u 2 are not well defined. As a result, the metric and the resulting geodesics become rather unstable in flat, spherical or saddle‐like surfaces (see Figure left). Pottmann‐Kovacs metric: Pottmann et al [PSH*04] and Kovacs et al [KMZ10] used a norm with a variable anisotropy, for some choice of β. The norm is well‐defined at an umbilical or saddle point, where g x ( v ) becomes an isotropic norm .…”

“…Pottmann‐Kovacs metric: Pottmann et al [PSH*04] and Kovacs et al [KMZ10] used a norm with a variable anisotropy, for some choice of β. The norm is well‐defined at an umbilical or saddle point, where g x ( v ) becomes an isotropic norm .…”

“…The main challenge here is that the new edge lengths may not preserve triangle inequality, a requirement by all Euclidean geodesic algorithms. Previous works have found that invalid triangles, which fail triangle inequality, are common when the metric has large anisotropy [KMZ10, CHK13]. Campen et al [CHK13] restores triangle inequality by solving a least‐square optimization problem with inequality constraints.…”

Global parametrization of range image sets

“…While some degree of feature alignment is possible (see [Lee et al 1998;Marinov and Kobbelt 2005]), it is difficult to preserve features during simplification. Other methods use global harmonic or conformal parametrizations with singularities [Gu and Yau 2003;Dong et al 2006;Tong et al 2006;Ben-Chen et al 2008;Springborn et al 2008;Kovacs et al 2009]. While some of these methods offer a degree of control over the size and structure of the domain mesh (e.g., [Dong et al 2006]), feature alignment is limited to determining positions of parametrization singularities.…”