Showing papers on "Delaunay triangulation published in 1994"

Three-dimensional alpha shapes

Abstract: Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the “shape” of the set. For that purpose, this article introduces the formal notion of the family of a-shapes of a finite point set in R3. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter a e R controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size n in time 0(n2), worst case. A robust implementation of the algorithm is discussed, and several applications in the area of scientific computing are mentioned.

Efficient three‐dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints

Abstract: A method is described which constructs three-dimensional unstructured tetrahedral meshes using the Delaunay triangulation criterion. Several automatic point creation techniques will be highlighted and an algorithm will be presented which can ensure that, given an initial surface triangulation which bounds a domain, a valid boundary conforming assembly of tetrahedra will be produced. Statistics of measures of grid quality are presented for several grids. The efficiency of the proposed procedure reduces the computer time for the generation of realistic unstructured tetrahedral grids to the order of minutes on workstations of modest computational capabilities.

Three-dimensional alpha shapes

Abstract: Frequently, data in scientific computing is in its abstract form a finite point set in space, and it is sometimes useful or required to compute what one might call the ``shape'' of the set. For that purpose, this paper introduces the formal notion of the family of $\alpha$-shapes of a finite point set in $\Real^3$. Each shape is a well-defined polytope, derived from the Delaunay triangulation of the point set, with a parameter $\alpha \in \Real$ controlling the desired level of detail. An algorithm is presented that constructs the entire family of shapes for a given set of size $n$ in time $O(n^2)$, worst case. A robust implementation of the algorithm is discussed and several applications in the area of scientific computing are mentioned.

Optimality of the Delaunay triangulation in źd

Abstract: In this paper we present new optimality results for the Delaunay triangulation of a set of points in ?d. These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ?d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.

Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations

Abstract: Basic algorithms for unstructured mesh generation and fluid flow calculation are discussed. In particular the following are addressed: preliminaries of graphs and meshes; duality and data structures; basic graph operations important in CFD (Computational Fluid Dynamics); triangulation methods, including Varonoi diagrams and Delaunay triangulation; maximum principle analysis; finite volume schemes for scalar conservation law equations; finite volume schemes for the Euler and Navier-Stokes equations; and convergence acceleration for steady state calculations.

Computing a subgraph of the minimum weight triangulation

Abstract: Given a set S of n points in the plane, it is shown that the 2 -skeleton of S is a subgraph of the minimum weight triangulation of S. The β-skeletons are polynomially computable Euclidean graphs introduced by Kirkpatrick and Radke [8]. The 2 -skeleton of S is the β-skeleton of S for β = 2 .

Linear-size nonobtuse triangulation of polygons

Abstract: We give an algorithm for triangulating n-vertex polygonal regions (with holes) so that no angle in the final triangulation measures more than p/2. The number of triangles in the triangulation is only O(n), improving a previous bound of O(n2), and the worst-case running time is O(nlog2n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.

Incremental Delaunay triangulation

Abstract: Dani Lischinski 580 ETC Building Cornell University Ithaca, NY 14850, USA danix@graphics.cornell.edu } Introduction } This gem gives a simple algorithm for the incremental construction of the Delaunay triangulation (DT) and the Voronoi diagram (VD) of a set of points in the plane. A triangulation is called Delaunay if it satis es the empty circumcircle property: the circumcircle of a triangle in the triangulation does not contain any input points in its interior. DT is the straight-line dual of the Voronoi diagram of a point set, which is a partition of the plane into polygonal cells, one for each point in the set, so that the cell for point p consists of the region of the plane closer to p than to any other input point (Preparata and Shamos 1985,Fortune 1992). Delaunay triangulations and Voronoi diagrams, which can be constructed from them, are a useful tool for e ciently solving many problems in computational geometry (Preparata and Shamos 1985). DT is optimal in several respects. For example, it maximizes the minimum angle and minimizes the maximum circumcircle over all possible triangulations of the same point set (Fortune 1992). Thus, DT is an important tool for high quality mesh generation for nite elements (Bern and Eppstein 1992). It should be noted, however, that standard DT doesn't allow edges that must appear in the triangulation to be speci ed in the input. Thus, in order to mesh general polygonal regions the more complicated constrained DT should be used (Bern and Eppstein 1992). The incremental DT algorithm given in this gem was originally presented by Green and Sibson (Green and Sibson 1978), but the implementation is based entirely on the quad-edge data structure and the pseudocode from the excellent paper by Guibas and Stol (Guibas and Stol 1985). I will brie y describe the data structures and the algorithm, but the reader is referred to Guibas and Stol for more details.

Proximity Drawability: a Survey

Abstract: Increasing attention has been given recently to drawings of graphs in which edges connect vertices based on some notion of proximity. Among such drawings are Gabriel, relative neighborhood, Delaunay, sphere of influence, and minimum spanning drawings. This paper attempts to survey the work that has been done to date on proximity drawings, along with some of the problems which remain open in this area.

Graph-Theoretical Conditions for Inscribability and Delaunay Realizability.

Abstract: We present new graph-theoretical conditions for polyhedra of inscribable type and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is of inscribable type. These results have several consequences: • • All 4-connected polyhedra are of inscribable type. • • All simplicial polyhedra in which all vertex degrees are between 4 and 6, inclusive, are of inscribable type. • • All triangulations without chords or nonfacial triangles are realizable as combinatorially equivalent Delaunay triangulations. We also strengthen some earlier results about matchings in polyhedra of inscribable type. Specifically, we show that any nonbipartite polyhedron of inscribable type has a perfect matching containing any specified edge, and that any bipartite polyhedron of inscribable type has a perfect matching containing any two specified disjoint edges. We give examples showing that these results are best possible.

An approach to refining three‐dimensional tetrahedral meshes based on Delaunay transformations

Abstract: A technique for refining three-dimensional tetrahedral meshes is proposed in this paper. The proposed technique is capable of treating arbitrary unstructured tetrahedral meshes, convex or non-convex with multiple regions resulting in high quality constrained Delaunay triangulations. The tetrahedra generated are of high quality (nearly equilateral). Sliver tetrahedra, which present a real problem to many algorithms are not produced with the new method. The key to the generation of high quality tetrahedra is the iterative application of a set of topological transformations based on the Voronoi–Delaunay theory and a reposition of nodes technique. The computational requirements of the proposed technique are in linear relationship with the number of nodes and tetrahedra, making it ideal for direct employment in a fully automatic finite element analysis system for 3-D adaptive mesh refinement. Application to some test problems is presented to show the effectiveness and applicability of the new method.

Parallel terrain triangulation

Abstract: Digital Elevation Models are considered in relation to their use in a parallel computing environment. In particular, the problem of approximating terrain surface through a Triangulated Irregular Network (TIN) is analysed. A parallel algorithm is presented that builds a TIN based on Delaunay triangulation, by selecting a sparse subset of points from a dense regular grid of sampled data. An implementation of the algorithm on a CM-2 is described and experimental results are shown.

Approximating the minimum weight steiner triangulation

Abstract: We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. InO(n logn) time we can compute a triangulation withO(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher-dimensional triangulation problems. No previous polynomial-time triangulation algorithm was known to approximate the MWST within a factor better thanO(logn).

Hamilton Triangulations for Fast Rendering

Abstract: High-performance rendering engines in computer graphics are often pipelined, and their speed is bounded by the rate at which triangulation data can be sent into the machine. To reduce the data rate, it is desirable to order the triangles so that consecutive triangles share a face, meaning that only one additional vertex need be transmitted to describe each triangle. Such an ordering exists if and only if the dual graph of the triangulation contains a Hamiltonian path. In this paper, we consider several problems concerning triangulations with Hamiltonian duals and a related class of “sequential triangulations”.

A linear-time construction of the relative neighborhood graph from the Delaunay triangulation

Abstract: A very simple linear-time algorithm for constructing the relative neighborhood graph RNG( V ) for a finite set V of points in the plane from the Delaunay triangulation of V is presented. It is extended to include the construction of the so called β -skeletons (generalization of RNG( V )) in the spectrum 1 ⩽ β ⩽ 2 in linear time from the Delaunay triangulation under the metric L p for 1 p p = 2.

Range image shape measurement using hierarchical ordering of multiresolution mesh data from delaunay triangulation

Abstract: An image processing method and apparatus using the method allows a target object to be displayed in a 3-D manner. A number of types of shape data with different resolutions for displaying the object are provided and the resolution of shape data used for the 3-D image display is decided in accordance with a display condition. The object is displayed in the form of a 3-D image by employing the shape data with the decided resolution.

Adaptive Delaunay triangulation for attractor image coding

Abstract: The principle of attractor image coding presented in this paper is based on the theory of IFS (iterated function systems). The algorithm exploits piece-wise similarities between block of different sizes. To improve the algorithms based on regular and square blocks, we propose an adaptive Delaunay triangulation of the image support. The originality of the method is to map the triangles on specific parts of the image in order to have lots of similarities between them.

Grid adaptation using a distribution of sources applied to inviscid compressible flow simulations

Abstract: Unstructured tetrahedral grids are generated using a new, very efficient procedure based upon the Delaunay triangulation. The generation procedure is extremely fast, having the capability to generate large grids in minutes on workstations. To maximize this computational performance, a new form of adaptivity has been developed involving the use of sources placed within regions of the domain which require further grid point resolution. A source has a position and a specified grid point density. An error indicator is used to find the elements within the grid which require refinement. Within such elements sources are placed with specified grid point densities which are proportional to the amount of refinement required. The grid generation procedure is then invoked and a grid generated whose grid point density is controlled by the sources. The resulting grid is thus refined in the regions identified by the error indicator as requiring greater resolution. The paper discusses the generation process and emphasizes the new solution adaptation capability. Several examples of the approach are given, including aerospace compressible flow simulations over realistic configurations.

A Chain Decomposition Algorithm for the Proof of a Property on Minimum Weight Triangulations

Abstract: In this paper, a chain decomposition algorithm is proposed and studied. Using this algorithm, we prove a property on minimum weight triangulations of points in the Euclidean plane, which shows that a special kind of line segments must be in the minimum weight triangulations and in the greedy triangulation of a given point set.

Fast greedy triangulation algorithms

Abstract: The greedy triangulation of a set $S$ of $n$ points in the plane is the triangulation obtained by starting with the empty set and at each step adding the shortest compatible edge between two of the points, where a compatible edge is defined to be an edge that crosses none of the previously added edges. In this paper we present a simple, practical algorithm that computes the greedy triangulation in expected time $O(n \log n)$ and space $O(n)$ for points uniformly distributed over any convex shape. A variant of this algorithm should be fast for some other distributions. As part of this algorithm we give an edge compatiblity test that requires $O(n)$ time for both tests and updates to the underlying data structure. We also prove properties about the expected lengths of edges in greedy and Delaunay triangulations of uniformly distributed points.

Reliable Delaunay-based mesh generation and mesh improvement

Abstract: The paper describes a method for generating triangulations of two-dimensional domains. Firstly, a description is given of a reliable algorithm that creates a constrained Delaunay triangulation for a multiply-connected planer domain, without performing any explicit visibility computations. Then, several techniques to improve an existing triangulation are discussed. It turned out that a new variant of mesh relaxation was most effective in improving an existing triangulation. Several examples are included to illustrate the method's overall behaviour.

Image segmentation by directed region subdivision

Abstract: In this paper, an image segmentation method based on directed image region partitioning is proposed. The method consists of two separate stages: a splitting phase followed by a merging phase. The splitting phase starts with an initial coarse triangulation and employs the incremental Delaunay triangulation as a directed image region splitting technique. The triangulation process is accomplished by adding points as vertices one by one into the triangulation. A top-down point selection strategy is proposed for selecting these points in the image domain of grey-value and color images. The merging phase coalesces the oversegmentation, generated by the splitting phase, into homogeneous image regions. Because images might be negatively affected by changes in intensity due to shading or surface orientation change, the authors propose homogeneity criteria which are robust to intensity changes caused by these phenomena for both grey-value and color images. Performance of the image segmentation method has been evaluated by experiments on test images.