Showing papers on "Delaunay triangulation published in 2001"

Geometry and Topology for Mesh Generation

Abstract: 1. Delaunay triangulations 2. Triangle meshes 3. Combinatorial topology 4. Surface simplification 5. Delaunay tetrahedrizations 6. Tetrahedron meshes 7. Open problems.

Delaunay based shape reconstruction from large data

Abstract: Surface reconstruction provides a powerful paradigm for modeling shapes from samples. For point cloud data with only geometric coordinates as input, Delaunay based surface reconstruction algorithms have been shown to be quite effective both in theory and practice. However, a major complaint against Delaunay based methods is that they are slow and cannot handle large data. We extend the COCONE algorithm to handle supersize data. This is the first reported Delaunay based surface reconstruction algorithm that can handle data containing more than a million sample points on a modest machine.

Constant mean curvature surfaces with Delaunay ends

Abstract: In this paper we shall present a construction of complete surfaces M in R3 with finitely many ends and finite topology, and with nonzero constant mean curvature (CMC). This construction is parallel to the well-known original construction by Kapouleas [5], but we feel that ours is somewhat simpler analytically, and controls the resulting geometry more closely. On the other hand, the surfaces we construct have a rather different, and usually simpler, geometry than those of Kapouleas; in particular, all of the surfaces constructed here are noncompact, so we do not obtain any of his immersed compact examples. The method we use here closely parallels the one we developed recently [10] to study the very closely related problem of constructing Yamabe metrics on the sphere with k isolated singular points, just as Kapouleas’ construction parallels the earlier construction of singular Yamabe metrics by Schoen [18]. The original examples of noncompact CMC surfaces were those in the one-parameter family of rotationally invariant surfaces discovered by Delaunay in 1841 [2]. One extreme element of this family is the cylinder; the ‘Delaunay surfaces’ are periodic, and the embedded members of this family (which are called unduloids) interpolate between the cylinder and an infinite string of spheres arranged along a common axis. The family continues beyond this, but the elements now are immersed (and are called nodoids). The role of Delaunay surfaces in the theory of complete CMC surfaces is analogous to the role of catenoids (and planes) in the study of complete minimal surfaces of finite total curvature. For example, just as any complete minimal surface with two ends must be a catenoid [19], it was proved by Meeks [14] and Korevaar, Kusner and Solomon [8] that any Alexandrov embedded constant mean curvature surface with at most two ends is necessarily a Delaunay surface. A rather more remarkable theorem, paralleling the fact that any end of a complete minimal surface of finite total curvature must be asymptotic to a catenoid or a plane, is the fact that any embedded end of a CMC surface must be asymptotic to one of these rotationally symmetric Delaunay surfaces (and in particular, must be cylindrically bounded).

Application-layer multicast with Delaunay triangulations

Abstract: Recently, application-layer multicast has emerged as an attempt to support group applications without the need for a network-layer multicast protocol, such as IP multicast. In application-layer multicast, applications arrange themselves as a logical overlay network and transfer data within the overlay network. In this paper, Delaunay triangulations are investigated as an overlay network topology for application-layer multicast. An advantage of Delaunay triangulations is that each application can locally derive next-hop routing information without the need for a routing protocol in the overlay. A disadvantage of a Delaunay triangulation as an overlay topology is that the mapping of the overlay to the network-layer infrastructure may be suboptimal. It is shown that this disadvantage can be partially addressed with a hierarchical organization of Delaunay triangulations. Using network topology generators, the Delaunay triangulation is compared to other proposed overlay topologies for application-layer multicast.

Dynamic skin triangulation

Abstract: This paper describes an algorithm for maintaining an approximating triangulation of a deforming surface in R3. The triangulation adapts dynamically to changing shape, curvature, and topology of the surface.

Walking in a triangulation

Abstract: Given a triangulation in the plane or a tetrahedralization in 3-space, we investigate the efficiency of locating a point by walking in the structure with different strategies.

A point-placement strategy for conforming delaunay tetrahedralization

Abstract: A strategy is presented to find a set of points that yields a Conforming Delaunay tetrahedralization of a three-dimensional Piecewise-Linear complex (PLC). This algorithm is novel because it impose...

Generating well-shaped Delaunay meshed in 3D

Abstract: A triangular mesh in 3D is a decomposition of a given geometric domain into tetrahedra. The mesh is well-shaped if the aspect ratio of every of its tetrahedra is bounded from above by a constant. It is Delaunay if the interior of the circum-sphere of each of its tetrahedra does not contain any other mesh vertices. Generating a well-shaped Delaunay mesh for any 3D domain has been a long term outstanding problem. In this paper, we present an efficient 3D Delaunay meshing algorithm that mathematically guarantees the well-shape quality of the mesh, if the domain does not have acute angles. The main ingredient of our algorithm is a novel refinement technique which systematically forbids the formation of shivers, a family of bad elements that none of the previous known algorithms can cleanly remove, especially near the domain boundary — needless to say, that our algorithm ensure that there is no sliver near the boundary of the domain.

AUTOCLUST+ : Automatic clustering of point-data sets in the presence of obstacles

Abstract: Wide spread clustering algorithms use the Euclidean distance to measure spatial proximity. However, obstacles in other GIS data-layers prevent traversing the straight path between two points. AUTOCLUST+ clusters points in the presence of obstacles based on Voronoi modeling and Delaunay Diagrams. The algorithm is free of user-supplied arguments and incorporates global and local variations. Thus, it detects high-quality clusters (clusters of arbitrary shapes, clusters of different densities, sparse clusters adjacent to high-density clusters, multiple bridges between clusters and closely located high-density clusters) without prior knowledge. Consequently, it successfully supports correlation analyses between layers (requiring high-quality clusters) and more general locational optimization problems in the presence of obstacles. All this within O(n log n+ [m + R] log n) expected time, where n is the number of data points, m is the number of line-segments that determine the obstacles and R is the number of Delaunay edges intersecting some obstacles. A series of detailed performance evaluations illustrates the power of AUTOCLUST+ and confirms the virtues of our approach.

Nice point sets can have nasty Delaunay triangulations

Abstract: We consider the complexity of Delaunay triangulations of sets of points in R^3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of n points in R^3 with spread D has complexity Omega(min{D^3, nD, n^2}) and O(min{D^4, n^2}). For the case D = Theta(sqrt{n}), our lower bound construction consists of a uniform sample of a smooth convex surface with bounded curvature. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has near-quadratic complexity.

An Experimental Study of Sliver Exudation.

Abstract: We present results on a two-step improvement of mesh quality in three-dimensional Delaunay triangulations. The first step refines the triangulation by inserting sinks and eliminates tetrahedra with large circumradius over shortest edge length ratio. The second step assigns weights to the vertices to eliminate slivers. Our experimental findings pro- vide evidence for the practical effectiveness of sliver exu- dation.

Dynamic triangulations for efficient 3D simulation of granular materials

Abstract: Granular materials are omnipresent in many fields ranging from civil engineering to food, mining and pharmaceutical industries. Often considered a fourth state of matter, they exhibit specific phenomena such as segregation, arching effects, pattern formation, etc. Due to its potential capability of realistically rendering these behaviors, the Distinct Element Method (DEM) is a very enticing simulation technique. Indeed it makes it possible to analyze and observe phenomena that are barely if at all accessible experimentally. DEM works by tracking every particle in the system individually, maintaining for each a trajectory influenced by external factors such as gravitation or contacts with boundary objects and by the interactions with other grains. The mathematical problem of identifying pairs of grains that interact and locating precisely where the contact occurs is highly dependent on the shape of the grains. We focus in this thesis on 3D spherical grains and use dynamic weighted Delaunay triangulations to track the collisions. The triangulation is built on the centers of the grains and evolves to follow their motion. We prove that all potentially colliding pairs of spheres are adjacent in the triangulation. As there are 6n to 8n edges for n spheres in most practical cases, the complexity of the collision detection becomes linear instead of quadratic in the number of particles, with a small overhead in maintaining the triangulation with efficient local operations. For the physical problem of realistically rendering the collision in a numerical contact model suitable for computer simulation, we have used widely accepted theories such as the viscoelastic model of Cundall, but have also tested some recent, more sophisticated developments in the field. The collision detection and contact models have been implemented in a modular DEM simulation code with advanced features in data structures storing the triangulation, in numerical robustness of the geometric computations, and in parallel processing on shared memory computers. Optimal packing of powders is important in many industrial processes, yet no theoretical result exists when dealing with grains of different sizes. We have performed simulations of such cases and could compare our results with experimental data. Preliminary results have been obtained regarding the relation between the size and proportion of grains and the density of the packing. Other simulations have also been performed, such as the granular flow through an hourglass. As no efficient simulation method is currently known for non-spherical 3D grains, we propose an intermediate approach of gluing spheres together into arbitrary shaped clusters and show some examples based on this approach.

Dense point sets have sparse Delaunay triangulations

Abstract: The spread of a finite set of points is the ratio between the longest and shortest pairwise distances We prove that the Delaunay triangulation of any set of n points in R^3 with spread D has complexity O(D^3) This bound is tight in the worst case for all D = O(sqrt{n}) In particular, the Delaunay triangulation of any dense point set has linear complexity We also generalize this upper bound to regular triangulations of k-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces On the other hand, for any n and D=O(n), we construct a regular triangulation of complexity Omega(nD) whose n vertices have spread D

Delaunay conforming iso-surface, skeleton extraction and noise removal

Abstract: Iso-surfaces are routinely used for the visualization of volumetric structures. Further processing (such as quantitative analysis, morphometric measurements, shape description) requires volume representations. The skeleton representation matches these requirements by providing a concise description of the object. This paper has two parts. First, we exhibit an algorithm which locally builds an iso-surface with two significant properties: it is a 2-manifold and the surface is a subcomplex of the Delaunay tetrahedrization of its vertices. Secondly, because of the latter property, the skeleton can in turn be computed from the dual of the Delaunay tetrahedrization of the iso-surface vertices. The skeleton representation, although informative, is very sensitive to noise. This is why we associate a graph to each skeleton for two purposes: (i) the amount of noise can be identified and quantified on the graph and (ii) the selection of the graph subpart that does not correspond to noise induces a filtering on the skeleton. Finally, we show some results on synthetic and medical images. An application, measuring the thickness of objects (heart ventricles, bone samples) is also presented.

Fuzzy modeling with multivariate membership functions: gray-box identification and control design

Abstract: A novel framework for fuzzy modeling and model-based control design is described. The fuzzy model is of the Takagi-Sugeno (TS) type with constant consequents. It uses multivariate antecedent membership functions obtained by Delaunay triangulation of their characteristic points. The number and position of these points are determined by an iterative insertion algorithm. Constrained optimization is used to estimate the consequent parameters, where the constraints are based on control-relevant a priori knowledge about the modeled process. Finally, methods for control design through linearization and inversion of this model are developed. The proposed techniques are demonstrated by means of two benchmark examples: identification of the well-known Box-Jenkins gas furnace and inverse model-based control of a pH process. The obtained results are compared with results from the literature.

New Approach for Quantifying Process Feasibility: Convex and 1-D Quasi-Convex Regions

Abstract: Uncertainities in chemical plants come from numerous sources: internal like fluctu- atedalues of reaction constants and physical properties or external such as quality and flow rates of feedstreams. Accounting for uncertainty inarious stages of plant opera- tions was identified as one of the most important problems in chemical plant design and operations. A new approach proposed describes process's feasible region and a new metric for ealuating process flexibility based on the conex hull that is inscribed within the feasible region and determines itsolume based on Delaunay Triangulation. The two steps inoled are: 1. a series of simple optimization problems are soled to deter- mine points at the boundary of the feasible region; 2. gien the set of points at the boundary of the feasible region, the conex hull inscribed within the feasible region is determined. This is achieed by implementing the Quickhull algorithm, an incremental procedure for ealuating the conex hull, and then by computing a Delaunay Triangula- tion to determine theolume of the conex hull proiding a new metric for process flexibility. This approach not only proides another feasibility measure, but an accurate description of the feasible space of the process. It was applied to 1-D conex problems, and work is in progress to extend it to nonconex systems.

Efficient Generation of Plane Triangulations without Repetitions

Abstract: A "based" plane triangulation is a plane triangulation with one designated edge on the outer face. In this paper we give a simple algorithm to generate all biconnected based plane triangulations with at most n vertices. The algorithm uses O(n) space and generates such triangulations in O(1) time per triangulation without duplications. The algorithm does not output entire triangulations but the difference from the previous triangulation. By modifying the algorithm we can generate all biconnected based plane triangulation having exactly n vertices including exactly r vertices on the outer face in O(1) time per triangulation without duplications, while the previous best algorithm generates such triangulations in O(n2) time per triangulation. Also we can generate without duplications all biconnected (non-based) plane triangulations having exactly n vertices including exactly r vertices on the outer face in O(r2n) time per triangulation, and all maximal planar graphs having exactly n vertices in O(n3) time per graph.

Sink-insertion for mesh improvement

Abstract: We propose sink-insertion as a new technique to improve the mesh quali ty of Delaunay triangulations. We compare it with the conventional circumcenter-insertion technique under three scheduling regimes: incremental, in blocks, and in parallel. Justification for sink-insertion is given in terms of mesh quality, numerical robustness, running time, and ease of parallelization.

Nice point sets can have nasty Delaunay triangulations

Abstract: We consider the complexity of Delaunay triangulations of sets of point s in $\Real^3$ under certain practical geometric constraints. The \emph{spread} of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of $n$ points in~$\Real^3$ with spread $\Delta$ has complexity $\Omega(\min\set{\Delta^3, n\Delta, n^2})$ and $O(\min\set{\Delta^4, n^2})$. For the case $\Delta = \Theta(\sqrt{n})$, our lower bound construction consists of a uniform sample of a smooth convex surface with bounded curvature. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has near-quadratic complexity.

Static and kinetic geometric spanners with applications

Abstract: It is well known that the Delaunay Triangulation is a spanner graph of its vertices. In this paper we show that any bounded aspect ratio triangulation in two and three dimensions is a spanner graph of its vertices as well. We extend the notion of spanner graphs to environments with obstacles and show that both the Constrained Delaunay Triangulation and bounded aspect ratio conforming triangulations are spanners with respect to the corresponding visibility graph. We also show how to kinetize the Constrained Delaunay Triangulation. Using such time-varying triangulations we describe how to maintain sets of near neighbors for a set of moving points in both unconstrained and constrained environments. Such nearest neighbor maintenance is needed in many virtual environments where nearby agents interact. Finally, we show how to use the Constrained Delaunay Triangulation in order to maintain the relative convex hull of a set of points moving inside a simple polygon.

On simulated annealing and the construction of linear spline approximations for scattered data

Abstract: We describe a method to create optimal linear spline approximations to arbitrary functions of one or two variables, given as scattered data without known connectivity. We start with an initial approximation consisting of a fixed number of vertices and improve this approximation by choosing different vertices, governed by a simulated annealing algorithm. In the case of one variable, the approximation is defined by line segments; in the case of two variables, the vertices are connected to define a Delaunay triangulation of the selected subset of sites in the plane. In a second version of this algorithm, specifically designed for the bivariate case, we choose vertex sets and also change the triangulation to achieve both optimal vertex placement and optimal triangulation. We then create a hierarchy of linear spline approximations, each one being a superset of all lower-resolution ones.

Optimal Möbius Transformations for Information Visualization and Meshing

Abstract: We give linear-time quasiconvex programming algorithms for finding a Mobius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, element size control in conformal structured mesh generation, and brain flat mapping.

Terminal-edges Delaunay (small-angle based) algorithm for the quality triangulation problem

Abstract: The terminal-edge Delaunay algorithm, initially called Lepp–Delaunay algorithm, deals with the construction of size-optimal (adapted to the geometry) quality triangulation of complex objects In two dimensions, the algorithm can be formulated in terms of the Delaunay insertion of both midpoints of terminal edges (the common longest-edge of a pair of Delaunay triangles) and midpoints of boundary related edges in the current mesh For the processing of a small angled triangle in the current mesh, the terminal-edge is found as the final longest-edge of the finite chain of triangles that neighbor on a longest edge — the longest edge propagating path of the small angled triangle Three boundary-related point insertion operations, which prevent nonconvergence behavior, are discussed in detail The triangle improvement properties of the point insertion operations are used to prove that optimal-size triangulations, with smallest-angle greater than or equal to 30° are always produced

A strategy for analysis of (molecular) equilibrium simulations: Configuration space density estimation, clustering, and visualization

Abstract: We propose an approach for summarizing the output of long simulations of complex systems, affording a rapid overview and interpretation. First, multidimensional scaling techniques are used in conjunction with dimension reduction methods to obtain a low-dimensional representation of the configuration space explored by the system. A nonparametric estimate of the density of states in this subspace is then obtained using kernel methods. The free energy surface is calculated from that density, and the configurations produced in the simulation are then clustered according to the topography of that surface, such that all configurations belonging to one local free energy minimum form one class. This topographical cluster analysis is performed using basin spanning trees which we introduce as subgraphs of Delaunay triangulations. Free energy surfaces obtained in dimensions lower than four can be visualized directly using iso-contours and -surfaces. Basin spanning trees also afford a glimpse of higher-dimensional to...

Generating Well-Shaped d-dimensional Delaunay Meshes

Abstract: A d-dimensional simplicial mesh is a Delaunay triangulation if the circumsphere of each of its simplices does not contain any vertices inside. A mesh is well-shaped if the maximum aspect ratio of all its simplices is bounded from above by a constant. It is a long-term open problem to generate well-shaped d-dimensional Delaunay meshes for a given polyhedral domain. In this paper, we present a refinement-based method that generates well-shaped d-dimensional Delaunay meshes for any PLC domain with no small input angles. Furthermore, we show that the generated well-shaped mesh has O(n) d-simplices, where n is the smallest number of d-simplices of any almost-good meshes for the same domain. A mesh is almost-good if each of its simplices has a bounded circumradius to the shortest edge length ratio.

Comparison of existing triangulation methods for regularly and irregularly spaced height fields

Abstract: Over the last two decades, the Delaunay triangulation has been the only choice for most geographical information system (GIS) users and researchers to build triangulated irregular networks (TINs). The classical Delaunay triangulation for creating TINs only considers the 2D distribution of data points. Recent research efforts have been devoted to generating data-dependent triangulation which incorporate information on both distribution and values of input data in the triangulation process. This paper compares the traditional Delaunay triangulations with several variant data-dependent triangulations based on Lawson's local optimization procedure (LOP). Two USGS digital elevation models (DEMs) are used in the comparison. It is clear from the experiments that the quality of TINs not only depends on the vertex placement but also on the vertex connection. Traditonal two step processes for TIN construction, which separate point selection from the triangulation, generate far worse results than the methods which i...