Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation

Labelle, François; Shewchuk, Jonathan Richard

doi:10.1145/777819.777822

Cited by 53 publications

(93 citation statements)

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“…In this section we introduce the class of Voronoi diagrams of point and polygonal sites equipped with anisotropic norms in the plane. This generalizes [LS03] which introduced anisotropic norms for point sites only. We show how to compute these Voronoi diagrams as an application of the algorithm that we developed in the previous section.…”

Section: Anisotropic Voronoi Diagramssupporting

confidence: 58%

See 1 more Smart Citation

Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

Bennett

Papadopoulou

Yap

2016

Computer Graphics Forum

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Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results.

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Section: Anisotropic Voronoi Diagramssupporting

confidence: 58%

“…The well‐known multiplicatively weighted Voronoi diagrams of a point set is a special case. Labelle and Shewchuk [LS03] introduced anisotropic norms for point sites, but their main focus was the associated Delaunay triangulation. Our extension to polygonal sites will greatly increase applicability in areas such as robot motion planning.…”

Section: Introductionmentioning

confidence: 99%

Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

Bennett

Papadopoulou

Yap

2016

Computer Graphics Forum

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“…Many problems, such as those arising in fluid flow, have an anisotropic character with solution values changing more rapidly in one direction than in another. Efficient finite element treatment of such problems requires an anisotropic mesh in which elements have large aspect ratio and different orientation in different regions .…”

Section: Mesh Smoothing Methodsmentioning

confidence: 99%

Mesh improvement by minimizing a weighted sum of squared element volumes

Renka

2014

Int. J. Numer. Meth. Engng

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SummaryWe describe a new mesh smoothing method that consists of minimizing the sum of squared element volumes over the free vertex positions. To the extent permitted by the fixed vertices and mesh topology, the resulting mesh elements have uniformly distributed volumes. In the case of a triangulation, uniform volume implies well‐shaped triangles. If a graded mesh is required, the element volumes may be weighted by centroidal values of a sizing function, resulting in a mesh that conforms to the required vertex density. The method has both a local and a global implementation. In addition to smoothing, the method serves as a simple parameter‐free means of untangling a mesh with inverted elements. It applies to all types of meshes, but we present test results here only for planar triangle meshes. Our test results show the new method to be fast, superior in uniformity or conformity to a sizing function, and among the best methods in terms of triangle shape quality. We also present a new angle‐based method that is simpler and more effective than alternatives. This method is directly aimed at producing well‐shaped triangles and is particularly effective when combined with the volume‐based method. It also generalizes to anisotropic mesh smoothing. Copyright © 2014 John Wiley & Sons, Ltd.

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“…To overcome this drawback, we propose to estimate the volume element of point p with the anisotropic Voronoi diagram [LS03] defined on the local tangent space

T_{p}

\begin{matrix} true \\ right {Vor}_{ani} (p) = ({} x_{p} \in T_{p} false| d_{T_{p}} false(p, x false) \leq d_{T_{q^{i}}} false(q^{i}, x false)), \end{matrix}

where

d_{T_{q^{i}}} false(q^{i}, x false)

is the distance between the projections of two points

q^{i}

and x in the tangent plane

T_{q^{i}}

, and

x_{p}

is the projection of the point x in the tangent plane

T_{p}

. Compared with Equation , we can see that all sampled points x are first projected to two tangent planes

T_{p}

and

T_{q^{i}}

and then

x_{p}

is adaptively determined by first finding a proper x through comparing

d_{T_{p}} false(p, x false)

with

d_{T_{q^{i}}} false(q^{i}, x false)

.…”

Section: δM Approximation With Anisotropic Voronoi Diagrammentioning

confidence: 99%

“…Although the anisotropic distance measure implied by the anisotropic Voronoi diagram is more accurate, it is difficult to estimate the area of such a Voronoi cell in the local tangent space. The boundary of the Voronoi cell

V o r_{a n i} false(p false)

is composed of patches of a quadratic curve as shown in [LS03]. The complexity of computing an anisotropic Voronoi diagram is in

O (n^{2})

.…”

Section: δM Approximation With Anisotropic Voronoi Diagrammentioning

confidence: 99%

Laplace–Beltrami Operator on Point Clouds Based on Anisotropic Voronoi Diagram

Qin

Chen

Wang

et al. 2017

Computer Graphics Forum

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The symmetrizable and converged Laplace–Beltrami operator (ΔM) is an indispensable tool for spectral geometrical analysis of point clouds. The ΔM, introduced by Liu et al. [LPG12] is guaranteed to be symmetrizable, but its convergence degrades when it is applied to models with sharp features. In this paper, we propose a novel ΔM, which is not only symmetrizable but also can handle the point‐sampled surface containing significant sharp features. By constructing the anisotropic Voronoi diagram in the local tangential space, the ΔM can be well constructed for any given point. To compute the area of anisotropic Voronoi cell, we introduce an efficient approximation by projecting the cell to the local tangent plane and have proved its convergence. We present numerical experiments that clearly demonstrate the robustness and efficiency of the proposed ΔM for point clouds that may contain noise, outliers, and non‐uniformities in thickness and spacing. Moreover, we can show that its spectrum is more accurate than the ones from existing ΔM for scan points or surfaces with sharp features.

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Anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation

Cited by 53 publications

References 0 publications

Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

Mesh improvement by minimizing a weighted sum of squared element volumes

Laplace–Beltrami Operator on Point Clouds Based on Anisotropic Voronoi Diagram

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