Duality of constrained Voronoi diagrams and Delaunay triangulations

We introduce theconstrained Voronoi diagram of a planar straight-line graph containingn vertices or sites where the line segments of the graph are regarded as obstacles, and show that an extended version of this diagram is the dual of theconstrained Delaunay triangulation. We briefly discussO(n logn) algorithms for constructing the extended constrained Voronoi diagram.This work was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

A Note on Point Location in Delaunay Triangulations of Random Points

This short note considers the problem of point location in a Delaunay triangulation of n random points, using no additional preprocessing or storage other than a standard data structure representing the triangulation. A simple and easy-to-implement (but, of course, worst-case suboptimal) heuristic is shown to take expected time O(n ^1/3 ) . Received November 27, 1997; revised February 15, 1998.     

Two algorithms for constructing a Delaunay triangulation

This paper provides a unified discussion of the Delaunay triangulation. Its geometric properties are reviewed and several applications are discussed. Two algorithms are presented for constructing the triangulation over a planar set ofN points. The first algorithm uses a divide-and-conquer approach. It runs inO(N logN) time, which is asymptotically optimal. The second algorithm is iterative and requiresO(N ²) time in the worst case. However, its average case performance is comparable to that of the first algorithm.This work was supported in part by the National Science Foundation under grant MCS-76-17321 and the Joint Services Electronics Program under contract DAAB-07-72-0259.     

约束Delaunay三角化的研究与实现

针对在文献[1]中提出的边界边、边界面片在Delaunay三角化中的存在条件的3个 命题,给出了采用Voronoi图和邻域的概念的证明过程。并在此基础上,给出了约束Delaun a y三角化的实现算法,进一步丰富了Delaunay三角化的理论基础。     

Randomized incremental construction of Delaunay and Voronoi diagrams

In this paper we give a new randomized incremental algorithm for the construction of planar Voronoi diagrams and Delaunay triangulations. The new algorithm is more on-line than earlier similar methods, takes expected timeO(ngn) and spaceO(n), and is eminently practical to implement. The analysis of the algorithm is also interesting in its own right and can serve as a model for many similar questions in both two and three dimensions. Finally we demonstrate how this approach for constructing Voronoi diagrams obviates the need for building a separate point-location structure for nearest-neighbor queries.Leonidas Guibas and Micha Sharir wish to acknowledge the generous support of the DEC Systems Research Center in Palo Alto, California, where some of this work was carried out. Donald Knuth has been supported by NSF Grant CCR-86-10181. Micha Sharir has been supported by NSF Grant CCR-89-01484, ONR Grant N00014-K-87-0129, the U.S.-Israeli Binational Science Foundation, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

An efficient sweep-line Delaunay triangulation algorithm

This paper introduces a new algorithm for constructing a 2D Delaunay triangulation. It is based on a sweep-line paradigm, which is combined with a local optimization criterion—a characteristic of incremental insertion algorithms. The sweep-line status is represented by a so-called advancing front, which is implemented as a hash-table. Heuristics have been introduced to prevent the construction of tiny triangles, which would probably be legalized. This algorithm has been compared with other popular Delaunay algorithms and it is the fastest algorithm among them. In addition, this algorithm does not use a lot of memory for supporting data structure, it is easy to understand and simple to implement.     

基于凸壳技术的Delaunay三角网生成算法

该文提出了一种针对散乱点集的快速构建Delaunay的算法。该算法首先对散乱点按有向角进行排序,以排序后的点顺序为基础,利用凸壳特性快速将散乱点联结成三角网,最后利用拓扑结构快速将其优化为Delaunay三角网。在联网过程中,充分利用有序点子集的凸壳特性,避免了所有的交点测试,从而保证了对散乱点集生成Delaunay三角网的效率。     

有限元网格的三角划分

三角形网格划分在有限元和计算机图形学中有着广泛的应用,本文主要讨论平面有限元三角形划分的两种算法,后一种是前一种改进了的快速算法。这两种算法可以使得用计算机实现三角网格的划分变得更加容易。     

一致分布点集Delaunay三角化最佳期望时间算法

对文献(Dwyer R A.Higher-dimensional Voronoi diagrams in linear expected time.Discrete&Computational Geometry,1991,6(4):342-367)给出的对d≥2维空间站点集合构造Delaunay超三角形算法做了改进,提...     

面向并行的动态增量式Delaunay三角剖分算法

三角剖分是计算机图形学中的重要话题。并行三角剖分算法的发展对传统三角剖分算法提出了新需求,其中之一即是给定一个点数不断增大的点集,实现对该点集三角剖分的快速增量更新。虽然现今已有一些增量三角剖分算法,但都无法支持新增点落入原有三角剖分之外的情况。为解决此问题,提出了三角剖分的外扩技术,基于插入法设计了增量三角剖分算法TID。该算法能够支持任意次、任意数量、任意位置点的增量添加。TID算法能够对任意分布的点集均给出唯一三角剖分结果。对TID算法的性能评估表明,TID算法比现有算法具有更高的计算效率,且增量功能引入的额外开销较小。此外,该算法已成功作为局地三角剖分算法用于并行三角剖分算法中。     

“The big sweep”: On the power of the wavefront approach to Voronoi diagrams

We show that the wavefront approach to Voronoi diagrams (a deterministic line-sweep algorithm that does not use geometric transform) can be generalized to distance measures more general than the Euclidean metric. In fact, we provide the first worst-case optimal (O (n logn) time,O(n) space) algorithm that is valid for the full class of what has been callednice metrics in the plane. This also solves the previously open problem of providing anO (nlogn)-time plane-sweep algorithm for arbitraryL _k -metrics. Nice metrics include all convex distance functions but also distance measures like the Moscow metric, and composed metrics. The algorithm is conceptually simple, but it copes with all possible deformations of the diagram. Research partially supported by the Natural Sciences and Engineering Research Council of Canada. Research partially supported by the Deutsche Forschungsgemeinschaft, Grant No. Kl 655/2-1.     

生成Delaunay三角网的改进算法

算法以原“改进的自连接Delaunay三角网生成算法”为基础。其主要方法仍是三角网生长法,但同时采用了逐点插入法中的凸壳。在原封闭点的基础上提出了封闭的边界点的概念,并增加了对边界点和边界边的识别和处理,从而进一步提高了构网效率。另外,采用的用边的法向量对边的某侧的点进行判断的方法也简单实用。     

Voronoi tessellation of points with integer coordinates: Time-efficient implementation and online edge-list generation

The Voronoi tessellation in the plane can be computed in a particularly time-efficient manner for generators with integer coordinates, such as typically acquired from a raster image. The Voronoi tessellation is constructed line by line during a single scan of the input image, simultaneously generating an edge-list data structure (DCEL) suitable for postprocessing by graph traversal algorithms. In contrast to the generic case, it can be shown that the topology of the grid permits the algorithm to run faster on complex scenes. Consequently, in Computer Vision applications, the computation of the Voronoi tessellation represents an attractive alternative to raster-based techniques in terms of both computational complexity and quality of data structures.     

一种改进的高效Delaunay三角网的生成算法

Delaunay三角网在GIS／VR中具有很广泛的用途,而分而治之算法和逐点插入法是目前普遍用于生成Delaunay三角网的两种算法。本在研究了基于这两种算法的合成算法后,对其进行了修改和优化,形成了高效合成算法。高效合成算法中提出了通过确定点线关系来解决点的定位问题,优化了其LOP的算法,提高了算法的稳定性,使其执行效率得到很明显地提高,本算法的设计思想还可推广到三维空间。     

基于Delaunay三角化的指纹匹配方法

将计算几何的三角划分方法引入指纹匹配，研究了一种基于DT（Delaunay triangulation）网的指纹匹配方法．通过对细节点的拓扑结构进行DT划分，把空间上位置相近的细节点按照一定规则相连，得到三角形网格．然后基于该网格寻找若干参考点对，并根据获得的参考点对将两幅指纹图像进行姿势调整．最后使用获得的参考点对实现基于点模式的指纹匹配．算法在第1届中国生物特征识别竞赛指纹组的测试结果证明了有效性．     

三角剖分综述

多变形的三角剖分是计算几何中的基本问题,本文简述三角剖分的基本理论及应用,对三角剖分算法做简要的综述,为设计更好的三角剖分算法提供一定的依据。     

基于EMST的平面点集Delaunay三角剖分

提出一种基于欧几里德最小支撑树(EMST)的平面点集Delaunay三角剖分算法.该算法使用线性时间的随机算法求出平面点集的EMST,逐次加入一边构成三角网络,按照最小角最大化的三角化准则,通过局部变换得到平面点集的Delaunay三角剖分.采用的随机化算法有效节省了寻找EMST的计算时间,提高了整个算法的效率.