Date 25 July 2006
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3rd International Symposium on Voronoi Diagrams in Science and Engineering  Cover
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3rd International Symposium on Voronoi Diagrams in Science and Engineering  Title
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3rd International Symposium on Voronoi Diagrams in Science and Engineering  Copyright
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3rd International Symposium on Voronoi Diagrams in Science and Engineering  Table of contents
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Conference Organizational Structure
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Voronoi Diagram and Delaunay Triangulation: Applications and Challenges in Bioinformatics
Publication Year: 2006 , Page(s): 2  3This paper first covers the applications of Voronoi diagram and Delaunay triangulation based on the author's personal experience. On the successful part, it covers the history and development of the jumpandwalk algorithm, which is known as the first sublinear geometric algorithm nowadays and has been used in several famous software packages. On the bioinformatics applications, the paper focuses on three challenges: (1) contour interpolation, (2) protein folding, and (3) protein local structure alignment. Voronoi diagram and Delaunay triangulation plays important roles in handling these problems. View full abstract»

Recent Developments and Open Problems in Voronoi Diagrams
Publication Year: 2006 , Page(s): 4  5This paper deals with some open problems in computational geometry related to Voronoi diagrams. The Voronoi diagram and Dealunay triangulation of n points in Ropf^{d} has complexity Theta (n ^{[d/2]}) in the worst case. However, the complexity is not high if points are random. Statistical properties of Voronoi diagrams of random points have been studied for decades. The expected complexity of the Voronoi diagram of n random points in the threedimensional cube is O(n). If n points are generated uniformly at random in the unit ball in R^{d}, the Voronoi diagram has expected complexity d^{O(d)}n. View full abstract»

Voronoi and Delaunay Tilings for Lattices
Publication Year: 2006 , Page(s): 6Summary form only given. In this paper the author considers two outstanding problems in the theory of Delaunay and Voronoi tilings for lattices. There is a new classification problem that has arisen from a new structure theorem. Sergei Ryshkov proved that: The Minkowski sum of two Voronoi polytopes is a Voronoi polytope, if and only if the corresponding Delaunay tilings are commensurate. This result raises the question of classifying all Minkowski irreduciable Voronoi poloytopesthe dual description is to determine all edge forms in Voronoi's theory of lattice types. The author describes the first steps that have been taken in this direction. View full abstract»

Distance Trisector Curves in Regular Convex Distance Me
Publication Year: 2006 , Page(s): 8  17Given two points A and B in the plane, we are interested in separating them by two curves Ca and Cb such that Ca is equidistant from A and Cb, and Cb is equidistant from B and Ca Such curves generalize the familiar notion of a bisector curve, and form the basis of a new kind of Voronoi diagram called a Zone diagram. These curves, which are referred to as distance trisector curves, have been studied in the Euclidean metric where they exist, are unique, and admit efficient approximations. Nevertheless, they have no known expression in terms of elementary functions and are conjectured to be nonalgebraic. In this paper, we study distance trisector curves with respect to a parameterized family of distance metrics that provide arbitrarily close approximations to the Euclidean distance. The advantage of studying distance trisectors in this setting is that they have a simple piecewiselinear description and an efficient (exact) construction. We show that distance trisectors defined in this way provide a conceptually simple alternative proof of the existence and uniqueness of Euclidean trisector curves. View full abstract»

Angular Voronoi Diagram with Applications
Publication Year: 2006 , Page(s): 18  24
Cited by: Papers (2)Given a set of line segments in the plane, we define an angular Voronoi diagram as follows: a point belongs to a Voronoi region of a line segment if the visual angle of the line segment from the point is smallest among all line segments. The Voronoi diagram is interesting in itself and different from an ordinary Voronoi diagram for a point set. After introducing interesting properties, we present an efficient algorithms for finding a point to maximize the smallest visual angle. Some applications to mesh improvement are also mentioned. View full abstract»

On the Stretch Factor of the Constrained Delaunay Triangulation
Publication Year: 2006 , Page(s): 25  31Given a set P of n points in the plane and a set S of noncrossing line segments whose endpoints are in P, let CDT(P, S) be the constrained Delaunay triangulation of P with respect to S. Given any two visible points p,q isin P, we show that there exists a path from p to q in CDT(P, S), denoted SP _{CDT}(p, q) such that every edge in the path has length at most pq and the ratio SP_{CDT}(p, q)/pq is at most 4piradic3/9 (ap 2.42), thereby improving on the previously known bound of pi(ap1+radic5)/2 (ap5.08). View full abstract»

AspectRatio Voronoi Diagram with Applications
Publication Year: 2006 , Page(s): 32  39
Cited by: Papers (3)This paper considers a problem of finding an optimal point within a polygon P in the sense that when we connect the point to every vertex of P by straight line then the worst aspect ratio among all resulting triangles is optimized. This problem has an important application to triangular mesh improvement. We propose three different approaches toward this problem. The first one is based on some new Voronoi diagram defined by an aspect ratio, which is interesting in itself. The second approach is essentially a binary search defined by geometric intersection. The third one is gridbased heuristic, which might be practically best but has no theoretical guarantee on its performance. View full abstract»

Stable and TopologyPreserving Extraction of Medial Axes
Publication Year: 2006 , Page(s): 40  47The paper presents a simple method for extracting global medial axes in a stable manner. A new index, called the normalized boundary distance, is introduced in order to measure the degree of importance of a point on the conventional medial axis. This index has a remarkable property that the set of points whose index values are greater than an arbitrarily chosen threshold is topologically equivalent to the original figure. In the proposed method, first the boundary of a given figure is replaced with a dense set of points, next the Voronoi diagram for these points is constructed, then the approximation of the medial axis is extracted from the Voronoi diagram, and finally the global medial axis is constructed by pruning the branches according to the new index. The performance of the proposed method is also shown by examples. View full abstract»

On a Geometric Structure of Pure Multiqubit Quantum States and Its Applicability to a Numerical Computation
Publication Year: 2006 , Page(s): 48  53
Cited by: Papers (1)For onequbit pure quantum states, it is already proved that the Voronoi diagrams with respect to two distances  Euclidean distance and the quantum divergence  coincide. This fact is a support for a known method to calculate the Holevo capacity. To consider an applicability of this method to quantum states of a higher level system, it is essential to check if the coincidence of the Voronoi diagrams also occurs. In this paper, we show a negative result for that expectation. In other words, we mathematically prove that those diagrams no longer coincide in a higher dimension. That indicates that the method used in onequbit case to calculate the Holevo capacity might not be effective in a higher dimension. View full abstract»

Robust PointLocation in Generalized Voronoi Diagrams
Publication Year: 2006 , Page(s): 54  59We address the problem of robust pointlocation in a generalized ddimensional Voronoi diagram. The exact point location requires the solution for expressions of degree four. A natural question is what can be done using expression of smaller degree. We apply polyhedral metrics for this task. In general dimensions two Minkowski metrics can be used L_{1} (Manhattan metric) and L_{infin}. The approximation factor is radic(d) and the computation uses expressions of degree one. We also show that a polygonal metric can be applied in two dimensions. The computation involves only 0(lg k) calls of the algorithm ESSA for detecting the sign of a sum using floatingpoint arithmetic. View full abstract»

Spherebased Computation of Delaunay Diagrams on Points from 4d Grids
Publication Year: 2006 , Page(s): 60  65The Delaunay diagram in d dimensions is the dual of the Voronoi diagram of a set of input sites. If we assume no degeneracies in the input, i.e. no d + 2 sites are cospherical, then the diagram is a triangulation. Because this assumption is common, and can be enforced by symbolic perturbation, we often forget that Delaunay diagrams need not be triangulations. Input sets chosen from integer grids are common in scientific visualization applications, however, and these often have many degeneracies. Perturbation signifcantly increases the size of the Delaunay and dual Voronoi diagrams  a single 4D cube becomes 16 to 24 simplices, so one dual vertex becomes many. Our result is a spherebased algorithm for direct, incremental computation of the Delaunay diagram in 4D. For input with many degeneracies, its speed is comparable to our fastest Delaunay triangulation program, yet it computes the exact Delaunay diagram. View full abstract»

Voronoi Random Fields
Publication Year: 2006 , Page(s): 66  75We propose a random field regarded as a generalization of Voronoi diagrams for the case where the positions of generators are distributed probabilistically. Our approach is a kind of stochastic approaches to Voronoi diagrams; however our definition is different from usual random Voronoi diagrams such as Poisson Voronoi diagrams. As numerical examples, first we discuss the case where the positions of generators are mutually independent and their marginal distributions are uniform distributions on disks. Second, we discuss the case where the distributions are given in the form of digital pictures. View full abstract»

Kinetic Voronoi/Delaunay Drawing Tools
Publication Year: 2006 , Page(s): 76  84
Cited by: Papers (1)  Patents (1)We describe two reversible linedrawing methods for cartographic applications based on the kinetic (movingpoint) Voronoi diagram. Our objectives were to optimize the user's ability to draw and edit the map, rather than to produce the most efficient batch oriented algorithm for large data sets, and all our algorithms are based on local operations (except for basic point location). Because the deletion of individual points or line segments is a necessary part of the manual editing process, incremental insertion and deletion is used. The original concept used here is that, as a curve (line) is the locus of a moving point, then segments are drawn by maintaining the topology of a single moving point (MP, or the "pen") as it moves through the topological network (visualized as either the Voronoi diagram or Delaunay triangulation). The trailing line accumulates the adjacency relationships of MP. There are thus three parts to our method: the maintenance of MP in the DT/VD; the use of MP to draw the constrained edges in the Delaunay triangulation; and the use of MP to draw the line segment Voronoi diagram. In all cases deletion is the inverse of the original drawing: move MP so as to "roll up" the desired segment. This approach also has the interesting property that a "log file" of all operations may be preserved, allowing reversion to previous map states, or "dates", as required. View full abstract»

An Efficient Swarm Neighborhood Management for a 3D Tactical Simulator
Publication Year: 2006 , Page(s): 85  93The paper presents a unique utilization of the dynamic Delaunay triangulation (DT) to devise a highly efficient algorithm for the neighborhood adjacency management which is a crucial component of a swarmbased simulation. This algorithm allows fast computation of the swarm neighborhood which is required to implement Boids flocking rules. The method also provides an efficient mechanism to detect and manage object collisions. This unique utilization of DT based data structures was applied to a new 3D tactical swarm simulation called the Battle Swarm. The result was a very high speed of simulation, which made complex application of genetic evolution possible for goalbased swarms in a three dimensional space. The method's effective and complex emergent behavior was successfully demonstrated by the Battle Swarm Simulator. In addition, the experimental section confirms that the method was highly efficient and can be used in similar applications. View full abstract»

Geometrical Algorithms to Detect Patterns from a Set of Points
Publication Year: 2006 , Page(s): 94  101This article presents a new approach for detecting patterns, such as lines, parabolas and circles, from a two dimensional cloud of points S. This scheme transforms the problem of detecting the patterns from S, to the problem of detecting simpler patterns from a set of points S^{1} computed from the Voronoi and Delaunay diagrams of S. It is based on the differential properties of the Voronoi diagram that reflect the patterns to be retrieved. For example, in the case of parabolas, the patterns to be detected in S' are straight lines. The general idea of using the Voronoi diagram to detect patterns is an alternative and a complementary approach to the Hough Transform. It does not need to code a space of parameters and can be generalized to higher dimensions, with some adaptations. View full abstract»

CRYSTAL  A new densitybased fast and efficient clustering algorithm
Publication Year: 2006 , Page(s): 102  111
Cited by: Papers (6)In this paper, we present a fast O(nlogn) clustering algorithm based on Delaunay triangulation for identifying clusters of different shapes, not necessarily convex. The clustering result is similar to human perception of clusters. The novelty of our method is the growth model we follow in the cluster formation that resembles the natural growth of a crystal. Our algorithm is able to identify dense as well as sparse clusters and also clusters connected by bridges. We demonstrate clustering results on several synthetic datasets and provide a comparison with popular Kmeans based clustering methods. The clustering is based purely on proximity analysis in the Delaunay triangulation and avoids usage of global parameters. It is robust in the presence of noise. Finally, we demonstrate the capability of our clustering algorithm in handling very large datasets. View full abstract»

Clustering of 3D Spatial Points Using Maximum Likelihood Estimator over Voronoi Tessellations: Study of the Galaxy Distribution in Redshift Space
Publication Year: 2006 , Page(s): 112  121This paper describes an algorithm based on the 2D approach of Allard & Fraley that uses Voronoi tessellation and a nonparametric maximum likelihood estimator. We have designed a 3D version of this algorithm which detects multiple clusters of points immersed in background noise; its application to the detection of galaxy clusters in redshift space, using the astronomical database of the 2degree Field Galaxy Redshift Survey, is presented and discussed. Adopting as a benchmark a particular set of catalogued clusters of galaxies, we find that the proposed algorithm recognizes the location of ~67% of the clusters. Three variants of the algorithm were assessed to deal with the elongation of the clusters in the radial direction of observation introduced by the astronomical distance indicator; their merits and limitations are discussed. We address separately the detection of the galaxy cluster location and the detection of galaxy cluster members, both of them having an anisotropic space as their search domain. In the case of detection of galaxy cluster members, a second stage of detection was incorporated in order to improve the results. View full abstract»

Closed Curve Reconstruction from Unorganized Sample Points
Publication Year: 2006 , Page(s): 122  131This paper considers the problem for reconstructing closed curves from unorganized sample points, where the curve to be reconstructed consists of a finite number of pairwise disjoint components of simple closed curves. This paper formulates this problem as a linear integer programming problem, and proposes an algorithm, called ZEROONE, based on it. In addition, two other heuristic algorithms called PASTE and SCISSORS are proposed for reducing computational time. Experimental results show that these algorithms yield exact polygonal reconstruction if the density of the sample points is high enough. In addition, it is shown that SCISSORS can be modified for the reconstruction problem of curves with boundaries. View full abstract»

3D Facial Model Synthesis using Voronoi Approach
Publication Year: 2006 , Page(s): 132  137
Cited by: Papers (1)Construction and animation of realistic human facial models is an important research field of computer graphics. How to efficiently create an individualized facial model for animation is still a challenge. In this paper, we present a method for 3D facial model synthesis that combines the traditional freeform deformation (FFD) model with techniques of data interpolation based on Dirichlet/Voronoi diagrams. With 18 feature points extracted from 2D facial images in two orthogonal views, Dirichlet free form deformation (DFFD) is utilized for modifying a generic 3D face to produce the individual face. The main advantages of this method over former extensions of FFD is in removing the constraints on control lattice and control points location. In addition, by assigning different weights to those control points, we improve the DFFD algorithm to make it more adaptable to the facial deformation. The reconstructed 3D faces can be used to generate different facial animations. View full abstract»