Showing papers on "Voronoi diagram published in 1992"

Spatial Tessellations: Concepts and Applications of Voronoi Diagrams

Abstract: Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Voronoi diagrams and Delaunay triangulations

Abstract: The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams and Delaunay triangulations have been rediscovered or applied in many areas of math ematics and the natural sciences they are central topics in computational geometry with hundreds of papers discussing algorithms and extensions Section discusses the de nition and basic properties in the usual case of point sites in R with the Euclidean metric while section gives basic algorithms Some of the many extensions obtained by varying metric sites environment and constraints are discussed in section Section nishes with some interesting and nonobvious structural properties of Voronoi diagrams and Delaunay triangulations

Randomized incremental construction of Delaunay and Voronoi diagrams

Abstract: In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more “on-line” than earlier similar methods, takes expected timeO(nℝgn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.

Voronoi skeletons: theory and applications

Abstract: A novel method of robust skeletonization based on the Voronoi diagram of boundary points, which is characterized by correct Euclidean metries and inherent preservation of connectivity, is presented. The regularization of the Voronoi medial axis (VMA) in the sense of H. Blum's (1967) prairie fire analogy is done by attributing to each component of the VMA a measure of prominence and stability. The resulting Voronoi skeletons appear largely invariant with respect to typical noise conditions in the image and geometric transformations. Hierarchical clustering of the skeleton branches, the so-called skeleton pyramid, leads to further simplification of the skeleton. Several applications demonstrate the suitability of the Voronoi skeleton to higher-order tasks such as object recognition. >

Construction of the Voronoi diagram for 'one million' generators in single-precision arithmetic

Abstract: A numerically stable algorithm for constructing Voronoi diagrams in the plane is presented. In this algorithm higher priority is placed on the topological structure than on numerical values, so that, however large the numerical errors, the algorithm will never come across topological inconsistency and thus can always complete its task. The behavior of the algorithm is shown with examples, including one for as many as 10/sup 6/ generators. >

Properties of a three-dimensional poisson-voronoi tesselation: A Monte Carlo study

Abstract: A complete statistical description of the properties of a cellular microstructure generated by a three-dimensional Poisson-Voronoi tesselation has been obtained by a rigorous computer simulation involving several hundred thousand cells. A two-parameter gamma distribution is found to be a good fit to the cell's face, volume, and surface area distributions. For a sample size of several thousand cells or less, a lognormal distribution can also be used to approximate these distributions. The individual face, area, and edge length distributions are also obtained.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

Abstract: Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Delaunay triangulation in computational fluid dynamics

Abstract: A method, which utilises the Delaunay criterion, is described by which computational grids consisting of assemblies of triangles or tetrahedra can be constructed. An algorithm is briefly outlined to construct the triangulation and its dual, the Voronoi diagram. Issues related to how to construct boundary conforming grids from such a triangulation are addressed, and details are presented of how grid points within the domain can be generated automatically. The point generation algorithm utilises either the given boundary point distribution, or, for grid adaption, a background mesh. Computational aspects of constructing the triangulation in both 2 and 3 dimensions are covered. Examples of meshes and flow computations for a range of aerospace geometries are presented.

An algorithm for generating NC tool paths for arbitrarily shaped pockets with islands

Abstract: In this paper we describe algorithms for generating NC tool paths for machining of arbitrarily shaped 2 l/2 dimensional pockets with arbitrary islands. These pocketing algorithms are based on a new offsetting algorithm presented in this paper. Our offsetting algorithm avoids costly two-dimensional Boolean set operations, relatively expensive distance calculations, and the overhead of extraneous geometry, such as the Voronoi diagrams, used in other pocketing algorithms.

Establishment of a pair of concentric circles with the minimum radial separation for assessing roundness error

Abstract: The paper presents a computational-geometry-based method of determining the roundness error of a measured workpiece. A set of n points (obtained from the measured workpiece) in a plane being given, it is required that the center and the radii of a pair of concentric circles be found such that no point is exterior to the space bounded by the circles, with the condition that the radial separation between the circles is minimum. The paper addresses the mathematical formalization of the problem. The properties of convex-hull and Voronoi diagrams have been exploited to develop a faster algorithm for establishing the circles. The methodology has been implemented, and the results have been presented to validate the computational effectiveness of the approach.

Applications of random sampling to on-line algorithms in computational geometry

Abstract: This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time complexities when averaging over all permutations of the input data. The method is general and can be applied to various geometric problems. The power of the technique is illustrated by new efficient on-line algorithms for constructing convex hulls and Voronoi diagrams in any dimension, Voronoi diagrams of line segments in the plane, arrangements of curves in the plane, and others.

On dynamic Voronoi diagrams and the minimum Hausdorff distance for point sets under Euclidean motion in the plane

Abstract: We show that the dynamic Voronoi diagram of k sets of points in the plane, where each set consists of m points moving rigidly, has complexity O(n2k2ls(k)) for some fixed s, where ls(n) is the maximum length of a (n, s) Davenport-Schinzel sequence. This improves the result of Aonuma et al., who show an upper bound of O(n3k4 log* k) for the complexity of such Voronoi diagrams. We then apply this result to the problem of finding the minimum Hausdorff distance between two point sets in the plane under Euclidean motion. We show that this distance can be computed in time O((m + n)6 log (mn)), where the two sets contain m and n points respectively.

A vectorizable random lattice

Abstract: We describe a family of random lattices in which the connectivity is determined by the Voronoi construction while the vectorizability is not lost. We can continuously vary the degree of randomness so in a certain limit a regular lattice is recovered. Several statistical properties of the cells and bonds of these lattices are measured. We also study anisotropy effects on the numerical solution of the Laplace equation for varying degrees of randomness.

Low-dimensional lattices. VI : Voronoi reduction of three-dimensional lattices

Abstract: The aim of this paper is to describe how the Voronoi cell of a lattice changes as that lattice is continuously varied. The usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice. The present paper deals with dimensions n $\leq $ 3; a sequel will treat four-dimensional lattices. An elegant algorithm is given for the Voronoi reduction of a three-dimensional lattice, leading to a new proof of Voronoi's theorem that every lattice of dimension n $\leq $ 3 is of the first kind, and of Fedorov's classification of the three-dimensional lattices into five types. There is a very simple formula for the determinant of a three-dimensional lattice in terms of its conorms.

Analysis of the network topology in liquid water and hydrogen sulphide by computer simulation

Abstract: The analysis of Voronoi polyhedra for liquid water and hydrogen sulphide, at different temperatures, has been performed by using the molecular configurations generated by computer simulation of the liquids with realistic potential models. Some topological and metric properties of the Voronoi polyhedra have been calculated and their distributions are studied. In addition, the cross correlation between pairs of metric quantities are also investigated. The latter correlations are found to be more relevant for a clear distinction between the two systems examined here. In particular, the cross correlation between the potential energy of a molecule and the volume of the corresponding Voronoi polyhedron makes it clear that the interpretation of the anomalous physical properties of water in terms of local volume has to be revised.

Voronoi diagrams of set-theoretic solid models

Abstract: The definition of a Voronoi diagram is extended to arbitrary set-theoretic solid models. A method for approximating such diagrams using recursive subdivision is described. The method relies on octrees, which have been used for computing the distances between whole solid models. Two- and three-dimensional images generated using the algorithm are presented. >

A simple on-line randomized incremental algorithm for computing higher order Voronoi diagrams

Abstract: We present a simple algorithm for maintaining order-k Voronoi diagrams in the plane. By using a duality transform that is of interest in its own right, we show that the insertion or deletion of a site involves little more than the construction of a single convex hull in three-space. In particular, the order-k Voronoi diagram for n sites can be computed in time and optimal space by an on-line randomized incremental algorithm. The time bound can be improved by a logarithmic factor without losing much simplicity. For k≥log2 n, this is optimal for a randomized incremental construction; we show that the expected number of structural changes during the construction is ⊝(nk2). Finally, by going back to primal space, we obtain a dynamic data structure that supports k-nearest neighbor queries, insertions, and deletions in a planar set of sites. The structure promises easy implementation, exhibits a satisfactory expected performance, and occupies no more storage than the current order-k Voronoi diagram.

The Meaning of ``Neighbour''

Abstract: Traditional vector and raster spatial models as used in many computer systems are examined to determine what is meant by the term “neighbour”. The limitations are examined and the Voronoi spatial model is proposed as a consistent alternative to both. It is then asked whether this new computer model of space bears any resemblance to human spatial reasoning and perception processes. Examination of the work of Blum, Ahuja, Tuceryan, Serra and others suggests that the Voronoi model may indeed relate to visual perception and the concept of “neighbour”.

Voronoi and Delaunay cells of root lattices: classification of their faces and facets by Coxeter-Dynkin diagrams

Abstract: The Voronoi domains, their duals (Delaunay domains) and all their faces of any dimension are classified and described in terms of the Weyl group action on a representative of each type of face. The representative of a face type is specified by a decoration of the corresponding Coxeter-Dynkin diagram. The rules of domain description are uniform for root lattices of simple Lie groups of all types. An explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple Lie groups. The Coxeter-Dynkin diagrams required here are the diagrams extended by the highest short root. Each diagram is partitioned into two subdiagrams, one describing completely a d-face of the Voronoi domain, its complement completely describing the dual of the d-face. The applicability of the authors' classification method to generalized kaleidoscopes is explained.

Improvements on Geometric Pattern Matching Problems

Abstract: We consider the following geometric pattern matching problem: find the minimum Hausdorff distance between two point sets under translation with L1 or L∞ as the underlying metric Huttenlocher, Kedem, and Sharir have shown that this minimum distance can be found by constructing the upper envelope of certain Voronoi surfaces Further, they show that if the two sets are each of cardinality n then the complexity of the upper envelope of such surfaces is Ω(n3) We examine the question of whether one can get around this cubic lower bound, and show that under the L1 and L∞ metrics, the time to compute the minimum Hausdorff distance between two point sets is On2 log2n)

Contact and chord length distributions of the Poisson Voronoi tessellation

Abstract: This paper presents the form of some characteristics of the Voronoi tessellation which is generated by a stationary Poisson process in . Expressions are given for the spherical and linear contact distribution functions. These formulae lead to numerically tractable double-integral formulae for chord length probability density functions.

Voronoi Diagrams in a River

Abstract: A new generalized Voronoi diagram is defined on the surface of a river with uniform flow; a point belongs to the territory of a site if and only if a boat starting from the site can reach the point faster than a boat starting from any other site. If the river runs slower than the boat, the Voronoi diagram has the same topological structure as the ordinary Voronoi diagram, and hence can be constructed from the ordinary Voronoi diagram by a certain transformation. If the river runs faster than the boat, on the other hand, the topological structure of the diagram becomes different from the ordinary one, but it can be constructed by the plane sweep technique. Moreover, Fortune’s plane sweep algorithm for constructing the ordinary Voronoi diagram can be interpreted as the algorithm for constructing the Voronoi diagram in a river in which the water flows at the same speed as the boat.

Recognizing Voronoi Diagrams with Linear Programming

Abstract: Let P = {P1, …, Pn} be a finite set of distinct points in ℝd and let Ri be the set of points whose distance from Pi is less than or equal to the distance from every other point in P. The collection of regions R1, …, Rn is called the Voronoi diagram generated by P. Our main result is a polynomial time algorithm for recognizing whether or not a given tessellation of ℝd into polyhedra is a Voronoi diagram. This is accomplished by describing a linear program that has a solution if and only if the given tessellation is a Voronoi diagram. As a consequence, for each Ri of a Voronoi diagram, the set of points in Ri contained in some generating set P is either a singleton or the interior of a polyhedron. We also give a polynomial time algorithm for describing this set for each Ri. Finally, this leads to a second algorithm for recognizing Voronoi diagrams; this algorithm also relies on linear programming but, for fixed dimension d, it is strongly polynomial and has linear time complexity. INFORMS Journal on Computi...

A General Algorithm for Computing Voronoi Volumes - Application to the Hydrated Crystal of Myoglobin

Abstract: In this study is presented a general algorithm for computing Voronoi volumes of atoms or group of atoms in condensed phases. The method is essentially an extension of the Medvedev procedure to allow vertice determination for any Voronoi polyhedron, primitive or with degenerate vertices. The algorithm has been employed for computing time-averaged volumes in the hydrated crystal of met-myoglobin, using the data of a molecular dynamics simulation. The results, compared to previous volume determination in myoglobin, emphasize the fundamental role of solvent structure close to the protein surface in relation to the packing density properties of the residues

The farthest point Delaunay triangulation minimizes angles

Abstract: The planar dual to the Euclidean farthest point Voronoi diagram for the set of vertices of a convex polygon has the lexicographic minimum possible sequence of triangle angles, sorted from sharpest to least sharp. As a consequence, the sharpest angle determined by three vertices of a convex polygon can be found in linear time.

Computer Vision, Descriptive Geometry, and Classical Mechanics

Abstract: The medial-axis transform, also called skeleton, is a shape abstraction proposed by computer vision. The concept is closely related to cyclographic maps, a tool developed by descriptive geometry to investigate distance functions, and to the solution of the eikonal equation. We discuss these connections and their implications on techniques for computing the skeleton.

Multidimensional signal sets through the shell construction for parallel channels

Abstract: The authors derive a procedure to send r bits on M parallel channels. A decomposition of the best constellation in Z/sup M/+(1/2, . . ., 1/2) is given in terms of the cross-products of lower dimensional shells of points. The proposed scheme can be used with good known coset codes to provide an alternate method of coded modulation. The results indicate that one can get good shaping gains for low encoder complexity. The method is also generalized for channels with unequal gains. The authors also find a significant performance advantage at fixed shaping gain, in certain cases, with respect to the recent Voronoi constellations in terms of peak-to-average power and constellation expansion. >