Showing papers on "Constrained Delaunay triangulation published in 2014"
TL;DR: The feature rejection algorithm for meshing (FRAM) is introduced to generate a high quality conforming Delaunay triangulation of a three-dimensional discrete fracture network (DFN) by prescribing a minimum length scale and then restricting the generation of the network to only create features of that size and larger.
Abstract: We introduce the feature rejection algorithm for meshing (FRAM) to generate a high quality conforming Delaunay triangulation of a three-dimensional discrete fracture network (DFN). The geometric features (fractures, fracture intersections, spaces between fracture intersections, etc.) that must be resolved in a stochastically generated DFN typically span a wide range of spatial scales and make the efficient automated generation of high-quality meshes a challenge. To deal with these challenges, many previous approaches often deformed the DFN to align its features with a mesh through various techniques including redefining lines of intersection as stair step functions and distorting the fracture edges. In contrast, FRAM generates networks on which high-quality meshes occur automatically by constraining the generation of the network. The cornerstone of FRAM is prescribing a minimum length scale and then restricting the generation of the network to only create features of that size and larger. The process is f...
141 citations
TL;DR: Experimental results on public databases and security analysis show that the Delaunay quadrangle-based system with topology code can achieve better performance and higher security level than theDelaunay triangle-basedSystem, the Delaunaay quadRangle- based system without topologycode, and some other similar systems.
Abstract: Although some nice properties of the Delaunay triangle-based structure have been exploited in many fingerprint authentication systems and satisfactory outcomes have been reported, most of these systems operate without template protection. In addition, the feature sets and similarity measures utilized in these systems are not suitable for existing template protection techniques. Moreover, local structural change caused by nonlinear distortion is often not considered adequately in these systems. In this paper, we propose a Delaunay quadrangle-based fingerprint authentication system to deal with nonlinear distortion-induced local structural change that the Delaunay triangle-based structure suffers. Fixed-length and alignment-free feature vectors extracted from Delaunay quadrangles are less sensitive to nonlinear distortion and more discriminative than those from Delaunay triangles and can be applied to existing template protection directly. Furthermore, we propose to construct a unique topology code from each Delaunay quadrangle. Not only can this unique topology code help to carry out accurate local registration under distortion, but it also enhances the security of template data. Experimental results on public databases and security analysis show that the Delaunay quadrangle-based system with topology code can achieve better performance and higher security level than the Delaunay triangle-based system, the Delaunay quadrangle-based system without topology code, and some other similar systems.
76 citations
TL;DR: A two-step indoor space subdivided into navigable and non-navigable areas considering human perceptions of the environment and human behaviour is described and the guidelines for generation of the navigation network and verification of the proposed model are presented.
Abstract: There are a number of great attempts to develop an indoor navigation that provide the most optimal path and guidance. Finding a way in large buildings can be a challenging task. In order to represent the real situation to a maximum extent, a representation of the whole room as one single indivisible object is not enough as such representation is very abstract and this could make the navigation difficult and may result into inefficient route planning. In order to provide a smooth navigation path, the presence of humans within the indoor environment and the natural movement of individuals should be taken into consideration. In this paper a two-step indoor space subdivision for indoor navigation is described. Firstly, the indoor space is subdivided into navigable and non-navigable areas considering human perceptions of the environment and human behaviour. Secondly, the navigable space is subdivided applying a constrained Delaunay triangulation. Finally, the guidelines for generation of the navigation network and verification of the proposed model are presented.
55 citations
TL;DR: An investigation into pedestrians’ routing behaviours within an indoor environment under normal, non-panic situations is presented, a network-based method using constrained Delaunay triangulation is adopted, and a utility-based model employing dynamic programming is developed.
Abstract: Modelling and prediction of pedestrian routing behaviours within known built environments has recently attracted the attention of researchers across multiple disciplines, owing to the growing demand on urban resources and requirements for efficient use of public facilities. This study presents an investigation into pedestrians’ routing behaviours within an indoor environment under normal, non-panic situations. A network-based method using constrained Delaunay triangulation is adopted, and a utility-based model employing dynamic programming is developed. The main contribution of this study is the formulation of an appropriate utility function that allows an effective application of dynamic programming to predict a series of consecutive waypoints within a built environment. The aim is to generate accurate sequence waypoints for the pedestrian walking path using only structural definitions of the environment as defined in a standard CAD format. The simulation results are benchmarked against those from the A ∗ algorithm, and the outcome positively indicates the usefulness of the proposed method in predicting pedestrians’ route selection activities.
34 citations
TL;DR: In this article, the Steiner points on edges are removed more systematically following a specific sequence in an alternative manner rather than a random selection commonly adopted in practice; whereas for Steiner Points on a facet, a weight on the steiner point adjacency would lead to an optimal order to facilitate their removal.
31 citations
TL;DR: A robust approach to image matching based on Hessian affine region detector and local Delaunay triangulation and affine invariant geometric constraint and experimental results indicate that the proposed method can get higher correctness of image matching than RANSAC based method.
31 citations
TL;DR: An intuitive framework for analyzing Delaunay refinement algorithms is presented that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and helps to solve the difficult problem of meshing nonmanifold domains with small angles.
Abstract: Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles. Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles. Unfortunately, this problem is not always soluble. A compromise is necessary. A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30° or greater and no angle is smaller than arcsin [ ( 3 / 2 ) sin ( ϕ / 2 ) ] ∼ ( 3 / 4 ) ϕ , where ϕ ⩽ 60 ° is the smallest angle separating two segments of the input domain. New angles smaller than 30° appear only near input angles smaller than 60°. In practice, the algorithm's performance is better than these bounds suggest. Another new result is that Ruppert's analysis technique can be used to reanalyze one of Chew's algorithms. Chew proved that his algorithm produces no angle smaller than 30° (barring small input angles), but without any guarantees on grading or number of triangles. He conjectures that his algorithm offers such guarantees. His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 26.5°, Chew's algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal.
27 citations
16 Nov 2014
TL;DR: In this article, a distributed-memory scalable parallel Delaunay and Voronoi tessellation algorithm is proposed that automatically determines which neighbor points need to be exchanged among the sub domains of a spatial decomposition.
Abstract: Computing a Voronoi or Delaunay tessellation from a set of points is a core part of the analysis of many simulated and measured datasets: N-body simulations, molecular dynamics codes, and LIDAR point clouds are just a few examples. Such computational geometry methods are common in data analysis and visualization, but as the scale of simulations and observations surpasses billions of particles, the existing serial and shared memory algorithms no longer suffice. A distributed-memory scalable parallel algorithm is the only feasible approach. The primary contribution of this paper is a new parallel Delaunay and Voronoi tessellation algorithm that automatically determines which neighbor points need to be exchanged among the sub domains of a spatial decomposition. Other contributions include periodic and wall boundary conditions, comparison of our method using two popular serial libraries, and application to numerous science datasets.
27 citations
TL;DR: This work proposes a new algorithm for vertex insertion, given a new vertex to be inserted into a CDT, that guarantees a new CDT including that vertex, and modify these algorithms to robustly succeed in practice for polygons whose vertices deviate from exact coplanarity.
Abstract: Constrained Delaunay tetrahedralizations (CDTs) are valuable for generating meshes of nonconvex domains and domains with internal boundaries, but they are difficult to maintain robustly when finite-precision coordinates yield vertices on a line that are not perfectly collinear and polygonal facets that are not perfectly flat. We experimentally compare two recent algorithms for inserting a polygonal facet into a CDT: a bistellar flip algorithm of Shewchuk (Proc. 19th Annual Symposium on Computational Geometry, June 2003) and a cavity retriangulation algorithm of Si and Gartner (Proc. Fourteenth International Meshing Roundtable, September 2005). We modify these algorithms to succeed in practice for polygons whose vertices deviate from exact coplanarity.
20 citations
01 Sep 2014-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: This paper compute the high-quality remeshing results for surfaces with different types of topologies and compares the method with several state-of-the-art approaches in anisotropic surface meshing by using the standard measurement criteria.
Abstract: This paper introduces a parameterization-based approach for anisotropic surface meshing. Given an input surface equipped with an arbitrary Riemannian metric, this method generates a metric-adapted mesh with user-specified number of vertices. In the proposed method, the edge length of the input surface is directly adjusted according to the given Riemannian metric at first. Then the adjusted surface is conformally embedded into a parametric 2D domain and a weighted Centroidal Voronoi Tessellation and its dual Delaunay triangulation are computed on the parametric domain. Finally the generated Delaunay triangulation can be mapped from the parametric domain to the original space, and the triangulation exhibits the desired anisotropic property. We compute the high-quality remeshing results for surfaces with different types of topologies and compare our method with several state-of-the-art approaches in anisotropic surface meshing by using the standard measurement criteria.
18 citations
TL;DR: The method generates fine and coarse grids and offers an accurate and good-quality grid and can be employed in the pre- and postprocessing stages to various possible meshing algorithms.
Abstract: Fractured reservoirs are complex domains where discrete fractures are internal constraining boundaries. The discrete fractures are discretized into intersected edges during a grid-generation process, and Delaunay triangulations are often used to represent complex structures. However, a Delaunay triangulation of a fractured medium generally does not conform to the fracture; recovering the fracture elements may violate the Delaunay empty-circle (2D) criterion and may lead to a low-quality triangulation. Refining the triangulation is not a practical solution in complex fractured media. A new approach combines both Gabriel and Delaunay triangulations. A modified Gabriel condition of edge-empty-circle is introduced and locally employed to quantify the quality of the fracture edges in 2D. The fracture edges violating the modified Gabriel criterion are released in the first stage. After that, a Delaunay triangulation is generated considering the rest of the fracture constraints. The released fracture edges are then approximated by the edges of the Delaunay triangles. The final representation of fractures might be slightly different, but a very accurate approximation is always maintained. The method generates fine and coarse grids and offers an accurate and good-quality grid. Numerical examples are presented to assess the performance and efficiency of the proposed method. Finally, the method can be employed in the pre- and postprocessing stages to various possible meshing algorithms.
Posted Content•
TL;DR: The proof that proximal Delaunay triangulation regions are convex polygons is proved and it is proved that every Delauny triangulating region has a local Leader uniform topology.
Abstract: A main result in this paper is the proof that proximal Delaunay triangulation regions are convex polygons. In addition, it is proved that every Delaunay triangulation region has a local Leader uniform topology.
08 Jun 2014
TL;DR: This work describes a provably good algorithm that generates high-quality meshes that are constrained Delaunay triangulations, rather than purely Delaunays, and proves that most mesh edges have lengths proportional to the domain's minimum local feature size.
Abstract: Most algorithms for guaranteed-quality tetrahedral mesh generation create Delaunay meshes. Delaunay triangulations have many good properties, but the requirement that all tetrahedra be Delaunay often forces mesh generators to overrefine where boundary polygons meet at small angles---that is, they produce too many tetrahedra, making them too small. Relaxing the Delaunay property makes it possible both to reduce overrefinement and to obtain higher-quality tetrahedra. We describe a provably good algorithm that generates high-quality meshes that are constrained Delaunay triangulations, rather than purely Delaunay. Given a piecewise linear domain free of small angles, our algorithm is guaranteed to construct a mesh in which every tetrahedron has a radius-edge ratio of 2 √2/3 = 1.63 or less. This is a substantial improvement over the usual bound of 2; it is obtained by relaxing the conditions in which the algorithm subdivides boundary triangles. Given a domain with small angles, our algorithm produces a mesh in which the quality guarantee is compromised only in specific places near small domain angles. We prove that most mesh edges have lengths proportional to the domain's minimum local feature size; the exceptions span small domain angles or lie on domain edges that participate in small angles. Our algorithm tends to generate meshes with fewer tetrahedra than purely Delaunay methods because it uses the constrained Delaunay property, not just vertex insertions, to enforce the conformity of the mesh to the domain boundaries. An implementation demonstrates that our algorithm does not overrefine near small domain angles.
TL;DR: In this paper, the advantages of Delaunay triangulation to extend the capabilities of particle tracking in three areas: discriminating irregularly shaped objects, which allows one to track items other than point features, combining time and space to better connect missing frames in trajectories; and identifying shape backbone.
Abstract: Particle tracking, the analysis of individual moving elements in time series of microscopic images, enables burgeoning new applications, but there is need to better resolve conformation and dynamics. Here we describe the advantages of Delaunay triangulation to extend the capabilities of particle tracking in three areas: (1) discriminating irregularly shaped objects, which allows one to track items other than point features; (2) combining time and space to better connect missing frames in trajectories; and (3) identifying shape backbone. To demonstrate the method, specific examples are given, involving analyzing the time-dependent molecular conformations of actin filaments and λ-DNA. The main limitation of this method, shared by all other clustering techniques, is the difficulty to separate objects when they are very close. This can be mitigated by inspecting locally to remove edges that are longer than their neighbors and also edges that link two objects, using methods described here, so that the combination of Delaunay triangulation with edge removal can be robustly applied to processing large data sets. As common software packages, both commercial and open source, can construct Delaunay triangulation on command, the methods described in this paper are both computationally efficient and easy to implement.
TL;DR: An algorithm is presented that takes as input a finite point set in Euclidean space, and performs a perturbation that guarantees that the Delaunay triangulation of the resulting perturbed point set has quantifiable stability with respect to the metric and the point positions.
Abstract: We present an algorithm that takes as input a finite point set in Euclidean space, and performs a perturbation that guarantees that the Delaunay triangulation of the resulting perturbed point set has quantifiable stability with respect to the metric and the point positions. There is also a guarantee on the quality of the simplices: they cannot be too flat. The algorithm provides an alternative tool to the weighting or refinement methods to remove poorly shaped simplices in Delaunay triangulations of arbitrary dimension, but in addition it provides a guarantee of stability for the resulting triangulation.
TL;DR: The enriched meshing method is advantageous in generating a high resolving mesh and can be used for a wide range of thermal, hydraulic, mechanical, chemical simulations and their numerical coupling system in a fractured medium.
TL;DR: In this article, a modified octree approach is used to generate polyhedral Delaunay meshes, where the vertices of each polyhedron are co-spherical and each polyhedral circum- sphere is point-free.
TL;DR: The results show that the proposed parallel algorithm is efficient for constructing the Delaunay triangulation with a good speed-up.
Abstract: To increase the efficiency when processing large data sets, a novel parallel algorithm is proposed for constructing the Delaunay triangulation of a planar point set based on a twofold-divide-and-conquer scheme. This algorithm automatically divides the planar point set into several non-overlapping subsets along the x-axis and y-axis directions alternately, according to the number of points and their spatial distribution. Next, the Guibas–Stolfi divide-and-conquer algorithm is applied to construct Delaunay sub-triangulations in each subset. Finally, the sub-triangulations are merged based on the binary tree. All three sequential steps are processed using multitasking parallel technology. Our results show that the proposed parallel algorithm is efficient for constructing the Delaunay triangulation with a good speed-up.
02 Jul 2014
TL;DR: The sequence of adjacent nodes (graph walk) visited by a routing algorithm on a graph G between given source and target nodes s and t is a c-competitive route if its length is at most c times the length of the shortest path from s to t in G.
Abstract: The sequence of adjacent nodes (graph walk) visited by a routing algorithm on a graph G between given source and target nodes s and t is a c-competitive route if its length in G is at most c times the length of the shortest path from s to t in G. We present 21.766-, 17.982- and 15.479-competitive online routing algorithms on the Delaunay triangulation of an arbitrary given set of points in the plane. This improves the competitive ratio on Delaunay triangulations from the previous best of 45.749. We present a 7.621-competitive online routing algorithm for Delaunay triangulations of point sets in convex position.
TL;DR: This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms and gives explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh.
Abstract: Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.
TL;DR: This work presents some new and basic properties of shape Delaunay tessellations, concerning flipping, subgraph structures, and recognition.
TL;DR: It is found that the kd-tree insertion is more efficient in locating the base tetrahedron, but it is also more sensitive to the triangulation of non-uniform point distributions with a large amount of conflicting elongated Tetrahedra.
23 Jun 2014
TL;DR: An implicit quad-tree is constructed directly from the floating point bit patterns of the input points by sorting the corresponding Morton codes with a radix sorting procedure, which adapts elegantly to any (non-uniform distribution of input points and increases the accuracy of the merging calculations by grouping floating point values with similar bit patterns.
Abstract: We present an efficient implementation of a Dwyer-style Delaunay triangulation algorithm that runs in O(N) expected time. An implicit quad-tree is constructed directly from the floating point bit patterns of the input points by sorting the corresponding Morton codes with a radix sorting procedure. This unique structure adapts elegantly to any (non-)uniform distribution of input points and increases the accuracy of the merging calculations by grouping floating point values with similar bit patterns. Our implementation allows for easy parallelization and we demonstrate a record construction speed of one Billion Delaunay triangles in just 8s on a many-core SMP machine.
TL;DR: Two new sphere-packing algorithms for generating the colored triangulation are presented, and one may also take as input a Delaunay refinement mesh and color it arbitrarily, and these mesh non-convex planar domains with provable quality are extended.
TL;DR: Experimental results prove that the method based on the combination between Harris and Sift interest point detectors is practical and gives satisfying results without going through the propagation step.
Abstract: In this paper, we are interested in the problem of Euclidean 3D reconstruction of unknown objects by passive stereo vision method. Our method is based on the combination between Harris and Sift interest point detectors, to take advantage of the power of these two detectors, which will be useful when matching step, as a key step for 3D reconstruction, In order to have a sufficient number of matches distributed on the images. These matches will be used to estimate the 3D points (the projection matrices will be estimated after calibration using 3D Calibration Pattern). Finally, a 3D mesh is constructed by 3D Delaunay triangulation, applied to the 3D points reconstructed. Experimental results prove that this method is practical and gives satisfying results without going through the propagation step.
08 Jun 2014
TL;DR: In general, it is proved that the realization space of a Delaunay triangulation in Rd can have Ω(2d) connected components and also shows that the realizability problem for Delauny triangulations is polynomially equivalent to the existential theory of the reals.
Abstract: We study realization spaces of Delaunay triangulations and show that they can be arbitrarily complicated, and in particular disconnected. Our smallest example consists of two configurations of 29 labeled points in R25 whose Delaunay triangulations are combinatorially equivalent but yet there is no continuous transformation that maps one to the other without changing the triangulation. In general, we prove that the realization space of a Delaunay triangulation in Rd can have Ω(2d) connected components. Our proof uses Mnev's Universality Theorem and also shows that the realizability problem for Delaunay triangulations is polynomially equivalent to the existential theory of the reals.
01 Jan 2014
TL;DR: A new method to find voids that starting from local longest-edges in a Delaunay triangulation builds the largest possible empty or almost empty polygons around them, validated in 2D points with artificially generated circular and non-convex polygon voids.
Abstract: In astronomy, the objective determination of large empty spaces or voids in the spatial distribution of galaxies is part of the characterization of the large scale structure of the universe. This paper proposes a new method to find voids that starting from local longest-edges in a Delaunay triangulation builds the largest possible empty or almost empty polygons around them. A polygon is considered a void if its area is larger than a threshold value. The algorithm is validated in 2D points with artificially generated circular and non-convex polygon voids. Since the algorithm naturally extends to 3D, preliminary results in 3D are also shown.
TL;DR: Theoretical analysis and experimental results show that the proposed algorithm based on Delaunay triangulation of randomly distributed points in the Euclidean plane is an efficient method of generating Voronoi diagram.
Abstract: Voronoi diagram and its geometric dual, the Delaunay triangulation, both are practical geometric constructions which have been applied extensively in spatial analysis. Considering the low efficiency of the algorithm of indirectly building Voronoi diagram, this paper proposes an improved Voronoi diagram generation algorithm based on Delaunay triangulation of randomly distributed points in the Euclidean plane. In the process of building Delaunay triangulation, correlative edges of points and correlative trianagles of edges information is dynamically updated. Theoretical analysis and experimental results show that the proposed algorithm is an efficient method of generating Voronoi diagram.
Proceedings Article•
01 Jan 2014TL;DR: This work seeks to place a new point q such that the constrained Delaunay triangulation of P[fqg has the largest possible minimum angle, and develops a simpler cubic-time algorithm quite dierent from the ones already known.
Abstract: Given a set P of n points in the plane in general position, and a set of non-crossing segments with endpoints in P , we seek to place a new point q such that the constrained Delaunay triangulation of P[fqg has the largest possible minimum angle. The expected running time of our (randomized) algorithm is O(n 2 logn) on any input, improving the near-cubic time of the best previously known algorithm. Our algorithm is somewhat complex, and along the way we develop a simpler cubic-time algorithm quite dierent from the ones already known.
07 Oct 2014
TL;DR: A method to triangulate an elevation surface structured in a sparse 3D grid by using a fast search algorithm based on the 2D Delaunay triangulation, which shows that the initial shape of the surface is well preserved, and the topology of the output triangular mesh approximates the input surface of 3D point clouds.
Abstract: This paper proposes a method to triangulate an elevation surface structured in a sparse 3D grid by using a fast search algorithm based on the 2D Delaunay triangulation. After projecting the 3D point clouds onto a natural 2D grid in the x, y plane, we triangulate the surface (actually, we compute a Delaunay triangulation of the 2D point cloud taking advantage of its regular structure). The main novelty of our approach is that the neighboring points of an edge ei are searched in a rectangle supported by the edge ei under consideration. The obtained results show that our method could process very fast, the initial shape of the surface is well preserved, and the topology of the output triangular mesh approximates the input surface of 3D point clouds.