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Lp Centroidal Voronoi Tessellation and its Applications
 ACM TRANSACTIONS ON GRAPHICS 29, 4 (2010)
, 2010
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HOT: HodgeOptimized Triangulations
 ACM Trans. Graph
, 2011
"... We introduce Hodgeoptimized triangulations (HOT), a family of wellshaped primaldual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each ..."
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Cited by 17 (4 self)
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We introduce Hodgeoptimized triangulations (HOT), a family of wellshaped primaldual pairs of complexes designed for fast and accurate computations in computer graphics. Previous work most commonly employs barycentric or circumcentric duals; while barycentric duals guarantee that the dual of each simplex lies within the simplex, circumcentric duals are often preferred due to the induced orthogonality between primal and dual complexes. We instead promote the use of weighted duals (“power diagrams”). They allow greater flexibility in the location of dual vertices while keeping primaldual orthogonality, thus providing a valuable extension to the usual choices of dual by only adding one additional scalar per primal vertex. Furthermore, we introduce a family of functionals on pairs of complexes that we derive from bounds on the errors induced by diagonal Hodge stars, commonly used in discrete computations. The minimizers of these functionals, called HOT meshes, are shown to be generalizations of Centroidal Voronoi Tesselations and Optimal Delaunay Triangulations, and to provide increased accuracy and flexibility for a variety of computational purposes.
Advances in Studies and Applications of Centroidal Voronoi Tessellations
"... Centroidal Voronoi tessellations (CVTs) have become a useful tool in many applications ranging from geometric modeling, image and data analysis, and numerical partial differential equations, to problems in physics, astrophysics, chemistry, and biology. In this paper, we briefly review the CVT concep ..."
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Cited by 15 (4 self)
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Centroidal Voronoi tessellations (CVTs) have become a useful tool in many applications ranging from geometric modeling, image and data analysis, and numerical partial differential equations, to problems in physics, astrophysics, chemistry, and biology. In this paper, we briefly review the CVT concept and a few of its generalizations and wellknown properties. We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs. Whenever possible, we point out some outstanding issues that still need investigating.
GPUAssisted Computation of Centroidal Voronoi Tessellation
, 2011
"... Centroidal Voronoi tessellations (CVT) are widely used in computational science and engineering. The most commonly used method is Lloyd’s method, and recently the LBFGS method is shown to be faster than Lloyd’s method for computing the CVT. However, these methods run on the CPU and are still too s ..."
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Cited by 12 (6 self)
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Centroidal Voronoi tessellations (CVT) are widely used in computational science and engineering. The most commonly used method is Lloyd’s method, and recently the LBFGS method is shown to be faster than Lloyd’s method for computing the CVT. However, these methods run on the CPU and are still too slow for many practical applications. We present techniques to implement these methods on the GPU for computing the CVT on 2D planes and on surfaces, and demonstrate significant speedup of these GPUbased methods over their CPU counterparts. For CVT computation on a surface, we use a geometry image stored in the GPU to represent the surface for computing the Voronoi diagram on the surface. In our implementation a new technique is proposed for parallel regional reduction on the GPU for evaluating integrals over Voronoi cells.
Packing circles and spheres on surfaces
 TO APPEAR IN THE ACM SIGGRAPH CONFERENCE PROCEEDINGS
, 2009
"... Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate ci ..."
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Cited by 12 (4 self)
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Inspired by freeform designs in architecture which involve circles and spheres, we introduce a new kind of triangle mesh whose faces' incircles form a packing. As it turns out, such meshes have a rich geometry and allow us to cover surfaces with circle patterns, sphere packings, approximate circle packings, hexagonal meshes which carry a torsionfree support structure, hybrid trihex meshes, and others. We show how triangle meshes can be optimized so as to have the incircle packing property. We explain their relation to conformal geometry and implications on solvability of optimization. The examples we give confirm that this kind of meshes is a rich source of geometric structures relevant to architectural geometry.
Computing SelfSupporting Surfaces by Regular Triangulation
"... and their corresponding power cells are colored in orange. Top right: initial selfsupporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distribu ..."
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Cited by 10 (1 self)
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and their corresponding power cells are colored in orange. Top right: initial selfsupporting mesh. Spikes appear due to extremely small reciprocal areas. Bottom right: applying our smoothing scheme (5 iterations) improves mesh quality. The power diagrams (black) show how power cell area is distributed more evenly. Masonry structures must be compressively selfsupporting; designing such surfaces forms an important topic in architecture as well as a challenging problem in geometric modeling. Under certain conditions, a surjective mapping exists between a power diagram, defined by a set of 2D vertices and associated weights, and the reciprocal diagram that characterizes the force diagram of a discrete selfsupporting network. This observation lets us define a new and convenient parameterization for the space of selfsupporting networks. Based on it and the discrete geometry of this design space, we present novel geometry processing methods including surface smoothing and remeshing which significantly reduce the magnitude of force densities and homogenize their distribution.
Efficient computation of 3d clipped voronoi diagram
 In GMP conf. proc
, 2010
"... Abstract. The Voronoi diagram is a fundamental geometry structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact 3D domain (i.e. a finite 3D volume), some Voronoi cells of their Voronoi diagram are infinite, but in practice onl ..."
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Cited by 7 (3 self)
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Abstract. The Voronoi diagram is a fundamental geometry structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact 3D domain (i.e. a finite 3D volume), some Voronoi cells of their Voronoi diagram are infinite, but in practice only the parts of the cells inside the domain are needed, as when computing the centroidal Voronoi tessellation. Such a Voronoi diagram confined to a compact domain is called a clipped Voronoi diagram. We present an efficient algorithm for computing the clipped Voronoi diagram for a set of sites with respect to a compact 3D volume, assuming that the volume is represented as a tetrahedral mesh. We also describe an application of the proposed method to implementing a fast method for optimal tetrahedral mesh generation based on the centroidal Voronoi tessellation.
Fitting polynomial surfaces to triangular meshes with Voronoi squared distance minimization
 In 20th International Meshing Roundtable–IMR2011. 601–618
, 2012
"... This paper introduces Voronoi Squared Distance Minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoibased approximation of the overall squared distance function between the surface and the input mesh (SDM). This ob ..."
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Cited by 6 (1 self)
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This paper introduces Voronoi Squared Distance Minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoibased approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of Centroidal Voronoi Tesselation (CVT), and can be minimized by a quasiNewton solver. VSDM naturally adapts the orientation of the mesh to best approximate the input, without estimating any differential quantities. Therefore it can be applied to triangle soups or surfaces with degenerate triangles, topological noise and sharp features. Applications of fitting quad meshes and polynomial surfaces to input triangular meshes are demonstrated. 1
Robust modeling of constant mean curvature surfaces
 ACM Trans. Graph. (SIGGRAPH
, 2012
"... Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial scree ..."
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Cited by 4 (0 self)
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Copyright Notice Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profi t or direct commercial advantage and that copies show this notice on the fi rst page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, to redistribute to lists, or to use any component of this work in other works requires prior specifi c permission and/or a fee. Permissions may be
Computing 2D periodic centroidal Voronoi tessellation
 In Proceedings of the 2011 International Symposium on Voronoi Diagrams in Science and Engineering
, 2011
"... Abstract—In this paper, we propose an efficient algorithm to compute the centroidal Voronoi tessellation in 2D periodic space. We first present a simple algorithm for constructing the periodic Voronoi diagram (PVD) from a Euclidean Voronoi diagram. The presented PVD algorithm considers only a small ..."
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Abstract—In this paper, we propose an efficient algorithm to compute the centroidal Voronoi tessellation in 2D periodic space. We first present a simple algorithm for constructing the periodic Voronoi diagram (PVD) from a Euclidean Voronoi diagram. The presented PVD algorithm considers only a small set of periodic copies of the input sites, which is more efficient than previous approaches requiring full copies of the sites (9 in 2D and 27 in 3D). The presented PVD algorithm is applied in a fast Newtonbased framework for computing the centroidal Voronoi tessellation (CVT). We observe that fullhexagonal patterns can be obtained via periodic CVT optimization attributed to the convergence of the Newtonbased CVT computation. KeywordsPeriodic Voronoi diagram, Delaunay triangulation, centroidal Voronoi tessellation, hexagonal pattern