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。     

基于Voronoi 最小邻近点集的Delaunay 三角化方法

针对局部条件下网格生成的需求,提出一种基于节点的Delaunay 三角化 生成算法,该算法以Delaunay 三角形及其对偶Voronoi 图的局部性特征为基础,通过在局部 搜索最小Voronoi 邻近点集,来生成约束点附近的局部网格,通过建立背景索引网格,来提 高算法效率。给出算法的原理证明、程序实现、效率分析和测试结果,并给出了算法的应用 领域。     

三维约束Delaunay三角化的实现

分析了约束Delaunay三角化中存在的边界一致性问题,给出了约束Delaunay三角化的理论依据,重点探讨了三维约束Delaunay三角化的可行性条件和范围,同时,给出了三维有限域约束Delaunay三角化的实现方法及其在石油地质勘探数据和机械零件方面的网格剖分实例.这种算法在复杂对象的科学计算和工程分析中发挥了重要作用.     

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.     

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。     

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.     

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...     

二维复杂限定Delaunay三角化算法

针对包括曲线边界和内部带有曲线限定条件的二维Delaunay三角化问题,提出了一种细化算法.首先给出了曲线段的逼近边定义,以保证限定曲线在网格中的存在;然后证明了该算法的收敛性和最终曲线的逼近边集合与原曲线的拓扑一致性,并且生成的网格符合Delaunay优化准则;最后给出了算法的应用实例,验证了其有效性.     

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。     

基于Delaunay三角剖分生成Voronoi图算法

针对Delaunay三角网生长算法和间接生成Voronoi图算法构网效率不高的问题,提出了一种Delaunay三角网生长法间接生成Voronoi图的改进算法。该算法以点集凸壳上一边快速生成种子三角形,定义了半封闭边界点的概念,在三角形扩展过程中动态删除封闭点及半封闭边界点,加快Delaunay三角网生成速度。然后又定义了有序目标三角形的概念,该算法能迅速查找点的有序目标三角形,生成无射线的Voronoi图;考虑凸壳上点的特性,借助三个无穷点生成带射线的Voronoi图。通过实验结果分析表明,改进的算法执行效率有了很大提高。     

多尺度形态滤波及弹性匹配技术在类星体谱线识别中的应用

本文提出了一种类星体谱线证认方法。首先针对特征为极值点的信号，研究了多尺度膨胀（腐蚀）关于极值点数的两种重要特性及其应用。其一是单调率特性，根据它自动选择滤波器尺度，有效地滤除脉冲噪声；另一种是单调性，它是＂从粗到精＂策略来重新恢复极值特征位置的理论基础。根据这些性质，对光谱进行多尺度膨胀（腐蚀）和特征恢复，以滤除脉冲噪声而不影响谱线特征。然后研究弹性匹配技术应用于谱线证认，并指出了匹配方法中参量的物理意义。该方法对其他一些应用领域也行之有效     

“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.     

Delaunay三角网剖分中的约束边嵌入算法

约束边嵌入是解决D-三角网转变为CD-三角网的一种非常有效的方法,而CD-三角网才能真实地虚拟地形地貌。该文研究了约束边嵌入D-三角网的问题,介绍了约束边嵌入算法的一些基本概念,分析了现存算法的特点,提出了改进的约束边嵌入算法——“插入－交换”算法。该算法能有效地处理各种特殊情况,程序实现简单,符合工程需求。