Abstract: Computational Geometry Methods and Intelligent Computing.- Generalized Voronoi Diagrams: State-of-the-Art in Intelligent Treatment of Applied Problems.- Shapes of Delaunay Simplexes and Structural Analysis of Hard Sphere Packings.- The ?-Shape and ?-Complex for Analysis of Molecular Structures.- Computational Geometry Analysis of Quantum State Space and Its Applications.- Efficient Swarm Neighborhood Management Using the Layered Delaunay Triangulation.- Intelligent Solutions for Curve Reconstruction Problem.- A Methodology for Automated Cartographic Data Input, Drawing and Editing Using Kinetic Delaunay/Voronoi Diagrams.- Density-Based Clustering Based on Topological Properties of the Data Set.- Modeling Optimal Beam Treatment with Weighted Regions for Bio-medical Applications.- Advanced Treatment of Topics of Special Interest.- Constructing Centroidal Voronoi Tessellations on Surface Meshes.- Simulated Annealing and Genetic Algorithms in Quest of Optimal Triangulations.- Higher Order Voronoi Diagrams and Distance Functions in Art and Visualization.- Robust Point-Location in Generalized Voronoi Diagrams.- Conclusions and Future Trends in Intelligent Treatment of Applied Problems.

Abstract: We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.

Abstract: This paper introduces a new algorithm for constrained Delaunay triangulation, which is built upon sets of points and constraining edges. It has various applications in geographical information system (GIS), for example, iso-lines triangulation or the triangulation of polygons in land cadastre. The presented algorithm uses a sweep-line paradigm combined with Lawson's legalisation. An advancing front moves by following the sweep-line. It separates the triangulated and non-triangulated regions of interest. Our algorithm simultaneously triangulates points and constraining edges and thus avoids consuming location of those triangles containing constraining edges, as used by other approaches. The implementation of the algorithm is also considerably simplified by introducing two additional artificial points. Experiments show that the presented algorithm is among the fastest constrained Delaunay triangulation algorithms available at the moment.

Abstract: Fields as found in the geosciences have properties that are not usually found in other disciplines: the phenomena studied are often three-dimensional (3D), they tend to change continuously over time, and the collection of samples to study the phenomena is problematic, which often results in highly anisotropic distributions of samples. In the geographical information system (GIS) community, raster structures (voxels or octrees) are the most popular solutions, but, as we show in this paper, they have shortcomings for modelling and analysing 3D geoscientific fields. As an alternative to using rasters, we propose a new spatial model based on the Voronoi diagram (VD) and its dual the Delaunay tetrahedralisation (DT), and argue that they have many advantages over other tessellations. We discuss the main properties of the 3D VD/DT, present some GIS operations that are greatly simplified when the VD/DT is used, and, to analyse two or more fields, we also present a variant of the map algebra framework where all the operations are performed directly on VDs. The usefulness of this Voronoi-based spatial model is demonstrated with a series of potential applications.

Abstract: We develop the first ever fully functional three-dimensional guaranteed quality parallel graded Delaunay mesh generator. First, we prove a criterion and a sufficient condition of Delaunay-independence of Steiner points in three dimensions. Based on these results, we decompose the iteration space of the sequential Delaunay refinement algorithm by selecting independent subsets from the set of the candidate Steiner points without resorting to rollbacks. We use an octree which overlaps the mesh for a coarse-grained decomposition of the set of candidate Steiner points based on their location. We partition the worklist containing poor quality tetrahedra into independent lists associated with specific separated leaves of the octree. Finally, we describe an example parallel implementation using a publicly available state-of-the art sequential Delaunay library (Tetgen). This work provides a case study for the design of abstractions and parallel frameworks for the use of complex labor intensive sequential codes on multicore architectures.

Abstract: A method for generating constrained Voronoi grids in a plane with internal features and boundaries is disclosed. The disclosed method generally includes approximation of internal features and boundaries with polylines based on plane geometry. Protected polygons or points are generated around the polylines, and Delaunay triangulation of protected points or protected polygon vertices is constructed. Delaunay triangulation that honors protected polygons or points is generated in the rest of the gridding domain. The constrained Voronoi grid is then generated from the Delaunay triangulation, which resolves all of the approximated features and boundaries with the edges of Voronoi cells. Constrained Voronoi grids may be generated with adaptive cell sizes based on specified density criterion.

Abstract: We propose a new approach for computing in an efficient way polygonal approximations of generalized 2D/3D Voronoi diagrams. The method supports distinct site shapes (points, line-segments, curved-arc segments, polygons, spheres, lines, polyhedra, etc.), different distance functions (Euclidean distance, convex distance functions, etc.) and is restricted to diagrams with connected Voronoi regions. The presented approach constructs a tree (a quadtree in 2D/an octree in 3D) which encodes in its nodes and in a compact way all the information required for generating an explicit representation of the boundaries of the Voronoi diagram approximation. Then, by using this hierarchical data structure a reconstruction strategy creates the diagram approximation. We also present the algorithms required for dynamically maintaining under the insertion or deletion of sites the Voronoi diagram approximation. The main features of our approach are its generality, efficiency, robustness and easy implementation.

Abstract: We present an isosurface meshing algorithm, DelIso, based on the Delaunay refinement paradigm. This paradigm has been successfully applied to mesh a variety of domains with guarantees for topology, geometry, mesh gradedness, and triangle shape. A restricted Delaunay triangulation, dual of the intersection between the surface and the three-dimensional Voronoi diagram, is often the main ingredient in Delaunay refinement. Computing and storing three-dimensional Voronoi/Delaunay diagrams become bottlenecks for Delaunay refinement techniques since isosurface computations generally have large input datasets and output meshes. A highlight of our algorithm is that we find a simple way to recover the restricted Delaunay triangulation of the surface without computing the full 3D structure. We employ techniques for efficient ray tracing of isosurfaces to generate surface sample points, and demonstrate the effectiveness of our implementation using a variety of volume datasets.

Abstract: An assumption of nearly all algorithms in computational geometry is that the input points are given precisely, so it is interesting to ask what is the value of imprecise information about points. We show how to preprocess a set of n disjoint unit disks in the plane in O(n log n) time so that if one point per disk is specified with precise coordinates, the Delaunay triangulation can be computed in linear time. From the Delaunay, one can obtain the Gabriel graph and a Euclidean minimum spanning tree; it is interesting to note the roles that these two structures play in our algorithm to quickly compute the Delaunay.

Abstract: We study the problem of two-dimensional Delaunay triangulation in the self-improving algorithms model [1]. We assume that the n points of the input each come from an independent, unknown, and arbitrary distribution. The first phase of our algorithm builds data structures that store relevant information about the input distribution. The second phase uses these data structures to efficiently compute the Delaunay triangulation of the input. The running time of our algorithm matches the information-theoretic lower bound for the given input distribution, implying that if the input distribution has low entropy, then our algorithm beats the standard Ω(n log n) bound for computing Delaunay triangulations.Our algorithm and analysis use a variety of techniques: e-nets for disks, entropy-optimal point-location data structures, linear-time splitting of Delaunay triangulations, and information-theoretic arguments.

Abstract: Greedy routing protocols for wireless sensor networks (WSNs) are fast and efficient but in general cannot guarantee message delivery. Hence researchers are interested in the problem of embedding WSNs in low dimensional space (e.g., R2) in a way that guarantees message delivery with greedy routing. It is well known that Delaunay triangulations are such embeddings. We present the algorithm FindAngles, which is a fast, simple, local distributed algorithm that computes a Delaunay triangulation from any given combinatorial graph that is Delaunay realizable. Our algorithm is based on a characterization of Delaunay realizability due to Hiroshima et al. (IEICE 2000). When compared to the PowerDiagram algorithm of Chen et al. (SoCG 2007), our algorithm requires on average 1/7th the number of iterations, scales better to larger networks, and has a much faster distributed implementation. The PowerDiagram algorithm was proposed as an improvement on another algorithm due to Thurston (unpublished, 1988). Our experiments show that on average the PowerDiagram algorithm uses about 18% fewer iterations than the Thurston algorithm, whereas our algorithm uses about 88% fewer iterations. Experimentally, FindAngles exhibits well behaved convergence. Theoretically, we prove that with certain initial conditions the error term decreases monotonically. Taken together, these suggest our algorithm may have polynomial time convergence for certain classes of graphs. We note that our algorithm runs only on Delaunay realizable triangulations. This is not a significant concern because Hiroshima et al. (IEICE 2000) indicate that most combinatorial triangulations are indeed Delaunay realizable, which we have also observed experimentally.

Abstract: The Centroidal Voronoi Diagram (CVD) is a very versatile structure, well studied in Computational Geometry. It is used as the basis for a number of applications. This paper presents a deterministic algorithm, entirely computed using graphics hardware resources, based on Lloyd's Method for computing CVDs. While the computation of the ordinary Voronoi diagram on GPU is a well explored topic, its extension to CVDs presents some challenges that the present study intends to overcome.

Abstract: This paper proposes a path planning algorithm for determining an optimal path with respect to the costs of a dual graph on the Constrained Delaunay Triangulation (CDT) of an environment. The advantages of using triangles for environment expression are: less data storage required, available mature triangulation methods and consistent with a potential motion planning framework. First we represent the polygon environment as a planar straight line graph (PSLG) described as a collection of vertices and segments, and then we adopt the CDT to partition the environment into triangles. Then on this CDT of the environment, a dual graph is constructed following the target attractive principle in order to avoid the nonoptimal paths caused by the different geometric size of the triangles. Correspondingly, a path planning algorithm via A* search algorithm finds an optimal path on the real-time building dual graph. In addition, completeness and optimization analysis of the algorithm is given. The simulation results demonstrate the effectiveness and optimization of the algorithm.

Abstract: The Delaunay triangulation and its dual the Voronoi diagram are ubiquitous geometric complexes. From a topological standpoint, the connection has recently been made between these cell complexes and the Morse theory of distance functions. In particular, in the generic setting, algorithms have been proposed to compute the flow complex--the stable and unstable manifolds associated to the critical points of the distance function to a point set. As algorithms ignoring degenerate cases and numerical issues are bound to fail on general inputs, this paper develops the first complete and robust algorithm to compute the flow complex.First, we present complete algorithms for the flow operator, unraveling a delicate interplay between the degenerate cases of Delaunay and those which are flow specific. Second, we sketch how the flow operator unifies the construction of stable and unstable manifolds. Third, we discuss numerical issues related to predicates on cascaded constructions. Finally, we report experimental results with CGAL's filtered kernel, showing that the construction of the flow complex incurs a small overhead w.r.t. the Delaunay triangulation when moderate cascading occurs. These observations provide important insights on the relevance of the flow complex for (surface) reconstruction and medial axis approximation, and should foster flow complex based algorithms.In a broader perspective and to the best of our knowledge, this paper is the first one reporting on the effective implementation of a geometric algorithm featuring cascading.

Abstract: We present efficient algorithms for approximating a height field using a piecewise-linear triangulated surface. The algorithms attempt to minimize both the error and the number of triangles in the approximation. The methods we examine are variants of the greedy insertion algorithm. This method begins with a simple triangulation of the domain as an initial approximation. It then iteratively finds the input point with highest error in the current approximation and inserts it as a vertex in the triangulation. We describe optimized algorithms using both Delaunay and data-dependent triangulation criteria. The algorithms have typical costs of O((m + n) logm), where n is the number of points in the input height field and m is the number of vertices in the final approximation. We also present empirical comparisons of several variants of the algorithms on large digital elevation models. We have made a C++ implementation of our algorithms publicly available.

Abstract: A newly developed polynomial preserving gradient recovery technique is further studied. The results are twofold. First, error bounds for the recovered gradient are established on the Delaunay type mesh when the major part of the triangulation is made of near parallelogram triangle pairs with e-perturbation. It is found that the recovered gradient improves the leading term of the error by a factor e. Secondly, the analysis is performed for a highly anisotropic mesh where the aspect ratio of element sides is unbounded. When the mesh is adapted to the solution that has significant changes in one direction but very little, if any, in another direction, the recovered gradient can be superconvergent. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Abstract: In this paper we propose an unstructured hybrid tessellation of a scattered point set that minimally covers the proximal space around each point. The mesh is automati- cally obtained in a bounded period of time by transforming an initial Delaunay tessellation. Novel types of polygonal interpolants are used for interpolation applications and the geometric qualities of the elements make them also useful for discretization schemes. The approach proves to be super- ior to classical Delaunay one in a finite element context.

Abstract: It is a well-known fact that, under mild sampling conditions, the restricted Delaunay triangulation provides good topological approximations of 1- and 2-manifolds. We show that this is not the case for higher-dimensional manifolds, even under stronger sampling conditions. Specifically, it is not true that, for any compact closed submanifold M of R n , and any sufficiently dense uniform sampling L of M, the Delaunay triangulation of L restricted to M is homeomorphic to M, or even homotopy equivalent to it. Besides, it is not true either that, for any sufficiently dense set W of witnesses, the witness complex of L relative to W contains or is contained in the restricted Delaunay triangulation of L.

Abstract: We study the problem of computing Voronoi diagrams distributedly for a set of nodes of a network modeled as a Unit Disk Graph (UDG). We present an algorithm to solve this problem eciently, which has direct applications in wireless networks. Comparing with some existing algorithms, our algorithm correctly computes the complete Voronoi diagram and uses a signicantly smaller number of transmissions. Furthermore, useful geometric structures such as the Delaunay triangulation and the convex hull can be obtained through our algorithm.

Abstract: Geometric spanners can be used for efficient routing in wireless ad hoc networks. Computation of existing spanners for ad hoc networks primarily focused on geometric properties without considering network requirements. In this paper, we propose a new spanner called constrained Delaunay triangulation (CDT) which considers both geometric properties and network requirements. The CDT is formed by introducing a small set of constraint edges into local Delaunay triangulation (LDel) to reduce the number of hops between nodes in the network graph. We have simulated the CDT using network simulator (ns-2.28) and compared with Gabriel graph (GG), relative neighborhood graph (RNG), local Delaunay triangulation (LDel), and planarized local Delaunay triangulation (PLDel). The simulation results show that the minimum number of hops from source to destination is less than other spanners. We also observed the decrease in delay, jitter, and improvement in throughput.

Abstract: High quality clustering is impossible without a priori information about clustering criteria. The paper describes the development of new clustering technique based on chaotic neural networks that overcomes the indeterminacy about number and topology of clusters. Proposed method of weights computation via Delaunay triangulation allows to cut down computing complexity of chaotic neural network clustering.

Abstract: We are interested in the following mesh refinement problem: given an input set of points P in R, we would like to produce a good-quality triangulation by adding new points in P . Algorithms for mesh refinement are typically incremental: they compute the Delaunay triangulation of the input, and insert points one by one. However, retriangulating after each insertion can take linear time. In this work we develop a query structure that maintains the mesh without paying the full cost of retriangulating. Assuming that the meshing algorithm processes bad-quality elements in increasing order of their size, our structure allows inserting new points and computing a restriction of the Voronoi cell of a point, both in constant time. We develop an example of such a meshing algorithm, and show that it produces a provably small mesh in time as fast as sorting the input plus writing the output.

Abstract: In this paper, with first introduce a new extension of Voronoi diagram. We assume Voronoi sites to be fuzzy sets and then define Voronoi diagram for this kind of sites, and provide an algorithm for computing this diagram for fuzzy sites. In the next part of the paper we change sites from set of points to set of fuzzy circles. Then we define the fuzzy Voronoi diagram for such sets and introduce an algorithm for computing it.

Abstract: This paper analyses the probability that randomly deployed sensor nodes triangulate any point within the target area. Its major result is the probability of triangulation for any point given the number of nodes lying up to a specific distance (2 units) from it, employing a graph representation where an edge exists between any two nodes close than 2 units from one another. The expected number of un-triangulated coverage holes, i.e. uncovered areas which cannot be triangulated by adjacent nodes, in a finite target area is derived. Simulation results corroborate the probabilistic analysis with low error, for any node density. These results will find applications in triangulation-based or trilateration-based pointing analysis, or any computational geometry application within the context of triangulation.

Abstract: Triangulation of a 3D point from two or more views can be solved in several ways depending on how perturbations in the image coordinates are dealt with. A common approach is optimal triangulation which minimizes the total L2 reprojection error in the images, corresponding to finding a maximum likelihood estimate of the 3D point assuming independent Gaussian noise in the image spaces. Computational approaches for optimal triangulation have been published for the stereo case and, recently, also for the three-view case. In short, they solve an independent optimization problem for each 3D point, using relatively complex computations such as finding roots of high order polynomials or matrix decompositions. This paper discuss three-view triangulation and reports the following results: (1) the 3D point can be computed as multi-linear mapping (tensor) applied on the homogeneous image coordinates, (2) the set of triangulation tensors forms a 7-dimensional space determined by the camera matrices, (3) given a set of corresponding 3D/2D calibration points, the 3D residual L1 errors can be optimized over the elements in the 7-dimensional space, (4) using the resulting tensor as initial value, the error can be further reduced by tuning the tensor in a two-step iterative process, (5) the 3D residual L1 error for a set of evaluation points which lie close to the calibration set is comparable to the three-view optimal method. In summary, three-view triangulation can be done by first performing an optimization of the triangulation tensor and once this is done, triangulation can be made with 3D residual error at the same level as the optimal method, but at a much lower computational cost. This makes the proposed method attractive for real-time three-view triangulation of large data sets provided that the necessary calibration process can be performed.

Abstract: In this video, we first review the incremental algorithm to compute Delaunay triangulations in R3 Then we examine the case of the periodic space T3, focusing on the differences with R3