Kokichi Sugihara

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

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. >

A kernel density estimation method for networks, its computational method and a GIS-based tool

We develop a kernel density estimation method for estimating the density of points on a network and implement the method in the GIS environment. This method could be applied to, for instance, finding ‘hot spots’ of traffic accidents, street crimes or leakages in gas and oil pipe lines. We first show that the application of the ordinary two‐dimensional kernel method to density estimation on a network produces biased estimates. Second, we formulate a ‘natural’ extension of the univariate kernel method to density estimation on a network, and prove that its estimator is biased; in particular, it overestimates the densities around nodes. Third, we formulate an unbiased discontinuous kernel function on a network. Fourth, we formulate an unbiased continuous kernel function on a network. Fifth, we develop computational methods for these kernels and derive their computational complexity; and we also develop a plug‐in tool for operating these methods in the GIS environment. Sixth, an application of the proposed methods to the density estimation of traffic accidents on streets is illustrated. Lastly, we summarize the major results and describe some suggestions for the practical use of the proposed methods.

Some location problems for robot navigation using a single camera

Abstract The paper considers two classes of point location problems found in visual navigation of a mobile robot. The problems we consider are finding the location of a robot using a map of the room where the robot moves and an image taken by a camera carried by the robot. In the first class of problems, vertical edges in the image are given, and a possible location for the robot is investigated by establishing a correspondence between the edges in the images and vertical poles given in the map. In the second class of problems, the possible region for the robot is investigated under the assumption that vertical edges are distinguishable from each other, but only the order in which the edges are found when the image is swept from left to right is given. These problems and their variations are considered from a computational geometry point of view, and efficient algorithms for solving them are given.

Spatial Analysis along Networks: Statistical and Computational Methods

In the real world, there are numerous and various events that occur on and alongside networks, including the occurrence of traffic accidents on highways, the location of stores alongside roads, the incidence of crime on streets and the contamination along rivers. In order to carry out analyses of those events, the researcher needs to be familiar with a range of specific techniques. Spatial Analysis Along Networks provides a practical guide to the necessary statistical techniques and their computational implementation. Each chapter illustrates a specific technique, from Stochastic Point Processes on a Network and Network Voronoi Diagrams, to Network K-function and Point Density Estimation Methods, and the Network Huff Model. The authors also discuss and illustrate the undertaking of the statistical tests described in a Geographical Information System (GIS) environment as well as demonstrating the user-friendly free software package SANET.

Voronoi diagram of a circle set from Voronoi diagram of a point set: geometry

Abstract In this and the following papers, we present an algorithm to compute the exact Voronoi diagram of a circle set from the Voronoi diagram of a point set. The circles are located in a two dimensional Euclidean space, the radii of the circles are non-negative and not necessarily equal, and the circles are allowed to intersect each other. The idea of the algorithm is to use the topology of the point set Voronoi diagram as a seed so that the correct topology of the circle set Voronoi diagram can be obtained through a number of edge flipping operations. Then, the geometries of the Voronoi edges of the circle set Voronoi diagram are computed. In particular, this paper discusses the topology aspect of the algorithm, and the following paper discusses the geometrical aspect. The main advantages of the proposed algorithm are in its robustness, speed, and the simplicity in its concept as well as implementation. Since the algorithm is based on the result of the point set Voronoi diagram and the flipping operation is the only topological operation, the algorithm is always as stable as the Voronoi diagram construction algorithm of a point set. The worst-case time complexity of the proposed algorithm is O( n 2 ), where n is the number of generators. However, the experiment with several cases shows a strong linear increase of the computation time.

A Necessary and Sufficient Condition for a Picture to Represent a Polyhedral Scene

As Shapira pointed out, a theorem by the author on line drawings of polyhedral scenes was not accurate. The present paper shows that the validity of the theorem is attained by a slight revision of the formulation.

Nearest neighbourhood operations with generalized Voronoi diagrams: a review

Abstract An ordinary geographical information system has a collection of nearest neighbourhood operations, such as generating a buffer zone and searching for the nearest facility from a given location, and this collection serves as a useful tool box for spatial analysis. Computationally, these operations are undertaken through the ordinary Voronoi diagram. This paper extends this tool box by generalizing the ordinary Voronoi diagram. The tool box consists of 35 nearest neighbourhood operations based upon twelve generalized Voronoi diagrams: the order-fe Voronoi diagram, the ordered order-fc Voronoi diagram, the farthest-point Voronoi diagram, the kth-nearest-point Voronoi diagram, the weighted Voronoi diagram, the line Voronoi diagram, the area Voronoi diagram, the Manhattan Voronoi diagram, the spherical Voronoi diagram, the Voronoi diagram in a river, the polyhedral Voronoi diagram, and the network Voronoi diagram. Each operation is illustrated with examples and the literature of computational methods.

TWO GENERALIZATIONS OF AN INTERPOLANT BASED ON VORONOI DIAGRAMS

Recently, the authors found an interpolant using Voronoi diagrams that differs from Sibsons interpolant. This paper generalizes our interpolant in two directions: one is to general-dimensional data, and the other is to data distributed continuously on curves. The Minkowskis theorem is used as the basic principle in generalization.

Approximation of generalized Voronoi diagrams by ordinary Voronoi diagrams

Abstract A numerically robust algorithm for the ordinary Voronoi diagrams is applied to the approximation of various types of generalized Voronoi diagrams. The generalized Voronoi diagrams treated here include Voronoi diagrams for figures, additively weighted Voronoi diagrams, Voronoi diagrams in a river, Voronoi diagrams in a Riemannian plane, and Voronoi diagrams with respect to collision-avoiding shortest paths. The construction of these generalized Voronoi diagrams is reduced to the construction of the ordinary Voronoi diagrams. The methods proposed here can save much time which is otherwise necessary for writing a computer program for each type of generalized Voronoi diagram.

Three-dimensional beta-shapes and beta-complexes via quasi-triangulation

The proximity and topology among particles are often the most important factor for understanding the spatial structure of particles. Reasoning the morphological structure of molecules and reconstructing a surface from a point set are examples where proximity among particles is important. Traditionally, the Voronoi diagram of points, the power diagram, the Delaunay triangulation, and the regular triangulation, etc. have been used for understanding proximity among particles. In this paper, we present the theory of the @b-shape and the @b-complex and the corresponding algorithms for reasoning proximity among a set of spherical particles, both using the quasi-triangulation which is the dual of the Voronoi diagram of spheres. Given the Voronoi diagram of spheres, we first transform the Voronoi diagram to the quasi-triangulation. Then, we compute some intervals called @b-intervals for the singular, regular, and interior states of each simplex in the quasi-triangulation. From the sorted set of simplexes, the @b-shape and the @b-complex corresponding to a particular value of @b can be found efficiently. Given the Voronoi diagram of spheres, the quasi-triangulation can be obtained in O(m) time in the worst case, where m represents the number of simplexes in the quasi-triangulation. Then, the @b-intervals for all simplexes in the quasi-triangulation can also be computed in O(m) time in the worst case. After sorting the simplexes using the low bound values of the @b-intervals of each simplex in O(mlogm) time, the @b-shape and the @b-complex can be computed in O(logm+k) time in the worst case by a binary search followed by a sequential search in the neighborhood, where k represents the number of simplexes in the @b-shape or the @b-complex. The presented theory of the @b-shape and the @b-complex will be equally useful for diverse areas such as structural biology, computer graphics, geometric modelling, computational geometry, CAD, physics, and chemistry, where the core hurdle lies in determining the proximity among spherical particles.