Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

Abstract

Summary. This paper proposes a unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes. We present a consistent derivation of these first and second order differential properties using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method, and compare them to existing formulations. Building upon previous work in discrete geometry, these operators are closely related to the continuous case, guaranteeing an appropriate extension from the continuous to the discrete setting: they respect most intrinsic properties of the continuous differential operators. We show that these estimates are optimal in accuracy under mild smoothness conditions, and demonstrate their numerical quality. We also present applications of these operators, such as mesh smoothing, enhancement, and quality checking, and show results of denoising in higher dimensions, such as for tensor images.

Perceptual models of viewpoint preference

“…a 9 : Mean curvature. We compute the mean curvature on the surface of the object using Meyer et al [2002]. We consider curvature magnitudes to be relevant (not generated by noise) if they could be generated by a feature larger than 1% of the object's size.…”

“…For mesh saliency, we use the implementation of Lee et al [2005] to assign a value at every vertex as a preprocess. Similarly, we compute the mean and Gaussian curvatures of the surface using the method of Meyer et al [2002]. For any particular view we render a "color" buffer containing these values interpolated across faces and compute the appropriate quantities on the resulting rendered images.…”

Multi-scale freeform surface texture filtering using a mesh relaxation scheme

“…The odd vertices were defined as a set of independent vertices which are not directly connected by an edge. All the selected odd vertices are removed by a mesh simplification algorithm in the global down-sampling stage, and then predicted back using the prediction operator that relax the curvature based on the Meyer smoothing operator [34,35].Szczensa proposed a new multi-resolution analysis for irregular meshes using the lifting scheme, in which she proposed a new prediction operator using Voronoi cells in a local neighbourhood [36,37].Very recently, the authors have proposed a lifting based algorithm to filter the texture on freeform surface, this technique consisted of five major operators; split, predict, update, simplify (down-sampling) and merge (upsampling) [16].…”

Skinning Mesh Algorithm Improvement Based on Double Weights

“…It explains that Gaussian curvature can reflect the local bending degree of the local surface. We use following formula to estimate Gaussian curvature [7] .…”