Geometric accuracy analysis for discrete surface approximation

Dai, Jiajun; Luo, Wei; Jin, Miao; Zeng, Wei; He, Ying; Yau, Shing–Tung; Gu, Xianfeng

doi:10.1016/j.cagd.2007.04.004

Cited by 22 publications

(9 citation statements)

References 30 publications

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“…The existence of triangulation on geometric spaces can hardly be underestimated both in Pure and in Applied Mathematics, in particular that of certain special types (see, e.g. [8], [11], [42], [31], [37], [38], [7], and [36], [2], [40], [13], respectively).…”

Section: Introductionmentioning

confidence: 99%

Curvature based triangulation of metric measure spaces

Saucan¹

2011

Complex Analysis and Dynamical Systems IV

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We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role of the lower bound on Ricci curvature is replaced by the curvature-dimension condition CD(K, N ). We show also that for weighted Riemannian manifolds the triangulation can be improved to become a thick one and that, in consequence, such manifolds admit weight-sensitive quasimeromorphic mappings. An application of this last result to information manifolds is considered.Further more, we extend to weak CD(K, N ) spaces the results of Kanai regarding the discretization of manifolds, and show that the volume growth of such a space is the same as that of any of its discretizations.1 2 EMIL SAUCAN without any modifications to metric measure spaces (except, of course, the obvious necessary adaptations to the more general context).The reason behind this is dual: On one hand there exists a feeling in the community, that, although the tools and results developed by Gromov, Lott, Villani, Sturm and others are most elegant, they lack, so far, any concrete and efficient application. We wish, therefore, to further emphasize that the notions of curvatures for metric measure spaces are highly natural by giving an extension of a classical problem and its (also classical) solution, to the new context. On the other hand, it appears that there exists a real interest in the triangulation and representation of the information manifold, that is the space of parameterized probability measures (or the statistical model) equipped with the Riemannian metric induced by the Fischer information metric onto the Euclidean sphere (see Section 4.1.3 below).The reminder of this paper is structured as follows: As already mentioned above, we present, in Section 2, the construction of Grove and Petersen. (This and the following section are the most "didactic", thus we proceed rather slowly; however, afterwards the pace of the exposition will become more brisk.) Next, in Section 3, we introduce the necessary notions and results regarding curvatures of metric measure spaces. (Unfortunately, the style is rather technical, since we had to cover quite a large number of definitions and results.) Section 4 constitutes the heart of our paper, in the sense that we show therein how to extend Grove-Petersen construction to the metric measure setting. We follow, in Sections 5, with an extension, to weak CD(K, N ) spaces, of Kanai's results regarding the discretization of manifolds, and show, in particular, that the volume growth of such a space coincides with that of any of its discretizations. In a sense, this represents the second part of the present paper, related to, yet distinct, from the main triangulation problem considered in the previous sections. We conclude, in Section 6, with a few very brief remarks.

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Section: Introductionmentioning

confidence: 99%

Curvature based triangulation of metric measure spaces

Saucan¹

2011

Complex Analysis and Dynamical Systems IV

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“…However, we have not demonstrated that the new sampling conditions are sufficient to ensure that the iDt-mesh is a substructure of the 3D Delaunay tetrahedralization. Since good convergence properties of the iDt-mesh have been demonstrated [DLYG06], this would put it on a more or less equal footing with the rDt. It is known that the rDt and the iDt-mesh are not necessarily combinatorially equivalent, regardless of sampling density [DZM07].…”

Section: Discussionmentioning

confidence: 99%

Surface sampling and the intrinsic Voronoi diagram

Dyer¹,

Zhang²,

Möller³

2008

Computer Graphics Forum

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We develop adaptive sampling criteria which guarantee a topologically faithful mesh and demonstrate an improvement and simplification over earlier results, albeit restricted to 2D surfaces. These sampling criteria are based on functions defined by intrinsic properties of the surface: the strong convexity radius and the injectivity radius. We establish inequalities that relate these functions to the local feature size, thus enabling a comparison between the demands of the intrinsic sampling criteria and those based on Euclidean distances and the medial axis.

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“…1) The sampling is computed in R 3 , and triangulated using the volumetric Delaunay triangulation algorithms, such as [4] [5] [6] [7] [8] [9]. 2) The sampling and triangulation are directly computed on curved surfaces, such as [10] [11].…”

Section: Introductionmentioning

confidence: 99%

Surface Meshing with Curvature Convergence

Zeng

Morvan

et al. 2014

IEEE Trans. Visual. Comput. Graphics

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Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.

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Geometric accuracy analysis for discrete surface approximation

Cited by 22 publications

References 30 publications

Curvature based triangulation of metric measure spaces

Curvature based triangulation of metric measure spaces

Surface sampling and the intrinsic Voronoi diagram

Surface Meshing with Curvature Convergence

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