Showing papers on "Voronoi diagram published in 2006"

Movement-assisted sensor deployment

Abstract: -Adequate coverage is very important for sensor networks to fulfill the issued sensing tasks. In many working environments, it is necessary to make use of mobile sensors, which can move to the correct places to provide the required coverage. In this paper, we study the problem of placing mobile sensors to get high coverage. Based on Voronoi diagrams, we design two sets of distributed protocols for controlling the movement of sensors, one favoring communication and one favoring movement. In each set of protocols, we use Voronoi diagrams to detect coverage holes and use one of three algorithms to calculate the target locations of sensors it holes exist. Simulation results show the effectiveness of our protocols and give insight on choosing protocols and calculation algorithms under different application requirements and working conditions.

The spatial skyline queries

Abstract: In this paper, for the first time, we introduce the concept of Spatial Skyline Queries (SSQ) Given a set of data points P and a set of query points Q each data point has a number of derived spatial attributes each of which is the point's distance to a query point An SSQ retrieves those points of P which are not dominated by any other point in P considering their derived spatial attributes The main difference with the regular skyline query is that this spatial domination depends on the location of the query points Q SSQ has application in several domains such as emergency response and online maps The main intuition and novelty behind our approaches is that we exploit the geometric properties of the SSQ problem space to avoid the exhaustive examination of all the point pairs in P and Q Consequently, we reduce the complexity of SSQ search from O(|P|2|Q|) to O(|S|2|C|+√|P|), where |S| and |C| are the solution size and the number of vertices of the convex hull of Q, respectivelyWe propose two algorithms, B2S2 and VS2, for static query points and one algorithm, VCS2, for streaming Q whose points change location over time (eg, are mobile) VCS2 exploits the pattern of change in Q to avoid unnecessary re-computation of the skyline and hence efficiently perform updates Our extensive experiments using real-world datasets verify that both R-tree-based B2S2 and Voronoi-based VS2 out perform the best competitor approach in terms of processing time by a wide margin (4-6 times better in most cases)

Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations

Abstract: Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids (mass centers) of the corresponding Voronoi regions with respect to a given density function. Centroidal Voronoi tessellations may also be defined in more abstract and more general settings. Due to the natural optimization properties enjoyed by CVTs, they have many applications in diverse fields. The Lloyd algorithm is one of the most popular iterative schemes for computing the CVTs but its theoretical analysis is far from complete. In this paper, some new analytical results on the local and global convergence of the Lloyd algorithm are presented. These results are derived through careful utilization of the optimization properties shared by CVTs. Numerical experiments are also provided to substantiate the theoretical analysis.

Adaptive binning of X‐ray data with weighted Voronoi tessellations

Abstract: We present a technique to adaptively bin sparse data using weighted Voronoi tesselations (WVTs). WVT binning is a generalisation of Cappellari & Copin’s (2001) Voronoi binning algorithm, developed for integral field spectroscopy. WVT binning is applicable to many types of data and creates unbiased binning structures with compact bins that do not lead the eye. We apply the algorithm to simulated data, as well as several X-ray data sets, to create adaptively binned intensity images, hardness ratio maps and temperature maps with constant signal-to-noise ratio per bin. We also illustrate the separation of diffuse gas emission from contributions of unresolved point sources in elliptical galaxies. We compare the performance of WVT binning with other adaptive binning and adaptive smoothing techniques. We find that the CIAO tool csmooth creates serious artefacts and advise against its use to interpret diffuse X-ray emission.

Path Planning for Mobile Robot Navigation using Voronoi Diagram and Fast Marching

Abstract: This paper presents a new sensor based global Path Planner which operates in two steps. In the first step the safest areas in the environment are extracted by means of a Voronoi diagram. In the second step Fast Marching Method is applied to the Voronoi extracted areas in order to obtain the shortest path. In this way the trajectory obtained is the shortest between the safe possible ones. This two step method combines an extremely fast global planner operating on a simple sensor based environment modeling, while it operates at the sensor frequency. The main characteristics are speed and reliability, because the map dimensions are reduced to a unidimensional map and this map represents the safest areas in the environment for moving the robot.

Jump flooding in GPU with applications to Voronoi diagram and distance transform

Abstract: This paper studies jump flooding as an algorithmic paradigm in the general purpose computation with GPU. As an example application of jump flooding, the paper discusses a constant time algorithm on GPU to compute an approximation to the Voronoi diagram of a given set of seeds in a 2D grid. The errors due to the differences between the approximation and the actual Voronoi diagram are hardly noticeable to the naked eye in all our experiments. The same approach can also compute in constant time an approximation to the distance transform of a set of seeds in a 2D grid. In practice, such constant time algorithm is useful to many interactive applications involving, for example, rendering and image processing. Besides the experimental evidences, this paper also confirms quantitatively the effectiveness of jump flooding by analyzing the occurrences of errors. The analysis is a showcase of insights to the jump flooding paradigm, and may be of independent interests to other applications of jump flooding.

Curve and Surface Reconstruction: Algorithms with Mathematical Analysis

Abstract: Many applications in science and engineering require a digital model of a real physical object. Advanced scanning technology has made it possible to scan such objects and generate point samples on their boundaries. This book, first published in 2007, shows how to compute a digital model from this point sample. After developing the basics of sampling theory and its connections to various geometric and topological properties, the author describes a suite of algorithms that have been designed for the reconstruction problem, including algorithms for surface reconstruction from dense samples, from samples that are not adequately dense and from noisy samples. Voronoi- and Delaunay-based techniques, implicit surface-based methods and Morse theory-based methods are covered. Scientists and engineers working in drug design, medical imaging, CAD, GIS, and many other areas will benefit from this first book on the subject.

SANET: A Toolbox for Spatial Analysis on a Network

Abstract: This article shows a geographical information systems (GIS)-based toolbox for analysing spatial phenomena that occur on a network (e.g., traffic accidents) or almost along a network (e.g., fast-food stores in a downtown). The toolbox contains 13 tools: random point generation on a network, the Voronoi diagram, the K-function and cross K-function methods, the unconditional and conditional nearest-neighbor distance methods, the Hull model, and preprocessing tools. The article also shows a few actual analyses carried out with these tools.

The critical probability for random Voronoi percolation in the plane is 1/2

Abstract: We study percolation in the following random environment: let Z be a Poisson process of constant intensity on ℝ2, and form the Voronoi tessellation of ℝ2 with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p>1/2 then the union of the black cells contains an infinite component with probability 1, while if p<1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten's results for bond percolation in ℤ2. The result corresponding to Harris' Theorem for bond percolation in ℤ2 is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten's Theorem for ℤ2; we hope they will be applicable in other contexts as well.

Grid-partition index: a hybrid method for nearest-neighbor queries in wireless location-based services

Abstract: Traditional nearest-neighbor (NN) search is based on two basic indexing approaches: object-based indexing and solution-based indexing. The former is constructed based on the locations of data objects: using some distance heuristics on object locations. The latter is built on a precomputed solution space. Thus, NN queries can be reduced to and processed as simple point queries in this solution space. Both approaches exhibit some disadvantages, especially when employed for wireless data broadcast in mobile computing environments.In this paper, we introduce a new index method, called the grid-partition index, to support NN search in both on-demand access and periodic broadcast modes of mobile computing. The grid-partition index is constructed based on the Voronoi diagram, i.e., the solution space of NN queries. However, it has two distinctive characteristics. First, it divides the solution space into grid cells such that a query point can be efficiently mapped into a grid cell around which the nearest object is located. This significantly reduces the search space. Second, the grid-partition index stores the objects that are potential NNs of any query falling within the cell. The storage of objects, instead of the Voronoi cells, makes the grid-partition index a hybrid of the solution-based and object-based approaches. As a result, it achieves a much more compact representation than the pure solution-based approach and avoids backtracked traversals required in the typical object-based approach, thus realizing the advantages of both approaches.We develop an incremental construction algorithm to address the issue of object update. In addition, we present a cost model to approximate the search cost of different grid partitioning schemes. The performances of the grid-partition index and existing indexes are evaluated using both synthetic and real data. The results show that, overall, the grid-partition index significantly outperforms object-based indexes and solution-based indexes. Furthermore, we extend the grid-partition index to support continuous-nearest-neighbor search. Both algorithms and experimental results are presented.

Automated Thiessen polygon generation

Abstract: Data uncertainty research of rain gauge network requires generation of large numbers of Thiessen polygons. Despite its importance in hydrology, few studies on computational Thiessen polygons have been carried out, and there is little published information in the hydrological literature. This paper describes two automated approaches and the ways for their implementation in hydrological applications: triangulation method and grid method. Triangulation is a lossless method but suffers from complications in coding and slow computational speed with small numbers of gauges. Grid method is easy to implement, but a compromise must be made between the computational grid size, accuracy, and speed. This paper describes a procedure to derive the relationship between the catchment area, grid size, and accuracy indicator based on weighted mean error. The computational speed comparison between the two methods has been found to follow a logarithm curve, and the critical number of gauges could be found from this curve for deciding the method choice if the computational speed is the limiting factor in a project.

Interactive 3D distance field computation using linear factorization

Abstract: We present an interactive algorithm to compute discretized 3D Euclidean distance fields. Given a set of piecewise linear geometric primitives, our algorithm computes the distance field for each slice of a uniform spatial grid. We express the non-linear distance function of each primitive as a dot product of linear factors. The linear terms are efficiently computed using texture mapping hardware. We also improve the performance by using culling techniques that reduce the number of distance function evaluations using bounds on Voronoi regions of the primitives. Our algorithm involves no preprocessing and is able to handle complex deforming models at interactive rates. We have implemented our algorithm on a PC with NVIDIA GeForce 7800 GPU and applied it to models composed of thousands of triangles. We demonstrate its application to medial axis approximation and proximity computations between rigid and deformable models. In practice, our algorithm is more accurate and almost one order of magnitude faster as compared to previous distance computation algorithms that use graphics hardware.

Sensor-based coverage with extended range detectors

Abstract: Coverage path planning determines a path that passes a robot, a detector, or some type of effector over all points in the environment. Prior work in coverage tends to fall into one of two extremes: coverage with an effector the same size of the robot, and coverage with an effector that has infinite range. In this paper, we consider coverage in the middle of this spectrum: coverage with a detector range that goes beyond the robot, and yet is still finite in range. We achieve coverage in two steps: The first step considers vast, open spaces, where the robot can use the full range of its detector; the robot covers these spaces as if it were as big as its detector range. Here we employ previous work in using Morse cell decompositions to cover unknown spaces. A cell in this decomposition can be covered via simple back-and-forth motions, and coverage of the vast space is then reduced to ensuring that the robot visits each cell in the vast space. The second step considers the narrow or cluttered spaces where obstacles lie within detector range, and thus the detector "fills" the surrounding area. In this case, the robot can cover the cluttered space by simply following the generalized Voronoi diagram (GVD) of that space. In this paper, we introduce a hierarchical decomposition that combines the Morse decompositions and the GVDs to ensure that the robot indeed visits all vast, open, as well as narrow, cluttered, spaces. We show how to construct this decomposition online with sensor data that is accumulated while the robot enters the environment for the first time.

An algorithm for three‐dimensional Voronoi S‐network

Abstract: The paper presents an algorithm for calculating the three-dimensional Voronoi-Delaunay tessellation for an ensemble of spheres of different radii (additively-weighted Voronoi diagram). Data structure and output of the algorithm is oriented toward the exploration of the voids between the spheres. The main geometric construct that we develop is the Voronoi S-network (the network of vertices and edges of the Voronoi regions determined in relation to the surfaces of the spheres). General scheme of the algorithm and the key points of its realization are discussed. The principle of the algorithm is that for each determined site of the network we find its neighbor sites. Thus, starting from a known site of the network, we sequentially find the whole network. The starting site of the network is easily determined based on certain considerations. Geometric properties of ensembles of spheres of different radii are discussed, the conditions of applicability and limitations of the algorithm are indicated. The algorithm is capable of working with a wide variety of physical models, which may be represented as sets of spheres, including computer models of complex molecular systems. Emphasis was placed on the issue of increasing the efficiency of algorithm to work with large models (tens of thousands of atoms). It was demonstrated that the experimental CPU time increases linearly with the number of atoms in the system, O(n).

Centroidal Voronoi Tessellation-Based Reduced-Order Modeling of Complex Systems

Abstract: oindent A reduced-order modeling methodology based on centroidal Voronoi tessellations (CVTs) is introduced. CVTs are special Voronoi tessellations for which the generators of the Voronoi diagram are also the centers of mass (means) of the corresponding Voronoi cells. For discrete data sets, CVTs are closely related to the h-means and k-means clustering techniques. A discussion of reduced-order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced-order bases. Then, detailed descriptions of CVT-based reduced-order bases and how they can be constructed from snapshot sets and how they can be applied to the low-cost simulation of complex systems are given. Subsequently, some concrete incompressible flow examples are used to illustrate the construction and use of CVT-based reduced-order bases. The CVT-based reduced-order modeling methodology is shown to be effective for these examples.

Deformable spanners and applications

Abstract: For a set S of points in Rd, an s-spanner is a subgraph of the complete graph with node set S such that any pair of points is connected via some path in the spanner whose total length is at most s times the Euclidean distance between the points. In this paper we propose a new sparse (1 + e)-spanner with O(n/ed) edges, where e is a specified parameter. The key property of this spanner is that it can be efficiently maintained under dynamic insertion or deletion of points, as well as under continuous motion of the points in both the kinetic data structures setting and in the more realistic blackbox displacement model we introduce. Our deformable spanner succinctly encodes all proximity information in a deforming point cloud, giving us efficient kinetic algorithms for problems such as the closest pair, the near neighbors of all points, approximate nearest neighbor search (aka approximate Voronoi diagram), well-separated pair decompositions, and approximate k-centers.

Discrete Sibson interpolation

Abstract: Natural-neighbor interpolation methods, such as Sibson's method, are well-known schemes for multivariate data fitting and reconstruction. Despite its many desirable properties, Sibson's method is computationally expensive and difficult to implement, especially when applied to higher-dimensional data. The main reason for both problems is the method's implementation based on a Voronoi diagram of all data points. We describe a discrete approach to evaluating Sibson's interpolant on a regular grid, based solely on finding nearest neighbors and rendering and blending d-dimensional spheres. Our approach does not require us to construct an explicit Voronoi diagram, is easily implemented using commodity three-dimensional graphics hardware, leads to a significant speed increase compared to traditional approaches, and generalizes easily to higher dimensions. For large scattered data sets, we achieve two-dimensional (2D) interpolation at interactive rates and 3D interpolation (3D) with computation times of a few seconds.

Efficient Generation of Poisson-Disk Sampling Patterns

Abstract: Poisson-disk sampling patterns are of interest to the graphics community because their blue-noise properties are desirable in sampling patterns for rendering, illumination, and other computations. Until now, such patterns could only be generated by slow methods with poor convergence, or could only be approximated by jittering individual samples or tiling sets of samples. We present a simple and efficient randomized algorithm for generating true Poissondisk sampling patterns in a square domain, given a minimum radius R between samples. The algorithm runs in O(N log N) time for a pattern of N points. The method is based on the Voronoi diagram. Source code is available online.

Curved Voronoi diagrams

Abstract: Voronoi diagrams are fundamental data structures that have been extensively studied in Computational Geometry. A Voronoi diagram can be defined as the minimization diagram of a finite set of continuous functions. Usually, each of those functions is interpreted as the distance function to an object. The as- sociated Voronoi diagram subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. Affine di- agrams, i.e. diagrams whose cells are convex polytopes, are well understood. Their properties can be deduced from the properties of polytopes and they can be constructed efficiently. The situation is very different for Voronoi dia- grams with curved regions. Curved Voronoi diagrams arise in various contexts where the objects are not punctual or the distance is not the Euclidean dis- tance. We survey the main results on curved Voronoi diagrams. We describe in some detail two general mechanisms to obtain effective algorithms for some classes of curved Voronoi diagrams. The first one consists in linearizing the diagram and applies, in particular, to diagrams whose bisectors are algebraic hypersurfaces. The second one is a randomized incremental paradigm that can construct affine and several planar non-affine diagrams. We finally introduce the concept of Medial Axis which generalizes the concept of Voronoi diagram to infinite sets. Interestingly, it is possible to efficiently construct a certified approximation of the medial axis of a bounded set from the Voronoi diagram of a sample of points on the boundary of the set.

Fast proximity computation among deformable models using discrete Voronoi diagrams

Abstract: We present novel algorithms to perform collision and distance queries among multiple deformable models in dynamic environments. These include inter-object queries between different objects as well as intra-object queries. We describe a unified approach to compute these queries based on N-body distance computation and use properties of the 2nd order discrete Voronoi diagram to perform N-body culling. Our algorithms involve no preprocessing and also work well on models with changing topologies. We can perform all proximity queries among complex deformable models consisting of thousands of triangles in a fraction of a second on a high-end PC. Moreover, our Voronoi-based culling algorithm can improve the performance of separation distance and penetration queries by an order of magnitude.

Revisiting the Voronoi description of protein-protein interfaces.

Abstract: We developed a model of macromolecular interfaces based on the Voronoi diagram and the related alpha-complex, and we tested its properties on a set of 96 protein–protein complexes taken from the Protein Data Bank. The Voronoi model provides a natural definition of the interfaces, and it yields values of the number of interface atoms and of the interface area that have excellent correlation coefficients with those of the classical model based on solvent accessibility. Nevertheless, some atoms that do not lose solvent accessibility are part of the interface defined by the Voronoi model. The Voronoi model provides robust definitions of the curvature and of the connectivity of the interfaces, and leads to estimates of these features that generally agree with other approaches. Our implementation of the model allows an analysis of protein–water contacts that highlights the role of structural water molecules at protein–protein interfaces.

Quasi-triangulation and interworld data structure in three dimensions

Abstract: It is well-known that the Voronoi diagram of points and the power diagram for weighted points, such as spheres, are cell complexes, and their respective dual structures, i.e. the Delaunay triangulation and the regular triangulation, are simplicial complexes. Hence, the topologies of these diagrams are usually stored in their dual complexes using a very compact data structure of arrays. The topology of the Voronoi diagram of three-dimensional spheres in the Euclidean distance metric, on the other hand, is stored in a radial edge data structure which is not as compact as the data structure used for the Voronoi diagram of points and the power diagram for weighted points. In this paper, we define a dual structure of the Voronoi diagram of three-dimensional spheres called a quasi-triangulation and present its important properties. Based on the properties of a quasi-triangulation, we propose a data structure, called an interworld data structure, based on arrays to compactly store the topology of the quasi-triangulation with a guaranteed query performance.

Voronoi formulas on GL(n)

Abstract: In this paper, we give a new, simple, purely analytic proof of the Voronoi formula for Maass forms on GL(3) first derived by Miller and Schmid Our method is based on two lemmas of the first author and Thillainatesan which appear in their recent non-adelic proof of the converse theorem on GL(3) Using a different, even simpler method we derive Voronoi formulas on GL(n) twisted by additive characters of prime conductors We expect that this method will work in general In the final section of the paper Voronoi formulas on GL(n) are obtained, but in this case, the twists are by automorphic forms from lower rank groups

Fast nearest neighbor search on road networks

Abstract: Nearest neighbor (NN) queries have been extended from Euclidean spaces to road networks Existing approaches are either based on Dijkstra-like network expansion or NN/distance precomputation The former may cause an explosive number of node accesses for sparse datasets because all nodes closer than the NN to the query must be visited The latter, eg, the Voronoi Network Nearest Neighbor (VN3) approach, can handle sparse datasets but is inappropriate for medium and dense datasets due to its high precomputation and storage overhead In this paper, we propose a new approach that indexes the network topology based on a novel network reduction technique It simplifies the network by replacing the graph topology with a set of interconnected tree-based structures called SPIE’s An nd index is developed for each SPIE and our new (k)NN search algorithms on an SPIE follow a predetermined tree path to avoid costly network expansion By mathematical analysis and experimental results, our new approach is shown to be efficient and robust for various network topologies and data distributions

Acceleration schemes for computing centroidal Voronoi tessellations

Abstract: Centroidal Voronoi tessellations (CVT) have diverse applications in many areas of science and engineering. The development of efficient algorithms for their construction is a key to their success in practice. In this paper, we study some new algorithms for the numerical computation of the CVT, including the Lloyd–Newton iteration and the optimization based multilevel method. Both theoretical analysis and computational simulations are conducted. Rigorous convergence results are presented and significant speedup in computation is demonstrated through the comparison with traditional methods. Copyright © 2006 John Wiley & Sons, Ltd.