New Bounds on the Size of Optimal Meshes

Sheehy, Donald R.

doi:10.1111/j.1467-8659.2012.03168.x

Cited by 13 publications

(8 citation statements)

References 25 publications

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“…For a wide class of inputs, the feature size measure is O(n) [31]. For general inputs, the bound of O(n log Δ) in (1) above is well-known and can also be derived as a corollary to Lemma 4 below.…”

Section: Voronoi Aspect Ratios and Voronoi Refinementmentioning

confidence: 79%

A New Approach to Output-Sensitive Construction of Voronoi Diagrams and Delaunay Triangulations

Sheehy

2014

Discrete Comput Geom

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We describe a new algorithm for computing the Voronoi diagram of a set of n points in constant-dimensional Euclidean space. The running time of our algorithm is O( f log n log Δ) where f is the output complexity of the Voronoi diagram and Δ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.

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Section: Voronoi Aspect Ratios and Voronoi Refinementmentioning

confidence: 79%

A New Approach to Output-Sensitive Construction of Voronoi Diagrams and Delaunay Triangulations

Sheehy

2014

Discrete Comput Geom

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“…For a wide class of input domains, the feature size measure is O(n) [27]. For general inputs, the bound of O(n log ∆) mentioned above is well known and can even be derived as a corollary to Lemma 5 below.…”

Section: Orthoballs and Encroachmentmentioning

confidence: 95%

A new approach to output-sensitive voronoi diagrams and delaunay triangulations

Sheehy

2013

Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry

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We describe a new algorithm for computing the Voronoi diagram of a set of n points in constant-dimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.

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“…Delaunay Meshing. The most studied and most widely used algorithms to generate tetrahedral meshes are based on the Delaunay condition [Alliez et al 2005a;Aurenhammer 1991;Aurenhammer et al 2013;Bishop 2016;Boissonnat et al 2002;Boissonnat and Oudot 2005;Busaryev et al 2009;Chen and Xu 2004;Cheng et al 2008Cheng et al , 2012Chew 1989Chew , 1993Cohen-Steiner et al 2002;Du and Wang 2003;Jamin et al 2015;Murphy et al 2001;Remacle 2017;Ruppert 1995;Sheehy 2012;Shewchuk 1996Shewchuk , 1998Shewchuk , 1999Shewchuk , 2002Si 2015;Si and Gartner 2005;Si and Shewchuk 2014;Tournois et al 2009]. These methods are efficient and are widely used in commercial software.…”

Section: Tetrahedral Meshingmentioning

confidence: 99%

Tetrahedral meshing in the wild

et al. 2018

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5.0% 2.6% 1.2% 0.6% 0.4% 0.2% 9.6% 6.8% 4.8% 3.7% 2.7% 1.5% 54.5% 26.9% 8.1% 1.9% 0.5% 0.1% 8.7% 2.1% 0.6% 0.2% 0.1% >1m >2m >4m >8m >16m >32m 0% 10% 20% 30% 40% 50% TetGen CGAL TetWild Ours CGAL #T = 1 362 980 444s TetGen #T = 8 221 130 1705s TetWild #T = 459 626 1588s Ours #T = 278 997 291s Input #F = 392 040 Fig. 1. A mouse skull model (from micro-CT) tetrahedralized by fTetWild (right) compared with other popular tetrahedral meshing algorithms. The plot shows the percentage of models requiring more than a certain time for the different approaches over 4540 inputs (the subset of Thingi10k where all 4 algorithms succeed). Our algorithm successfully meshes 99.4% of the input models in less than 2 minutes, and processes all models within 32 minutes. The comparison has been done using the experimental setup of TetWild ] and selecting a similar target resolution for all methods. The CGAL surface approximation parameter has been selected to be comparable to the envelope size used for TetWild and for our method.We propose a new tetrahedral meshing technique, fTetWild, to convert triangle soups into high-quality tetrahedral meshes. Our method builds upon the TetWild algorithm, inheriting its unconditional robustness, but dramatically reducing its computation cost and guaranteeing the generation of a valid tetrahedral mesh with floating point coordinates. This is achieved by introducing a new problem formulation, which is well suited for a pure floating point implementation and naturally leads to a parallel implementation. Our algorithm produces results qualitatively and quantitatively similar to TetWild, but at a fraction of the computational cost, with a running time comparable to Delaunay-based tetrahedralization algorithms. ACM Reference format:

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New Bounds on the Size of Optimal Meshes

Cited by 13 publications

References 25 publications

A New Approach to Output-Sensitive Construction of Voronoi Diagrams and Delaunay Triangulations

A New Approach to Output-Sensitive Construction of Voronoi Diagrams and Delaunay Triangulations

A new approach to output-sensitive voronoi diagrams and delaunay triangulations

Tetrahedral meshing in the wild

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