Topic
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.
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27 Oct 1996
TL;DR: This paper proposes a method, based on object blending, that visually softens the transition between two levels of Delaunay triangulation, and presents an algorithm that establishes, in an off-line process, a correspondence between two given polygonal objects.
Abstract: The representation of a scene at different levels of detail is necessary to achieve real-time rendering. In aerial views, only the part of the scene that is close to the viewing point needs to be displayed with a high level of detail, while more distant parts can be displayed with a low level of detail. However, when a sequence of images is generated and displayed in real-time, the transition between different levels of detail causes noticeable temporal aliasing. In this paper, we propose a method, based on object blending, that visually softens the transition between two levels of Delaunay triangulation. We present an algorithm that establishes, in an off-line process, a correspondence between two given polygonal objects. The correspondence enables on-line blending between two representations of an object, so that one representation (level of detail) progressively evolves into the other.
76 citations
TL;DR: An efficient algorithm to compute the clipped Voronoi diagram for a set of sites with respect to a compact 2D region or a 3D volume is presented and the proposed method is applied to optimal mesh generation based on the centroidal Vor onoi tessellation.
Abstract: The Voronoi diagram is a fundamental geometric structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact domain (i.e. a bounded and closed 2D region or a 3D volume), some Voronoi cells of their Voronoi diagram are infinite or partially outside of the domain, but in practice only the parts of the cells inside the domain are needed, as when computing the centroidal Voronoi tessellation. Such a Voronoi diagram confined to a compact domain is called a clipped Voronoi diagram. We present an efficient algorithm to compute the clipped Voronoi diagram for a set of sites with respect to a compact 2D region or a 3D volume. We also apply the proposed method to optimal mesh generation based on the centroidal Voronoi tessellation.
76 citations
TL;DR: The study reveals that specific triangles in text areas can be clustered together and identified as text body and Experiments show the method is also very efficient.
76 citations
TL;DR: A robust and fast algorithm for performing astrometry and source cross-identification on lists of two-dimensional points, such as between a catalog and an astronomical image, or between two images, tailored to work efficiently on wide fields with a large number of sources and significant nonlinear distortions.
Abstract: We present a robust and fast algorithm for performing astrometry and source cross-identification on lists of two-dimensional points, such as between a catalog and an astronomical image, or between two images. The method is based on minimal assumptions: the lists can be rotated, magnified, and inverted with respect to each other in an arbitrary way. The algorithm is tailored to work efficiently on wide fields with a large number of sources and significant nonlinear distortions, as long as the distortions can be approximated with linear transformations locally over the scale length of the average distance between the points. The procedure is based on symmetric point matching in a newly defined continuous triangle space that consists of triangles generated by extended Delaunay triangulation. Our software implementation performed at the 99.995% success rate on ∼260,000 frames taken by the HATNet project.
76 citations
TL;DR: This paper proves that the stretch factor of the Delaunay triangulation is less than $\rho = 1.998$, significantly improving the current best upper bound of 2 and improving the upper bound that can be achieved by a plane spanner of a Euclidean graph.
Abstract: Let $S$ be a finite set of points in the Euclidean plane. Let $D$ be a Delaunay triangulation of $S$. The stretch factor (also known as dilation or spanning ratio) of $D$ is the maximum ratio, among all points $p$ and $q$ in $S$, of the shortest path distance from $p$ to $q$ in $D$ over the Euclidean distance $||pq||$. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long-standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation is less than $\rho = 1.998$, significantly improving the current best upper bound of 2.42 by Keil and Gutwin [``The Delaunay triangulation closely approximates the complete Euclidean graph,” in Proceedings of the 1st Workshop on Algorithms and Data Structures (WADS), 1989, pp. 47--56]. Our bound of 1.998 also improves the upper bound of the best stretch factor that can be achieved by a plane spanner of a Euclidean graph (the current best upper bound is 2). Our result has a direc...
75 citations