Abstract: Let S be a finite set of points in the Euclidean plane. Let D be a Delaunay triangulation of S. The stretch factor (also known as dilation or spanning ratio) of D is the maximum ratio, among all points p and q in S, of the shortest path distance from p to q in D over the Euclidean distance ||pq||. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long standing open problem in computational geometry.In this paper we prove that the stretch factor of the Delaunay triangulation of a set of points in the plane is less than ρ = 1.998, improving the previous best upper bound of 2.42 by Keil and Gutwin (1989).

Abstract: We propose a new algorithm that preprocess a set of n disjoint unit disks to be able to compute the Delaunay triangulation in O(n) expected time. Conversely to previous similar results, our algorithm is actually faster than a direct computation in O(n log n) time.

Abstract: In surface metrology, morphological filters, which evolved from the envelope filtering system (E-system) work well for functional prediction of surface finish in the analysis of surfaces in contact. The naive algorithms are time consuming, especially for areal data, and not generally adopted in real practice. A fast algorithm is proposed based on the alpha shape. The hull obtained by rolling the alpha ball is equivalent to the morphological opening/closing in theory. The algorithm depends on Delaunay triangulation with time complexity O(nlogn). In comparison to the naive algorithms it generates the opening and closing envelope without combining dilation and erosion. Edge distortion is corrected by reflective padding for open profiles/surfaces. Spikes in the sample data are detected and points interpolated to prevent singularities. The proposed algorithm works well both for morphological profile and area filters. Examples are presented to demonstrate the validity and superiority on efficiency of this algorithm over the naive algorithm.

Abstract: This paper revisits the k-nearest-neighbor (k-NN) Voronoi diagram and presents the first output-sensitive paradigm for its construction. It introduces the k-NN Delaunay graph, which corresponds to the graph theoretic dual of the k-NN Voronoi diagram, and uses it as a base to directly compute the k-NN Voronoi diagram in R2. In the L1, L∞ metrics this results in O((n + m) log n) time algorithm, using segment-dragging queries, where m is the structural complexity (size) of the k-NN Voronoi diagram of n point sites in the plane. The paper also gives a tighter bound on the structural complexity of the k-NN Voronoi diagram in the L∞ (equiv. L1) metric, which is shown to be O(min{k(n - k), (n - k)2}).

Abstract: A simple and efficient method is presented in this paper to reliably reconstruct 2D polygonal curves and 3D triangular surfaces from discrete points based on the respective clustering of Delaunay circles and spheres. A Delaunay circle is the circumcircle of a Delaunay triangle in the 2D space, and a Delaunay sphere is the circumsphere of a Delaunay tetrahedron in the 3D space. The basic concept of the presented method is that all the incident Delaunay circles/spheres of a point are supposed to be clustered into two groups along the original curve/surface with satisfactory point density. The required point density is considered equivalent to that of meeting the well-documented r -sampling condition. With the clustering of Delaunay circles/spheres at each point, an initial partial mesh can be generated. An extrapolation heuristic is then applied to reconstructing the remainder mesh, often around sharp corners. This leads to the unique benefit of the presented method that point density around sharp corners does not have to be infinite. Implementation results have shown that the presented method can correctly reconstruct 2D curves and 3D surfaces for known point cloud data sets employed in the literature.

Abstract: We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality. Our comparison-based algorithm runs in O(n log n + m) time, where n is the input size and m is the output size, and with constants depending only on the dimension and the desired element quality. It can terminate early in O(n log n) time returning a O(n) size Voronoi diagram of a superset of P, which again matches the known lower bounds.The previous best results in the comparison model depended on the log of the spread of the input, the ratio of the largest to smallest pairwise distance. We reduce this dependence to O(log n) by using a sequence of e-nets to determine input insertion order into a incremental Voronoi diagram. We generate a hierarchy of well-spaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.

Abstract: We give a definition of the Delaunay triangulation of a point set in a closed Euclidean d-manifold, i.e. a compact quotient space of the Euclidean space for a discrete group of isometries (a so-called Bieberbach group or crystallographic group). We describe a geometric criterion to check whether a partition of the manifold actually forms a triangulation (which subsumes that it is a simplicial complex). We provide an algorithm to compute the Delaunay triangulation of the manifold for a given set of input points, if it exists. Otherwise, the algorithm returns the Delaunay triangulation of a finitely-sheeted covering space of the manifold. The algorithm has optimal randomized worst-case time and space complexity.Whereas there was prior work for the special case of the flat torus, as far as we know this is the first result for general closed Euclidean d-manifolds. This research is motivated by application fields, like computational biology for instance, showing a need to perform simulations in quotient spaces of the Euclidean space by more general groups of isometries than the groups generated by d independent translations.

Abstract: Delaunay triangulation is a common mesh generation method in scientific computation. This parallel 3D Delaunay triangulation method uses domain-decomposition approach. With the properties of Delaunay triangulation, this method devise algorithm when merge block triangulations. To reduce the communications between processors, it finds the 3D affected zone that may be modified during the merge of two sub-block triangulations. The merging triangulation can be generated with the search just on the boundary of block triangulations.

Abstract: We analyze, implement, and evaluate a distribution- sensitive point location algorithm based on the classical Jump & Walk, called Keep, Jump, & Walk. For a batch of query points, the main idea is to use previous queries to improve the current one. In practice, Keep, Jump, & Walk is actually a very competitive method to locate points in a triangulation. Regarding point location in a Delaunay triangulation, we show how the Delaunay hierarchy can be used to answer, under some hypotheses, a query q with a O(log#(pq)) randomized expected complexity, where p is a previously located query and #(s) indicates the number of simplices crossed by the line segment s. The Delaunay hierarchy has O(n log n) time complexity and O(n) memory complexity in the plane, and under certain realistic hypothe- ses these complexities generalize to any finite dimension. Fi- nally, we combine the good distribution-sensitive behavior of Keep, Jump, & Walk, and the good complexity of the Delaunay hierarchy, into a novel point location algorithm called Keep, Jump, & Climb. To the best of our knowl- edge, Keep, Jump, & Climb is the first practical distribution- sensitive algorithm that works both in theory and in practice for Delaunay triangulations.

Abstract: Over the past decade, the kinetic-data-structures framework has become the standard in computational geometry for dealing with moving objects. A fundamental assumption underlying the framework is that the motions of the objects are known in advance. This assumption severely limits the applicability of KDSs. We study KDSs in the black-box model, which is a hybrid of the KDS model and the traditional time-slicing approach. In this more practical model we receive the position of each object at regular time steps and we have an upper bound on dmax, the maximum displacement of any point in one time step. We study the maintenance of the convex hull and the Delaunay triangulation of a planar point set P in the black-box model, under the following assumption on dmax: there is some constant k such that for any point p ∑ P the disk of radius dmax contains at most k points. We analyze our algorithms in terms of ∑k , the so-called k-spread of P. We show how to update the convex hull at each time step in O(k∑k log2 n) amortized time. For the Delaunay triangulation our main contribution is an analysis of the standard edge-flipping approach; we show that the number of flips is O(k2 ∑k2) at each time step.

Abstract: The theoretical complexity of vertex removal in a Delaunay triangulation is often given in terms of the degree d of the removed point, with usual results O(d), O(dlogd), or O(d^2). In fact, the asymptotic complexity is of poor interest since d is usually quite small. In this paper we carefully design code for small degrees 3=

Abstract: We present a heuristic for the Euclidean Steiner tree problem in R d for d≥2. The algorithm utilizes the Delaunay triangulation to generate candidate Steiner points for insertion, the minimum spanning tree to identify the Steiner points to remove, and second-order cone programming to optimize the location of the remaining Steiner points. Unlike other ESTP heuristics relying upon Delaunay triangulation, we insert Steiner points probabilistically into Delaunay triangles to achieve different subtrees on subsets of terminal points. We govern this neighbor generation procedure with a local search framework that extends effectively into higher dimensions. We present computational results on benchmark test problems in R d for 2≤d≤5.

Abstract: This article presents an algorithm to construct constrained Delaunay tetrahedralizations of geometric domains bounded by piecewise smooth surfaces. Meshes are built from the bottom-up by first discretizing the boundary curves and then by sampling the smooth surfaces. The sampling procedure refines the Delaunay triangulation restricted to these surfaces, targeting topological violations and poor quality triangles. Unlike previously published algorithms adopting a similar approach, we propose to sample each smooth surface patch independently. This obviates the need for a boundary protection scheme around small dihedral angles in the input and can also lead to coarser constraining triangulations. Starting from a Delaunay tetrahedralization of the point samples, a combination of mesh reconfigurations and vertex insertions is then used to obtain a tetrahedralization constrained to the boundary surfaces. The algorithm is designed to produce tetrahedralizations that can be used in conjunction with a Delaunay refinement algorithm implemented on a Bowyer-Watson framework.

Abstract: We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree. Both algorithms run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by Krznaric and Levcopolous [40] and Buchin and Mulzer [10]. Our main tool for the second algorithm is the well-separated pair decomposition (WSPD) [13], a structure that has been used previously to find Euclidean minimum spanning trees in higher dimensions [27]. We show that knowing the WSPD (and a quadtree) suffices to compute a planar EMST in linear time. With the EMST at hand, we can find the Delaunay triangulation in linear time [21].As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay triangulations, such as splitting planar Delaunay triangulations [19, 20], preprocessing imprecise points for faster Delaunay computation [9, 42], and transdichotomous Delaunay triangulations [10, 15, 16].

Abstract: Clustering is an essential tool in data mining that has drawn enormous attention. In this paper, we present a new clustering algorithm with the help of Voronoi diagram. Here the clusters are formed by considering the neighboring Voronoi cells. The points belong to the closer Voronoi cells are merged to form the clusters. The similarity of the points is measured based on Euclidean distance of the neighboring points and hence it is not necessary to compare the distances from one point to all other points of the given set. We perform various experiments using many synthetic and biological data sets. The experimental results demonstrate the significance of the proposed method.

Abstract: An unstructured triangulation approach, new to our knowledge, is proposed to apply triangular meshes for representing and rendering a scene on a cubic panorama (CP). It sophisticatedly converts a complicated three-dimensional triangulation into a simple three-step triangulation. First, a two-dimensional Delaunay triangulation is individually carried out on each face. Second, an improved polygonal triangulation is implemented in the intermediate regions of each of two faces. Third, a cobweblike triangulation is designed for the remaining intermediate regions after unfolding four faces to the top/bottom face. Since the last two steps well solve the boundary problem arising from cube edges, the triangulation with irregular-distribution feature points is implemented in a CP as a whole. The triangular meshes can be warped from multiple reference CPs onto an arbitrary viewpoint by face-to-face homography transformations. The experiments indicate that the proposed triangulation approach provides a good modeling for the scene with photorealistic rendered CPs.

Abstract: In the given paper the algorithm describing original and universal principles of a triangulation of a smooth molecular surface: solvent excluding solvent (SES), received by primary and secondary rolling, and solvent accessible surface (SAS) is presented. These surfaces are a boundary between molecule and solvent. Originality of the given paper consists in creation of the universal and adaptive algorithm of a triangulation. Universality of algorithm of a triangulation consists that it is suitable for not only for a surface, received by rolling and consisting of fragments of torus and sphere, but for any smooth surface, including any level surface. Adaptability of this algorithm consists in facts that the mesh size of a triangulation can vary depending on its location; reflecting even small, but smooth features of a surface; preventing "jump" to close, but not neighbor sites of the surface, excepting "cut off" of narrow necks and channels. It is reached by either decreasing triangulation lattice step to value smaller than two principal radiuses of curvature of the molecular surface or decreasing triangulation lattice step close to the active centre - closed, but not neighbor sites of the surface. The received triangulated surface can be used for the demonstration purposes in molecular editors (the algorithm is applicable for a triangulation of any smooth surface, for example, level surfaces) together with for calculation of solvation energy and its gradients for continual models of solvent.

Abstract: An important objective in the choice of a triangulation is that the smallest angle becomes as large as possible. In the straight-line case, it is known that the Delaunay triangulation is optimal in this respect. We propose and study the concept of a circular arc triangulation--a simple and effective alternative that offers flexibility for additionally enlarging small angles--and discuss its applications in graph drawing.

Abstract: Given an image or an image Delaunay Triangulation diagram representation, the purpose of this project is to identify a near-duplicate in an image database. This is done by computing the corner points of the image under some default settings, using the Voronoi diagram to adjust these settings in order to get better adjusted corner points for the image, and then using the Delaunay Triangulation of the image to identify whether there are possible duplicates in the database. The project investigates Voronoi Diagrams based on a default Harris Corner Detector as an image segmentation technique for adjusting the threshold for the Harris Corner Detector, as well as Delaunay Triangulation Diagrams as a measure of similarity between images. Ultimately, the project finds image nearduplicates in a database with some likelihood.

Abstract: A simple correspondence approach in camera calibration is proposed based on Delaunay triangulation to automatically match the image coordinates with the world coordinates for feature points with regular grid network structure.The topological structure of the feature points was generated by Delaunay triangulation and pretreatment which removes the redundant triangles.The correspondence between the image coordinates and world coordinates of feature points was implemented by using the topological structure.The experimental results show that the proposed approach is simple,robust,and not sensitive to lens distortion.The approach can still effectively implement the feature point correspondence when the pattern is viewed from an oblique angle and under occlusion condition.It is most suitable for on-line camera calibration.

Abstract: As modern high speed Package Circuit Boards (PCBs) and Integrated Circuit (IC) packages become more and more complicated, the electromagnetic simulation inside the PCBs and IC packages becomes more and more difficult. The most challenge work is that complicated multiple layer shapes inside PCBs and IC packages need very huge amount of high quality meshes for electromagnetic simulation, which lead to huge memory requirement for solving the linear matrix. For most personal computers, that requirement usually lead to out of memory in matrix solution. To overcome this problem without losing the simulation precision, this article presents an algorithm to simplify complex 2-D polygons based on Delaunay triangulation. By checking the smallest angles of triangles which contain an edge of the polygons, the algorithm simplifies polygons adaptively and updates the simplified polygons' Delaunay triangulation. Different from other simplifications which try to decrease the count of vertices, this algorithm tries to delete redundant short edges, leading to greatly reduced mesh points during the process of generating the constrained and quality Delaunay triangulation. The algorithm presented here has been applied for mesh generation for finite-element analysis of complicated PCBs and IC packages. Test results show that the proposed algorithm can reduce more than 75% in total nodes in comparison with the approach without simplification, resulting in significant reduction of computer resources in numerical computation.

Abstract: In this paper we present a new method for lossy compression of volumetric data that is based on data dependent triangulation. We have extended an approach that has previously been successfully applied in the case of two dimensional images. In our method we first select significant points in the data, and using them, a three dimensional Delaunay triangulation is created. The tetrahedrons existing in the triangulation are used as cells for a linear interpolation spline that gives an approximation of the original image. The compression is done by storing the positions and values of the nodes of the tetrahedrons instead of the entire data set. We compare our compression technique to JPG 2000 3D which is a de-facto standard for compression of volumetric data. Tests are done on different classes of data sets, on which we compare the bits per voxel needed to achieve the same level of peak signal to noise ration. We show that our algorithm performs significantly different than wavelet based compression, as in the implementation of JPG 2000 3D, and in case of data that is smooth outperforms it.

Abstract: We show how to compute Delaunay triangulations and Voronoi diagrams of a set of points in hyperbolic space in a very simple way. While the algorithm follows from [CCCG92], we elaborate on arithmetic issues, observing that only rational computations are needed. This allows an exact and efficient implementation.

Abstract: The power Voronoi diagrams are difficult to construct because of their complicated structures. In traditional algorithm, production process which is based on the Delaunay diagram was extremely complex. While dynamic algorithm is only concerned with positions of generators, so it is effective for constructing Voronoi diagrams with complicated shapes of Voronoi polygons. It can be applied to power Voronoi diagram with any generators, and can get over most shortcomings of traditional algorithm. So it is more useful and effective. Model is constructed with dynamic algorithm. And the application example shows that the algorithm is both simple and practicable and of high potential value in practice.