Showing papers on "Delaunay triangulation published in 2000"
01 Sep 2000
TL;DR: This work defines the sampling criteria to sample the surface and guarantee a topologically correct mesh after surface reconstruction for such a sampled surface, and presents a new algorithm to find the normal at a vertex when the surface is sampled according to the given criteria.
Abstract: We present a fast, memory efficient algorithm that generates a manifold triangular mesh S passing through a set of unorganized points P 3 Nothing is assumed about the geometry, topology or presence of boundaries in the data set except that P is sampled from a real manifold surface The speed of our algorithm is derived from a projection-based approach we use to determine the incident faces on a point We define our sampling criteria to sample the surface and guarantee a topologically correct mesh after surface reconstruction for such a sampled surface We also present a new algorithm to find the normal at a vertex, when the surface is sampled according our given criteria We also present results of our surface reconstruction using our algorithm on unorganized point clouds of various models
247 citations
01 Jan 2000
TL;DR: In this article, the authors study several known volume computation algorithms for convex d-polytopes by classifying them into two classes, triangulation methods and signed-decomposition methods.
Abstract: We study several known volume computation algorithms for convex d-polytopes by classifying them into two classes, triangulation methods and signed-decomposition methods. By incorporating the detection of simplicial faces and a storing/reusing scheme for face volumes we propose practical and theoretical improvements for two of the algorithms. Finally we present a hybrid method combining advantages from the two algorithmic classes. The behaviour of the algorithms is theoretically analysed for hypercubes and practically tested on a wide range of polytopes, where the new hybrid method proves to be superior.
218 citations
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TL;DR: It is shown that, if the Delaunay triangulation has the ratio property introduced in Miller et al.
Abstract: A silver is a tetrahedon whose four vertices lie close to a plane and whose orthogonal projection to that plane is a convex quadrilateral with no short edge. Silvers are notoriously common in 3-dimensional Delaunay triangulations even for well-spaced point sets. We show that, if the Delaunay triangulation has the ratio property introduced in Miller et al. [1995], then there is an assignment of weights so the weighted Delaunay traingulation contains no silvers. We also give an algorithm to compute such a weight assignment.
161 citations
01 May 2000
TL;DR: For a sufficiently dense set of points in any closed Riemannian manifold, it is proved that a unique Delannay triangulation exists that has the same properties as in Euclidean space.
Abstract: For a sufficiently dense set of points in any closed Riemannian manifold, we prove that a unique Delannay triangulation exists. This triangulation has the same properties as in Euclidean space. Algorithms for constructing these triangulations will also be described.
149 citations
TL;DR: In this article, an efficient and robust algorithm to localize the intergrid boundaries for the overset unstructured grid method is proposed, where neighbor-to-neighbor jump search algorithm is efficiently utilized in the method.
Abstract: The use of the overset concept for the unstructured grid method is relatively unexplored. However, the overset approach can extend the applicability of the unstructured grid method for real engineering problems without much need for code development. The multiple moving-body problem is one of those applications. Improvement in local resolution for Euler/Navier-Stokes computations on unstructured grids is another use of the overset concept. An efficient and robust algorithm to localize the intergrid boundaries for the overset unstructured grid method is proposed. Simplicity and automation in the intergrid-boundary definition are realized using the wall distance as a basic parameter. The neighbor-to-neighbor jump search algorithm is efficiently utilized in the method. The robustness and efficiency of the search is improved by the use of subsidiary grids that are generated as a byproduct of the Delaunay triangulation method. The basic procedure of the present method is described for a multielement airfoil problem. The effects of the overset method on the solution accuracy and the convergence are tested by ONERA M6-wing
132 citations
13 Jun 2000
TL;DR: An algorithm is presented that starts with an initial rough triangulation and refines the triangulations until it obtains a surface that best accounts for the images of the object and is able to overcome the surface ambiguity problem.
Abstract: Given a set of 3D points that we know lie on the surface of an object, we can define many possible surfaces that pass through all of these points. Even when we consider only surface triangulations, there are still an exponential number of valid triangulations that all fit the data. Each triangulation will produce a different faceted surface connecting the points. Our goal is to overcome this ambiguity and find the particular surface that is closest to the true object surface. We do not know the true surface but instead we assume that we have a set of images of the object. We propose selecting a triangulation based on its consistency with this set of images of the object. We present an algorithm that starts with an initial rough triangulation and refines the triangulation until it obtains a surface that best accounts for the images of the object. Our method is thus able to overcome the surface ambiguity problem and at the same time capture sharp corners and handle concave regions and occlusions. We show results for a few real objects.
126 citations
TL;DR: In this article, a cell centred finite volume method based on Roe's approximate Riemann solver across the edges of the cells is presented and the results are compared for first-and second-order accuracy.
Abstract: Numerical modelling of shallow water flow in two dimensions is presented in this work with the results obtained in dam break tests Free surface flow in channels can be described mathematically by the shallow-water system of equations These equations have been discretized using an approach based on unstructured Delaunay triangles and applied to the simulation of two-dimensional dam break flows A cell centred finite volume method based on Roe's approximate Riemann solver across the edges of the cells is presented and the results are compared for first- and second-order accuracy Special treatment of the friction term has been adopted and will be described The scheme is capable of handling complex flow domains as shown in the simulation corresponding to the test cases proposed, ie that of a dam break wave propagating into a 45° bend channel (UCL) and in a channel with a constriction (LNEC-IST) Comparisons of experimental and numerical results are shown
124 citations
30 Jul 2000
TL;DR: The results show that the similarity retrieval accuracy of the TVPAS method is as good as the other methods, while it has the lowest computational cost for generating the shape signatures of the objects and is the only method that has direct support for RST query types.
Abstract: Besides traditional applications (e.g. CAD/CAM and trademark registry), new multimedia applications, such as structured video, animation and the MPEG-7 standard, require the storage and management of well-defined objects. We focus on shape-based object retrieval and conduct a comparison study on four such techniques: Fourier descriptors (FD), the grid-based (GB) method, Delaunay triangulation (DT) and MBC-TPVAS (minimum bounding circles/touch-point vertex-angle sequence). Our results show that the similarity retrieval accuracy of our method (TVPAS) is as good as the other methods, while it has the lowest computational cost for generating the shape signatures of the objects. Moreover, it has low storage requirements and a comparable computation cost to compute the similarity between two shape signatures. In addition, TPVAS requires no normalization of the objects and is the only method that has direct support for RST (rotation/scaling/translation) query types. In this paper, we also introduce a new shape description taxonomy.
124 citations
TL;DR: In this paper, an indirect method for meshing parametric surfaces conforming to a user-specifiable size map is presented, based on the intrinsic properties of the surface, the Riemannian structure is induced into the parametric space.
Abstract: An indirect method for meshing parametric surfaces conforming to a user-specifiable size map is presented. First, from this size specification, a Riemannian metric is defined so that the desired mesh is one with unit length edges with respect to the related Riemannian space (the so-called ‘unit mesh’). Then, based on the intrinsic properties of the surface, the Riemannian structure is induced into the parametric space. Finally, a unit mesh is generated completely inside the parametric space such that it conforms to the metric of the induced Riemannian structure. This mesh is constructed using a combined advancing-front—Delaunay approach applied within a Riemannian context. The proposed method can be applied to mesh composite parametric surfaces. Several examples illustrate the efficiency of our approach. Copyright © 2000 John Wiley & Sons, Ltd.
110 citations
TL;DR: This paper connects the predominantly combinatorial work in classical computational geometry with the numerical interest in mesh generation with the two- and three-dimensional case and covers results obtained during the twentieth century.
Abstract: The Delaunay triangulation of a finite point set is a central theme in computational geometry. It finds its major application in the generation of meshes used in the simulation of physical processes. This paper connects the predominantly combinatorial work in classical computational geometry with the numerical interest in mesh generation. It focuses on the two- and three-dimensional case and covers results obtained during the twentieth century.
99 citations
01 May 2000
TL;DR: This paper shows that such a point set permits a small perturbation whose Delaunay triangulation contains no slivers, and gives deterministic algorithms that compute the perturbations of n points in time O(n logn) with one processor and inTime O(log n) with O( n) processors.
Abstract: A sliver is a tetrahedron whose four vertices lie close to a plane and whose perpendicular projection to that plane is a convex quadrilateral with no short edge. Slivers axe both undesirable and ubiquitous in 3-dimensional Delaunay triangulations. Even when the point-set is well-spaced, slivers may result. This paper shows that such a point set permits a small perturbation whose Delaunay triangulation contains no slivers. It also gives deterministic algorithms that compute the perturbation of n points in time O(n logn) with one processor and in time O(log n) with O(n) processors.
TL;DR: An algorithm is proposed that takes as input a generic set of unorganized points, sampled on a real object, and returns a closed interpolating surface that generates a closed 2‐manifold surface made of triangular faces, without limitations on the shape or genus of the original solid.
Abstract: In this paper an algorithm is proposed that takes as input a generic set of unorganized points, sampled on a real object, and returns a closed interpolating surface. Specifically, this method generates a closed 2-manifold surface made of triangular faces, without limitations on the shape or genus of the original solid. The reconstruction method is based on generation of the Delaunay tetrahedralization of the point set, followed by a sculpturing process constrained to particular criteria. The main applications of this tool are in medical analysis and in reverse engineering areas. It is possible, for example, to reconstruct anatomical parts starting from surveys based on TACs or magnetic resonance.
TL;DR: In this article, an algorithm for automatic regridding, able to fit the local resolution to the available raypaths, which is based on Delaunay triangulation and Voronoi tessellation, is described.
Abstract: The solutions of traveltime inversion problems are often not unique because of the poor match between the raypath distribution and the tomographic grid. However, by adapting the local resolution iteratively, by means of a singular value analysis of the lomographic matrix, we can reduce or eliminate the null space influence on our earth image: in this way, we get a much more reliable estimate of the velocity field of seismic waves. We describe an algorithm for an automatic regridding, able to fit the local resolution to the available raypaths, which is based on Delaunay triangulation and Voronoi tessellation. It increases the local pixel density where the null space energy is low or the velocity gradient is large, and reduces it elsewhere. Consequently, the tomographic image can reveal the boundaries of complex objects, but is not affected by the ambiguities that occur when the grid resolution is not adequately supported by the available raypaths.
01 May 2000
TL;DR: Algorithms for constructing constrained Delaunay triangulations (CDTs) in dimensions higher than two are discussed, yielding an efficient way to delete a vertex from a CDT.
Abstract: I discuss algorithms for constructing constrained Delaunay triangulations (CDTs) in dimensions higher than two. If the CDT of a set of vertices and constraining simplices exists, it can be constructed in time, where is the number of input vertices and is the number of output -simplices. In practice, the running time is likely to be in all but the most pathological cases. The CDT of a star-shaped polytope can be constructed in time, yielding an efficient way to delete a vertex from a CDT.
18 Dec 2000
TL;DR: There exists a routing algorithm for arbitrary triangulations that has no memory and uses no randomization, and there is no competitive online routing algorithm under the Euclidean distance metric in arbitraryTriangulations.
Abstract: We consider online routing algorithms for finding paths between the vertices of plane graphs. We show (1) there exists a routing algorithm for arbitrary triangulations that has no memory and uses no randomization, (2) no equivalent result is possible for convex subdivisions, (3) there is no competitive online routing algorithm under the Euclidean distance metric in arbitrary triangulations, and (4) there is no competitive online routing algorithm under the link distance metric even when the input graph is restricted to be a Delaunay, greedy, or minimum-weight triangulation.
TL;DR: A family of simple incremental algorithms for constructing short paths on the Delaunay graph is defined and potential applications to routeing in mobile communication networks are discussed.
Abstract: Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.
Posted Content•
TL;DR: The Delaunay Density Estimator Method (DDEEM) as mentioned in this paper is based on the stochastic geometric concept of the DELaunay tessellation generated by the point set.
Abstract: Here we introduce the Delaunay Density Estimator Method. Its purpose is rendering a fully volume-covering reconstruction of a density field from a set of discrete data points sampling this field. Reconstructing density or intensity fields from a set of irregularly sampled data is a recurring key issue in operations on astronomical data sets, both in an observational context as well as in the context of numerical simulations. Our technique is based upon the stochastic geometric concept of the Delaunay tessellation generated by the point set. We shortly describe the method, and illustrate its virtues by means of an application to an N-body simulation of cosmic structure formation. The presented technique is a fully adaptive method: automatically it probes high density regions at maximum possible resolution, while low density regions are recovered as moderately varying regions devoid of the often irritating shot-noise effects. Of equal importance is its capability to sharply and undilutedly recover anisotropic density features like filaments and walls. The prominence of such features at a range of resolution levels within a hierarchical clustering scenario as the example of the standard CDM scenario is shown to be impressively recovered by our scheme.
TL;DR: A new computational method of fully automated anisotropic triangulation of a trimmed parametric surface by defining proximity-based interacting forces among the ellipsoids and finding a force-balancing configuration using dynamic simulation.
Abstract: This paper describes a new computational method of fully automated anisotropic triangulation of a trimmed parametric surface. Given as input: (1) a domain geometry and (2) a 3×3 tensor field that specifies a desired anisotropic node-spacing, this new approach first packs ellipsoids closely in the domain by defining proximity-based interacting forces among the ellipsoids and finding a force-balancing configuration using dynamic simulation. The centers of the ellipsoids are then connected by anisotropic Delaunay triangulation for a complete mesh topology. Since a specified tensor field controls the directions and the lengths of the ellipsoids' principal axes, the method generates a graded anisotropic mesh whose elements conform precisely to the given tensor field.
TL;DR: In this article, a method for the generation of highly stretched grids suitable for Reynolds-averaged Navier-Stokes (RANS) calculations is presented, which has the advantages of not requiring any type of surface recovery, not requiring extra passes or work to mesh concave ridges/corners, and guarantees a final mesh, an essential requirement for industrial environments.
Abstract: A procedure for the generation of highly stretched grids suitable for Reynolds-averaged Navier–Stokes (RANS) calculations is presented. In a first stage, an isotropic (Euler) mesh is generated. In a second stage, this grid is successively enriched with points in order to achieve highly stretched elements. The element reconnection is carried out using a constrained Delaunay approach. Points are introduced from the regions of lowest stretching towards the regions of highest stretching. The procedure has the advantages of not requiring any type of surface recovery, not requiring extra passes or work to mesh concave ridges/corners, and guarantees a final mesh, an essential requirement for industrial environments. Given that point placement and element quality are highly dependent for the Delaunay procedure, special procedures were developed in order to obtain optimal point placement. Copyright © 2000 John Wiley & Sons, Ltd.
TL;DR: The robustness and versatility of the approach, as well as the good quality of the resulting meshes, is demostrated with the aid of selected examples.
12 Sep 2000
TL;DR: AUTOCLUST+ clusters points in the presence of obstacles based on Voronoi modeling and Delaunay Diagrams and detects high-quality clusters without prior knowledge and successfully supports correlation analyses between layers and more general locational optimization problems in the absence of obstacles.
Abstract: Wide spread clustering algorithms use the Euclidean distance to measure spatial proximity. However, obstacles in other GIS data-layers prevent traversing the straight path between two points. AUTOCLUST+ clusters points in the presence of obstacles based on Voronoi modeling and Delaunay Diagrams. The algorithm is free of usersupplied arguments and incorporates global and local variations. Thus, it detects high-quality clusters (clusters of arbitrary shapes, clusters of different densities, sparse clusters adjacent to high-density clusters, multiple bridges between clusters and closely located high-density clusters) without prior knowledge. Consequently, it successfully supports correlation analyses between layers (requiring high-quality clusters) and more general locational optimization problems in the presence of obstacles. All this within O(n log n+[m+R] log n) expected time, where n is the number of data points, m is the number of line-segments that determine the obstacles and R is the number of Delaunay edges intersecting some obstacles. A series of detailed performance evaluations illustrates the power of AUTOCLUST+ and confirms the virtues of our approach.
TL;DR: A mathematical formalism defining the shape of a finite point set which is a finite set of points which positions variation allows A -shape to generate a family of graphs extracted from Delaunay triangulation.
TL;DR: The proposed method involves an iterative evolution of a piecewise-linear approximation of the shape skeleton by using a minimum spanning tree-based self-organizing map (SOM) and the adjacency relationships between regions in the shape are detected and used in the evolution of the skeleton.
Abstract: This paper presents a method for computing the skeleton of planar shapes and objects which exhibit sparseness (lack of connectivity), within their image regions. Such sparseness in images may occur due to poor lighting conditions, incorrect thresholding or image sub-sampling. Furthermore, in document image analysis, sparse shapes are characteristic of texts faded due to aging and/or poor ink quality. Given the pixel distribution for a shape, the proposed method involves an iterative evolution of a piecewise-linear approximation of the shape skeleton by using a minimum spanning tree-based self-organizing map (SOM). By constraining the SOM to lie on the edges of the Delaunay triangulation of the shape distribution, the adjacency relationships between regions in the shape are detected and used in the evolution of the skeleton. The SOM, on convergence, gives the final skeletal shape. The skeletonization is invariant to Euclidean transformations. The potential of the method is demonstrated on a variety of sparse shapes from different application domains.
Journal Article•
TL;DR: The presented technique is a fully adaptive method: automatically it probes high density regions at maximum possible resolution, while low density regions are recovered as moderately varying regions devoid of the often irritating shot-noise effects.
Abstract: Here we introduce the Delaunay Density Estimator Method. Its purpose is rendering a fully volume-covering reconstruction of a density field from a set of discrete data points sampling this field. Reconstructing density or intensity fields from a set of irregularly sampled data is a recurring key issue in operations on astronomical data sets, both in an observational context as well as in the context of numerical simulations. Our technique is based upon the stochastic geometric concept of the Delaunay tessellation generated by the point set. We shortly describe the method, and illustrate its virtues by means of an application to an N-body simulation of cosmic structure formation. The presented technique is a fully adaptive method: automatically it probes high density regions at maximum possible resolution, while low density regions are recovered as moderately varying regions devoid of the often irritating shot-noise effects. Of equal importance is its capability to sharply and undilutedly recover anisotropic density features like filaments and walls. The prominence of such features at a range of resolution levels within a hierarchical clustering scenario as the example of the standard CDM scenario is shown to be impressively recovered by our scheme.
TL;DR: The Dorfmeister–Pedit–Wu construction is used to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics, which can have both unduloidal and nodoidal ends.
Abstract: We use the Dorfmeister–Pedit–Wu construction to present three new classesof immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane. An appendix, by the third author, describes the DPW potentials that appear to give CMC punctured spheres with k Delaunay ends (k-noids): the evidence is experimental at present. These can have both unduloidal and nodoidal ends.
TL;DR: An incremental method is presented to generate automatically boundary-fitted Delaunay triangulations of the global ocean that takes into account Earth curvature and allows local mesh refinement in order to resolve topological or dynamical features like midocean ridges or western boundary currents.
TL;DR: The aim of the first part is to analyse the local kinematics variables obtained in a numerical simulation using the DEM and the analysis of two different approaches to define the strain tensor from local variables based on the Delaunay triangulation.
Abstract: This paper analyses the link between local kinematics and strain tensor defined at a macroscale. The aim of the first part is to analyse the local kinematics variables obtained in a numerical simulation using the DEM. The second part is focused on the analysis of two different approaches to define the strain tensor from local variables. The first approach which only considers relative displacements at the contacts is well known. It is shown that this approach is not usually valid. The second approach is based on the Delaunay triangulation. A decomposition of the local strain is then proposed and it allows the influence of the different mechanisms of the local kinematics (kinematics at contact and displacement of neighbouring particles) to be demonstrated. Copyright © 2000 John Wiley & Sons, Ltd.
01 Jan 2000
TL;DR: A variety of multi-resolution visualisation methods have been designed to serve as tools for interactive visualisation of large datasets and different approaches have been presented to solve the outstanding continuity problem, i.e., to avoid cracks in the adaptive isosurfaces.
Abstract: A variety of multi-resolution visualisation methods have been designed to serve as tools for interactive visualisation of large datasets. The local resolution of the generated visual objects, such as isosurfaces, is thereby steered by error indicators which measure the error due to a locally coarser approximation of the data. On one hand, post-processing methods can be applied to already extracted surfaces and can turn them into multi-resolutional objects, which can then be interactively inspected [1–4]. On the other hand, we can also adaptively extract the considered isosurfaces from the 3D dataset. Thereby, starting at a coarse approximation of the data, we recursively add details in areas where some error indicator points out a local error with respect to the exact data values. If the error is below a user prescribed threshold, the algorithm locally stops the successive refinement and extracts the surface on the current level. Different approaches have been presented to solve the outstanding continuity problem, i.e., to avoid cracks in the adaptive isosurfaces. In the Delaunay approach by Cignoni et al. [5] and the nested mesh method by Grosso et al. [6], the successive remeshing during the refinement guarantees the continuity. On the other hand, Shekhar et al. [7] rule out hanging nodes by inserting additional points on faces with a transition from finer to coarser elements due to an adaptive stopping criterion.
TL;DR: In this paper, a Delaunay background mesh is defined over which natural neighbour interpolation, a neighbourhood-based interpolation scheme, is used to generate a sizing function, which may be used to interpolate element sizes, anisotropic stretching parameters or other surface characteristics required during surface meshing.
Abstract: A method is presented for controlling element sizes on the interior of areas during surface meshing. A Delaunay background mesh is defined over which natural neighbour interpolation, a neighbourhood-based interpolation scheme, is used to generate a sizing function. The sizing function may be used to interpolate element sizes, anisotropic stretching parameters or other surface characteristics required during meshing. A brief description of natural neighbour interpolation is included and the sizing results obtained from this interpolation method are compared to those obtained using linear interpolation. Three specific applications are presented that utilize the sizing function, namely boundary layer meshing, surface curvature refinement and anisotropic mesh generation. For these applications, criteria used for augmenting the sizing function based on insertion of additional interior vertices into the background mesh are discussed. Copyright © 2000 John Wiley & Sons, Ltd.
Patent•
06 Apr 2000TL;DR: In this paper, a method for creating amorphous patterns based on a constrained Voronoi tesselation of 2-space that can be tiled is presented, where the tiling feature is accomplished by modifying only the nucleation point poriton of the algorithm.
Abstract: The present invention provides a method for creating amorphous patterns based on a constrained Voronoi tesselation of 2-space that can be tiled. There are three basic steps required to generate a constrained Voronoi tesselation of 2-space: 1) nucleation point placement; 2) Delauney triangulation of the nucleation points; and 3) polygon extraction from the Delauney triangulated space. The tiling feature is accomplished by modifying only the nucleation point poriton of the algorithm. The method of the present invention, for creating an amorphous two-dimensional pattern of interlocking two-dimensional geometrical shapes having at least two opposing edges which can be tiled together, comprises the steps of: (a) specifying the width xmax of the pattern measured in direction x between the opposing edges; (b) adding a computational border region of width B to the pattern along one of the edges located at the x distance xmax; (c) computationally generating (x, y) coordinates of a nucleation point having x coordinates between 0 and xmax; (d) selecting nucleation points having x coordinates between 0 and B and copying them into the computational border region by adding xmax to their x coordinate value; (e) comparing both the computationally generated nucleation point and the corresponding copied nucleation point in the computational border against all previously generated nucleation points; and (f) repeating steps (c) through (e) until the desired number of nucleation points has been generated. To complete the pattern formation process, the additional steps of: (g) performing a Delaunay triangulation on the nucleation points; and (h) performing a Voronoi tessellation on the nucleation points to form two-dimensional geometrical shapes are included. Patterns having two pairs of opposing edges which may be tiled together may be generated by providing computional borders in two mutually orthogonal coordinate directions.