Delaunay triangulation

About: Delaunay triangulation is a research topic. Over the lifetime, 5816 publications have been published within this topic receiving 126615 citations. The topic is also known as: Delone triangulation.

Engineering a compact parallel delaunay algorithm in 3D

Abstract: We describe an implementation of a compact parallel algorithm for 3D Delaunay tetrahedralization on a 64-processor shared-memory machine Our algorithm uses a concurrent version of the Bowyer-Watson incremental insertion, and a thread-safe space-efficient structure for representing the mesh Using the implementation we are able to generate significantly larger Delaunay meshes than have previously been generated—10 billion tetrahedra on a 64 processor SMP using 200GB of RAMThe implementation makes use of a locality based relabeling of the vertices that serves three purposes—it is used as part of the space efficient representation, it improves the memory locality, and it reduces the overhead necessary for locks The implementation also makes use of a caching technique to avoid excessive decoding of vertex information, a technique for backing out of insertions that collide, and a shared work queue for maintaining points that have yet to be inserted

On Approximation Behavior of the Greedy Triangulation for Convex Polygons

Abstract: We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n 2) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).

Matching and Interpolation of Shapes using Unions of Circles

Abstract: While the notion of shape of an object is very intuitive, its precise definition is very elusive, and defining a useful metric for the shape distance between objects is a difficult endeavor. At the same time many successful techniques have been developed which interpolate between two objects, so in essence interpolate between shapes. We present here work which uses a representation of objects as union of circles to define a distance between two objects and to base a method to interpolate between the two. This method can be used in a totally automatic fashion (that is, without any user intervention), or users can guide a pre-registration phase as well as a segmentation phase, after which the matched segments are interpolated pair-wise. The union of circles representation of the two objects is obtained from the Delaunay triangulation of their boundary points. The circles can be simplified to obtain smaller data sets. The circles are then optimally matched according to a distance metric between circles which is a function of their position, size, and feature, that is, a local configuration of circles. The interpolation between the two objects is then obtained by interpolating between the matched pairs of circles (the interpolations can be affine or non-affine). Examples with simple and more complez objects show how the technique can give results which correspond closely to the human notion of shape interpolation. The interpolations shown include some between a calf and a cow and between a cow and a giraffe. The ezamples given are in two dimensions, but all the steps except the segmentation have been implemented as well for three dimensional objects. We also show the results of computation of distances between the objects used in our examples.

Algorithm 872: Parallel 2D constrained Delaunay mesh generation

Abstract: Delaunay refinement is a widely used method for the construction of guaranteed quality triangular and tetrahedral meshes We present an algorithm and a software for the parallel constrained Delaunay mesh generation in two dimensions Our approach is based on the decomposition of the original mesh generation problem into N smaller subproblems which are meshed in parallel The parallel algorithm is asynchronous with small messages which can be aggregated and exhibits low communication costs On a heterogeneous cluster of more than 100 processors our implementation can generate over one billion triangles in less than 3 minutes, while the single-node performance is comparable to that of the fastest to our knowledge sequential guaranteed quality Delaunay meshing library (the Triangle)