Mesh Statistics for Robust Curvature Estimation

Abstract

While it is usually not difficult to compute principal curvatures of a smooth surface of sufficient differentiability, it is a rather difficult task when only a polygonal approximation of the surface is available, because of the inherent ambiguity of such representation. A number of different approaches has been proposed in the past that tackle this problem using various techniques. Most papers tend to focus on a particular method, while an comprehensive comparison of the different approaches is usually missing. We present results of a large experiment, involving both common and recently proposed curvature estimation techniques, applied to triangle meshes of varying properties. It turns out that none of the approaches provides reliable results under all circumstances. Motivated by this observation, we investigate mesh statistics, which can be computed from vertex positions and mesh connectivity information only, and which can help in deciding which estimator will work best for a particular case. Finally, we propose a meta‐estimator, which makes a choice between existing algorithms based on the value of the mesh statistics, and we demonstrate that such meta‐estimator, despite its simplicity, provides considerably more robust results than any existing approach.

A Survey of Surface‐Based Illustrative Rendering for Visualization

“…The calculation of the principal curvature directions (PCDs) and their curvatures can be carried out by fitting higher order polynomials to the mesh [CP05, GI04] or by calculating the normal curvatures along the edges and then estimating the shape operator [CS92, HS03, MDSB02, PKS*01, Tau95a]. We focus on another category of methods that estimate the shape operator directly [ACSD*03, CSM03, Rus04, HP11], see also [VVP*16] for an analysis of curvature estimations. In this section, we provide the curvature estimation according to Rusinkiewicz [Rus04].…”

A Geometric Optimization Approach for the Detection and Segmentation of Multiple Aneurysms

“…Various schemes for the approximation of the curvatures of a surfaces from an approximating triangle mesh have been proposed [CSM03, Rus04, PWY*07, HP11]. For a recent quantitative evaluation curvature estimation scheme, we refer to [VVP*16]. For our experiments, we use the technique by Rusinkiewicz [Rus04], which yields an approximation of the two principal curvatures, κ 1 , κ 2 , at every triangle of a mesh.…”

Sketch‐based shape‐constrained fireworks simulation in head‐mounted virtual reality

“…In particular, the method in Reference is also called the Voronoi method since it uses average Voronoi cells. Based on the research study by Vasa et al, the Voronoi method is optimal for estimating various curvatures of triangle mesh surfaces.…”

“…[22][23][24][25] In particular, the method in Reference 25 is also called the Voronoi method since it uses average Voronoi cells. Based on the research study by Vasa et al, 26 the Voronoi method is optimal for estimating various curvatures of triangle mesh surfaces. The basic idea of the Voronoi method is to regard the smooth surface as the limit or linear approximation of a cluster of triangular patches.…”