Showing papers on "Delaunay triangulation published in 1997"

Voronoi diagrams and Delaunay triangulations

Abstract: The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams and Delaunay triangulations have been rediscovered or applied in many areas of math ematics and the natural sciences they are central topics in computational geometry with hundreds of papers discussing algorithms and extensions Section discusses the de nition and basic properties in the usual case of point sites in R with the Euclidean metric while section gives basic algorithms Some of the many extensions obtained by varying metric sites environment and constraints are discussed in section Section nishes with some interesting and nonobvious structural properties of Voronoi diagrams and Delaunay triangulations

Delaunay refinement mesh generation

Abstract: : Delaunay refinement is a technique for generating unstructured meshes of triangles or tetrahedral suitable for use in the finite element method or other numerical methods for solving partial differential equations. Popularized by the engineering community in the mid-1980s, Delaunay refinement operates by maintaining a Delaunay triangulation or Delaunay tetrahedralization, which is refined by the insertion of additional vertices. The placement of these vertices is chosen to enforce boundary conformity and to improve the quality of the mesh. Pioneering papers by L. Paul Chew and Jim Ruppert have placed Delaunay refinement on firm theoretical ground. The purpose of this thesis is to further this progress by cementing the foundations of two-dimensional Delaunay refinement, and by extending the technique and its analysis to three dimensions.

Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagram on the surface of a sphere

Abstract: STRIPACK is a Fortran 77 software package that employs an incremental algorithm to construct a Delaunay triangulation and, optionally, a Voronoi diagram of a set of points (nodes) on the surface of the unit sphere. The triangulation covers the convex hull of the nodes, which need not be the entire surface, while the Voronoi diagram covers the entire surface. The package provides a wide range of capabilities including an efficient means of updating the triangulation with nodal additions or deletions. For N nodes, the storage requirement for the triangulation is 13N integer storage locations in addition to 3N nodal corrdinates. Using an off-line algorithm and work space of size 3N, the triangulation can be constructed with time complexity O(NlogN).

Guaranteed-quality Delaunay meshing in 3D (short version)

Abstract: The main contribution of this paper is a new mesh generation technique for producing 3D tetrahedral meshes. Like many existing techniques, this one is based on the Delaunay triangulation (DT). Unlike existing techniques, thk is the first Delaunay-based method that is mathematically guaranteed to avoid slivers. A sliver is a tetrahedral mesh-element that is almost completely flat. For example, imagine the tetrahedron created as the (3D) convex hull of the four corners of a square; th~ tetrahedron has nicely shaped faces — all faces are 45 degree right-triangles — but the tetrahedron has zero volume. Slivers in the mesh generally lead to poor numerical accuracy in a finite element analysis. The Delaunay triangulation (DT) has been widely used for mesh generation. In 21), the DT maximizes the minimum angle for a given point set; thus, small angles are avoided. There is no analogous property involving angles in 3D. We make use of the Empty Circle Property for the DT of a set of point sites: the circumcircle of each triangle is empty of all other sites. In 3D, the analogous property holds: the circumsphere of each tetrahedron is empty of all other sites. The Empty Circle Property can be used as the definition of the DT. There is a vsst literature on mesh generation with most of the material emanating from the various applications communities. We refer the reader to the excellent survey by Bern and Eppstein [BE92]. We consider here only work related to the topic of mesh generation with mathematical quality guarantees. Chew [Che89] showed how to use the DT to triangulate any 2D region with smooth boundaries and no sharp corners to attain a mesh of uniform density in which all angles are greater than 30 degrees. An optimality theorem for meshes of nonuniform density was developed by Bern, Eppstein and Gilbert [BEG94] using a quadtree-based approach. Ruppert [Ru93] later showed that a modification of Chew’s algorithm could also attain the same strong results

Sphere packings, I

Abstract: We describe a program to prove the Kepler conjecture on sphere packings. We then carry out the first step of this program. Each packing determines a decomposition of space into Delaunay simplices, which are grouped together into finite configurations called Delaunay stars. A score, which is related to the density of packings, is assigned to each Delaunay star. We conjecture that the score of every Delaunay star is at most the score of the stars in the face-centered cubic and hexagonal close packings. This conjecture implies the Kepler conjecture. To complete the first step of the program, we show that every Delaunay star that satisfies a certain regularity condition satisfies the conjecture.

A comparison of sequential Delaunay triangulation algorithms

Abstract: This paper presents an experimental comparison of a number of different algorithms for computing the Delaunay triangulation. The algorithms examined are: Dwyer's divide and conquer algorithm, Fortune's sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri and Murota, a new bucketing-based algorithm described in this paper, and Devillers's version of a Delaunay-tree based algorithm that appears in LEDA), an algorithm that incrementally adds a correct Delaunay triangle adjacent to a current triangle in a manner similar to gift wrapping algorithms for convex hulls, and Barber's convex hull based algorithm. Most of the algorithms examined are designed for good performance on uniformly distributed sites. However, we also test implementations of these algorithms on a number of non-uniform distributions. The experiments go beyond measuring total running time, which tends to be machine-dependent. We also analyze the major high-level primitives that algorithms use and do an experimental analysis of how often implementations of these algorithms perform each operation.

A practical algorithm for finding optimal triangulations

Abstract: An algorithm called QUICKTREE is developed for finding a triangulation T of a given undirected graph G such that the size of T's maximal clique is minimum and such that no other triangulation of G is a subgraph of T. We have tested QUICKTREE on graphs of up to 100 nodes for which the maximum clique in an optimal triangulation is of size 11. This is the first algorithm that can optimally triangulate graphs of such size in a reasonable time frame. This algorithm is useful for constraint satisfaction problems and for Bayesian inference through the clique tree inference algorithm.

Aspects of 2-d delaunay mesh generation

Abstract: SUMMARY This paper aims to outline the dierent phases necessary to implement a Delaunay-type automatic mesh generator. First, it summarizes this method and then describes a variant which is numerically robust by mentioning at the same time the problems to solve and the dierent solutions possible. The Delaunay insertion process by itself, the boundary integrity problem, the way to create the eld points as well as the optimization procedures are discussed. The two-dimensional situation is described fully and possible extensions to the three-dimensional case are briey indicated. ? 1997 by John Wiley & Sons, Ltd. There exist a large number of papers dealing with 2-D-mesh generation using the Delaunay method (see Reference 1). Nevertheless, we propose a new scheme with ambition to optimize, if possible, the dierent steps used in the practical application of the algorithm. We have tried to develop for each step an optimal solution. The method has been implemented and results show the performance of the algorithm. So we can construct a well-shaped mesh constituted by a million of triangles in a few minutes on a HP 735 workstation. The 2-D case is fully detailed for clarity but most of the results can be extended without diculty in three dimensions. The problem to be solved is to construct a consistent mesh of a domain essentially from its boundary data segments. The resulting mesh will be the support of a nite element computation which can indicate if it is adapted or must be adapted by any method (it is not the goal of this paper). Nevertheless, it is important to begin with a well-shaped mesh knowing that this request can only be based on geometric considerations, since the sole known information is of geometric nature. Thus, the shape and the size of the elements must be consistent with these data.

A spatial data mining method by Delaunay triangulation

Abstract: It becomes an important task to discover significant pattern or characteristics which may implicitly exist in huge spatial dntabases, such as geographical or medical databases. In this paper, we present a spatial data mining method named SMiYN (Spatial data Mining by Triangulated Irregular Network), which is based on Delaunay Triangulation. Sh47ZN demonstrates important advantages over the previous works. First, it discovers even sophisticated pattern like nested doughnuts, and hierarchical structure of cluster distribution. Second, in order to execute SMTIN, we do not need to know a priori the nature of distribution, for example the number of clusters, which is indispensable to other methods. Third, experiments show that SMTIN requires less CPU processing time than other methods such as BIRCH and CLARANS. Finally it is not ordering sensitive and handles efticiently outliers.

An accelerated triangulation method for computing the skeletons of free-form solid models

Abstract: Shape skeletons are powerful geometric abstractions that provide useful intermediate representations for a number of geometric operations on solid models including feature recognition, shape decomposition, finite element mesh generation, and shape design As a result there has been significant interest in the development of effective methods for skeleton generation of general free-form solids In this paper we describe a method that combines Delaunay triangulation with local numerical optimization schemes for the generation of accurate skeletons of 3D implicit solid models The proposed method accelerates the slow convergence of Voronoi diagrams to the skeleton, which, without optimization, would require impraelically large sample point sets and resulting meshes to attain acceptable accuracy The Delaunay triangulation forms the basis for generating the topological structure of the skeleton The optimization step of the process generates the geometry of the skeleton patches by moving the vertices of Delaunay tetrahedra and relocating their centres to form maximally inscribed spheres The computational advantage of the optimization scheme is that it involves the solution of one small optimization problem per tetrahedron and its complexity is therefore only linear (O(n)) in the number of points used for the skeleton approximation We demonstrate the effectiveness of the method on a number of representative solid models

Computing exact geometric predicates using modular arithmetic with single precision

Abstract: We propose an efficient method that determines the sign of a multivariate polynomial expression with integer coefficients. This is a central operation on which the robustness of many geometric algorithms depends. Our method relies on modular computations, for which comparisons are usually thought to require multiprecision. Our novel technique of {\it recursive relaxation of the moduli} enables us to carry out sign determination and comparisons by using only floating point computations in single precision. This leads us to propose a hybrid symbolic-numeric approach to exact arithmetic. The method is highly parallelizable and is the fastest of all known multiprecision methods \from a complexity point of view. As an application, we show how to compute a few geometric predicates that reduce to matrix determinants and we discuss implementation efficiency, which can be enhanced by arithmetic filters. We substantiate these claims by experimental results and comparisons to other existing approaches. Our method can be used to generate robust and efficient implementations of geometric algorithms (convex hulls, Delaunay triangulations, arrangements) and numerical computer algebra (algebraic representation of curves and points, symbolic perturbation, Sturm sequences and multivariate resultants).

Constructing solid models using implicit functions defining connectivity relationships among layers of an object to be modeled

Abstract: A solid model is constructed from surface point data that represent layers of an object. The model is represented as the level set of an implicit function that is fitted to the surface point data. In the two-dimensional application of the technique, a Delaunay triangulation is performed for each layer. In this step, surface points are connected to form Delaunay triangles; the data points are the vertices of the Delaunay triangles. A circumcircle is then created around each Delaunay triangle, passing through the three vertices of the triangle. To decimate the circumcircle data, overlapping circumspheres are merged according to a merging criterion. A pseudo-union of implicit functions for the reduced number of circumcircles provides an initial implicit function for the layer. Errors in the implicit function are substantially reduced by optimizing the position and/or radii of the circumcircles. The implicit functions for a plurality of adjacent layers are blended to define an implicit function for the object that is used for reconstruction or modeling of the object. The technique is generally extended to n dimensional objects by using simplices instead of the Delaunay triangles and hyperspheres instead of the circumcircles. The method is capable of constructing solid models with highly localized surface curvature.

Combining region splitting and edge detection through guided Delaunay image subdivision

Abstract: In this paper, an adaptive split-and-merge segmentation method is proposed. The splitting phase of the algorithm employs the incremental Delaunay triangulation competent of forming grid edges of arbitrary orientation, and position. The tessellation grid, defined by the Delaunay triangulation, is adjusted to the semantics of the image data by combining similarity and difference information among pixels. Experimental results on synthetic images show that the method is robust to different object edge orientations, partially weak object edges and very noisy homogeneous regions. Experiments on a real image indicate that the method yields good segmentation results even when there is a quadratic sloping of intensities particularly suited for segmenting natural scenes of man-made objects.

Properties of the Delaunay triangulation

Abstract: Some of the most well-known names in Computational Geometry are those of two prominent Russian mathematicians: Georgy F. Voronoi (1868 – 1908) and Boris N. Delaunay (1890 1980). Their considerable contribution to the Number Theory and Geometry is well known to the specialists in these fields. Surprisingly, their names (their works remained unread and later re-discovered) became the most popular not among “pure” mathematician, but among the researchers who used geometric applications. Such terms as “ Voronoi diagram” and “ Delaunay triangulation” are very important not only for Computational Geometry, but also for Geometric Modeling, Image Processing, CAD, GIS etc. Delaunay triangulation is used in numerous applications. It is widely used in plane and 3D case. A natural question may arise: why th~ triangulation is better than the others. Usually the advantages of Delaunay triangulation are rationalized by the max-min angle criterion and other properties [1,2,5,10,11,12]. The max-min angle criterion requires that the diagonal of every convex quadrilateral occurring in the triangulation “should be well chosen” [12], in the sense that replacement of the chosen diagonal by the alternative one must not increase the minimum oft he six angles in the two triangles making up the quadrilateral. Thus the Delaunay triangulation of a planar point set maximizes the minimum angle in any triangle. More specifically, the sequence of triangle angles, sorted from sharpest to leaat sharp, is lexicographlcally maximized over all such sequences constructed from triangulation of S. We defined several functional on the set of all triangulations of the finite system of sites in Rd attaining global minimum on the Delaunay triangulation (DT). First we consider a so called “parabolic” functional and prove that it attains its minimum on DT in all dimensions. It could be used as an equivalent definition for DT. Secondly we treat “mean radius” functiorral(the mean of circumradii of triangles) for planar triangulations. Thirdly we treat a so called “harmonic” functional. For a triangle this functional equals the ration of the sum of squaresof sides over area. Finally, we consider a discrete anidogue of the Dirichlet functional. Actually in all these cases the optimality of DT in 2D directly follow from flipping (swapping) aIgorithm: after each flip the corresponding functional decrease until Delaunay triangulation is reached. In 2D case all of these functional on triagles are Iexicographically minimised over all such sequences constructed from triangulation of S like for the max-min angle criterion. If d >2 then Delaunay triangulation is not optimal for the functional “mean radius”, “harmonic” and “ Dirichlet”. ~l?rom this point of view the usage of DT in dimensions d >2 may be nonappropriate. Thus the problem of finding” good” triangulations for this functional in higher dimensions is opened and more detailed consideration is necessary.

Using longest‐side bisection techniques for the automatic refinement of delaunay triangulations

Abstract: In this paper we discuss, study and compare two linear algorithms for the triangulation refinement problem: the known longest-side (triangle bisection) refinement algorithm, as well as a new algorithm that uses longest side bisection techniques for refining Delaunay triangulations. We show that the automatic point insertion criterion, taken from the fractal property of optimal (linear) longest-side bisection algorithms, assures the construction of good quality Delaunay triangulations in linear time. Numerical evidence, showing that the practical behaviour of the new algorithm is in complete agreement with the theory, is included. © 1997 by John Wiley & Sons, Ltd.

A new approach to protein fold recognition based on Delaunay tessellation of protein structure.

Abstract: We propose new algorithms for sequence-structure compatibility (fold recognition) searches in multi-dimensional sequence-structure space. Individual amino acid residues in protein structures are represented by their C alpha atoms; thus each protein is described as a collection of points in three-dimensional space. Delaunay tessellation of a protein generates an aggregate of space-filling, irregular tetrahedra, or Delaunay simplices. Statistical analysis of quadruplet residue compositions of all Delaunay simplices in a representative dataset of protein structures leads to a novel four body contact residue potential expressed as log likelihood factor q. The q factors are calculated for native 20 letter amino acid alphabet and several reduced alphabets. Two sequence-structure compatibility functions are computed as (i) the sum of q factors for all Delaunay simplices in a given protein, or (ii) 3D-1D Delaunay tessellation profiles where the individual residue profile value is calculated as the sum of q factors for all simplices that share this vertex residue. Both threading functions have been implemented in structure-recognizes-sequence and sequence-recognizes-structure protocols for protein fold recognition. We find that both profile and total score based threading functions can distinguish both the native fold from incorrect folds for a sequence, and the native sequence from non-native sequences for a fold.

Implementation and evaluation of an efficient parallel Delaunay triangulation algorithm

Abstract: Abstract : This paper describes the derivation of an empirically efficient parallel two-dimensional Delaunay triangulation program from a theoretically efficient CREW PRAM algorithm. Compared to previous work, the resulting implementation is not limited to datasets with a uniform distribution of points, achieves significantly better speedups over good serial code, and is widely portable due to its use of MPI as a communication mechanism. Results are presented for a loosely-coupled cluster of workstations, two distributed-memory multicomputers, and a shared-memory multiprocessor. The Machiavelli toolkit used to transform the nested data parallelism inherent in the divide-and-conquer algorithm into achievable task and data parallelism is also described and compared to previous techniques.

Linear-Time Reconstruction of Delaunay Triangulations with Applications

Abstract: Many of the computational geometers' favorite data structures are planar graphs, canonically determined by a set of geometric data, that take Θ(n log n) time to compute. Examples include 2-d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3-d convex hulls. Given such a structure, one can determine a permutation of the data in O(n) time such that the data structure can be reconstructed from the permuted data in O(n) time by a simple incremental algorithm.

Topographic triangulation in reduced time

Abstract: Method and system for rapid triangulation of a region into an array of triangles that can be used for a GIS triangulation A dataset S of three or more distinct points is set down, and an array including a sequence of triangles is constructed, using these points as vertices A triangle is removed from the array if at least a triangle included angle is greater than a selected threshold angle value (such as 90°) or if the ratio of triangle height to triangle width is too large A first triangle is replaced in the array by one or more other triangles if a first triangle included angle is less than an included angle for a second triangle, formed by replacing the first triangle included angle vertex by another point in the dataset S The new method provides an acceptable triangulation, with computation time equal to about 40 percent of the time required for a Delaunay triangulation using the same dataset S The dataset and/or the array of triangles can be displayed and manipulated

The big sweep : On the power of the wavefront approach to Voronoi diagrams

Abstract: We show that the wavefront approach to Voronoi diagrams (a deterministic line-sweep algorithm that does not use geometric transform) can be generalized to distance measures more general than the Euclidean metric. In fact, we provide the first worst-case optimal (O (n logn) time,O(n) space) algorithm that is valid for the full class of what has been callednice metrics in the plane. This also solves the previously open problem of providing anO (nlogn)-time plane-sweep algorithm for arbitraryL k -metrics. Nice metrics include all convex distance functions but also distance measures like the Moscow metric, and composed metrics. The algorithm is conceptually simple, but it copes with all possible deformations of the diagram.

A new theoretically motivated higher order upwind scheme on unstructured grids of simplices

Abstract: Many approaches exist to define a cell-centered upwind finite volume scheme of higher order on an unstructured grid of simplices. However, real theoretical motivation in the form of a convergence result does not exist for these approaches. Furthermore, some theoretical results of convergence exist for higher order finite volume methods, where no description of the numerical implementation is given to realize the necessary requirements for the convergence theory. Therefore we present in this paper a new limiter function which is motivated by these requirements and ensures a convergent scheme in the theoretical context: The approximated solution converges to the entropy solution in the case of scalar conservation laws in two space dimensions. This new limiter function is combined with a typical class of reconstruction functions very efficiently, which is illustrated by several test examples for scalar conservation laws as well as systems of such laws. In connection with the requirements to be fulfilled, a proof of a maximum principle of the finite volume scheme applied to simplices and dual cells is given. So for the approach of the higher order upwind finite volume scheme on dual cells, as used in several papers, a missing proof is now given. The ideas in this proof are also applied to the discontinuous Galerkin method, so that an existing maximum principle can be improved considerably. The main advantage comes from the fact that no requirements on the discretization of the domain are necessary: no B-triangulations or Delaunay triangulation are needed.