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60
On Bregman Voronoi Diagrams
 in "Proc. 18th ACMSIAM Sympos. Discrete Algorithms
, 2007
"... The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi ..."
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Cited by 61 (28 self)
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The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a wellshaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a byproduct, Bregman Voronoi diagrams allow one to define informationtheoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other
Anisotropic centroidal Voronoi tessellations and their applications
 SIAM J. SCI. COMPUT
, 2005
"... In this paper, we introduce a novel definition of the anisotropic centroidal Voronoi tessellation (ACVT) corresponding to a given Riemann metric tensor. A directional distance function is used in the definition to simplify the computation. We provide algorithms to approximate the ACVT using the Llo ..."
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Cited by 47 (7 self)
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In this paper, we introduce a novel definition of the anisotropic centroidal Voronoi tessellation (ACVT) corresponding to a given Riemann metric tensor. A directional distance function is used in the definition to simplify the computation. We provide algorithms to approximate the ACVT using the Lloyd iteration and the construction of anisotropic Delaunay triangulation under the given Riemannian metric. The ACVT is applied to optimization of two dimensional anisotropic Delaunay triangulation, to the generation of surface CVT and high quality triangular mesh on general surfaces. Various numerical examples demonstrate the effectiveness of the proposed method.
Lp Centroidal Voronoi Tessellation and its Applications
 ACM TRANSACTIONS ON GRAPHICS 29, 4 (2010)
, 2010
"... ..."
Gaussian limits for random measures in geometric probability
 Ann. Appl. Probab
, 2005
"... We establish Gaussian limits for measures induced by binomial and Poisson point processes in ddimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general central limit theorems are applied to measures induced by random gr ..."
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Cited by 27 (7 self)
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We establish Gaussian limits for measures induced by binomial and Poisson point processes in ddimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general central limit theorems are applied to measures induced by random graphs (nearest neighbor, Voronoi, and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth growth models), and statistics of germgrain models. 1
Anisotropic Delaunay mesh adaptation for unsteady simulations
"... Anisotropic mesh adaptation is a key feature in many numerical simulations to capture the physical behavior of a complex phenomenon at a reasonable computational cost. It is a challenging problem, especially when dealing with time dependent and interface capturing or tracking problems. Here, we des ..."
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Cited by 25 (4 self)
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Anisotropic mesh adaptation is a key feature in many numerical simulations to capture the physical behavior of a complex phenomenon at a reasonable computational cost. It is a challenging problem, especially when dealing with time dependent and interface capturing or tracking problems. Here, we describe an extension of the Delaunay kernel for creating anisotropic mesh elements based on adequate metric tensors. The accuracy and efficiency of the method is assessed on various numerical examples of complex threedimensional simulations.
S.: Local distance functions: A taxonomy, new algorithms, and an evaluation
 In: Proc. ICCV (2009
"... We present a taxonomy for local distance functions where most existing algorithms can be regarded as approximations of the geodesic distance defined by a metric tensor. We categorize existing algorithms by how, where and when they estimate the metric tensor. We also extend the taxonomy along each ax ..."
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Cited by 24 (0 self)
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We present a taxonomy for local distance functions where most existing algorithms can be regarded as approximations of the geodesic distance defined by a metric tensor. We categorize existing algorithms by how, where and when they estimate the metric tensor. We also extend the taxonomy along each axis. How: We introduce hybrid algorithms that use a combination of dimensionality reduction and metric learning to ameliorate overfitting. Where: We present an exact polynomial time algorithm to integrate the metric tensor along the lines between the test and training points under the assumption that the metric tensor is piecewise constant. When: We propose an interpolation algorithm where the metric tensor is sampled at a number of references points during the offline phase, which are then interpolated during online classification. We also present a comprehensive evaluation of all the algorithms on tasks in face recognition, object recognition, and digit recognition. 1.
Lecture Notes on Delaunay Mesh Generation
, 1999
"... purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ..."
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Cited by 23 (0 self)
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purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the
Anisotropic mesh adaptation for evolving triangulated surfaces
 In: Proc. 15th International Meshing Roundtable. (2006
, 1989
"... Summary. Dynamic surfaces arise in many applications, such as free surfaces in multiphase ows and moving interfaces in uidsolid interactions. In many applications, an explicit surface triangulation is used to track the dynamic surfaces, posing signi cant challenges in adapting their meshes, especia ..."
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Cited by 17 (2 self)
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Summary. Dynamic surfaces arise in many applications, such as free surfaces in multiphase ows and moving interfaces in uidsolid interactions. In many applications, an explicit surface triangulation is used to track the dynamic surfaces, posing signi cant challenges in adapting their meshes, especially if large curvatures and sharp features may dynamically appear or vanish as the surfaces evolve. In this paper, we present an anisotropic mesh adaptation technique to meet these challenges. Our technique strives for optimal aspect ratios of the triangulation to reduce interpolation errors and to capture geometric features based on a novel extension of the quadricbased surface analysis. Our adaptation algorithm combines the operations of vertex redistribution, edge ipping, edge contraction, and edge splitting. Experimental results demonstrate the e ectiveness of our anisotropic adaptation techniques for static and dynamic surfaces. Key words: Mesh adaptation; anisotropic meshes; dynamic surfaces; feature preservation 1
Manifoldbased Approach to Semiregular Remeshing
, 2006
"... This paper describes a method for semiregular remeshing of arbitrary shapes. The proposed approach is based on building a parameterization map which is smooth with respect to a differential structure built on the base domain. A global parametric energy functional is introduced and optimized in orde ..."
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Cited by 16 (0 self)
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This paper describes a method for semiregular remeshing of arbitrary shapes. The proposed approach is based on building a parameterization map which is smooth with respect to a differential structure built on the base domain. A global parametric energy functional is introduced and optimized in order to establish a globally smooth parameterization. The proposed approach avoids using metamesh construction during the parameterization and resampling stages which allows for an easier implementation. A simple extension of the method is proposed to improve the approximation properties of the resulting remesh. 2 1
M.: Locally uniform anisotropic meshing
 In: Proceedings of the 24th Annu. ACM Sympos. Comput. Geom
, 2008
"... Anisotropic meshes are triangulations of a given domain in the plane or in higher dimensions, with elements elongated along prescribed directions. Anisotropic triangulations have been shown to be particularly well suited for interpolation of functions or numerical modeling. We propose a new approach ..."
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Cited by 16 (6 self)
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Anisotropic meshes are triangulations of a given domain in the plane or in higher dimensions, with elements elongated along prescribed directions. Anisotropic triangulations have been shown to be particularly well suited for interpolation of functions or numerical modeling. We propose a new approach to anisotropic mesh generation, relying on the notion of locally uniform anisotropic mesh. A locally uniform anisotropic mesh is a mesh such that the star around each vertex v coincides with the star that v would have if the metric on the domain was uniform and equal to the metric at v. This definition allows to define a simple refinement algorithm which relies on elementary predicates, and provides, after completion, an anisotropic mesh in dimensions 2 and 3. A practical implementation has been done in the 2D case. 1