Orphan-Free Anisotropic Voronoi Diagrams

Canas, Guillermo D.; Gortler, Steven J.

doi:10.1007/s00454-011-9372-6

Cited by 10 publications

(23 citation statements)

References 8 publications

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“…where an asymmetric -net is simply a weaker form ofnet defined on non-symmetric functions D, which can be computed with the iterative algorithm of [10], and the metric variation σ is a Lipschitz-type condition on Q [4]. The above condition is known to be conservative, and there may be simpler conditions to achieve orphan-freedom.…”

Section: Final Resultsmentioning

confidence: 99%

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Duals of orphan-free anisotropic voronoi diagrams are embedded meshes

Canas

Gortler

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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Given an anisotropic Voronoi diagram, we address the fundamental question of when its dual is embedded. We show that, by requiring only that the primal be orphan-free (have connected Voronoi regions), its dual is always guaranteed to be an embedded triangulation. Further, the primal diagram and its dual have properties that parallel those of ordinary Voronoi diagrams: the primal's vertices, edges, and faces are connected, and the dual triangulation has a simple, closed boundary. Additionally, if the underlying metric has bounded anisotropy (ratio of eigenvalues), the dual is guaranteed to triangulate the convex hull of the sites. These results apply to the duals of anisotropic Voronoi diagrams of any set of sites, so long as their Voronoi diagram is orphan-free. By combining this general result with existing conditions for obtaining orphan-free anisotropic Voronoi diagrams, a simple and natural condition for a set of sites to form an embedded anisotropic Delaunay triangulation follows.

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Section: Final Resultsmentioning

confidence: 99%

“…In practice, we may combine our results with those of [4] to conclude that duals of anisotropic Voronoi diagrams of appropriate -nets are always embedded triangulations (Cor. 6.4).…”

Section: Introductionmentioning

confidence: 82%

Duals of orphan-free anisotropic voronoi diagrams are embedded meshes

Canas

Gortler

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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“…Note that more basic shape requirements include disctopology (the later being already not direct for anisotropic partitions [14,2,5]) and convexity.…”

Section: Previous Workmentioning

confidence: 99%

“…Conformity is however not optimized for when using the Lloyd iteration as the optimized energy does not relate to conformity. In addition, anisotropy is in general limited to low aspect ratios or to metrics with smooth grading to avoid partitioning defects [5]. Another drawback of such relaxation methods is that control over sizing is only relative such that the optimized cells are sized not strictly in accordance to the input metric but rather proportionally.…”

Section: Previous Workmentioning

confidence: 99%

Anisotropic Rectangular Metric for Polygonal Surface Remeshing

Pellenard

Morvan

Alliez

2013

Proceedings of the 21st International Meshing Roundtable

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Summary. We propose a new method for anisotropic polygonal surface remeshing. Our algorithm takes as input a surface triangle mesh. An anisotropic rectangular metric, defined at each triangle facet of the input mesh, is derived from both a user-specified normal-based tolerance error and the requirement to favor rectangleshaped polygons. Our algorithm uses a greedy optimization procedure that adds, deletes and relocates generators so as to match two criteria related to partitioning and conformity.

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“…The algorithm is, however, very complicated and no implementation has been reported. Although some progress has been recently reported in Canas and Gortler [2011], it remains unclear under which conditions an anisotropic Voronoi diagram admits a dual embedded triangulation for 3 or higher-dimensional domains.…”

Section: Introductionmentioning

confidence: 99%

Anisotropic Delaunay Meshes of Surfaces

Boissonnat

Shi

Tournois³

et al. 2015

ACM Trans. Graph.

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Anisotropic simplicial meshes are triangulations with elements elongated along prescribed directions. Anisotropic meshes have been shown well suited for interpolation of functions or solving PDEs. They can also significantly enhance the accuracy of a surface representation. Given a surface S endowed with a metric tensor field, we propose a new approach to generate an anisotropic mesh that approximates S with elements shaped according to the metric field. The algorithm relies on the well-established concepts of restricted Delaunay triangulation and Delaunay refinement and comes with theoretical guarantees. The star of each vertex in the output mesh is Delaunay for the metric attached to this vertex. Each facet has a good aspect ratio with respect to the metric specified at any of its vertices. The algorithm is easy to implement. It can mesh various types of surfaces like implicit surfaces, polyhedra, or isosurfaces in 3D images. It can handle complicated geometries and topologies, and very anisotropic metric fields.

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Orphan-Free Anisotropic Voronoi Diagrams

Cited by 10 publications

References 8 publications

Duals of orphan-free anisotropic voronoi diagrams are embedded meshes

Duals of orphan-free anisotropic voronoi diagrams are embedded meshes

Anisotropic Rectangular Metric for Polygonal Surface Remeshing

Anisotropic Delaunay Meshes of Surfaces

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