Showing papers on "Constrained Delaunay triangulation published in 2011"
TL;DR: An adaptive spatial clustering algorithm based on Delaunay triangulation (ASCDT for short) that can automatically discover clusters of complicated shapes, and non-homogeneous densities in a spatial database, without the need to set parameters or prior knowledge is proposed.
95 citations
TL;DR: This paper proposes a practical algorithm based on the construction of a constrained Delaunay tetrahedralization for a set of constraints (segments and facets) that adds additional points (so‐called Steiner points) on segments only.
56 citations
TL;DR: The vertex set of the Capacity-Constrained Delaunay Triangulation (CCDT) is shown to have good blue noise characteristics, comparable in quality to those of state-of-the-art methods, achieved at a fraction of the runtime.
44 citations
TL;DR: A Conforming Delaunay Triangulation (CDT) algorithm based on maximal Poisson disk sampling that works well in practice, and has the blue-noise property, and is also fast and uses little memory.
Abstract: We present a Conforming Delaunay Triangulation (CDT) algorithm based on maximal Poisson disk sampling. Points are unbiased, meaning the probability of introducing a vertex in a disk-free subregion is proportional to its area, except in a neighborhood of the domain boundary. In contrast, Delaunay refinement CDT algorithms place points dependent on the geometry of empty circles in intermediate triangulations, usually near the circle centers. Unconstrained angles in our mesh are between 30? and 120?, matching some biased CDT methods. Points are placed on the boundary using a one-dimensional maximal Poisson disk sampling. Any triangulation method producing angles bounded away from 0? and 180? must have some bias near the domain boundary to avoid placing vertices infinitesimally close to the boundary.Random meshes are preferred for some simulations, such as fracture simulations where cracks must follow mesh edges, because deterministic meshes may introduce non-physical phenomena. An ensemble of random meshes aids simulation validation. Poisson-disk triangulations also avoid some graphics rendering artifacts, and have the blue-noise property.We mesh two-dimensional domains that may be non-convex with holes, required points, and multiple regions in contact. Our algorithm is also fast and uses little memory. We have recently developed a method for generating a maximal Poisson distribution of n output points, where n = ? ( Area / r 2 ) and r is the sampling radius. It takes O ( n ) memory and O ( n log n ) expected time; in practice the time is nearly linear. This, or a similar subroutine, generates our random points. Except for this subroutine, we provably use O ( n ) time and space. The subroutine gives the location of points in a square background mesh. Given this, the neighborhood of each point can be meshed independently in constant time. These features facilitate parallel and GPU implementations. Our implementation works well in practice as illustrated by several examples and comparison to Triangle. Highlights? Conforming Delaunay triangulation algorithm based on maximal Poisson-disk sampling. ? Angles between 30? and 120?. ? Two-dimensional non-convex domains with holes, planar straight-line graphs. ? O ( n ) space, E ( n log n ) time; efficient in practice. Background squares ensure all computations are local.
40 citations
TL;DR: This work shows how to leverage the knowledge of ℛ for faster Delaunay computation and optimally handles a wide variety of inputs, e.g., overlapping disks of different sizes and fat regions.
Abstract: Suppose we want to compute the Delaunay triangulation of a set P whose points are restricted to a collection ℛ of input regions known in advance. Building on recent work by Loffler and Snoeyink, we show how to leverage our knowledge of ℛ for faster Delaunay computation. Our approach needs no fancy machinery and optimally handles a wide variety of inputs, e.g., overlapping disks of different sizes and fat regions.
39 citations
TL;DR: In this paper, an enhanced Steiner point suppression procedure and a small polyhedron reconnection operation are combined to achieve a complete three-dimensional constrained boundary recovery algorithm, and its effectiveness is demonstrated in various numerical examples.
32 citations
01 Jan 2011
TL;DR: This paper presents a robust parallel algorithm for computing Delaunay triangulations in three dimensions and is the first method that parallelizes point removals, an operation that significantly slows refinement down.
Abstract: Delaunay meshing is a popular technique for mesh generation. Usually, the mesh has to be refined so that certain fidelity and quality criteria are met. Delaunay refinement is achieved by dynamically inserting and removing points in/from a Delaunay triangulation. In this paper, we present a robust parallel algorithm for computing Delaunay triangulations in three dimensions. Our triangulator offers fully dynamic parallel insertions and removals of points and is thus suitable for mesh refinement. As far as we know, ours is the first method that parallelizes point removals, an operation that significantly slows refinement down. Our shared memory implementation makes use of a custom memory manager and light-weight locks which greatly reduce the communication and synchronization cost. We also employ a contention policy which is able to accelerate the execution times even in the presence of high number of rollbacks. Evaluation on synthetic and real data shows the effectiveness of our method on widely used multi-core SMPs.
25 citations
TL;DR: A new symbolic perturbation is presented that allows to always define these triangulations in a unique way, as soon as the points are not all coplanar.
Abstract: The Delaunay triangulation and the weighted Delaunay triangulation are not uniquely defined when the input set is degenerate. We present a new symbolic perturbation that allows to always define these triangulations in a unique way, as soon as the points are not all coplanar. No flat tetrahedron exists in the defined triangulation. The perturbation scheme is easy to code. It is implemented in cgal, and guarantees that both vertex insertion and vertex removal are fully robust.
24 citations
TL;DR: It is shown that sliver tetrahedra can be eliminated from constrained Delaunay tetrahedralizations solely by refinement, and the algorithm can consistently generate meshes with dihedral angles between 18circ and 154∘.
Abstract: This paper studies the practical performance of Delaunay refinement tetrahedral mesh generation algorithms By using non-standard quality measures to drive refinement, we show that sliver tetrahedra can be eliminated from constrained Delaunay tetrahedralizations solely by refinement Despite the fact that quality guarantees cannot be proven, the algorithm can consistently generate meshes with dihedral angles between 18circ and 154∘ Using a fairer quality measure targeting every type of bad tetrahedron, dihedral angles between 14∘ and 154∘ can be obtained The number of vertices inserted to achieve quality meshes is comparable to that needed when driving refinement with the standard circumradius-to-shortest-edge ratio We also study the use of mesh improvement techniques on Delaunay refined meshes and observe that the minimum dihedral angle can generally be pushed above 20∘, regardless of the quality measure used to drive refinement The algorithm presented in this paper can accept geometric domains whose boundaries are piecewise smooth Copyright © 2011 John Wiley & Sons, Ltd
19 citations
TL;DR: This paper proves that in the special case when the points in S are in convex position the stretch factor of the Delaunay triangulation of S is at most @r=2.33.
Abstract: Let S be a finite set of points in the Euclidean plane. Let G be a geometric graph in the plane whose point set is S. The stretch factor of G is the maximum ratio, among all points p and q in S, of the length of the shortest path from p to q in G over the Euclidean distance |pq|. Keil and Gutwin in 1989 [11] proved that the stretch factor of the Delaunay triangulation of a set of points S in the plane is at most 2@p/(3cos(@p/6))~2.42. Improving on this upper bound remains an intriguing open problem in computational geometry. In this paper we consider the special case when the points in S are in convex position. We prove that in this case the stretch factor of the Delaunay triangulation of S is at most @r=2.33.
19 citations
TL;DR: A new algorithm is proposed that preprocess a set of n disjoint unit disks to be able to compute the Delaunay triangulation in O(n) expected time.
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.
TL;DR: The presented method can correctly reconstruct 2D curves and 3D surfaces for known point cloud data sets employed in the literature and shows the unique benefit that point density around sharp corners does not have to be infinite.
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.
TL;DR: A novel robust mapping determination algorithm is proposed to speed up the matching process while the accuracy is maintained and the main idea is that the order of three visible feature points in the 3D are seen must be the same independently of the camera position.
26 May 2011
TL;DR: In this paper, a parallel 3D Delaunay triangulation method using domain-decomposition approach is proposed to reduce the communication between processors, which finds the 3D affected zone that may be modified during the merge of two sub-block triangulations.
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.
13 Jun 2011
TL;DR: 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: 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.
TL;DR: The developed algorithm for the reconstruction of connections in a surface is suitable to be used for simulations of biological cells because of the inherent conservation of the number of particles attributed to the membrane.
01 Jan 2011
TL;DR: A technique to create a Delaunay triangulation for terrain visualization on a virtual globe and a level of detail approach for large scale out-of-core spherical terrain rendering for virtual globes is presented using the previously created TIN.
Abstract: A technique to create a Delaunay triangulation for terrain visualization on a virtual globe is presented. This method can be used to process large scale elevation datasets with billions of points by using little RAM during data processing. All data is being transformed to a global spatial reference system. If grid based elevation data is used as input, a reduced TIN can be calculated. Furthermore, a level of detail approach for large scale out-of-core spherical terrain rendering for virtual globes is presented using the previously created TIN.
TL;DR: A heuristic for the Euclidean Steiner tree problem in ℜd for d≥2.5 that inserts Steiner points probabilistically into Delaunay triangles to achieve different subtrees on subsets of terminal points and governs this neighbor generation procedure with a local search framework that extends effectively into higher dimensions.
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.
TL;DR: This paper carefully design code for small degrees 3= in Delaunay triangulation, with usual results O(d), O(Dlogd), or O( d^2).
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=
TL;DR: This work shows how the challenges of localized refinement of volumes can be met with simple steps while retaining provable guarantees, and that the algorithm scales many folds better than a state‐of‐the‐art meshing tool provided by CGAL.
Abstract: Delaunay refinement, recognized as a versatile tool for meshing a variety of geometries, has the deficiency that it does not scale well with increasing mesh size. The bottleneck can be traced down to the memory usage of 3D Delaunay triangulations. Recently an approach has been suggested to tackle this problem for the specific case of smooth surfaces by subdividing the sample set in an octree and then refining each subset individually while ensuring termination and consistency. We extend this to localized refinement of volumes, which brings about some new challenges. We show how these challenges can be met with simple steps while retaining provable guarantees, and that our algorithm scales many folds better than a state-of-the-art meshing tool provided by CGAL.
13 Jun 2011
TL;DR: In this article, the authors 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.
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.
TL;DR: An unstructured triangulation approach, new to the authors' knowledge, is proposed to apply triangular meshes for representing and rendering a scene on a cubic panorama (CP) and provides a good modeling for the scene with photorealistic rendered CPs.
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.
TL;DR: An algorithm to construct constrained Delaunay tetrahedralizations of geometric domains bounded by piecewise smooth surfaces by first discretizing the boundary curves and then by sampling the smooth surfaces, which obviates the need for a boundary protection scheme.
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.
23 Jan 2011
TL;DR: In this paper, it was shown that Delaunay triangulations and compressed quadtrees are equivalent structures and that they can be computed in deterministic linear time on a pointer machine.
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].
21 Sep 2011
TL;DR: The concept of a circular arc triangulation is proposed--a simple and effective alternative that offers flexibility for additionally enlarging small angles--and its applications in graph drawing are discussed.
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.
28 Jun 2011
TL;DR: An intrinsic algorithm for isotropic mesh simplification that computes the geodesic Delaunay triangulation with regard to the sample set and iteratively evolves the Delaunays such that theDelaunay edges become almost equal in length.
Abstract: In this paper, we present an intrinsic algorithm for isotropic mesh simplification. Starting with a set of unevenly distributed samples on the surface, our method computes the geodesic Delaunay triangulation with regard to the sample set and iteratively evolves the Delaunay triangulation such that the Delaunay edges become almost equal in length. Finally, our method outputs the simplified mesh by replacing each curved Delaunay edge with a line segment. We conduct experiments on numerous real-world models of complicated geometry and topology. The promising experimental results demonstrate that the proposed method is intrinsic and insensitive to initial mesh triangulation.
01 Jan 2011
06 Apr 2011
TL;DR: It is shown how the triangulation topology follows the arithmetic decomposition of the line slope as well as its combinatorial decomposition (splitting formula) and a byproduct is a linear time algorithm for computing the Delaunay triangulating and the Voronoi diagram of such sets.
Abstract: We present new results concerning the Delaunay triangulation of the set of points of pieces of digital straight lines More precisely, we show how the triangulation topology follows the arithmetic decomposition of the line slope as well as its combinatorial decomposition (splitting formula) A byproduct is a linear time algorithm for computing the Delaunay triangulation and the Voronoi diagram of such sets
TL;DR: It is shown that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delahan triangulations, the maximum number is 2^n^-^3.
01 Jan 2011
TL;DR: The project investigates Voronoi Diagrams based on a default Harris Corner Detector as an image segmentation technique for adjusting the threshold for the Harris Cornerdetector, as well as Delaunay TriangulationDiagrams as a measure of similarity between images.
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.