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.

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.

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.

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.

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.

Abstract: The currently most efficient algorithm for inference with a probabilistic network builds upon a triangulation of a network's graph. In this paper, we show that pre-processing can help in finding good triangulations for probabilistic networks, that is, triangulations with a minimal maximum clique size. We provide a set of rules for stepwise reducing a graph. The reduction allows us to solve the triangulation problem on a smaller graph. From the smaller graph's triangulation, a triangulation of the original graph is obtained by reversing the reduction steps. Our experimental results show that the graphs of some well-known real-life probabilistic networks can be triangulated optimally just by pre-processing; for other networks, huge reductions in size are obtained.

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

Abstract: Minimum spanning tree problem is a very important problem in VLSI CAD. Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least ω(n2) time. More efficient approaches find a minimum spanning tree only among edges in the Delaunay triangulation of the points. However, Delaunay triangulation is not well defined in rectilinear distance. In this paper, we first establish a framework for minimum spanning tree construction which is based on a general concept of spanning graphs. A spanning graph is a natural definition and not necessarily a Delaunay triangulation. Based on this framework, we then design an O(n log n) sweep-line algorithm to construct a rectilinear minimum spanning tree without using Delaunay triangulation.

Abstract: This study introduces a new Triangulated Irregular Network(TIN) compression method and a progressive visualization technique using Delaunay triangulation. The compression strategy is based on the assumption that most triangulated 2.5-dimensional terrains are very similar to their Delaunay triangulation. Therefore, the compression algorithm only needs to maintain a few edges that are not included in the Delaunay edges. An efficient encoding method is presented for the set of edges by using vertex reordering and a general bracketing method. In experiments, the compression method examined several sets of TIN data with various resolutions, which were generated by five typical terrain simplification algorithms. By exploiting the results, the connecting structures of common terrain data are compressed to 0.17 bits per vertex on average, which is superior to the results of previous methods. The results are shown by a progressive visualization method for web-based GIS.

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...

Abstract: A key step in the finite element method is to generate well-shaped meshes in 3D. A mesh is well-shaped if every tetrahedron element has a small aspect ratio. It is an old outstanding problem to generate well-shaped Delaunay meshes in three or more dimensions. Existing algorithms do not completely solve this problem, primarily because they can not eliminate all slivers. A sliver is a tetrahedron whose vertices are almost coplanar and whose circumradius is not much larger than its shortest edge length. We present two new algorithms to generate sliver-free Delaunay meshes. The first algorithm locally moves the vertices of an almost-good mesh, whose tetrahedra have small circumradius to shortest edge length ratio. We show that the Delaunay triangulation of the perturbed mesh vertices is still almost good. Furthermore, most slivers disappear after a mild perturbation of the mesh vertices. The remaining slivers migrate to the boundary where they can be peeled off or can be treated with boundary enforcement heuristics. The second algorithm adds points to generate well-shaped meshes. It is based on the following observations. Any tetrahedron will disappear from the Delaunay triangulation if a point is added inside the circumsphere of the tetrahedron. Among the tetrahedra created by inserting this new point there could be tetrahedra with large radius-edge ratios, or slivers, or both. However, the new point is incident to every new tetrahedron. We first eliminate tetrahedra with large radius-edge ratios. We then select the point that avoids creating any small slivers when inserting point inside the circumsphere of slivers. We show that the algorithm will not introduce short edges to the Delaunay triangulation. A simple volume argument implies that the algorithm terminates and generates a well-shaped Delaunay mesh. The generated mesh has a good grading. The number of mesh elements is within a small constant factor of any almost-good mesh for that given domain. We also describe some variations of this refinement-based algorithm. In particular, we show that inserting points near sinks instead of circumcenters of bad tetrahedra also generates sliver-free Delaunay meshes.

Abstract: Delaunay triangulation has been much used in such applications as volume rendering, shape representation, terrain modeling and so on. The main disadvantage of Delaunay triangulation is large computation time required to obtain the triangulation on an input points sets. This time can be reduced by using more than one processor, and several parallel algorithms for Delaunay triangulation have been proposed. In this paper, we propose an improved parallel algorithm for Delaunay triangulation, which partitions the bounding convex region of the input points set into a number of regions by using Delaunay edges and generates Delaunay triangles in each region by applying an incremental construction approach. Partitioning by Delaunay edges makes it possible to eliminate merging step required for integrating subresults. It is shown from the experiments that the proposed algorithm has good load balance and is more efficient than Cignoni et al.'s algorithm and our previous algorithm.

Abstract: We describe an algorithm which generates tetrahedral decomposition of a general solid body, whose surface is given as a collection of triangular facets. The principal idea is to modify the constraints in such a way as to make them appear in an unconstrained triangulation of the vertex set apriori. The vertex set positions are randomized to guarantee existence of a unique triangulation which satisfies the Delaunay empty-sphere property. (Algorithms for robust, parallelized construction of such triangulations are available.) In order to make the boundary of the solid appear as a collection of tetrahedral faces, we iterate two operations, edge flip and edge split with the insertion of additional vertex, until all of the boundary facets are present in the tetrahedral mesh. The outcome of the vertex insertion is another triangulation of the input surfaces, but one which is represented as a subset of the tetrahedral faces. To determine if a constraining facet is present in the unconstrained Delaunay triangulation of the current vertex set, we use the results of Rajan which re-formulate Delaunay triangulation as a linear programming problem.

Abstract: Unstructured mesh generation exposes highly irregular computation patterns, which imposes a challenge in implementing triangulation algorithms on parallel machines. This paper reports on an efficient parallel implementation of near Delaunay triangulation with High Performance Fortran (HPF). Our algorithm exploits embarrassing parallelism by performing sub‐block triangulation and boundary merge independently at the same time. The sub‐block triangulation is a divide & conquer Delaunay algorithm known for its sequential efficiency, and the boundary triangulation is an incremental construction algorithm with low overhead. Compared with prior work, our parallelization method is both simple and efficient. In the paper, we also describe a solution to the collinear points problem that usually arises in large data sets. Our experiences with the HPF implementation show that with careful control of the data distribution, we are able to parallelize the program using HPF's standard directives and extrinsic procedures. Experimental results on several parallel platforms, including an IBM SP2 and a DEC Alpha farm, show that a parallel efficiency of 42–86% can be achieved for an eight‐node distributed memory system. We also compare efficiency of the HPF implementation with that of a similarly hand‐coded MPI implementation. Copyright © 2004 John Wiley & Sons, Ltd.

Abstract: This paper presents a novel algorithm for volumetric reconstruction of objects from planar sections using Delaunay triangulation, which solves the main problems posed to models defined by reconstruction, particularly from the viewpoint of producing meshes that are suitable for interaction and simulation tasks. The requirements for these applications are discussed here and the results of the method are presented. Additionally, it is compared to another commonly used reconstruction algorithm based on Delaunay triangulation, showing the advantages of the reconstructions obtained by our technique.

Abstract: The starting point of the analysis in this paper is the following situation: "In a bounded domain in ℝ2, let a finite set of points be given. A triangulation of that domain has to be found, whose vertices are the given points and which is `suitable' for the linear conforming Finite Element Method (FEM)." The result of this paper is that for the discrete Poisson equation and under some weak additional assumptions, only the use of Delaunay triangulations preserves the maximum principle.

Abstract: Computing the Delaunay triangulation of n points is well known to have an Ω(n log n) lower bound. Researchers have attempted to break that bound in special cases where additional information is known.

Abstract: Facial features determination is essential in many applications such as personal identification, 3D face modeling and model based video coding. Fast and accurate facial feature extraction is still a filed to be explored. In this paper, an automatic extracting algorithm is developed to locate "key points" of facial features. The Delaunay Triangulation/Voronoi Diagram technique well known in computational geometric is applied on the edge enhanced binarized facial image. Facial features are classified and extracted in terms of various types of Delaunay triangles and the dual of a subset of the Delaunay triangles, Voronoi edges form the skeleton of facial skin. That is, facial feature's shape is described by Delaunay Triangulation/Voronoi Diagram. Furthermore, the facial features can be identified. The method succeeds in locating facial features in the facial region exactly and is insensitive to face deformation. The method is executable in a reasonably short time.

Abstract: This paper presents simple point insertion and deletion operations in Voronoi diagrams and Delaunay triangulations which may be useful for a wide variety of applications, either where interactivity is important, or where local modification of the topology is preferable to global rebuilding. While incremental point insertion has been known for many years, point deletion is relatively unknown. The robustness and efficiency of a new algorithm are described. A wide variety of potential applications are summarized, and the included computer program may be used as the basis for many new projects.

Abstract: Recently, Aichholzer introduced the remarkable concept of the so-called triangulation path (of a triangulation with respect to a segment), which has the potential of providing efficient counting of triangulations of a point set, and efficient representations of all such triangulations. Experiments support such evidence, although – apart from the basic uniqueness properties – little has been proved so far. In this paper we provide an algorithm which enumerates all triangulation paths (of all triangulations of a given point set with respect to a given segment) in time O(tn3logn) and O(n) space, where n denotes the number of points and t is the number of triangulation paths. For the algorithm we introduce the notion of flips between such paths, and define a structure on all paths such that the reverse search approach can be applied. We also refute Aichholzer's conjecture that points in convex position maximize the number of such paths. There are configurations that allow Ω(2 2n−Θ( log n) ) paths.

Abstract: The triangulation of constrained data set is widely used in Geographic Information System(GIS),geo-science,computational geometry,multi-resolution and high precision DTM, et al This paper researches the triangulation of constrained data set and briefly analyses some existing algorithms A new iterative algorithm and deleting algorithm for triangulating constrained data is proposed

Abstract: In this paper we discuss the kinetic maintenance of the Euclidean Voronoi diagram and its dual, the Delaunay triangulation, for a set of moving disks. The most important aspect in our approach is that we can maintain the Voronoi diagram even in the case of intersecting disks. We achieve that by augmenting the Delaunay triangulation with some edges associated with the disks that do not contribute to the Voronoi diagram. Using the augmented Delaunay triangulation of the set of disks as the underlying structure, we discuss how to maintain, as the disks move, (1) the closest pair, (2) the connectivity of the set of disks and (3) in the case of non-intersecting disks, the near neighbors of a disk.

Abstract: This paper reports on efficient parallel implementations of two-dimensional Delaunay triangulation in High Performance Fortran (HPF) and in Message Passing Interface (MPI). Our parallelization algorithm performs subblock triangulation and boundary merge independently at the same time. The sub-block triangulation is by a divide & conquer Delaunay algorithm known for its sequential efficiency, and the boundary triangulation is by an incremental construction algorithm with low overhead. Compared to prior work, our parallelization method is both simple and efficient. In the paper we also describe a solution to the collinear points problem that usually arises in large data sets for divide & conquer algorithm. Our experiences with the HPF implementation show that with careful control of the data distribution, we are able to parallelize the program using HPF's standard directives and extrinsic procedures. Experimental results on several parallel platforms, including an IBM SP2 and a DEC Alpha farm, show that a parallel efficiency of 31%-59% can be achieved for an 8-node distributed memory system. We also compare efficiency of the HPF implementation to that of a similarly hand-coded MPI implementation.

Abstract: This paper studies the multiplicatively weighted crystal-growth Voronoi diagram, which describes the partition of the plane into crystals with different growth speeds. This type of the Voronoi diagram is defined, and its basic properties are investigated. An approximation algorithm is proposed. This algorithm is based on a finite difference method, called a fast marching method, for solving a special type of a partial differential equation. The proposed algorithm is applied to the planning of a collision-free path for a robot avoiding enemy attacks.

Abstract: In this paper, the boundary conformability of 3D constrained Delaunay triangulation is analyzed. Based on the theory of Delaunay triangulation, the feasibility of 3D constrained Delaunay triangulation is discussed emphatically. The algorithm of 3D constrained Delaunay triangulation of finite domain is designed. Applications in oil and geological exploration and mechanical part design are illustrated. The algorithm plays an important role in the computation and analysis of complicated objects.

Abstract: The paper brings results of post-optimization of Delaunay tetrahedrization. Delaunay triangulation is a very popular method to create 2D meshes, but in 3D its properties are not as good as in 2D and can be improved. For this improvement an already existing method was used: local transformations, so called flips, performed in order to improve different geometrical properties of tetrahedra and applied to the finished 3D Delaunay triangulation. We examined this method by using various criteria to compare their benefits and losses in the area of the tetrahedra shape or time demand. We found out that the greatest benefit comes from the so called compound criteria and from the criterion which minimizes the numbers of tetrahedra. Other criteria have no positive influence on mesh improvement, they rather degrade the quality of the Delaunay mesh. The question of time is not so important, because all criteria are fast enough (they take at most 10 percent of time needed to construct a Delaunay mesh).