Showing papers on "Voronoi diagram published in 1988"

Parallel computational geometry

Abstract: We present efficient parallel algorithms for several basic problems in computational geometry: convex hulls, Voronoi diagrams, detecting line segment intersections, triangulating simple polygons, minimizing a circumscribing triangle, and recursive data-structures for three-dimensional queries.

A method for generating irregular computational grids in multiply connected planar domains

Abstract: A method for generating irregular triangular computational grids in two-dimensional multiply connected domains is described. A set of points around each body is defined using a simple grid generation technique appropriate to the geometry of each body. The Voronoi regions associated with the resulting global point distribution are constructed from which the Delaunay triangulation of the set of points is thus obtained. The definition of Voronoi regions ensures that the triangulation produces triangles of reasonable aspect ratios given a grid point distribution. The approach readily accommodates local clustering of grid points to facilitate variable resolution of the domain. The technique is generally applicable and has been used with success in computing triangular grids in multiply connected planar domains. The suitability of such grids for flow calculations is demonstrated using a finite element method for solution of the inviscid transonic flow over two- dimensional high-lift aerofoil configurations.

Texture segmentation using Voronoi polygons

Abstract: The authors have developed a texture-segmentation algorithm based on the Voronoi tessellation. The algorithm first builds the Voronoi tessellation of the tokens that make up the image. It then computes a set of features from the Voronoi polygons. Finally, it performs a probabilistic relaxation labeling on the tokens, to identify the interior and the border regions of the textures. The algorithm has successfully segmented images containing textures whose primitives have identical second order statistics. >

Abstract Voronoi diagrams and their applications (extended abstract)

Abstract: Given a set S of n points in the plane, and for every two of them a separating Jordan curve, the abstract Voronoi diagram V(S) can be defined, provided that the regions obtained as the intersections of all the “halfplanes” containing a fixed point of S are path-connected sets and together form an exhaustive partition of the plane. This definition does not involve any notion of distance. The underlying planar graph, \(\hat V\)(S), turns out to have O(n) edges and vertices. If S=L ∪ R is such that the set of edges separating L-faces from R-faces in \(\hat V\)(S) does not contain loops then \(\hat V\)(L) and \(\hat V\)(R) can be merged within O(n) steps giving \(\hat V\)(S). This result implies that for a large class of metrics d in the plane the d-Voronoi diagram of n points can be computed within optimal O(n log n) time. Among these metrics are, for example, the symmetric convex distance functions as well as the metric defined by the city layout of Moscow or Karlsruhe.

Optimal parallel algorithms for polygon and point-set problems

Abstract: In this paper we give parallel algorithms for a number of problems defined on polygons and point sets. All of our algorithms have optimal T(n) *P(n) products, where T(n) is the time complexity and P(n) is the number of processors used, and are for the EREW PRAM or CREW PRAM models. In addition, our algorithms provide parallel analogues to well known phenomena from sequential computational geometry, such as the fact that problems for polygons can oftentimes be solved more efficiently that point-set problems, and that one can solve nearest-neighbor problems without explicitly constructing a Voronoi diagram.

The distribution of interparticle distance and its application in finite-element modelling of composite materials

Abstract: The distribution of interparticle distance, based on a Voronoi tessellation, is found approximately for a hard-core Gibbs process. The moments of this distribution are then used as input for finite-element analysis of the region surrounding a single filler sphere within a composite material. Statistical analysis provides close bounds for overall elastic properties of the material. Results from finite-element analysis can therefore be applied to real composite materials; reasonable agreement is found with a particular set of experimental data.

Voronoi Diagrams Based on General Metrics in the Plane

Abstract: Voronoi diagrams based on metrics others than the Euclidean metric or convex distance functions have recently received considerable interest in robotics and in computational geometry. Since the number of relevant metrics is large (and likely to increase, as new applications come up) a general theory should be developed, leading to results on the structure and on the computation of Voronoi diagrams that hold for large classes of metrics, rather than investigating each case separately.

Structure of simple liquids as a percolation problem on the Voronoi network

Abstract: Analysis of the mutual arrangements of Delaunay simplices is formulated as a site-correlated percolation problem on the Voronoi network. Typical cluster configurations of simplices of tetrahedral and octahedral form (the main structural elements of simple liquids) are found and percolation thresholds for them are obtained.

Computing the volume of the union of spheres

Abstract: OnO(n 2) exact algorithm is given for computing the volume of a set ofn spheres in space. The algorithm employs the Laguerre Voronoi (power) diagram and a method for computing the volume of the intersection of a simplex and a sphere exactly. We give a new proof of a special case of a conjecture, popularized by Klee, concerning the change in volume as the centres of the spheres become further apart.

Geometric relations among Voronoi diagrams

Abstract: Two general classes of Voronoi diagrams are introduced and, along with their modifications to higher order, are shown to be geometrically related. This geometric background, on the one hand, serves to analyse the size and combinatorial structure and, on the other, implies general and efficient methods of construction for various important types of Voronoi diagrams considered in the literature.

Average-case analysis of algorithms for convex hulls and Voronoi diagrams

Abstract: This thesis addressed the design and analysis of fast-on-average algorithms for two classic problems of computational geometry: the construction of convex hulls and Voronoi diagrams of finite point sets in Euclidean d-space. The main contributions of the thesis are: (1) A new algorithm for enumerating ther vertices of a convex hull that requires between $\Theta$(n) and $\Theta$(n$\sp2$) time on average for a set of n independent and identically distributed (i.i.d.) points. The exact running time depends on the input distribution. This algorithm is a useful preprocessing step for algorithms for the facet-enumeration and facial-lattice versions of the convex-hull problem. (2) A new method for bounding the expected number of vertices of the convex hull of random points, new results on the asymptotic behavior of the expected number of vertices and facets of the convex hull of n i.i.d. points drawn from any of a wide variety of input distributions in d dimensions, and application of these results to the analysis of existing convex-hull algorithms. The distributions considered are: spherically symmetric distributions with algebraic, exponential, and truncated tails; certain uniform product distributions; and uniform distributions in d-polytopes. For all but the product distributions, it is shown that two well-known convex-hull algorithms require o(n$\sp2$) time on average; for some of the distributions, linear time suffices. (3) A new method for analyzing the expected combinatorial complexity and other properties of a random Voronoi diagram in d dimensions, and new results on the expected complexity of the Voronoi diagram of n points chosen from the uniform distribution in the unit d-ball. (4) A new linear-expected-time algorithm for constructing the Voronoi diagram of n points from the uniform distribution in the unit d-ball. (5) A practical improvement to the classic divide-and-conquer algorithm for the two-dimensional Voronoi diagram that decreases its average running time from $\Theta$(n log n) to $\Theta$(n log log n) for n points from the uniform distribution in the unit square and similar distributions.

Computational morphology : a computational geometric approach to the analysis of form

Abstract: Computational Complexity of Restricted Polygon Decompositions (A. Aggarwal, S.K. Ghosh, R.K. Shyamasundar). Computing Monotone Simple Circuits in the Plane (D. Avis, D. Rappaport). Circular Separability of Planar Point Sets (B.K. Bhattacharya). Symmetry Finding Algorithms (P. Eades). Computing the Relative Neighbour Decomposition of a Simple Polygon (H.A. ElGindy, G.T. Toussaint). Polygonal Approximations of a Curve - Formulations and Algorithms (H. Imai, M. Iri). On Polygonal Chain Approximation (A.A. Melkman, J. O'Rourke). Uniqueness of Orthogonal Connect-the-Dots (J. O'Rourke). On the Shape of a Set of Points (J.D. Radke). Ortho-Convexity and Its Generalizations (G.J.E. Rawlins, D. Wood). Guard Placement in Rectilinear Polygons (J.R. Sack, G.T. Toussaint). Realizability of Polyhedrons from Line Drawings (K. Sugihara). Voronoi and Related Neighbours on Digitized Two-Dimensional Space with Applications to Texture Analysis (J. Toriwaki, S. Yokoi). A Graph-Theoretical Primal Sketch (G.T. Toussaint).

Putting the best face on a Voronoi polyhedron

Abstract: Etude des configurations extremales dans l'empilement compact de spheres identiques dans R 3

Random packing and random tessellation in relation to the dimension of space

Abstract: SUMMARY The random packing of non-overlapping objects and the random tessellation of space by objects are discussed with relation to the dimension of space. The random packing discussed here is random ‘sequential’ packing and random tessellation is defined in this paper as the Voronoi tessellation of Poisson point processes. The conjecture by Palasti is examined for homothetic packing of cubes by using the existing data up to four dimensions; the generalized Palasti conjecture for spheres is also examined from the results of Monte Carlo simulations for two- and three-dimensional space. As for the random tessellation, Kiang's conjecture is examined using the data for two- and three-dimensional space. The results of the study indicate that all of the conjectures are false. Some details are given of the simulation methods.

3-Dimensional Shortest Paths in the Presence of Polyhedral Obstacles

Abstract: We consider the problem of finding a minimum length path between two points in 3-dimensional Euclidean space which avoids a set of (not necessarily convex) polyhedral obstacles; we let n denote the number of the obstacle edges and k denote the number of "islands" in the obstacle space An island is defined to be a maximal convex obstacle surface such that for any two points contained in the interior of the island, a minimal length path between these two points is strictly contained in the interior of the island; for example, a set of i disconnected convex polyhedra forms a set of i islands, however, a single non-convex polyhedron will constitute more that one island Prior to this work, the best known algorithm required double-exponential time We present an algorithm that runs in \(n^{k^{0(1)} }\) time and also one that runs in O(nlog(k)) space

An optimal expected-time parallel algorithm for Voronoi diagrams

Abstract: We present a parallel algorithm which constructs the Voronoi diagram of a planar n-point set within a square window. When the points are independently drawn from a uniform distribution, the algorithm runs in O(log n) expected time on CRCW PRAM with O(n/log n) processors. The fast operation of the algorithm results from the efficiency of a new multi-level bucketing technique convenient in processor assignment. The concurrent write is used only for the distribution of points in their home buckets in the bottom level.

Hierarchical Figure-Based Shape Description for Medical Imaging

Abstract: Medical imaging has long needed a good method of shape description, both to quantitate shape and as a step toward object recognition. Despite this need none of the shape description methods to date have been sufficiently general, natural, and noise-insensitive to be useful. We have developed a method that is automatic and appears to have great hope in describing the shape of biological objects in both 0 and 3D.

Image representation using Voronoi tessellation: adaptive and secure

Abstract: An image is represented by the Voronoi tessellation generated from selected sampling points. Using a multiresolution approach, the density of the sampling points can be adaptive to image properties: smoother regions will have fewer sampling points than more detailed regions. The adaptation property results in better image quality than nonadaptive Voronoi representations, while preserving the property that only the holder of the seed of the pseudorandom number generator can reconstruct the original image. >

Voronoi and Related Neighbors on Digitized two-Dimensional Space with Applications to Texture Analysis

Abstract: In this article, we present neighbor relations among figures on a 2-D digitized picture. First those relations (the Voronoi neighbor, the relative neighbor, and the Gabriel neighbor) are extended to a set of connected components (arbitrary shapes of figures). Second, several procedures to obtain those neighbor relations from a given binary image are described with estimation of the amount of computation required. Finally applications of the adjacency graphs induced from the above neighbor relations to texture analysis are described with the results of experiments.

The Voronoi region of E*.:KC 7

Abstract: The Voronoi region and covering radius of the lattice $E_7^*$ are determined, and the normalised second moment is calculated, confirming the estimate given by Conway and Sloane [Voronoi regions of lattices, second moments of polytopes, and quantization, IEEE Trans. Inform. Theory, 28 (1982), pp. 211–226].

Greedy Triangulation acn be Efficiently Implemented in the Average Case (Extended Abstract)

Abstract: Let S be a set of n points uniformly distributed in a unit square. We show that the greedy triangulation of S can be computed in O(nlog1.5n) expected time (without bucketing). The best previously known upper-bound on the expected-time performance of an algorithm for the greedy triangulation was O(n2).

A continuous set covering problem as a quasidifferentiable optimization problem

Abstract: In this paper we present an algorithm to solve a family of finite covering problems in . Given a compact, finitely convex decomposable set and an integer we are looking for the centers and the minimal radius of m balls with the property . It will be shown that this problem can be reduced to the computation of Dirichtlet tessellations (Voronoi sets) and the computation of minima of quasidifferentiable optimization problems.

A survey of paralle computational geometry algorithms

Abstract: We survey computational geometry algorithms developed for various models of parallel computation including the PRAM, hypercube, mesh-of-processors, linear processor array, mesh of trees, and pyramid.

A new generalized voronoi diagram in the plane

Abstract: In this thesis, a new generalized Voronoi diagram in the plane, called a bounded Voronoi diagram, is defined. This generalization is one of the natural extensions of the previously existing Voronoi diagrams in the plane. Three important cases of bounded Voronoi diagrams are discussed. They are: (1) The bounded Voronoi diagram for a monotone chain. (2) The bounded Voronoi diagram for a simple polygon. (3) The bounded Voronoi diagram for a set of non-crossing line segments. Algorithms for constructing these bounded Voronoi diagrams are presented. All of them are optimal. Bounded Voronoi diagrams are powerful tools which help to efficiently solve many geometric problems.