New Bounds on the Size of Optimal Meshes
The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents...
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| Vydáno v: | Computer graphics forum Ročník 31; číslo 5; s. 1627 - 1635 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Oxford, UK
Blackwell Publishing Ltd
01.08.2012
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| ISSN: | 0167-7055, 1467-8659 |
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| Abstract | The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing ϕ, then the number of vertices in an optimal mesh will be O(ϕdn), where d is the input dimension. We give a new analysis of this integral showing that the output size is only θ(n+nlogϕ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n). |
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| AbstractList | The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing ϕ, then the number of vertices in an optimal mesh will be O(ϕ
d
n
), where d is the input dimension. We give a new analysis of this integral showing that the output size is only θ(
n
+
n
logϕ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(
n
). The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing phi , then the number of vertices in an optimal mesh will be O( phi dn), where d is the input dimension. We give a new analysis of this integral showing that the output size is only [thetas](n+nlog phi ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n). The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing , then the number of vertices in an optimal mesh will be O(dn), where d is the input dimension. We give a new analysis of this integral showing that the output size is only [theta](n+nlog). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n). [PUBLICATION ABSTRACT] The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al. showed that if an ordered point set has pacing ϕ, then the number of vertices in an optimal mesh will be O(ϕdn), where d is the input dimension. We give a new analysis of this integral showing that the output size is only θ(n+nlogϕ). The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size O(n). |
| Author | Sheehy, Donald R. |
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| Cites_doi | 10.3390/a2041327 10.1137/S0097539796314124 10.1145/1137856.1137876 10.1145/218013.218078 10.1145/1998196.1998252 10.1145/355483.355487 10.1145/225058.225286 10.1016/S0022-0000(05)80059-5 10.1145/1810959.1811006 10.1137/1.9781611973068.113 10.1016/S0304-3975(02)00437-1 10.1007/s10208-011-9098-0 10.1016/j.comgeo.2008.06.002 10.1007/s00454-004-1089-3 10.1145/1073204.1073238 10.1007/978-3-540-75103-8_19 10.1145/2010324.1964998 10.1006/jagm.1995.1021 10.1145/378583.378636 |
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| Notes | ark:/67375/WNG-DFD2RKFS-V ArticleID:CGF3168 istex:E4E9CE791060A97C2B4FD83570F7BAD16D886055 This work was partially supported by the National Science Foundation under grant number CCF‐1065106, by GIGA grant ANR‐09‐BLAN‐0331‐01, and by the European project CG‐Learning No. 255827. SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
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| References_xml | – reference: Chazal F., Cohen-Steiner D., Mérigot Q.: Geometric inference for probability measures. Foundations of Computational Mathematics 11 (2011), 733-751. 4. – reference: Ruppert J.: A Delaunay refinement algorithmfor quality 2-dimensional mesh generation. J. Algorithms 18, 3 (1995), 548-585. 2. – reference: Erickson J.: Dense point sets have sparse delaunay triangulations or "... but not too nasty". Discrete & Computational Geometry 33 (2005), 83-115. 8. – reference: Mitchell S. A., Vavasis S. A.: Quality mesh generation in higher dimensions. SIAM J. Comput. 29, 4 (2000), 1334-1370 (electronic). 3. – reference: Cheng S.-W., Dey T. K., Edelsbrunner H., Facello M. A., Teng S.-H.: Sliver exudation. JACM: Journal of the ACM 47 (2000). 3. – reference: ÜNgör A.: Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed Delaunay triangulations. Comput. Geom. 42, 2 (2009), 109-118. 2. – reference: Shewchuk J. R.: Unstructured Mesh Generation. CRC Press, 2011, ch. 10, pp. 259-298. 1. – reference: Cheng S.-W., Dey T. K.: Quality meshing with weighted Delaunay refinement. SICOMP: SIAM Journal on Computing 33 (2004). 3. – reference: Mullen P., Memari P., De Goes F., Desbrun M.: HOT: Hodge-optimized triangulations. ACM Trans. Graph 30, 4 (2011). 2. – reference: Dey T. K., Levine J. A.: Delaunay meshing of piecewise smooth complexes without expensive predicates. Algorithms 2, 4 (2009), 1327-1349. 8. – reference: Bern M. W., Eppstein D., Gilbert J. R.: Provably good mesh generation. J. Computer & Systems Sciences 48, 3 (June 1994), 384-409. Special issue for 31st FOCS. 2, 5. – reference: Li X.-Y.: Generating well-shaped d-dimensional Delaunay meshes. Theor. Comput. Sci. 296, 1 (2003), 145-165. 3. – reference: Boissonnat J.-D., Wormser C., Yvinec M.: Anisotropic Delaunay Mesh Generation. Rapport de recherche RR-7712, INRIA, Aug. 2011. URL: http://hal.inria.fr/inria-00615486/en/. 8. – reference: Alliez P., Cohen-Steiner D., Yvinec M., Desbrun M.: Variational tetrahedral meshing. ACM Transactions on Graphics 24, 3 (2005), 617-625. 2. – start-page: 277 year: 2010 end-page: 286 – start-page: 321 year: 2011 end-page: 330 – year: 2009 – start-page: 259 year: 2011 end-page: 298 – start-page: 683 year: 1995 end-page: 692 – start-page: 305 year: 1995 end-page: 312 – start-page: 96 year: 2001 end-page: 105 – volume: 18 start-page: 548 issue: 3 year: 1995 end-page: 585 article-title: A Delaunay refinement algorithmfor quality 2‐dimensional mesh generation publication-title: J. Algorithms – start-page: 2 year: 1997 – start-page: 335 year: 2007 end-page: 346 – volume: 48 start-page: 384 issue: 3 year: June 1994 end-page: 409 article-title: Provably good mesh generation publication-title: J. Computer & Systems Sciences – volume: 47 year: 2000 article-title: Sliver exudation publication-title: JACM: Journal of the ACM – volume: 296 start-page: 145 issue: 1 year: 2003 end-page: 165 article-title: Generating well‐shaped ‐dimensional Delaunay meshes publication-title: Theor. Comput. Sci. – volume: 11 start-page: 733 year: 2011 end-page: 751 article-title: Geometric inference for probability measures publication-title: Foundations of Computational Mathematics – volume: 24 start-page: 617 issue: 3 year: 2005 end-page: 625 article-title: Variational tetrahedral meshing publication-title: ACM Transactions on Graphics – volume: 2 start-page: 1327 issue: 4 year: 2009 end-page: 1349 article-title: Delaunay meshing of piecewise smooth complexes without expensive predicates publication-title: Algorithms – volume: 33 year: 2004 article-title: Quality meshing with weighted Delaunay refinement publication-title: SICOMP: SIAM Journal on Computing – year: 2006 – volume: 42 start-page: 109 issue: 2 year: 2009 end-page: 118 article-title: Off‐centers: A new type of Steiner points for computing size‐optimal quality‐guaranteed Delaunay triangulations publication-title: Comput. Geom. – year: Aug. 2011 – start-page: 175 year: 2008 end-page: 178 – start-page: 4 year: 2011 – volume: 29 start-page: 1334 issue: 4 year: 2000 end-page: 1370 article-title: Quality mesh generation in higher dimensions publication-title: SIAM J. Comput. – volume: 33 start-page: 83 year: 2005 end-page: 115 article-title: Dense point sets have sparse delaunay triangulations or “… but not too nasty” publication-title: Discrete & Computational Geometry – volume: 30 issue: 4 year: 2011 article-title: HOT: Hodge‐optimized triangulations publication-title: ACM Trans. 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| SubjectTerms | Computer graphics F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems-Geometrical problems and computations Initiatives Integrals Lower bounds Matching Mathematical analysis Mathematical functions Mathematical models Optimization Sizing Studies |
| Title | New Bounds on the Size of Optimal Meshes |
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