Surface meshing with curvature convergence

Handle URI:
http://hdl.handle.net/10754/563570
Title:
Surface meshing with curvature convergence
Authors:
Li, Huibin; Zeng, Wei; Morvan, Jean-Marie; Chen, Liming; Gu, Xianfengdavid
Abstract:
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. © 2014 IEEE.
KAUST Department:
Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Publisher:
Institute of Electrical and Electronics Engineers (IEEE)
Journal:
IEEE Transactions on Visualization and Computer Graphics
Issue Date:
Jun-2014
DOI:
10.1109/TVCG.2013.253
Type:
Article
ISSN:
10772626
Sponsors:
This work was supported under the Grants ANR 2010 INTB 0301 01, NSF DMS-1221339, NSF Nets-1016829, NSF CCF-1081424 and NSF CCF-0830550.
Appears in Collections:
Articles; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorLi, Huibinen
dc.contributor.authorZeng, Weien
dc.contributor.authorMorvan, Jean-Marieen
dc.contributor.authorChen, Limingen
dc.contributor.authorGu, Xianfengdaviden
dc.date.accessioned2015-08-03T11:54:42Zen
dc.date.available2015-08-03T11:54:42Zen
dc.date.issued2014-06en
dc.identifier.issn10772626en
dc.identifier.doi10.1109/TVCG.2013.253en
dc.identifier.urihttp://hdl.handle.net/10754/563570en
dc.description.abstractSurface 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. © 2014 IEEE.en
dc.description.sponsorshipThis work was supported under the Grants ANR 2010 INTB 0301 01, NSF DMS-1221339, NSF Nets-1016829, NSF CCF-1081424 and NSF CCF-0830550.en
dc.publisherInstitute of Electrical and Electronics Engineers (IEEE)en
dc.subjectconformal parameterizationen
dc.subjectconvergenceen
dc.subjectcurvature measuresen
dc.subjectDelaunay refinementen
dc.subjectMeshingen
dc.subjectnormal cycleen
dc.titleSurface meshing with curvature convergenceen
dc.typeArticleen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalIEEE Transactions on Visualization and Computer Graphicsen
dc.contributor.institutionDepartment of Mathematics and Informatics, Ecole Centrale Lyon, 36 Av. Guy de Collongue, Bat E6, 2ème Etage, Ecully, Lyon, 69134, Franceen
dc.contributor.institutionSchool of Computing and Information Sciences, Florida International University, 11200 SW 8th Street, Miami, FL 33199, United Statesen
dc.contributor.institutionDepartement de Mathématiques, Université Claude Bernard Lyon 1, Batiment Jean Braconnier, 43 blvd du 11 Novembre 1918, Villeurbanne Cedex, Lyon, 69622, Franceen
dc.contributor.institutionDepartment of Computer Science, State University of New York at Stony Brook, 2425 CSE Building, Stony Brook, NY 11794-4400, United Statesen
kaust.authorMorvan, Jean-Marieen
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