Abstract: Given a boundary surface mesh (a set of triangular facets) of a polyhedron, the problem of deciding whether or not a triangulation exists is reported to be NP-hard. In this paper, an algorithm to triangulate a general polyhedron is presented which makes use of a classical Delaunay triangulation algorithm, a phase for recovering the missing boundary facets by means of facet partitioning, and a final phase that makes it possible to remove the additional points defined in the previous step. Following this phase, the resulting mesh conforms to the given boundary surface mesh. The proposed method results in a discussion of theoretical interest about existence and complexity issues. In practice, however, the method should provide what we call ‘ultimate’ robustness in mesh generation methods. Copyright © 2003 John Wiley & Sons, Ltd.

Abstract: Though Delaunay triangulations are very well known geometric data structures, the problem of the robust removal of a vertex in a three-dimensional Delaunay triangulation is still a problem in practice.We propose a simple method that allows to remove any vertex even when the points are in very degenerate configurations. The solution is available in CGAL.

Abstract: The Delaunay triangulation is a widely appreciated and investigated mathematical model for topographic surface represen- tation. After a brief theoretical description, six possible basic algorithms to construct a Delaunay triangulation are analyzed and properties that can be exploited for multibeam echosounder data processing are investigated. Two concepts will be treated in more depth: the divide-and-conquer construction algorithm and the incremental method. The calculation speed of the divide-and-conquer method makes it an ideal candidate to construct the initial triangulation of multibeam data. Its runtime performance is compared to that of the incremental algorithm to demonstrate this. The algorithm's merge step appears to be useful also in replacing triangulated areas of existing triangula- tions by new data. The incremental algorithm does not seem an effective construction method but it can easily be adapted to accommodate insertion of individual vertices into an existing triangulation and as such it is useful for editing purposes.

Abstract: This paper introduces a new algorithm for constructing a 2D Delaunay triangulation. It belongs to the class of incremental insertion algorithms, which are known as less demanding from the implementation point of view. The most time consuming step of the incremental insertion algorithms is locating the triangle containing the next point to be inserted. In this paper, this task is transformed to the nearest point problem, which is solved by a two-level uniform subdivision acceleration technique. Dependencies on the distribution of the input points are reduced using this technique. The algorithm is compared with other popular triangulation algorithms: two variants of Guibas, Knuth, and Sharir's incremental insertion algorithm, two different implementations of Mucke's algorithm, Fortune's sweep-line algorithm, and Lee and Schachter's divide and conquer algorithm. The following point distributions are used for tests: uniform, regular, Gaussian, points arranged in clusters, and real data sets from a GIS databas...

Abstract: The key step in the construction of the Delaunay triangulation of a finite set of planar points is to establish correctly whether a given point of this set is inside or outside the circle determined by any other three points. We address the problem of formulating the in-circle test when the coordinates of the planar points are given only up to a given precision, which is usually the case in practice. By modelling imprecise points as rectangles, and using the idea of partial disc, we construct a reliable in-circle test that provides the best possible Delaunay triangulation with the imprecise input data given by rectangles.

Abstract: It is well known that the complexity of the Delaunay triangulation of $n$ points in $\RR ^d$, i.e., the number of its simplices, can be $\Omega (n^{\lceil {d}/{2}\rceil })$. In particular, in $\RR ^3$, the number of tetrahedra can be quadratic. Put another way, if the points are uniformly distributed in a cube or a ball, the expected complexity of the Delaunay triangulation is only linear. The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface. In this paper we bound the complexity of the Delaunay triangulation of points distributed on the boundary of a given polyhedron. Under a mild uniform sampling condition, we provide deterministic asymptotic bounds on the complexity of the three-dimensional Delaunay triangulation of the points when the sampling density increases. More precisely, we show that the complexity is $O(n^{1.8})$ for general polyhedral surfaces and $O(n\sqrt{n})$ for convex polyhedral surfaces. Our proof uses a geometric result of independent interest that states that the medial axis of a surface is well approximated by a subset of the Voronoi vertices of the sample points.

Abstract: This chapter introduces the two basic approaches to the generation of unstructured grids, Delaunay triangulation, and the Advancing Front method. The presentation in the chapter is confined to two dimensions, but the extension of these methods to three dimensions is straightforward in principle. In most cases, results for triangles in two dimensions can be generalized to tetrahedra in three dimensions. The principal objective of the chapter is to represent the two-dimensional solution domain of a problem by a set of triangles. Delaunay triangulation, the method discussed in the chapter, was first presented by Dirichlet in terms of connecting an arbitrary set of points together, thus producing a set of triangles, in such a way that the resulting triangulation was as near uniformly equilateral as possible. An important feature of a Delaunay triangulation is the Circumcircle Property: this guarantees that in a Delaunay triangulation, none of the points (vertices) of a triangle can lie within the circumcircle of any other triangle. The advancing front technique is an unstructured grid generation method that preserves boundary integrity and has the capacity to create the clustering of high aspect-ratio triangles in boundary-layer regions.

Abstract: We show that the traditional criterion for a simplex to belong to the Delaunay triangulation of a point set is equivalent to a criterion which is a priori weaker. The argument is quite general; as well as the classical Euclidean case, it applies to hyperbolic and hemispherical geometries and to Edelsbrunner's weighted Delaunay triangulation. In spherical geometry, we establish a similar theorem under a genericity condition. The weak definition finds natural application in the problem of approximating a point-cloud data set with a simplical complex.

Abstract: The randomized incremental insertion algorithm of Delaunay triangulation in E3 is very popular due to its simplicity and stability. This paper describes a new parallel algorithm based on this approach. The goals of the proposed parallel solution are not only to make it efficient but also to make it simple. The algorithm is intended for computer architectures with several processors and shared memory. Several versions of the proposed method were tested on workstations with up to eight processors and on datasets of up to 200000 points with favorable results.

Abstract: The Voronoi diagram is a classical problem in the area of computational geometry. Given a plane and a set of n seed points, the objective is to divide the plane into tiles (or subsets of points), such that the set of points in a tile are closer to a particular seed point than to any other seed. This problem has become increasingly important in the area of geographic information systems (GIS) as Voronoi diagrams are used in GIS for computing zonal statistics. Current or timely data for GIS is acquired via remotely-sensed devices providing data in raster(regular or gridded) format. This paper describes parallel algorithms to find the Voronoi Diagram, the furthest site Voronoi diagram and the order-k Voronoi diagram. Prototype implementations in Parallaxes are outlined.

Abstract: This paper describes a newly proposed simple and efficient parallel algorithm for the construction of the Delaunay triangulation (DT) in E2 by randomized incremental insertion. The construction of the DT is one of the fundamental problems in computer graphics. The proposed algorithm is designed for parallel systems with shared memory and several processors. Such hardware (especially with two-processors) became available in the last few years thanks to low prices and at present, there is still a lack of parallel algorithms that are simple to implement and efficient enough to be an attractive alternative to long existing serial algorithms. The designed algorithm incorporates new method for synchronization among PEs based on the simple geometric test (i.e. if no other points lie in the circum-circle of accessed triangle, this triangle can be modified independently on others PEs). We implemented the algorithm in C++ and tested it on workstations up to four processors where we reached relatively good speed-up to our serial implementation. When only two processors were used we reached even super-linear speed-up.

Abstract: This paper presents a dynamic algorithm for the construction of the Euclidean Voronoi diagram of a set of convex objects in the plane. We consider first the Voronoi diagram of smooth convex objects forming pseudo-circles set. A pseudo-circles set is a set of bounded objects such that the boundaries of any two objects intersect at most twice. Our algorithm is a randomized dynamic algorithm. It does not use a conflict graph or any sophisticated data structure to perform conflict detection. This feature allows us to handle deletions in a relatively easy way. In the case where objects do not intersect, the randomized complexity of an insertion or deletion can be shown to be respectively O(^2 n) and O(^3 n). Our algorithm can easily be adapted to the case of pseudo-circles sets formed by piecewise smooth convex objects. Finally, given any set of convex objects in the plane, we show how to compute the restriction of the Voronoi diagram in the complement of the objects' union.

Abstract: A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or 'particles') of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples - based on the Voronoi and Delaunay tessellations - where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

Abstract: Zou and Yan have recently developed a skeletonization algorithm of digital shapes based on a regularity/singularity analysis; they use the polygon whose vertices are the boundary pixels of the image to compute a constrained Delaunay triangulation (CDT) in order to find local symmetries and stable regions. Their method has produced good results but it is slow since its complexity depends on the number of contour pixels. This paper presents an extension of their technique to handle arbitrary polygons, not only polygons of short edges. Consequently, not only can we achieve results as good as theirs for digital images, but we can also compute skeletons of polygons of any number of edges. Since we can handle polygonal approximations of figures, the skeletons are more resilient to noise and faster to process.

Abstract: Given a triangulation T of n points in the plane, we are interested in the minimal set of edges in T such that T can be reconstructed from this set (and the vertices of T) using constrained Delaunay triangulation. We show that this minimal set is precisely the set of non locally Delaunay edges, and that its cardinality is less than or equal to n+i/2 (if i is the number of interior points in T), which is a tight bound.

Abstract: Surface reconstruction algorithms produce piece-wise linear approximations of a surface S from a finite, sufficiently dense, subset of its points. In this paper, we present a fast algorithm for surface reconstruction from scattered data sets. This algorithm is inspired of an existing numerical convection scheme developed by Zhao, Osher and Fedkiw. Unlike this latter, the result of our algorithm does not depend on the precision of a (rectangular- ) grid. The reconstructed surface is simply a set of oriented faces located into the 3D Delaunay triangulation of the points. It is the result of the evolution of an oriented pseudo-surface. The representation of the evolving pseudo-surface uses an appropriate data structure together with operations that allow deformation and topological changes of it. The presented algorithm can handle complicated topologies and, unlike most of the others schemes, it involves no heuristic. The complexity of that method is that of the 3D Delaunay triangulation of the points. We present results of this algorithm which turned out to be efficient even in presence of noise.

Abstract: This paper introduces procedures involving the recursive construction of Voronoi diagrams and Delaunay tessellations. In such constructions, Voronoi and Delaunay concepts are used to tessellate an object space with respect to a given set of generators and then the construction is repeated every time with a new generator set, which comprises members selected from the previous generator set plus features of the current tessellation. Such constructions are shown to provide an integrating conceptual framework for a number of disparate procedures, as well as extending the existing functionality of the basic Voronoi and Delaunay procedures to variable spatial resolutions. Further, because they are shown to be fractal in nature, it is suggested that this characteristic can be exploited in the development of new strategies for spatial modelling.

Abstract: In this paper the presentation of the ball-packing method is reviewed, and a scheme to generate mesh for complex 3-D geometric models is given, which consists of 4 steps: (1) create nodes in 3-D models by ball-packing method, (2) connect nodes to generate mesh by 3-D Delaunay triangulation, (3) retrieve the boundary of the model after Delaunay triangulation, (4) improve the mesh.

Abstract: The paper is the first report on approximation algorithms for computing the maximum weight triangulation of a set of n points in the plane. We prove an Ω( √ n) lower bound on the approximation factor for several heuristics: maximum greedy triangulation, maximum greedy spanning tree triangulation and maximum spanning tree triangulation. We then propose the Spoke Triangulation algorithm, which always approximates the maximum weight triangulation for points in general position within a factor of six and can be computed in O(n log n) time. We also prove that Spoke Triangulation approximates the maximum weight triangulation of a convex polygon within a factor of two.

Abstract: Terrain landform features play major roles in such fields as geomorphology type recognition, relief map generalization, DEM construction and hydrology analysis. This paper presents an automatic method to extract terrain landform features and organize drainage system into tree structure based on bend assessment using Delaunay triangulation model. Compared with traditional DEM or TIN based methods, this pure vector approach obtains not only the topological structure of drainage system in planar graph, but also the valley distribution polygon range. Depending on geometrical computation and judgment of vector line, polygon, the structured properties in drainage representation is enhanced, avoiding the case of noise disturbance in DEM based method. The core algorithm makes use of the ability of Delaunay triangulation in detecting hierarchical structure of each contour line. Three kinds of tree structure organization are discussed: the hierarchical binary tree representing bend inclusion relationship contained in single contour line, the plane structure tree representing valley topological relationship, the semantic hierarchical tree representing valley join level from the point of view of hydrology. This paper gives systemically experiment and detailed comparative analysis.

Abstract: In this paper, we propose a symmetry-based method for face boundary extraction from a binarized facial image, which contains a number of blobs after thresholding. The salient facial features such as eyes, mouth, and face borders are among the blobs. In our method, the symmetric axis transform is conducted on the exterior contours of the blobs in a binarized facial image by a Constrained Delaunay triangulation. A face shape is decomposed into a few components represented by the chain of three types of Delaunay triangles. The facial features are identified by the symmetry related distance analysis and then the face boundaries are traced with a group of connected Delaunay edges. With this method, face boundary with arbitrary shape can be traced precisely and small broken edges can be linked without obviously distortion.

Abstract: Advanced 3D scanning technologies enable us to obtain dense and accurate surface sample point sets. From sufficiently dense sample point set, Crust algorithm, which is based on Voronoi diagram and its dual Delaunay triangulation, can reconstruct a triangle mesh that is topologically valid and convergent to the original surface. However, the algorithm is restricted in the practical application because of its long running time. Based on the fact that we do not need dense sample in featureless area for successful reconstruction, we propose a nonuniformly sampling method to resample the input data set according to the local feature size before reconstruction. In this way, we increase the speed of reconstruction without losing the details we need.

Abstract: Based on the analysis of common Delaunay triangulations methods,especially the divide-and-conquer method,a faster algorithm for constructing Delaunay triangulations is presented.It divides the point set by self-adaptive grid,constructs and merges the sub-triangulations.The key problems of merging sub-triangulations,dealing with the terrain features and the flat triangles are described.The experiments show that the expected time of the algorithm is O(n).

Abstract: The problem of determining whether a polyhedron has a con- strained Delaunay tetrahedralization is NP-complete. How- ever, if no five vertices of the polyhedron lie on a common sphere, the problem has a polynomial-time solution. Con- strained Delaunay tetrahedralization has the unusual status (for a small-dimensional problem) of being NP-hard only for degenerate inputs.

Abstract: The problem of computing a d-dimensional Euclidean Voronoi diagram of spheres is relevant to many areas, including computer simulation, motion planning, CAD, and computer graphics. This paper presents a new algorithm based on the explicit computation of the coordinates and radii of Euclidean Voronoi diagram vertices for a set of spheres. The algorithm is further applied to compute the Voronoi diagram with a specified precision in a fixed length floating-point arithmetic. The algorithm is implemented using the ECLibrary (Exact Computation Library) and tested on the example of a 3-dimensional Voronoi diagram of a set of spheres.