Showing papers on "Delaunay triangulation published in 1996"

Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

Abstract: Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

Incremental topological flipping works for regular triangulations

Abstract: A set ofn weighted points in general position in źd defines a unique regular triangulation. This paper proves that if the points are added one by one, then flipping in a topological order will succeed in constructing this triangulation. If, in addition, the points are added in a random sequence and the history of the flips is used for locating the next point, then the algorithm takes expected time at mostO(nlogn+n[d/2]). Under the assumption that the points and weights are independently and identically distributed, the expected running time is between proportional to and a factor logn more than the expected size of the regular triangulation. The expectation is over choosing the points and over independent coin-flips performed by the algorithm.

Fast randomized point location without preprocessing in two- and three-dimensional Delaunay triangulations

Abstract: This paper studies the point location problem in Delaunay triangulations without preprocessing and additional storage. The proposed procedure finds the query point by simply “walking through” the triangulation, after selecting a “good starting point” by random sampling. The analysis generalizes and extends a recent result for dD 2 dimensions by proving this procedure takes expected time close to O.n 1=.dC1/ / for point location in Delaunay triangulations of n random points indD 3 dimensions. Empirical results in both two and three dimensions show that this procedure is efficient in practice. © 1999 Elsevier Science B.V. All rights reserved.

Delaunay Tessellation of Proteins: Four Body Nearest Neighbor Propensities of Amino Acid Residues

Abstract: Delaunay tessellation is applied for the first time in the analysis of protein structure. By representing amino acid residues in protein chains by Cα atoms, the protein is described as a set of points in three-dimensional space. Delaunay tessellation of a protein structure generates an aggregate of space-filling irregular tetrahedra, or Delaunay simplices. The vertices of each simplex define objectively four nearest neighbor Cα atoms, i.e., four nearest-neighbor residues. A simplex classification scheme is introduced in which simplices are divided into five classes based on the relative positions of vertex residues in protein primary sequence. Statistical analysis of the residue composition of Delaunay simplices reveals nonrandom preferences for certain quadruplets of amino acids to be clustered together. This nonrandom preference may be used to develop a four-body potential that can be used in evaluating sequence–structure compatibililty for the purpose of inverted structure prediction.

Marching triangles: range image fusion for complex object modelling

Abstract: A new surface based approach to implicit surface polygonisation is introduced. This is applied to the reconstruction of 3D surface models of complex objects from multiple range images. Geometric fusion of multiple range images into an implicit surface representation was presented in previous work. This paper introduces an efficient algorithm to reconstruct a triangulated model of a manifold implicit surface, a local 3D constraint is derived which defines the Delaunay surface triangulation of a set of points on a manifold surface in 3D space. The 'marching triangles' algorithm uses the local 3D constraint to reconstruct a Delaunay triangulation of an arbitrary topology manifold surface. Computational and representational costs are both a factor of 3-5 lower than previous volumetric approaches such as marching cubes.

Shape description by medial surface construction

Abstract: The medial surface is a skeletal abstraction of a solid that provides useful shape information, which compliments existing model representation schemes. The medial surface and its associated topological entities are defined, and an algorithm for computing the medial surface of a large class of B-rep solids is then presented. The algorithm is based on the domain Delaunay triangulation of a relatively sparse distribution of points, which are generated on the boundary of the object. This strategy is adaptive in that the boundary point set is refined to guarantee a correct topological representation of the medial surface.

Fractal image compression based on Delaunay triangulation and vector quantization

Abstract: Presents a new scheme for fractal image compression based on adaptive Delaunay triangulation. Such a partition is computed on an initial set of points obtained with a split and merge algorithm in a grey level dependent way. The triangulation is thus fully flexible and returns a limited number of blocks allowing good compression ratios. Moreover, a second original approach is the integration of a classification step based on a modified version of the Lloyd algorithm (vector quantization) in order to reduce the encoding complexity. The vector quantization algorithm is implemented on pixel histograms directly generated from the triangulation. The aim is to reduce the number of comparisons between the two sets of blocks involved in fractal image compression by keeping only the best representative triangles in the domain blocks set. Quality coding results are achieved at rates between 0.25-0.5 b/pixel depending on the nature of the original image and on the number of triangles retained.

A new method for accurate estimation of velocity field statistics

Abstract: We introduce two new methods to obtain reliable velocity field statistics from N-body simulations, or indeed from any general density and velocity fluctuation field sampled by discrete points, These methods, the Voronoi tessellation method and Delaunay tessellation method, are based on the use of the Voronoi and Delaunay tessellations of the point distribution defined by the locations at which the velocity held is sampled. In the Voronoi method the velocity is supposed to be uniform within the Voronoi polyhedra, whereas the Delaunay method constructs a velocity field by linear interpolation between the four velocities at the locations defining each Delaunay tetrahedron. The most important advantage of these methods is that they provide an optimal estimator for determining the statistics of volume-averaged quantities, as opposed to the available numerical methods that mainly concern mass-averaged quantities. As the major share of the related analytical work on velocity field statistics has focused on volume-averaged quantities, the availability of appropriate numerical estimators is of crucial importance for checking the validity of the analytical perturbation calculations. In addition, it allows us to study the statistics of the velocity field in the highly non-linear clustering regime. Specifically we describe in this paper how to estimate, in both the Voronoi and the Delaunay methods, the value of the volume-averaged expansion scalar theta = H(-1)del .upsilon (the divergence of the peculiar velocity, expressed in units of the Hubble constant H), as well as the value of the shear and the vorticity of the velocity field, at an arbitrary position. The evaluation of these quantities on a regular grid leads to an optimal estimator for determining the probability distribution function (PDF) of the volume-averaged expansion scalar, shear and vorticity. Although in most cases both the Voronoi and the Delaunay methods lead to equally good velocity field estimates, the Delaunay method may be slightly preferable. In particular it performs considerably better at small radii. Note that it is more CPU-time intensive while its requirement for memory space is almost a factor 8 lower than the Voronoi method. As a test we here apply our estimator to that of an N-body simulation of such structure formation scenarios. The PDF:; determined from the simulations agree very well with the analytical predictions. An important benefit of the present method is that, unlike previous methods, it is capable of probing accurately the velocity field statistics in regions of very low density, which in N-body simulations are typically sparsely sampled, In a forthcoming paper we will apply the newly developed tool to a plethora of structure formation scenarios, of both Gaussian and non-Gaussian initial conditions, in order to see to what extent the velocity field PDFs are sensitive discriminators, highlighting fundamental physical differences between the scenarios.

Hamiltonian triangulations for fast rendering

Abstract: High-performance rendering engines are often pipelined; their speed is bounded by the rate at which triangulation data can be sent into the machine. An ordering such that consecutive triangles share a face, which reduces the data rate, exists if and only if the dual graph of the triangulation contains a Hamiltonian path. We (1) show thatany set ofn points in the plane has a Hamiltonian triangulation; (2) prove that certain nondegenerate point sets do not admit asequential triangulation; (3) test whether a polygonP has a Hamiltonian triangulation in time linear in the size of its visibility graph; and (4) show how to add Steiner points to a triangulation to create Hamiltonian triangulations that avoid narrow angles.

Multiresolution models for topographic surface description

Abstract: Multiresolution terrain models describe a topographic surface at various levels of resolution. Besides providing a data compression mechanism for dense topographic data, such models enable us to analyze and visualize surfaces at a variable resolution. This paper provides a critical survey of multiresolution terrain models. Formal definitions of hierarchical and pyramidal models are presented. Multiresolution models proposed in the literature (namely, surface quadtree, restricted quadtree, quaternary triangulation, ternary triangulation, adaptive hierarchical triangulation, hierarchical Delaunay triangulation, and Delaunay pyramid) are described and discussed within such frameworks. Construction algorithms for all such models are given, together with an analysis of their time and space complexities.

Algorithm 751: TRIPACK: a constrained two-dimensional Delaunay triangulation package

Abstract: TRIPACK is a Fortran 77 software package that employs an incremental algorithm to construct a constrained Delaunay traingulation of a set of points in the plane (nodes). The triangulation covers the convex hull of the nodes but may include polygonal constraint regions whose triangles are distinguishable from those in the remainder of the triangulation. This effectively allows for a nonconvex or multiply connected triangulation (the complement of the union of constraint regions) while retaining the efficiency of searching and updating a convex triangulation. 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 is 13N integer storage locations in addition to the 2N nodal coordinates.

A boundary recovery algorithm for Delaunay tetrahedral meshing

Abstract: A method for automatic generation of unstructured grids comprised of tetrahedra is discussed. Delaunay approach for tetrahedral grid generation is used. Particular attention is given to the boundary constraining problem. A simple and robust algorithm for the boundary constraining by successive use of boundary edge swapping, tetrahedral edge swapping and direct subdivision of tetrahedra is used. Small modifications allow to apply the method for viscous grid generation as well. Grid examples demonstrate efficiency of the method.

Optimal delaunay point insertion

Abstract: SUMMARY An efficient algorithm for Delaunay triangulation of a given set of points in d dimensions is presented. Various steps of the point insertion algorithm are reviewed and many acceleration procedures are implemented to speed up the triangulation process. New features include the search for a neighbouring point by a layering scheme, locating the containing simplex by a random walk, formulas of important geometrical quantities of a new simplex based on those of an old one, a novel approach in establishing the adjacency relationship using connection matrices. The resulting scheme seems to be one of the fastest triangulation algorithms known, which enables us to generate tetrahedra in R3 with a linear generation rate of 15 OOO tetrahedra per second for randomly generated points on an HP 735 workstation.

Quasi-greedy triangulations approximating the minimum weight triangulation

Abstract: This article settles the following two longstanding open problems:?What is the worst case approximation ratio between the greedy triangulation and the minimum weight triangulation??Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum?The answer to the first question is that the known lower bound is tight. The second question is answered in the affirmative by using a slight modification of anO(nlogn) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds.

Flipping edges in triangulations

Abstract: In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least \(\lceil (n-4)/2 \rceil\) edges that can be flipped. We also prove that O(n + k2) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n2) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.

Temporal continuity of levels of detail in Delaunay triangulated terrain

Abstract: The representation of a scene at different levels of detail is necessary to achieve real-time rendering. In aerial views, only the part of the scene that is close to the viewing point needs to be displayed with a high level of detail, while more distant parts can be displayed with a low level of detail. However, when a sequence of images is generated and displayed in real-time, the transition between different levels of detail causes noticeable temporal aliasing. In this paper, we propose a method, based on object blending, that visually softens the transition between two levels of Delaunay triangulation. We present an algorithm that establishes, in an off-line process, a correspondence between two given polygonal objects. The correspondence enables on-line blending between two representations of an object, so that one representation (level of detail) progressively evolves into the other.

Simulation of static and dynamic wrinkles of skin

Abstract: Wrinkles are an extremely important contribution for enhancing the realism of human figure models. We present an approach to generate static and dynamic wrinkles on human skin. For the static model, we consider micro and macro structures of the skin surface geometry. For the wrinkle dynamics, an approach using a biomechanical skin model is employed. The tile texture patterns in the micro structure of skin surface are created using planar Delaunay triangulation. Functions of barycentric coordinates are applied to simulate the curved ridges. The visible (macro) flexure lines which may form wrinkles are predefined edges on the micro structure. These lines act as constraints for the hierarchical triangulation process. Furthermore, the dynamics of expressive wrinkles-controlling their depth and fold-is modeled according to the principal strain of the deformed skin surface. Bump texture mapping is used for skin rendering.

Anisotropic adaptive mesh generation in two dimensions for CFD

Abstract: This paper describes the extension of the classical Delaunay method in the case where anisotropic meshes are required such as in CFD when the modelized physic is strongly directional. The way in which such a mesh generation method can be incorporated in an adaptative loop of CFD as well as the case of multicriterium adaptation are discussed. Several concrete application examples are provided to illustrate the capabilities of the proposed method.

Anisotropic Mesh Generation with Particles.

Abstract: : Many important real world problems require meshing, that is the approximation of a given geometry by a set of simpler elements such as triangles or quadrilaterals in two dimensions, and tetrahedra or hexahedra in three dimensions. Applications include finite element analysis and computer graphics. This work focuses on the former. A physically based model of interacting 'particles' is introduced to uniformly spread points over a 2-dimensional polygonal domain. The set of points is triangulated to form a triangle mesh. Delaunay triangulation is used because it guarantees a low computational cost and reasonably well shaped elements. Several particle interaction (repulsion and attraction) models are investigated ranging from Gaussian energy potentials to Laplacian smoothing. Particle population control mechanisms are introduced to make the size of the mesh elements converge to the desired size. In most applications spatial mesh adaptivity is desirable. Triangles should not only adapt in size but also in shape, to better fit the function to approximate. Computational fluid dynamics simulations typically require triangles stretched in the direction of the flow. A metric tensor is introduced to quantify the stretching. The triangulation procedure is changed to generate 'Delaunay' meshes in the Riemannian space defined by the metric. This new approach to mesh generation appears quite promising.

Algorithms for proximity problems in higher dimensions

Abstract: We present algorithms for five interdistance enumeration problems that take as input a set S of n points in R d (for a fixed but arbitrary dimension d ) and as output enumerate pairs of points in S satisfying various conditions. We present: an O ( n log n + k ) time and O ( n ) space algorithm that takes as additional input a distance δ and outputs all k pairs of points in S separated by a distance of δ or less; an O ( n log n + k log k ) time and O ( n + k ) space algorithm that enumerates in non-decreasing order the k closest pairs of points in S ; an O ( n log n + k ) time algorithm for the same problem without any order restrictions; an O ( nk log n ) time and O ( n ) space algorithm that enumerates in nondecreasing order the nk pairs representing the k nearest neighbors of each point in S ; and an O ( n log n + kn ) time algorithm for the same problem without any order restrictions. The algorithms combine a modification of the planar approach of Dickerson, Drysdale, and Sack [11] with the method of Bern, Eppstein, and Gilbert [3] for augmenting a point set to have a linear size bounded degree Delaunay triangulation. Thus, in addition to providing new solutions to these problems, the paper also shows how the Delaunay triangulation can be used as the underlying data structure in a unified approach to proximity problems even in higher dimensions.

System and method for rapidly generating an optimal mesh model of a 3D object or surface

Abstract: A system and method for the rapid creation of a mesh model depicting a real world object, terrain or other three-dimensional surface. The system inserts points into the mesh incrementally, building the mesh point by point. Before incremental building, the system orders the points so that each next point is a near neighbor to the previously inserted point. This ordering procedure optimizes mesh construction by guaranteeing a minimal time for locating the area on the mesh into which the next point will be inserted. The present invention also provides a system and method to ensure an optimal quality of mesh at any level of insertion or deletion, following systematized checking function to maintain quality such as that required in Delaunay triangulation. The system and method can also incorporate a history file to store data concerning the results of the checking to substantially reduce processing time in mesh regeneration applications.

A (usually?) connected subgraph of the minimum weight triangulation

Abstract: that computes a subgraph of the minimum weight triangulation (A4WL”) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give two variants of our algorithm that produce a more complete subgraph of the MWT: an extended LMT-skeleton requiring worst case 0(n6) time and O(n3) space; and a modified LMT-skeleton requiring 0(n2) time and 0( TZ1”5) space in the expected case for uniform distributions.

Matching and Interpolation of Shapes using Unions of Circles

Abstract: While the notion of shape of an object is very intuitive, its precise definition is very elusive, and defining a useful metric for the shape distance between objects is a difficult endeavor. At the same time many successful techniques have been developed which interpolate between two objects, so in essence interpolate between shapes. We present here work which uses a representation of objects as union of circles to define a distance between two objects and to base a method to interpolate between the two. This method can be used in a totally automatic fashion (that is, without any user intervention), or users can guide a pre-registration phase as well as a segmentation phase, after which the matched segments are interpolated pair-wise. The union of circles representation of the two objects is obtained from the Delaunay triangulation of their boundary points. The circles can be simplified to obtain smaller data sets. The circles are then optimally matched according to a distance metric between circles which is a function of their position, size, and feature, that is, a local configuration of circles. The interpolation between the two objects is then obtained by interpolating between the matched pairs of circles (the interpolations can be affine or non-affine). Examples with simple and more complez objects show how the technique can give results which correspond closely to the human notion of shape interpolation. The interpolations shown include some between a calf and a cow and between a cow and a giraffe. The ezamples given are in two dimensions, but all the steps except the segmentation have been implemented as well for three dimensional objects. We also show the results of computation of distances between the objects used in our examples.

Triangulations intersect nicely

Abstract: We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimum-weight triangulation which can be computed in polynomial time by matching and network-flow techniques. We exhibit an easy-to-recognize class of point sets for which the minimum-weight triangulation coincides with the greedy triangulation.

Voronoi diagrams and Morse theory of the distance function

Abstract: We consider the (minimal) distance function of a point in the plane to a set P of N points in the plane. The locus of non-dierentiability of this distance function consists (besides of the points of P) exactly of the Voronoi diagram of P. We show that the number of minima (m), maxima (M) and `saddle points' (s) of the distance function satisfy: m - s + M = 1. This is similar to the Morse type of statements for dierentiable functions. The saddle points occur exactly where a Delaunay edge cuts the corresponding Voronoi edge in its interior. The set of those edges form a subgraph of the Delaunay graph, which connects all minima and saddle points. This graph devides the plane into regions. In each of the compact regions, there is exactly one maximum, the non compact regions don't contain a local maximum. At the end we classify all those graphs if P contains of 3 or 4 points.

Developing a practical projection-based parallel Delaunay algorithm

Abstract: Guy E. Blelloch Gary L. Miller Dafna Talmor Computer Science Department Carnegie Mellon University {blelloch, glmiller, talmor}@cs. emu. edu In this paper we are concerned with developing a practical parallel algorithm for Delaunay triangulation that works well on general distributions, particularly those that arise in Scientific Computation. Although there have been many theoretical algorithms for the problem, and some implementations based on bucketing that work well for uniform distributions, there has been little work on implementations for general distributions. We use the well known reduction of 2D Delaunay triangulation to 3D convex hull of points on a sphere or paraboloid. A variant of the Edelsbrunner and Shi 3D convex hull is used, but for the special case when the point set lies on either a sphere or a paraboloid. Our variant greatly reduces the constant costs from the 3D convex hull algorithm and seems to be a more promising for a practical implementation than other parallel approaches. We have run experiments on the algorithm using a variety of distributions that are motivated by various problems that use Delaunay triangulations. Our experiments show that for these distributions we are within a factor of approximately two in work from the best sequential algorithm.