Showing papers on "Delaunay triangulation published in 1998"

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 new Voronoi-based surface reconstruction algorithm

Abstract: Author(s): Amenta, Nina; Bern, Marshall; Kamvysselis, Manolis | Abstract: We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in IR3. The algorthim is the first for this problem with provable guarantees. Given a ''good sample'' from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling sensity increases. The definition of a good sample is itself interesting: the required sampling density varies locally, rigorously capturing the intuitive notion that featureless areas can be reconstructed from fewer samples. The output mesh interpolates, rather than approximates, the input points. Our algorithm is based on the three-dimensional Voronoi diagram. Given a good program for thsi fundamental subroutine, the algorithm is quite easy to implement.

Surface reconstruction by Voronoi filtering

Abstract: We give a simple combinatorial algorithm that computes a piecewise-linear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on a local feature size function, the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We briefly describe an implementation of the algorithm and show example outputs.

Tetrahedral mesh generation by Delaunay refinement

Abstract: Given a complex of vertices, constraining segments, and planar straight-line constraining facets in E3, with no input angle less than 90’. an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradius-to-shortest edge ratios are no greater than two. The sizes of the tetrahedra can provably gr

A practical numerical approach for large deformation problems in soil

Abstract: A practical method is presented for numerical analysis of problems in solid (in particular soil) mechanics which involve large strains or deformations. The method is similar to what is referred to as ‘arbitrary Lagrangian–Eulerian’, with simple infinitesimal strain incremental analysis combined with regular updating of co-ordinates, remeshing of the domain and interpolation of material and stress parameters. The technique thus differs from the Lagrangian or Eulerian methods more commonly used. Remeshing is accomplished using a fully automatic remeshing technique based on normal offsetting, Delaunay triangulation and Laplacian smoothing. This technique is efficient and robust. It ensures good quality shape and distribution of elements for boundary regions of irregular shape, and is very quick computationally. With remeshing and interpolation, small fluctuations appeared initially in the load-deformation results. In order to minimize these, different increment sizes and remeshing frequencies were explored. Also, various planar linear interpolation techniques were compared, and the unique element method found to work best. Application of the technique is focused on the widespread problem of penetration of surface foundations into soft soil, including deep penetration of foundations where soil flows back over the upper surface of the foundation. Numerical results are presented for a plane strain footing and an axisymmetric jack-up (spudcan) foundation, penetrating deeply into soil which has been modelled as a simple Tresca or Von Mises material, but allowing for increase of the soil strength with depth. The computed results are compared with plasticity solutions for bearing capacity. The numerical method is shown to work extremely well, with potential application to a wide range of soil–structure interaction problems. © 1998 John Wiley & Sons, Ltd.

Fast Polygonal Approximation of Terrains and Height Fields

Abstract: Several algorithms for approximating terrains and other height fields using polygonal meshes are described, compared, and optimized. These algorithms take a height field as input, typically a rectangular grid of elevation data H(x, y), and approximate it with a mesh of triangles, also known as a triangulated irregular network, or TIN. The algorithms attempt to minimize both the error and the number of triangles in the approximation. Applications include fast rendering of terrain data for flight simulation and fitting of surfaces to range data in computer vision. The methods can also be used to simplify multi-channel height fields such as textured terrains or planar color images. The most successful method we examine is the greedy insertion algorithm. It begins with a simple triangulation of the domain and, on each pass, finds the input point with highest error in the current approximation and inserts it as a vertex in the triangulation. The mesh is updated either with Delaunay triangulation or with data-dependent triangulation. Most previously published variants of this algorithm had expected time cost of O(mn) or O(n logm+m), where n is the number of points in the input height field and m is the number of vertices in the triangulation. Our optimized algorithm is faster, with an expected cost of O((m+n) logm). On current workstations, this allows one million point terrains to be simplified quite accurately in less than a minute. We are releasing a C++ implementation of our algorithm.

DeWall: A fast divide and conquer Delaunay triangulation algorithm in Ed

Abstract: The paper deals with Delaunay Triangulations (DT) in Ed space. This classic computational geometry problem is studied from the point of view of the efficiency, extendibility to any dimensionality, and ease of implementation. A new solution to DT is proposed, based on an original interpretation of the well-known Divide and Conquer paradigm. One of the main characteristics of this new algorithm is its generality: it can be simply extended to triangulate point sets in any dimension. The technique adopted is very efficient and presents a subquadratic behaviour in real applications in E’, although its computational complexity does not improve the theoretical bounds reported in the literature. An evaluation of the performance on a number of datasets is reported, together with a comparison with other DT algorithms. 0 1998 Published by Elsevier Science Ltd. All rights reserved.

Computer Animation'97 Dirichlet Free-Form Deformations and their Application to Hand Simulation

Abstract: We present a generalized method for free-form deformations that combines the traditional freeform deformation model with techniques of scattered data interpolation based on Delaunay and Dirichlet/Voronoi diagrams. This technique offers many advantages over traditional FFDs, including simple control of local deformations. It also keeps all the capabilities of FFD extensions, such as extended free-form deformations and direct FFDs. The deformation model has much potential for 3D modeling and animation. We choose to illustrate this with a nontrivial human simulation task: hand animation. We implement a multi-layer deformation model where DFFDs are used to simulate the intermediate layer between the skeleton and the skin.

Voronoi polyhedra and Delaunay simplexes in the structural analysis of molecular-dynamics-simulated materials

Abstract: Voronoi and Delaunay tessellations are applied to pattern recognition of atomic environments and to investigation of the nonlocal order in molecular-dynamics ~MD!-simulated materials. The method is applicable also to materials generated using other computer techniques such as Monte Carlo. The pattern recognition is based on an analysis of the shapes of the Voronoi polyhedron ~VP!. A procedure for contraction of short edges and small faces of the polyhedron is presented. It involves contraction to vertices of all edges shorter than a certain fraction x of the average edge length, with concomitant contraction of the associated faces. Thus, effects of fluctuations are eliminated, providing ‘‘true’’ values of the geometric coordination numbers f , both local and averaged over the material. Nonlocal order analysis involves geometric relations between Delaunay simplexes. The methods proposed are used to analyze the structure of MD-simulated solid lead @J. Rybicki, W. Alda, S. Feliziani, and W. Sandowski, in Proceedings of the Conference on Intermolecular Interactions in Matter , edited by K. Sangwal, E. Jartych, and J. M. Olchowik ~Technical University of Lublin, Lublin, 1995! ,p . 57; J. Rybicki, R. Laskowski, and S. Feliziani, Comput. Phys. Commun. 97, 185 ~1997!# and germianium dioxide @T. Nanba, T. Miyaji, T. Takada, A. Osaka, Y. Minura, and I. Yosui, J. Non-Cryst. Solids 177, 131 ~1994!#. For Pb the contraction results are independent of x. For the open structure of GeO2 there is an x dependence of the contracted structure, so that using several values of x is preferable. In addition to removing effects of thermal perturbation, in open structures the procedure also cleans the resulting VP from faces contributed by the second neighbors. The analysis can be combined with that in terms of the radial distribution g(R), making possible comparison of geometric coordination numbers with structural ones @W. Brostow, Chem. Phys. Lett. 49, 285 ~1977!#. @S0163-1829~98!05721-X#

Hybrid Prismatic/Tetrahedral Grid Generation for Viscous Flow Applications

Abstract: A method for the automatic generation of unstructured grids composed of tetrahedra and prisms is proposed. The prismatic semistructured grid is generated around viscous boundary surfaces and covers viscous regions, whereas the tetrahedral grid covers the rest of the computational domain. The Delaunay approach for tetrahedral grid generation is used. The proposed prismatic grid is structured in directions normal to the boundary faces, but the number of prisms generated from one boundary face is variable from face to face. Unlike conventional prismatic grid generators, this technique works well even in regions of cavities and gaps. The Delaunay background grid generated for surface nodes serves as an efficient data structure to check possible intersections of prisms. Particular attention is given to the boundary-constraining problem. A robust algorithm for the boundary recovery by edge swapping followed by a direct subdivision of tetrahedra is used. Grid examples for internal and external flow problems of complex shapes demonstrate the efficiency of the method

Control Volume Meshes Using Sphere Packing

Abstract: We present a sphere-packing technique for Delaunay-based mesh generation, refinement and coarsening. We have previously established that a bounded radius of ratio of circumscribed sphere to smallest tetrahedra edge is sufficient to get optimal rates of convergence for approximate solutions of Poisson's equation constructed using control volume (CVM) techniques. This translates to Delaunay meshes whose dual, the Voronoi cells diagram, is well-shaped. These meshes are easier to generate in 3D than finite element meshes, as they allow for an element called a sliver.

Shape Reconstruction with Delaunay Complex

Abstract: The reconstruction of a shape or surface from a finite set of points is a practically significant and theoretically challenging problem. This paper presents a unified view of algorithmic solutions proposed in the computer science literature that are based on the Delaunay complex of the points.

Numerical Conformal Mapping Using Cross-Ratios and Delaunay Triangulation

Abstract: We propose a new algorithm for computing the Riemann mapping of the unit disk to a polygon, also known as the Schwarz--Christoffel transformation. The new algorithm, CRDT (for cross-ratios of the Delaunay triangulation), is based on cross-ratios of the prevertices, and also on cross-ratios of quadrilaterals in a Delaunay triangulation of the polygon. The CRDT algorithm produces an accurate representation of the Riemann mapping even in the presence of arbitrary long, thin regions in the polygon, unlike any previous conformal mapping algorithm. We believe that CRDT solves all difficulties with crowding and global convergence, although these facts depend on conjectures that we have so far not been able to prove. We demonstrate convergence with computational experiments. The Riemann mapping has applications in two-dimensional potential theory and mesh generation. We demonstrate CRDT on problems in long, thin regions in which no other known algorithm can perform comparably.

A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations

Abstract: Let X be a complex of vertices and piecewise linear constraining facets embedded in Ed. Say that a simplex is strongly Delaunay if its vertices are in X and there exists a sphere that passes through its vertices but passes through and encloses no other vertex. Then X has a d-dimensional constrained Delaunay triangulation if each k-dimensional constraining facet in X with k d 2 is a union of strongly Delaunay k-simplices. This theorem is especially useful in E3 for forming tetrahedralizations that respect specified planar facets. If the bounding segments of these facets are subdivided so that the subsegments are strongly Delaunay, then a constrained tetrahedralization exists. Hence, fewer vertices are needed than in the most common practice in the literature, wherein additional vertices are inserted in the relative interiors of facets to form a conforming (but unconstrained) Delaunay tetrahedralization.

A Note on Point Location in Delaunay Triangulations of Random Points

Abstract: This short note considers the problem of point location in a Delaunay triangulation of n random points, using no additional preprocessing or storage other than a standard data structure representing the triangulation. A simple and easy-to-implement (but, of course, worst-case suboptimal) heuristic is shown to take expected time O(n 1/3 ) .

Improved incremental randomized Delaunay triangulation

Abstract: We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the levels below. Point location is done by marching in a triangulation to determine the nearest neighbor of the query at that level, then the march restarts from that neighbor at the level below. Using a small sample (3 %) allows a small memory occupation; the march and the use of the nearest neighbor to change levels quickly locate the query.

Distribution of Points on a Sphere with Application to Star Catalogs

Abstract: A new framework is introduced to analyze the spatial distribution of stars in a catalog, namely the Voronoi diagram/Delaunay triangulation. The Voronoi diagram is a partition of the celestial sphere into polygonal cells, one for each star, so that the cell for star P consists of the region on the sphere closer to P than to any other star. The Delaunay triangulation is the topological dual with the important property that each spherical cap circumscribing a triangle contains no stars in its interior. Measures of the uniformity in star density and geometric spacing based on the Voronoi diagram/Delaunay triangulation are presented and compared with existing measures. Methods to generate uniformly distributed points on the sphere, which serve as usefiil test cases for stellar attitude determination analysis, are formulated and compared. One such method, based on a spherical spiral, is easy to implement and yields a very uniform distribution of points. Finally the Voronoi density reduction method is introduced to select stars for an on-board catalog from a larger candidate set. The candidate with the smallest Voronoi cell is removed and the Voronoi diagram of the reduced set is constructed. This process is repeated until the desired number of stars remains.

Finite Element Approximation of the Diffusion Operator on Tetrahedra

Abstract: Linear Galerkin finite element discretizations of the Laplace operator produce nonpositive stiffness coefficients for internal element edges of two-dimensional Delaunay triangulations. This property, also called the positive transmissibility (PT) condition, is a prerequisite for the existence of an M-matrix and ensures that nonphysical local extrema are not present in the solution. For tetrahedral elements, it has already been shown that the linear Galerkin approach does not in general satisfy the PT condition. We propose a modification of the three-dimensional Galerkin scheme that, if a Delaunay triangulation is used, satisfies the PT condition for internal edges and, if further conditions on the boundary are specified, yields an $M$-matrix. The proposed approach can also be extended to the general diffusion operator with nonconstant or anisotropic coefficients.

Shelling Polyhedral 3-Balls and 4-Polytopes

Abstract: There is a long history of constructions of nonshellable triangulations of three-dimensional (topological) balls. This paper gives a survey of these constructions, including Furch's 1924 construction using knotted curves, which also appears in Bing's 1962 survey of combinatorial approaches to the Poincare conjecture, Newman's 1926 explicit example, and M. E. Rudin's 1958 nonshellable triangulation of a tetrahedron with only 14 vertices (all on the boundary) and 41 facets. Here an (extremely simple) new example is presented: a nonshellable simplicial three-dimensional ball with only 10 vertices and 21 facets. It is further shown that shellings of simplicial 3 -balls and 4 -polytopes can ``get stuck'': simplicial 4 -polytopes are not in general ``extendably shellable.'' Our constructions imply that a Delaunay triangulation algorithm of Beichl and Sullivan, which proceeds along a shelling of a Delaunay triangulation, can get stuck in the three-dimensional version: for example, this may happen if the shelling follows a knotted curve.

Object skeletons from sparse shapes in industrial image settings

Abstract: Presents a method for computing the shape skeleton of planar objects in presence of noise occurring inside the image regions. Such noise may be due to poor control of lighting conditions, incorrect thresholding or image subsampling. Binary images of objects with such noise exhibit sparseness (lack of connectivity), within their image regions. Such non-contiguity may also be observed in thresholded images of objects which consist of regions having varying albedo. The problem of obtaining the skeletal description of sparse shapes is ill posed in the sense of conventional skeletonization techniques. We propose a skeletonization method which is based on obtaining the shape skeleton by evolving an approximation of the principal curve of the shape distribution. Our method is implemented as a batch mode Kohonen self-organizing map algorithm and involves iterating the following two steps: (1) Voronoi tessellation of the data, (2) kernel smoothing on the Voronoi centroids. Adjacency relationships between the Voronoi regions are obtained by computing a Delaunay triangulation of the centroids. The Voronoi centroids are connected by a minimum spanning tree after each iteration. The final shape skeleton is obtained by joining centroids which are disjoint in the spanning tree, but have adjacent Voronoi regions. The skeletal descriptions obtained with the method are invariant to translation, rotation, and scale changes of the shape. The potential of the method is demonstrated on industrial objects having varying shape complexity under different imaging conditions.

Nested parallel 2D Delaunay triangulation method

Abstract: A nested parallel implementation of 2D triangulation method recursively sub-divides processors of a parallel computer into asynchronous processor teams. Each of the teams uses data parallel operations to compute a partitioning of the collection of points distributed to it. When each team has a single processor as a result of the recursive partitioning steps, the processors switch to a serial version of the 2D triangulation method. The nested parallel implementation has two levels of recursion: 1) one to partition a collection of points into two new sets; and 2) a second layer nested in the first to compute convex hulls used to form a border around the two new sets of points. In each layer of recursion the implementation sub-divides processors into teams and assigns a control parallel function to each team. Within each team, the processors perform data parallel operations on the collection of points distributed to the processors in the team.

A triangulation method of an arbitrary point set for biomagnetic problems

Abstract: A new triangulation method has been developed for extracting isosurface from volume data. The nodes for triangulation can be selected arbitrarily from the surface of the object of interest. The Voronoi polygons for nodes are searched on the surface and triangulation is accomplished by connecting the neighboring Voronoi areas. The method is basically Delaunay triangulation using geodesic distances instead of Euclidean ones. In areas where the curvature of the surface is low, the Delaunay criteria are fulfilled. When the curvature is high, the geometry of the object is described more accurately than in Euclidean Delaunay methods. Since geodesic distances are utilized, i.e., the surface information is used in triangulation, the topology of the object can be preserved more easily than in the Euclidean cases. Our fully automatic method has been developed for boundary element modeling and it has been successfully applied in magnetocardiographic and electrocardiographic forward and inverse studies. However, the method can be utilized in any triangulation problem if the surface description is provided.

A Robust Implementation for Three-Dimensional Delaunay Triangulations

Abstract: This paper presents an implementation for Delaunay triangulations of three-dimensional point sets. The code uses a variant of the randomized incremental flip algorithm and employs symbolic perturbation to achieve robustness. The algorithm's theoretical time complexity is quadratic in n, the number of input points, and this is optimal in the worst case. However, empirical running times are proportional to the number of triangles in the final triangulation, which is typically linear in n. Even though the symbolic perturbation scheme relies on exact arithmetic, the resulting code is efficient in practice. This is due to a careful implementation of the geometric primitives and the arithmetic module. The source code is available on the Internet.

Delaunay triangulation for image object indexing: a novel method for shape representation

Abstract: Recent research on image databases has been aimed at the development of content-based retrieval techniques for the management of visual information Compared with such visual information as color, texture, and spatial constraints, shape is an important feature Associated with those image objects of interest, shape alone may be sufficient to identify and classify an object completely and accurately This paper presents a novel method, based on feature point histogram indexing for object shape representation in image databases In this scheme, the feature point histogram is obtained by discretizing the angles produced by the Delaunay triangulation of a set of unique feature points, which characterize object shape in context, and then counting the number of times each discrete angle occurs in the resulting triangulation The proposed shape representation technique is translation, scale, and rotation independent Our various experiments concluded that the Euclidean distance performs well as the similarity measure function, in combination with the feature point histogram computed by counting the two largest angles of each individual Delauney triangle Through further experiments, we also found evidence that an image object representation, using a feature point histogram, provides an effective cue for image object discrimination

Novel method to detect a motif of local structures in different protein conformations.

Abstract: In order to detect a motif of local structures in different protein conformations, the Delaunay tessellation is applied to protein structures represented by C(alpha) atoms only. By the Delaunay tessellation the interior space of the protein is uniquely divided up into Delaunay tetrahedra whose vertices are the C(alpha) atom positions. Some edges of the tetrahedra are virtual bonds connecting adjacent residues' C(alpha) atoms along the polypeptide chain and others indicate interactions between residues nearest neighbouring in space. The rules are proposed to assign a code, i.e., a string of digits, to each tetrahedron to characterize the local structure constructed by the vertex residues of one relevant tetrahedron and four surrounding it. Many sets comprised of the local structures with the same code are obtained from 293 proteins, each of which has relatively low sequence similarity with the others. The local structures in each set are similar enough to each other to represent a motif. Some of them are parts of secondary or supersecondary structures, and others are irregular, but definite structures. The method proposed here can find motifs of local structures in the Protein Data Bank much more easily and rapidly than other conventional methods, because they are represented by codes. The motifs detected in this method can provide more detailed information about specific interactions between residues in the local structures, because the edges of the Delaunay tetrahedra are regarded to express interactions between residues nearest neighbouring in space.