Random Laguerre tessellations

Lautensack, Claudia; Zuyev, Sergei

doi:10.1239/aap/1222868179

Cited by 121 publications

(102 citation statements)

References 64 publications

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“…Concerning materials science, great interest has been devoted to the application of convex tessellation models, in which the grains are convex polyhedra; see, e.g., Lyckegaard et al (2011). The simplicity of these models allows a simple evaluation of size and shape characteristics of the grains (Lautensack and Zuyev, 2008) and relatively fast and accurate fitting to empirical data (Spettl et al, 2016). However, in connection with recent progress in microscopic research, the interest of scientists has significantly increased regarding more general tessellations with curved boundaries, which can better describe real grain shapes.…”

Section: Introductionmentioning

confidence: 99%

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Description of the 3d Morphology of Grain Boundaries in Aluminum Alloys Using Tessellation Models Generated by Ellipsoids

Šedivý

Dake²,

Krill³

et al. 2017

Image Anal Stereol

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Parametric tessellation models are often used to approximate complex grain morphologies of polycrystalline microstructures. A big advantage of such models is the substantial reduction in disk space required to store large, three-dimensional data sets, especially when compared with voxel-based alternatives. By selection of an appropriate tessellation model, a reasonably small loss of information on the real grain shapes can usually be achieved. Special attention has recently been devoted to models based on ellipsoidal approximations fitted to each grain. Faces of these tessellations are portions of quadric surfaces whose parameters can be derived easily. In this paper, we deal with geometric features of the structure, notably curvatures and dihedral angles, which are closely related to the kinetics of grain growth. These characteristics are computed for ellipsoidbased tessellations fitted to two different aluminum alloys with nominal composition Al-3 wt% Mg-0.2 wt% Sc and Al-1 wt% Mg. The results are then compared with estimations based on meshed empirical data. We observe that the model offers more consistent estimations of grain shape characteristics than do the meshed empirical data. Precise description of grain boundaries by the model is also promising with respect to possible applications of these tessellations in stochastic space-time modeling of grain growth.

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Section: Introductionmentioning

confidence: 99%

“…1 is a function of both the points' locations and their additional parameters (marks). A first extension of the Voronoi tessellation is the Laguerre tessellation or power diagram; see Lautensack and Zuyev (2008). Its generating marked point pattern is P = {x i , w i } ⊆ R 3 × R, and the distance measure is given as…”

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confidence: 99%

Description of the 3d Morphology of Grain Boundaries in Aluminum Alloys Using Tessellation Models Generated by Ellipsoids

Šedivý

Dake²,

Krill³

et al. 2017

Image Anal Stereol

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“…An important generalization is the Laguerre tessellation, also called power diagram, which employs weighted generator points and uses the power distance to measure the proximity of points; see, e.g., [1,2,3]. It has been shown that many convex tilings in three or more dimensions are Laguerre tessellations (see [3]), and also in two dimensions Laguerre tessellations are common.…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown that many convex tilings in three or more dimensions are Laguerre tessellations (see [3]), and also in two dimensions Laguerre tessellations are common. In addition, Voronoi tessellations in a number of nonEuclidean geometries can be represented as Laguerre tessellations (see [4]).…”

Section: Introductionmentioning

confidence: 99%

Inverting Laguerre Tessellations

Duan

Kroese

Brereton³

et al. 2014

The Computer Journal

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A Laguerre tessellation is a generalization of a Voronoi tessellation where the proximity between points is measured via a power distance rather than the Euclidean distance.Laguerre tessellations have found significant applications in materials science, providing improved modeling of (poly)crystalline microstructures and grain growth. There exist efficient algorithms to construct Laguerre tessellations from given sets of weighted generator points, similar to methods used for Voronoi tessellations. The purpose of this paper is to provide theory and methodology for the inverse construction; that is, to recover the weighted generator points from a given Laguerre tessellation. We show that, unlike the Voronoi case, the inverse problem is in general non-unique: different weighted generator points can create the same tessellation. To recover pertinent generator points we formulate the inversion problem as a multimodal optimization problem and apply the cross-entropy method to solve it.

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“…Cowan [43], [44], using ergodic theory, analyzed mosaic processes and Poisson point processes. Lautensack and Zuyev [45] evaluated Laguerre tessellations, which are weighted generalizations of Voronoi tessellations. To the best of our knowledge, the properties of random Voronoi diagrams for discrete graphs have not been investigated previously.…”

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confidence: 99%

Balancing Graph Voronoi Diagrams

Honiden

Houle

Sommer

2009

2009 Sixth International Symposium on Voronoi Diagrams

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Abstract-Many facility location problems are concerned with minimizing operation and transportation costs by partitioning territory into regions of similar size, each of which is served by a facility. For many optimization problems, the overall cost can be reduced by means of a partitioning into balanced subsets, especially in those cases where the cost associated with a subset is superlinear in its size. In this paper, we consider the problem of generating a Voronoi partition of a discrete graph so as to achieve balance conditions on the region sizes. Through experimentation, we first establish that the region sizes of randomly-generated graph Voronoi diagrams vary greatly in practice. We then show how to achieve a balanced partition of a graph via Voronoi site resampling. For bounded-degree graphs, where each of the n nodes has degree at most d, and for an initial randomly-chosen set of s Voronoi nodes, we prove that, by extending the set of Voronoi nodes using an algorithm by Thorup and Zwick, each Voronoi region has size at most 4dn/s + 1 nodes, and that the expected size of the extended set of Voronoi nodes is at most 2s log n.

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Random Laguerre tessellations

Cited by 121 publications

References 64 publications

Description of the 3d Morphology of Grain Boundaries in Aluminum Alloys Using Tessellation Models Generated by Ellipsoids

Description of the 3d Morphology of Grain Boundaries in Aluminum Alloys Using Tessellation Models Generated by Ellipsoids

Inverting Laguerre Tessellations

Balancing Graph Voronoi Diagrams

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