Fast computation of generalized voronoi diagrams using graphics hardware kenneth e. Pdf centroidal power diagrams, lloyds algorithm, and. Spherical centroidal voronoi tessellations scvts offer one potential path toward the development of a robust, multiresolution climate system model components. The clustering provides an efficient way to construct uniform tessellations, and therefore leads to uniform coarsening of polygonal meshes, when the output triangulation has many fewer elements than the input mesh. For each seed there is a corresponding region consisting of all points of the plane closer to that seed than to any other. Fast computation of generalized voronoi diagrams using. The voronoi diagram and centroidal voronoi tessellation cvt are defined and their properties explored. It is based on the iterative approach of lloyds method while having good considerations to address the two major challenges of achieving fast convergence with few iterations, and at the same time achieving fast computation within each iteration. Centroidal voronoi tessellation cvt a centroidal voronoi tessellation cvt of a set of sites x fx i gn 1 is a special voronoi diagram for which each site x icoincides with the centroid i. Centroidal voronoi tessellation based algorithms for. Pdf lp centroidal voronoi tessellation and its applications. Machine learning is a subfield of soft computing within computer science that evolved from the study of pattern recognition and computational learning theory in artificial intelligence. Centroidal jensenbregman voronoi diagrams for the burg and shannon js entropies. A sample of the problems addressed by this technique include closest pair, all nearest.
The voronoi diagram vd aurenhammer 1991 is a fundamental geometric data structure. Mesh clustering by approximating centroidal voronoi. Approximated centroidal voronoi diagrams for uniform. Pdf 2d centroidal voronoi tessellations with constraints. The dots are the voronoi generators and the circles are the centroids of the corresponding voronoi regions. With a given voronoi diagram, one can also find the largest empty circle amongst a set of points, and in an enclosing polygon. Up until this point, we have been exploring oronoiv diagrams and their properties. How can i have the centroidal voronoi tessellation according to lloyds algorithm. The voronoi diagram of p is the subdivision of the plane into n cells, one for each site. Xiaoqiang wang abstract a new method for the simpli. It can be viewed as an optimal partition corresponding to an optimal distribution of generators. A voronoi diagram divides the space into voronoi cells, regp for some p if regp is a strange shape, hard to figure out if the query is inside regp fortunately, as the picture suggests, voronoi cells have simple. Algorithms and criteria for volumetric centroidal voronoi.
Clipped 3d voronoi cells are accurately computed, at a subfacet precision f. Hoff iii, tim culver, john keyser, ming lin, dinesh manocha university of north carolina at chapel hill, dept. The latter formulation is more suitable for computation. The dual of a voronoi diagram vd is a delaunaytriangulationdt,whichhas the propert y that the outcircle of every triangle does not contain any ot her site when considering the 2d plane. Computing centroidal voronoi tessellation using the gpu. We apply a remeshing technique based on approximated centroidal voronoi diagrams 15 16 to obtain a much more uniform mesh with a suitable resolution figure 4c. A centroidal voronoi tessellation cvt of a shape can be viewed. To distin guish it with the classic cvts, we call it an edge.
Fast methods for computing centroidal voronoi tessellations. We present a novel clustering algorithm for polygonal meshes which approximates a centroidal voronoi diagram construction. A centroidal voronoi tessellation is a voronoi tessellation whose generating points are the centroids centers of mass of the corresponding voronoi regions. Segment voronoi cells colors are approximated by the union of sampled points voronoi cells thin lines, right half of d. Ncar summer colloquium on dynamical cores, june 2, 2008.
The dots are simultaneously the generators for the voronoi tessellation and the centroids of the voronoi regions. Every point on the plane that is not a vertex or part of an edge is a point in a distinct voronoi region. Pdf we tackle the problem of constructing 2d centroidal voronoi tessellations. Someone who is located at a position q within the city. Voronoi diagrams voronoi diagrams of line segments higherorder voronoi diagrams motivation properties construction beach line thebeach lineseparates the known and unknown part of the voronoi diagram, it is. Resulting centroidal voronoi diagram, a closer look at the gulf stream region.
The centroidal voronoi tessellation cvt is a special case of the. Centroidal voronoi diagrams for isotropic surface remeshing. This decomposition has the property that an arbitrary point p within the region ri is closer to point i than any other point. Xlgorithm expanding load 1 level load 2 levels load 3 levels. Acentroidal voronoi tessellation cvt is a voronoi tessellation where each generator zi coincides with the centroid of each voronoi region vi. The complexity of our algorithm in terms of calculations and memory requirements is low, allowing the processing of large meshes, as shown in the results section, where processing meshes with up to 1 million triangles within a minute. The region of influence is called a voronoi region and the collection of all the voronoi regions is the voronoi diagram. Gpuassisted computation of centroidal voronoi tessellation.
Spatial tessellations wiley series in probability and. Probabilistic methods for centroidal voronoi tessellations. In the simplest case, these objects are just finitely many points in the plane called seeds, sites, or generators. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the. Pdf in this paper, we propose an efficient algorithm to compute the centroidal voronoi tessellation in 2d periodic space. Starting from a centroidal voronoi tessellation of the simulation domain we optimize the mesh by minimizing a carefully designed energy functional that e ectively removes the major reason for numerical instabilitiesshort edges in the voronoi diagram. We show that critical points of the energy are centroidal power diagrams, which are generalizations of centroidal voronoi tessellations, and that they can be approximated by a generalization of lloyds algorithm lloyds algorithm is a common method for finding centroidal voronoi tessellations. Scvts allow for the generation of highquality voronoi diagrams and delaunay triangulations responsible editor. Figure 1 relaxes under lloyds algorithm to become figure 2. We present a new approach for computing generalized voronoi diagrams in two and three dimensions using. Surface sampling, error diffusion, centroidal voronoi diagram, constrained delaunay triangulation, parameterization, optimal cutting, polygonal. Definition of voronoi diagram letp be a set of n distinct points sites in the plane. Centroidal voronoi tesselation of line segments and graphs. It turns on that the optimal placement of the schools is at thecentroidsof a cvt of the city with respect to a given density function of the student population.
On the right, a 10point centroidal voronoi tessellation. I believe this is related to lloyds method for constructing blue noise points. Centroidal voronoi tessellation based algorithms for vector. Most existing applications of centroidal voronoi tessellations cvts lack con sideration of the length of the cluster boundaries. Optimizing voronoi diagrams for polygonal finite element. In this paper, we propose a generalization of centroidal. We show that critical points of the energy are centroidal power diagrams, which are generalizations of centroidal voronoi tessellations, and that they can be approximated by a. In geometry, a centroidal voronoi tessellation cvt is a special type of voronoi tessellation or voronoi diagram. This updating scheme improves the quality of the output coarse mesh.
Centroidal voronoi tessellation cvt is a particular type of voronoi tessellation that has many applications in computational sciences and engineering. The centroidal voronoi tessellation cvt is a special case of the voronoi diagram, where every site coincides with the centroid of its voronoi cell. Introduction to voronoi diagrams and delaunay triangulations. The voronoi cells adjust themselves to the motion of the voronoi seeds.
Centroidal voronoi tesselations 16 places the seeds for voronoi in the center of the mass of the voronoi polygons. Our main contribution is an extension of centroidal voronoi tesselation to line segments, using approximated voronoi diagrams of segments e. Centroidal voronoi diagrams a centroidal voronoi diagram is a voronoi diagram where each voronoi site zi is also the mass centroid of its voronoi region. Voronoi diagrams of music alex mclean february 2006 1 introduction the voronoi diagram is a mathematical tool applied across many scienti c disciplines, but somehow rarely in the eld of music. A centroidal voronoi diagram cvd is a voronoi diagram where each voronoi site z. Pdf computing 2d periodic centroidal voronoi tessellation.
The voronoi diagram of a discrete set of points x decomposes the space around each point xi into a region of influence ri. Grid generation and optimization based on centroidal voronoi tessellations q qiang du, max gunzburger departmentofmathematics,iowastateuniversity,400carverhall,ames,ia50012064,usa abstract centroidal voronoi tessellations cvts are voronoi tessellations of a region such. Voronoi tessellations, which are uniform and regular volumetric tessellations. Centroidal voronoi diagrams and gpu computation the voronoi diagram is a fundamental geometric data structure. Generating centroidal voronoi diagrams lloyds method is an iterative algorithm to generate a centroidal voronoi diagram from any set of generating points. Voronoi diagrams, the pseudodual of the constrained delaunay triangulation. A point q lies in the cell corresponding to a site pi. This book will appeal equally to those whose interests in voronoi diagrams are theoretical, practical or both. Voronoi diagrams for their use in polygonal nite element computations. Hyperbolic centroidal voronoi tessellation the university of texas. Probabilistic methods for centroidal voronoi tessellations and their parallel implementations. The diagram is described in this essay along with some of its mathematical properties which have found many uses since the 17th century. It can be used to represent free volume of the polymer. Pdf this paper introduces lpcentroidal voronoi tessellation lpcvt, a generalization of cvt that minimizes a higherorder moment of the coordinates.
A centroidal voronoi tessellation cvt is a voronoi tessellation where each generator zi coincides with the centroid of each voronoi region vi. In geometry, a centroidal voronoi tessellation cvt is a special type of voronoi tessellation or. Hyperbolic centroidal voronoi tessellation guodong rong university of texas at dallas. The dual of a voronoi diagram is a delaunay triangulation, which has the property that the outcircle of every triangle does not contain any other site. The text will appeal to students and researchers studying spatial data in a number of areas, in particular, applied probability, computational geometry, and geographic information science gis. We prove that the algorithm is energy decreasing and prove a convergence theorem.
A voronoi tessellation is called centroidal when the generating point of each voronoi cell is also its centroid, i. We propose a novel algorithm to compute centroidal voronoi tessellation using the gpu. Gpuassisted computation of centroidal voronoi tessellation guodong rong, yang liu, wenping wang, xiaotian yin, xianfeng david gu, and xiaohu guo abstract centroidal voronoi tessellations cvt are widely used in computational science and engineering. Triangulated irregular networks 17 are often used for map representation. In this paper we propose a new model and algorithms to produce segmentations which would minimize the total energy a sum of the classic cvt energy and the weighted length of cluster boundaries. The convergence of lloyds algorithm to a centroidal voronoi diagram has been proven for the onedimensional case and higher dimensions. A multiresolution method for climate system modeling. Cvt is a fundamental notion that has a wide spectrum of. A weighted centroidal voronoi diagram cvd provides a structure that is space.
417 1372 448 1090 5 869 991 402 1347 864 552 1522 1092 483 1537 120 366 655 561 329 472 414 825 280 396 291 458 1030 465 617 164 337 702 215 1213 1484 653 250 441 1209 860 396 1169