Showing papers on "Constrained 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.

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.

Skeletonization of ribbon-like shapes based on regularity and singularity analyses

Abstract: A major problem with traditional skeletonization algorithms is that their results do not always conform to human perceptions since they often contain unwanted artifacts. This paper presents an indirect skeletonization method to reduce these artifacts. The method is based on analyzing regularities and singularities of shapes. A shape is first partitioned into a set of triangles using the constrained Delaunay triangulation technique. Then, regular and singular regions of the shape are identified from the partitioning. Finally, singular regions are stabilized to produce a better result. Experiments show that skeletons obtained from the proposed method closely resemble human perceptions of the underlying shapes.

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

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.

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.

Structural Lines, TINs, and DEMs

Abstract: The standard method of building compact triangulated surface approximations to terrain surfaces (TINs) from dense digital elevation models (DEMs) adds points to an initial sparse triangulation or removes points from a dense initial mesh. Instead, we find structural lines to act as the initial skeleton of the triangulation. These lines are based on local curvature of the surface, not on the flow of water. We build TINs from DEMs with points and structural lines. These experiments show that initializing the TIN with structural lines at the correct scale creates a TIN with fewer points given a particular approximation error. Structural lines are especially effective for small numbers of points and correspondingly rougher approximations.

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

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...

Cartoon image vectorization based on shape subdivision

Abstract: This paper presents a non-pixel-based skeletonization method for vectorizing cartoon images. The constrained Delaunay triangulation technique is applied to subdivide a shape into a set of non-overlapping triangles. Then, certain triangles in the triangulation are merged to remove artifacts. The skeleton of a shape is obtained from the skeletons of its constituent parts. Experimental results show that the proposed method is more accurate and efficient than a typical thinning method.

Sliver-free three dimensional delaunay mesh generation

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.

A TIN compression method 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.

Efficient minimum spanning tree construction without Delaunay triangulation

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.

An improved parallel algorithm for delaunay triangulation on distributed memory parallel computers

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.

Variational Delaunay approach to the generation of tetrahedral finite element meshes

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.

Splitting a Delaunay Triangulation in Linear Time

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.

A Novel Approach for Delaunay 3D Reconstruction with a Comparative Analysis in the Light of Applications

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.

Triangulated irregular network optimization from contour data using bridge and tunnel edge removal

Abstract: This paper proposes an optimization methodology associated with the conversion process from a contour-based elevation model to a TIN model and two algorithms that are combined to reach an optimized result. The first algorithm uses a constrained Delaunay triangulation, in which each contour is used as a non-crossing break line. The second algorithm is designed to eliminate edges known as bridge and tunnel edges by inserting a point using parabolic interpolation at the middle of such features.

Facial feature location with Delaunay triangulation/Voronoi diagram calculation

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.

Dynamic Voronoi/Delaunay Methods and Applications

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.

On exclusion regions for optimal triangulations

Abstract: An exclusion region for a triangulation is a region that can be placed around each edge of the triangulation such that the region cannot contain points from the set on both sides of the edge. We survey known exclusion regions for several classes of triangulations, including Delaunay, Greedy, and Minimum Weight triangulations. We then show an exclusion region of larger area than was previously known for the minimum weight triangulation, which significantly speeds up an algorithm of Beirouti and Snoeyink. We also show that no exclusion region exists for the general class of locally optimal triangulations, in which every triangulation edge optimally triangulates the region determined by its two incident triangles.

Multicriteria-optimized triangulations

Abstract: Triangulation of a given set of points in a plane is one of the most commonly solved problems in computer graphics and computational geometry. Because they are useful in many applications, triangulations must provide well-shaped triangles. Many criteria have been developed to provide such meshes, namely weight and angular criteria. Each criterion has its pros and cons, some of them are difficult to compute, and sometimes even the polynomial algorithm is not known. By any of the existing deterministic methods, it is not possible to compute a triangulation which satisfies more than one criterion or which contains parts developed according to several criteria. We explain how such a mixture can be generated using genetic optimization.

Three-dimensional unstructured mesh generation

Abstract: An unstructured mesh generation method for three dimensional region is described The boundary surfaces are triangulated by a two dimensional anisotropic triangulation method Then the three dimensional meshes are generated by means of a constrained Delaunay algorithm with automatic field point creation Methods for mesh quality optimization are also discussed Several examples are given to show the applications of the method