Filtering Relocations on a Delaunay Triangulation

Castro, Pedro Machado Manhães de; Tournois, Jane; Alliez, Pierre; Devillers, Olivier

doi:10.1111/j.1467-8659.2009.01523.x

Cited by 11 publications

(3 citation statements)

References 38 publications

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“…We use the Regular_triangulation_2 package from CGAL [cga]. It is tempting to try to avoid recomputing the whole power diagram for every evaluation of the function Φ by using the same approach that was used in [MMd‐CTAD09] to maintain the Delaunay triangulation. However, as shown in Figure 3(a), the topology of the power diagram keeps changing until the very last steps of the optimization, thus discarding this approach.…”

Section: Methodsmentioning

confidence: 99%

A Multiscale Approach to Optimal Transport

Mérigot

2011

Computer Graphics Forum

240

283

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In this paper, we propose an improvement of an algorithm of Aurenhammer, Hoffmann and Aronov to find a least square matching between a probability density and finite set of sites with mass constraints, in the Euclidean plane. Our algorithm exploits the multiscale nature of this optimal transport problem. We iteratively simplify the target using Lloyd's algorithm, and use the solution of the simplified problem as a rough initial solution to the more complex one. This approach allows for fast estimation of distances between measures related to optimal transport (known as Earth-mover or Wasserstein distances). We also discuss the implementation of these algorithms, and compare the original one to its multiscale counterpart.

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Section: Methodsmentioning

confidence: 99%

A Multiscale Approach to Optimal Transport

Mérigot

2011

Computer Graphics Forum

240

283

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“…To this end, we simply check each edge of the triangulation and flip it if the Delaunay property is not satisfied. A more efficient implementation would be possible by filtering Delaunay relocations, as described in [24]. Note that even before this step the mesh is already almost a Delaunay triangulation, since the geometric optimization leads to very regular triangulations with well-shaped faces.…”

Section: Mesh Optimizationmentioning

confidence: 99%

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Sieger

Alliez

Botsch

2010

Proceedings of the 19th International Meshing Roundtable

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Summary. We present a 2D mesh improvement technique that optimizes Voronoi diagrams for their use in polygonal finite element computations. Starting from a centroidal Voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that effectively removes the major reason for numerical instabilities-short edges in the Voronoi diagram. We evaluate our method on a 2D Poisson problem and demonstrate that our simple but effective optimization achieves a significant improvement of the stiffness matrix condition number.

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“…De Castro et al [5] describe how to easily determine a tolerance region for each point, such that as long as the point remains within its tolerance region we do not have to check its certificates. They then give experimental results showing that for fairly stable Delaunay triangulations this filtering method is faster than traditional KDSs.…”

Section: Introductionmentioning

confidence: 99%

Kinetic convex hulls and delaunay triangulations in the black-box model

Berg

Roeloffzen

Speckmann

2011

Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry

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Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step.We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the blackbox model, under the following assumption on dmax: there is some constant k such that for any point p ∈ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ∆ k , the so-called k-spread of P . We show how to update the convex hull at each time step in O(k∆ k log 2 n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k 2 ∆ 2 k ) at each time step.

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Filtering Relocations on a Delaunay Triangulation

Cited by 11 publications

References 38 publications

A Multiscale Approach to Optimal Transport

A Multiscale Approach to Optimal Transport

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

Kinetic convex hulls and delaunay triangulations in the black-box model

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