Abstract: A general and direct computational scheme to locate the surface separating arbitrarily shaped domains made up of molecules (or any other particles) has been developed and is described and illustrated for several, both artificial and physical examples. The proposed scheme consists of two modules: (i) triangulation and (ii) assignment of simplices to domains. Three different triangulation methods are employed, viz., the Delaunay triangulation, regular triangulation, and quasi-triangulation. In the triangulated system, the assignment step is carried out in two different ways, one based on the characteristic metric of a particular triangulation procedure and the other on the concept of a touching sphere. Some of the combinations of the triangulation and assignment steps lead to methods already used by others to find interfacial or surface molecules, namely the alpha-shape-based method of Usabiaga nad Duque [Phys. Rev. E 79 (2009) 046709] and GITIM of Sega et al. [J. Chem. Phys. 138 (2013) 044110]. The resulting surface is defined not only as a discrete set of particles, but it is build up of facets of the triangulation forming a broken line in two dimensions or a polyhedral surface in three dimensions. Individual molecular layers are identified in a very straightforward manner, starting with the interfacial layer itself and proceeding into the interior of the phase. The proposed scheme is illustrated first by identifying border molecules of pre-sampled domains of several shapes in a plane and then applied to five physically meaningful examples: thin films, near critical water, liquid water slab in an electric field, liquid water at a solid wall, and water at condition of electric-field-induced jetting. Performance of the considered methods is critically assessed. Treatment of domains forming percolating clusters through periodic boundary conditions is also described along with the determination of their periodicity and dimensionality.

Abstract: Geometry-based localization algorithms come to the forefront with its unique superiority. This paper proposes a Delaunay triangulation based localization scheme (DBLS), which takes use of received signal strength indicator(RSSI) from anchor nodes. Based on RSSI, the Delaunay area is generated. The unknown node determines which Delaunay area it belongs to and then record it down. After several iteration, the overlapping region of all Delaunay areas is identified as the possible region where sensor resides in. By fixing the Delaunay area, we restrains the influence of noise on the result. The simulation result shows that DBLS has a better performance than Voronoi diagrams based localization scheme (VBLS) and centroid algorithm.

Abstract: In this paper, we propose a new graphics processing unit GPU method able to compute the 2D constrained Delaunay triangulation CDT of a planar straight-line graph consisting of points and segments. All existing methods compute the Delaunay triangulation of the given point set, insert all the segments, and then finally transform the resulting triangulation into the CDT. To the contrary, our novel approach simultaneously inserts points and segments into the triangulation, taking special care to avoid conflicts during retriangulations due to concurrent insertion of points or concurrent edge flips. Our implementation using the Compute Unified Device Architecture programming model on NVIDIA GPUs improves, in terms of running time, the best known GPU-based approach to the CDT problem.

Abstract: We present a data‐parallel algorithm for the construction of Delaunay triangulations on the sphere. Our method combines a variant of the classical Bowyer–Watson point insertion algorithm with the recently published parallelization technique by Jacobsen et al. It resolves a breakdown situation of the latter approach and is suitable for practical implementation because of its compact formulation. Some complementary aspects are discussed such as the parallel workload and floating‐point arithmetics. In a second step, the generated triangulations are reordered by a stripification algorithm. This improves cache performance and significantly reduces data‐read operations and indirect addressing in multi‐threaded stencil loops. This paper is an extended version of our Parallel Processing and Applied Mathematics conference contribution. Copyright © 2016 John Wiley & Sons, Ltd.

Abstract: We propose the first working GPU algorithm for the 2D Delaunay refinement problem. Our algorithm adds Steiner points to an input planar straight line graph (PSLG) to generate a constrained Delaunay mesh with triangles having no angle smaller than an input θ. It is shown to run from a few times to an order of magnitude faster than the well-known Triangle software, which is the fastest CPU Delaunay mesh generator. Our implementation handles degeneracy and is numerically robust. It is proven to terminate with finite output size for an input PSLG with no angle smaller than 60° and θ ≥ 20.7°. In addition, we notice meshes generated by our algorithm are of similar sizes to that by Triangle, which has incorporated good consideration in keeping output small in size.

Abstract: Surface reconstruction of scattered data points is one of the challenging area where the main purpose is to produce a smooth surface. In this research, Delaunay triangulation method was used to construct surface of scattered data points for six different test functions. In certain cases some surface producing holes after scanning where it becomes difficulty to produce a smooth surface. This research intends to test the accuracy of Delaunay triangulation in generating different surface when the points of scattered data were removed. The points removed were according to the percentage of points and the new surface was generated for every removing point. The result of the study shows interpolating surface produced by the removing points and the total absolute error with mean absolute error was calculated and compared.

Abstract: The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Rieman-nian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This structure has been implemented and was shown to lead to good triangulations in R 2 and on surfaces embedded in R 3 as detailed in our experimental companion paper. In this paper, we study theoretical aspects of our structure. Given a finite set of points P in a domain Ω equipped with a Riemannian metric, we compare the discrete Riemannian Voronoi diagram of P to its Riemannian Voronoi diagram. Both diagrams have dual structures called the discrete Riemannian Delaunay and the Riemannian Delaunay complex. We provide conditions that guarantee that these dual structures are identical. It then follows from previous results that the discrete Riemannian Delaunay complex can be embedded in Ω under sufficient conditions, leading to an anisotropic triangulation with curved simplices. Furthermore, we show that, under similar conditions, the simplices of this triangulation can be straightened. 1998 ACM Subject Classification Computational Geometry and Object Modeling 1 Introduction Anisotropic triangulations are triangulations whose elements are elongated along prescribed directions. Anisotropic triangulations are known to be well suited when solving PDE's [10, 19, 24]. They can also significantly enhance the accuracy of a surface representation if the anisotropy of the triangulation conforms to the curvature of the surface [15]. Many methods to generate anisotropic triangulations are based on the notion of Rieman-nian metric and create triangulations whose elements adapt locally to the size and anisotropy prescribed by the local geometry. The numerous theoretical and practical results [1] of the Euclidean Voronoi diagram and its dual structure, the Delaunay triangulation, have pushed authors to try and extend these well-established concepts to the anisotropic setting. La-belle and Shewchuk [17] and Du and Wang [12] independently introduced two anisotropic Voronoi diagrams whose anisotropic distances are based on a discrete approximation of the Riemannian metric field. Contrary to their Euclidean counterpart, the fact that the dual of these anisotropic Voronoi diagrams is an embedded triangulation is not immediate, and, despite their strong theoretical foundations, the anisotropic Voronoi diagrams of Labelle and

Abstract: . In this work, we devise an efficient method for the land-use optimization problem based on Laguerre Voronoi diagram. Previous Voronoi diagram-based methods are more efficient and more suitable for interactive design than discrete optimization-based method, but, in many cases, their outputs do not satisfy area constraints. To cope with the problem, we propose a force-directed graph drawing algorithm, which automatically allocates generating points of Voronoi diagram to appropriate positions. Then, we construct a Laguerre Voronoi diagram based on these generating points, use linear programs to adjust each cell, and reconstruct the diagram based on the adjustment. We adopt the proposed method to the practical case study of Chiang Mai University’s allocated land for a mixed-use complex. For this case study, compared to other Voronoi diagram-based method, we decrease the land allocation error by 62.557 %. Although our computation time is larger than the previous Voronoi-diagram-based method, it is still suitable for interactive design.

Abstract: Computational geometry is a mathematical knowlege in the field related to the design and analysis of algorithm to solve geometry problems. Its can be applicated in the fields of mapping, robotics, geometry and so forth. A method can be used is Voronoi diagram. Voronoi diagram is a method of deviding the area to a smaller area based on the principle of the nearest neighboring. This method only used in 1-order voronoi diagram. In voronoi diagram there is a new variation named Highest Order Voronoi Diagram (HSVD). HSVD can be used for all orders voronoi diagram. However, these methods have disadvantage that accessing fragment use linear search. Consequently make data fragment searches to find the region to be slow and takes a long time. Therefore, in this paper will present a index structure that incoperates Highest Order Voronoi Diagrams into Quadtree. Quadtree index used is capable of cutting more than half of the original data. This algorithm makes the search regions faster than before.

Abstract: Computational geometry is concerned with the design and analysis of algorithms for geometrical problems. In addition, other more practically oriented, areas of computer science-such as computer graphics, computer-aided design, robotics, pattern recognition, and operation research-give rise to problems that inherently are geometrical. A method that is commonly used is Voronoi diagram. Voronoi diagram is a diagram that divides plane based on nearest neighbour approach. This method does not require checking objects one by one. In Voronoi diagram there is a new variation named Highest Order Voronoi Diagram. Highest order voronoi diagram can be used directly to identify the region for various type of spatial queries and can be used for all order Voronoi Diagram. From work related there is a method named FLIP. Deficiency from work related is data structure for construction are too high which makes construction of Highest Order Voronoi Diagram is heavy. Therefore, in this paper will present Left with Least-Angle Movement (LAM) method for construction Highest Order Voronoi Diagram that change data structure to scale up data that can be processed.

Abstract: Recently, methods have been proposed to reconstruct a 2-manifold surface from a sparse cloud of points estimated from an image sequence. Once a 3D Delaunay triangulation is computed from the points, the surface is searched by growing a set of tetrahedra whose boundary is maintained 2-manifold. Shelling is a step that adds one tetrahedron at once to the growing set. This paper surveys properties that helps to understand the shelling performances: shelling provides most tetrahedra enclosed by the final surface but it can " get stuck " or block in unexpected cases.

Abstract: We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges. We show that a phase transition occurs for sufficiently large activities and for sufficiently large potential parameter proving an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transition manifests itself in the mixed site-bond percolation of the corresponding random-cluster model. Our proofs rely mainly on geometric properties of Delaunay tessellations in $\mathbb{R}^2 $ and on recent studies [DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is a uniform bound on the number of connected components in the Delaunay graph which provides a novel approach to Delaunay Widom Rowlinson models based on purely geometric arguments. The interaction potential ensures that shorter Delaunay edges are more likely to be open and thus offsets the possibility of having an unbounded number of connected components.

Abstract: Summary The paper illustrates a parallel and distributed scheme for computing a planar Delaunay triangulation using a divide-and-conquer strategy in Cloud environment, which combines the incremental insertion algorithm and the divide-and-conquer method. The proposed hybrid algorithm for Delaunay triangulation construction is easy to be parallelized due to the dynamic pruned characteristic of the binary tree model used. Moreover, the Cloud platform decreases the communication overhead and improves data locality by making use of a data partitioning and integrating scheme offered by the map-reduce architecture. The implementation of the parallel and distributed version of the algorithm relied on a robust data structure called quad-edge, which implies the geometric relationship among the edges and vertexes adjacent. More importantly, the data are serialized easily and transmitted efficiently between different Cloud nodes; the algorithm is executed conveniently on PC clusters. We tested the parallel version of the algorithm on GeoKSCloud, a geographical knowledge service Cloud developed by our research team. Experimental results show that the proposed hybrid algorithm is efficient and competitive; it can be easily migrated and deployed in distributed and parallel computing environment, such as grid and Cloud. The parallel implementation of the hybrid algorithm has a good speed-up, while data communication is the crucial factor for the efficiency of the parallel version. Overall, the parallel version outperforms both the sequential divide-and-conquer algorithm and the sequential incremental insertion algorithm.

Abstract: In one procedure for finding the maximal prime decomposition of a Bayesian network or undirected graphical model, the first step is to create a minimal triangulation of the network, and a common and straightforward way to do this is to create a triangulation that is not necessarily minimal and then thin this triangulation by removing excess edges We show that the algorithm for thinning proposed in several previous publications is incorrect A different version of this algorithm is available in the R package gRbase, but its correctness has not previously been proved We prove that this version is correct and provide a simpler version, also with a proof We compare the speed of the two corrected algorithms in three ways and find that asymptotically their speeds are the same, neither algorithm is consistently faster than the other, and in a computer experiment the algorithm used by gRbase is faster when the original graph is large, dense, and undirected, but usually slightly slower when it is directed

Abstract: A R T I C L E I N F O A B S T R A C T Article history: Received: 18 May, 2017 Accepted: 15 June, 2017 Online: 11 July, 2017 In this paper a novel approach to extract 2D skeleton information (skeletonization) from natural image is proposed. The work presented here is the extension of our previous paper presented at the International Sympsosium on Multimedia 2016. In the past work, a threshold based method is utilized. Here the algorithm is further improved by using a better edge points detection and skeleton extraction. Furthermore the proposed method is compared with the Skeleton Strength Map (SSM) and shows better result visually and numerically (F-measure comparison).

Abstract: The randomized incremental construction (RIC) for building geometric data structures has been analyzed extensively, from the point of view of worst-case distributions. In many practical situations however, we have to face nicer distributions. A natural question that arises is: do the usual RIC algorithms automatically adapt when the point samples are nicely distributed. We answer positively to this question for the case of the Delaunay triangulation of e-nets. e-nets are a class of nice distributions in which the point set is such that any ball of radius e contains at least one point of the net and two points of the net are distance at least e apart. The Delaunay triangulations of e-nets are proved to have linear size; unfortunately this is not enough to ensure a good time complexity of the randomized incremental construction of the Delaunay triangulation. In this paper, we prove that a uniform random sample of a given size that is taken from an e-net has a linear sized Delaunay triangulation in any dimension. This result allows us to prove that the randomized incremental construction needs an expected linear size and an expected O(n log n) time. Further, we also prove similar results in the case of non-Euclidean metrics, when the point distribution satisfies a certain bounded expansion property; such metrics can occur, for example, when the points are distributed on a low-dimensional manifold in a high-dimensional ambient space.

Abstract: This paper investigates the scalability of the Delaunay triangulation (DT) based diversity preservation technique for solving many-objective optimization problems (MaOPs). Following the NSGA-II algorithm, the proposed optimizer with DT based density measurement (NSGAII-DT) determines the density of individuals according to the DT mesh built on the population in the objective space. To reduce the computing time, the population is projected onto a plane before building the DT mesh. Experimental results show that NSGA-II-DT outperforms NSGA-II on WFG problems with 4, 5 and 6 objectives. Two projection strategies using a unit plane and a least-squares plane in the objective space are investigated and compared. Our results also show that the former is more effective than the latter.

Abstract: Consider a weighted graph G whose vertices are points in the plane and edges are line segments between pairs of points whose weight is the Euclidean distance between its endpoints. A routing algorithm on G sends a message from any vertex s to any vertex t in G. The algorithm has a competitive ratio of c if the length of the path taken by the message is at most c times the length of the shortest path from s to t in G. It has a routing ratio of c if the length of the path is at most c times the Euclidean distance from s to t. The algorithm is online if it makes forwarding decisions based on (1) the k-neighborhood in G of the message’s current position (for constant k > 0) and (2) limited information stored in the message header.

Abstract: We present a method for reconstructing a 3D surface triangulation from an input point set The main component of the method is an algorithm that computes the restricted Voronoi diagram In our specific case, it corresponds to the intersection between the 3D Voronoi diagram of the input points and a set of disks centered at the points and orthogonal to the estimated normal directions The method does not require coherent normal orientations (just directions) Our algorithm is based on a property of the restricted Voronoi cells that leads to an embarrassingly parallel implementation We experimented our algorithm with scanned point sets with up to 100 million vertices that were processed within few minutes on a standard computer The complete implementation is provided

Abstract: In this paper, we propose a new graphics processing unit (GPU) method able to compute the 2D constrained Delaunay triangulation (CDT) of a planar straight-line graph consisting of points and segments. All existing methods compute the Delaunay triangulation of the given point set, insert all the segments, and then finally transform the resulting triangulation into the CDT. To the contrary, our novel approach simultaneously inserts points and segments into the triangulation, taking special care to avoid conflicts during retriangulations due to concurrent insertion of points or concurrent edge flips. Our implementation using the Compute Unified Device Architecture programming model on NVIDIA GPUs improves, in terms of running time, the best known GPU-based approach to the CDT problem.

Abstract: This paper presents a fine grain parallel version of the 3D Delaunay Kernel procedure using the OpenMP (Open Multi-Processing) API. A set S={p1,,pn} of n points is taken as input. S is initially sorted along a space-filling curve so that two points that are close in the insertion order are also close geometrically. The sorted set of points is then divided into M subsets Si, 1iM of equal size n/M. The multithreaded version of the Delaunay kernel inserts M points at a time in the triangulation. OpenMP barriers provide the required synchronization that is needed after each multiple insertion in order to avoid data races. This simple approach exhibits two standard problems of parallel computing: load imbalance and parallel overheads. Those two issues are addressed using a two-level version of the multithreaded Delaunay kernel. Tests show that triangulations of about a billion tetrahedra can be generated on a 32 core machine (Intel Xeon E5-4610 v2 @ 2.30GHz with 128GB of memory) in less that 3 minutes of wall clock time, with a speedup of 18 compared to the single-threaded implementation. A fine grain parallel Delaunay kernel algorithm is proposed.The method that is proposed allows to generate meshes with more than a billion tetrahedra in about two minutes.The implementation uses simple OpenMP constructs.