Hadoop下面元加权Voronoi图并行算法及应用

面元加权Voronoi图是生成元为面元的加权Voronoi图。针对大规模数据情况下面元加权Voronoi图存在的计算效率不高问题,结合面元边界点提取方法,提出一种基于Hadoop云平台的面元加权Voronoi图的并行生成算法,进行了单机和集群实验。实验结果表明,算法能有效处理大规模栅格数据,明显提高面元加权Voronoi图的生成速度。还可应用于城市绿地设计规划,为绿地设计提供决策依据。     

移动点Voronoi图拓扑维护策略的研究

移动环境下基于Voronoi图的最近邻查询必须要解决随时间不断改变的移动点Voronoi图的拓扑结构的维护问题。通过一组离散的,有限的事件序列对其对偶图Delaunay图拓扑改变过程的模拟来实现对移动点Voronoi图拓扑结构的维护。把带有事件驱动机制的移动数据结构（Kinetic Data Structure,KDS）模型作为移动点的运动模型,给出了KDS模型对其对偶图Delaunay图拓扑结构改变维护的具体策略,并对移动环境下动态插入或删除移动点时Voronoi图的拓扑维护问题进行了研究。最后给出了移动环境下基于Voronoi图的近邻查询的数据库实现模型。     

一种基于点集自适应分组构建Voronoi 图的并行算法

论文提出一种基于点集自适应分组构建Voronoi 图的并行算法,其基本思 路是采用二叉树分裂的方法将平面点集进行自适应分组,将各分组内的点集独立生成 Voronoi 图,称为Voronoi 子图;提取所有分组内位于四边的边界点,对边界点集构建Voronoi 图,称为边界点Voronoi 图;最后,针对每个边界点,提取其位于Voronoi 子图和边界点Voronoi 图内所对应的两个多边形,进行Voronoi 多边形的合并,最终实现子网的合并。考虑到算法 耗时主要在分组点集的Voronoi 图生成,而各分组的算法实现不受其他分组影响,采用并行 计算技术加速分组点集的Voronoi 图生成。理论分析和测试表明,该算法是一个效率较高的 Voronoi 图生成并行算法。     

广义Voronoi图求解多机器人运动规划

在拥挤环境中,由于障碍物的边界形状比较复杂,需要使用广义Voronoi图表示空间环境。且在多移动机器人的运动规划过程中,需要协调多个机器人的运动,必须得到Voronoi图通道的宽度。为此提出了一种计算拥挤障碍物环境中生成的广义Voronoi图及其通道宽度的算法。并在生成的Voronoi图上利用A*算法对多个机器人进行路径规划,并利用分布式方法协调多个机器人运动。对协调两个机器人运动的过程进行了仿真,仿真结果表明利用提出的算法生成的具有通道宽度信息的Voronoi图能够满足多移动机器人运动规划的需要。     

Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations

The combinatorial complexities of (1) the Voronoi diagram of moving points in 2D and (2) the Voronoi diagram of lines in 3D, both under the Euclidean metric, continues to challenge geometers because of the open gap between the Ω(n²) lower bound and the O(n3+?) upper bound. Each of these two combinatorial problems has a closely related problem involving Minkowski sums: (1′) the complexity of a Minkowski sum of a planar disk with a set of lines in 3D and (2′) the complexity of a Minkowski sum of a sphere with a set of lines in 3D. These Minkowski sums can be considered “cross-sections” of the corresponding Voronoi diagrams. Of the four complexity problems mentioned, problems (1′) and (2′) have recently been shown to have a nearly tight bound: both complexities are O(n2+?) with lower bound Ω(n²).In this paper, we determine the combinatorial complexities of these four problems for some very simple input configurations. In particular, we study point configurations with just two degrees of freedom (DOFs), exploring both the Voronoi diagrams and the corresponding Minkowski sums. We consider the traditional versions of these problems to have 4 DOFs. We show that even for these simple configurations the combinatorial complexities have upper bounds of either O(n²) or O(n2+?) and lower bounds of Ω(n²).     

Voronoi diagrams with barriers and on polyhedra for minimal path planning

Two generalizations of the Voronoi diagram in two dimensions (E²) are presented in this paper. The first allows impenetrable barriers that the shortest path must go around. The barriers are straight line segments that may be combined into polygons and even mazes. Each region of the diagram delimits a set of points that have not only the same closest existing point, but have the same topology of shortest path. The edges of this diagram, which has linear complexity in the number of input points and barrier lines, may be hyperbolic sections as well as straight lines. The second construction considers the Voronoi diagram on the surface of a convex polyhedron, given a set of fixed source points on it. Each face is partitioned into regions, such that the shortest path to any goal point in a given region from the closest fixed source point travels over the same sequence of faces to the same closest point.This material is based upon work supported by the National Science Foundation under grants ECS-8021504 and ECS-8351942. The second author is also supported in part by a Fulbright scholarship     

A parallel algorithm for constructing Voronoi diagrams based on point‐set adaptive grouping

This paper presents a parallel algorithm for constructing Voronoi diagrams based on point‐set adaptive grouping. The binary tree splitting method is used to adaptively group the point set in the plane and construct sub‐Voronoi diagrams for each group. Given that the construction of Voronoi diagrams in each group consumes the majority of time and that construction within one group does not affect that in other groups, the use of a parallel algorithm is suitable. After constructing the sub‐Voronoi diagrams, we extracted the boundary points of the four sides of each sub‐group and used to construct boundary site Voronoi diagrams. Finally, the sub‐Voronoi diagrams containing each boundary point are merged with the corresponding boundary site Voronoi diagrams. This produces the desired Voronoi diagram. Experiments demonstrate the efficiency of this parallel algorithm, and its time complexity is calculated as a function of the size of the point set, the number of processors, the average number of points in each block, and the number of boundary points. Copyright © 2013 John Wiley & Sons, Ltd.     

Region-expansion for the Voronoi diagram of 3D spheres

Given a set of spheres in 3D, constructing its Voronoi diagram in Euclidean distance metric is not easy at all even though many mathematical properties of its structure are known. This Voronoi diagram has been known for many important applications from science and engineering. In this paper, we characterize the Voronoi diagram of spheres in three-dimensional Euclidean space, which is also known as an additively weighted Voronoi diagram, and propose an algorithm to construct the diagram. Starting with the ordinary Voronoi diagram of the centers of the spheres, the proposed region-expansion algorithm constructs the desired diagram by expanding the Voronoi region of each sphere, one after another. We also show that the whole Voronoi diagram of n spheres can be constructed in O(n³) time in the worst case.     

基于Voronoi图的点群目标普适综合算法

点要素综合算法的目的是在点数减少的情况下尽量正确地传输包含在点群中的信息，但是目前提出的两种算法均不能达到此要求，如为居民地选取的增长算法不能很好处理拓扑信息，而基于Voronoi的算法又没有考虑点的重要性程度(即点包含的专题信息)。为克服这些缺点，提出了一个新的算法。该算法采用以下两种方法确保不同信息的正确传输：(1)根据基本选取法则确保点数的正确；(2)反复构造剩余点的Voronoi图，并根据一个点与其周围点重要性程度的比较来确定其删除与否，从而使拓扑、专题和几何信息能正确传输。该算法的缺点是没有考虑点的符号化，由此可能导致地图上符号的压盖和重叠。     

A general framework for image fusion based on multi-scale transform and sparse representation

In image fusion literature, multi-scale transform (MST) and sparse representation (SR) are two most widely used signal/image representation theories. This paper presents a general image fusion framework by combining MST and SR to simultaneously overcome the inherent defects of both the MST- and SR-based fusion methods. In our fusion framework, the MST is firstly performed on each of the pre-registered source images to obtain their low-pass and high-pass coefficients. Then, the low-pass bands are merged with a SR-based fusion approach while the high-pass bands are fused using the absolute values of coefficients as activity level measurement. The fused image is finally obtained by performing the inverse MST on the merged coefficients. The advantages of the proposed fusion framework over individual MST- or SR-based method are first exhibited in detail from a theoretical point of view, and then experimentally verified with multi-focus, visible-infrared and medical image fusion. In particular, six popular multi-scale transforms, which are Laplacian pyramid (LP), ratio of low-pass pyramid (RP), discrete wavelet transform (DWT), dual-tree complex wavelet transform (DTCWT), curvelet transform (CVT) and nonsubsampled contourlet transform (NSCT), with different decomposition levels ranging from one to four are tested in our experiments. By comparing the fused results subjectively and objectively, we give the best-performed fusion method under the proposed framework for each category of image fusion. The effect of the sliding window’s step length is also investigated. Furthermore, experimental results demonstrate that the proposed fusion framework can obtain state-of-the-art performance, especially for the fusion of multimodal images.     

An optimization algorithm for shape analysis of regular polygons

Machine vision has the potential to impact both quality and productivity significantly in computer integrated manufacturing due to its versatility, flexibility, and relative speed. Unfortunately, algorithm development has not kept pace with the advances in vision-hardware technology, particularly in the areas of analysis and decision making. The specific subject of this investigation is the development of a machine-vision algorithm for the dimensional checking, pose estimation, and overall shape verification of regular polygonal objects (e.g., surface-mounted electronic components and fastener heads). Algorithmically, the image boundary data is partitioned inton segments, and then a non-ordinary least squares technique is used to find the best fitting polygon. The algorithm is well-suited for online implementation in an automated environment due to its flexibility and demonstrated speed.     

A Shape Metric for Blum Ribbons

A Blum ribbon is a figure whose boundary is the envelope of a family of circles the centers of which lie along a single unbranched curve called its medial axis. Define a Blum ribbon to be simple if its medial axis is the line segment joining points (0,0) and (1,0). Any Blum ribbon can be made simple by flexing the medial axis, rotating, then dilating, all of which are basic transformations in Blum's geometry of shape. The Lie group SL(2, R) acts on circles centered on the x-axis by linear fractional transformations. By means of this action it is possible to associate to any simple Blum ribbon a curve in SL(2, R). A distance between corresponding points lying on such curves is defined using the bi-invariant metric on SL(2, R). Resulting scale-invariant metrics on the set of figures defined as Blum ribbons are described and it is shown that these metrics can provide effective measures of shape difference.     

Topology-oriented incremental computation of Voronoi diagrams of circular arcs and straight-line segments

We introduce an algorithm for computing Voronoi diagrams of points, straight-line segments and circular arcs in the two-dimensional Euclidean plane. Based on a randomized incremental insertion, we achieve a Voronoi algorithm that runs in expected time O(nlogn) for a total of n points, segments and arcs, if at most a constant number of segments and arcs is incident upon every point. Our theoretical contribution is a careful extension of the topology-oriented approach by Sugihara and Iri in order to make the incremental insertion applicable to circular arcs.Our main practical contribution is the extension of Held’s Voronoi code Vroni to circular arcs. We discuss implementational issues such as the computation of the Voronoi nodes. As demonstrated by test runs on several thousands of synthetic and real-world data sets, this circular-arc extension of Vroni is reliable and exhibits the average-case time complexity predicted by theory. As a service to the community, all circular-arc data sets (except for proprietary data) have been made public.To our knowledge, this enhanced version of Vroni constitutes the first implementation that computes Voronoi diagrams of genuine circular arcs on a standard floating-point arithmetic reliably and efficiently, without resorting to some form of approximation or sampling of the circular arcs.     

A framework of Voronoi diagram for planning multiple paths in free space

Most of the emphasis in path planning, a topic of much interest in several domains, has been on finding the optimal path or at most k optimal paths. However, in domains such as adversarial planning, one of the agents might deliberately take less optimal paths to confuse the opponent, and by the same token an agent, for inferring opponent's intent, has to consider all possible paths that the opponent might take. We introduce the notion of representative paths in free space (2D) and study the problem of computing all representative paths with different properties, such as all representative paths with at most L loops, among polygonal regions using a framework of Voronoi diagram. We prove three properties: (1) the upper and lower bounds to the number of simple paths in a Voronoi graph (2) given any path, a homotopic path can always be obtained from the Voronoi diagram of the regions and (3) all representative paths with a given property might not be always obtainable from the Voronoi graph even after searching the graph exhaustively and present an algorithm to work around this limitation. We also show how our findings can be applied for efficient entity re-identification, a problem involving a large number of dynamic entities and obstacles in the military domain.     

Generalized Voronoi diagrams for a ladder: II. Efficient construction of the diagram

We present a collection of algorithms, all running in timeO(n ² logn (n) ^o((n)3)) for some fixed integers(where (n) is the inverse Ackermann's function), for constructing a skeleton representation of a suitably generalized Voronoi diagram for a ladder moving in a two-dimensional space bounded by polygonal barriers consisting ofn line segments. This diagram, which is a two-dimensional subcomplex of the dimensional configuration space of the ladder, is introduced and analyzed in a companion paper by the present authors. The construction of the diagram described in this paper yields a motion-planning algorithm for the ladder which runs within the same time bound given above.Work on this paper has been supported in part by Office of Naval Research Grant N00014-82-K-0381, and by grants from the Digital Equipment Corporation, the Sloan Foundation, the System Development Foundation, the IBM corporation, and by National Science Foundation CER Grant No. DCR-8320085. Work by the second author has also been supported in part by a grant from the US-Israeli Binational Science Foundation.     

MATHSM: medial axis transform toward high speed machining of pockets

The pocketing operation is a fundamental procedure in NC machining. Typical pocketing schemes compute uniform successive offsets or parallel cuts of the outline of the pocket, resulting in a toolpath with C¹ discontinuities. These discontinuities render the toolpath quite impractical in the context of high speed machining (HSM).This work addresses and fully resolves the need for a C¹ continuous toolpath in HSM and offers MATHSM, a C¹ continuous toolpath for arbitrary C¹ continuous pockets. MATHSM automatically generates a C¹ continuous toolpath that consists of primarily circular arcs while maximizing the radii of the generated arcs and, therefore, minimizing the exerted radial acceleration. MATHSM is especially suited for machining of elongated narrow pockets.     

A simple GPU-based approach for 3D Voronoi diagram construction and visualization

In this paper we propose a simple GPU-based approach for discrete incremental approximation of 3D Voronoi diagram. By constructing region maps via GPU. Nearest sites, space clustering, and shortest distance query can be quickly answered by looking up the region map. In addition, we propose another representation of the 3D Voronoi diagram for visualization.     

Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane

Abstract. We present an optimal parallel randomized algorithm for the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane. Our algorithm runs in O(log n) time with high probability using O(n) processors on a CRCW PRAM. This algorithm is optimal in terms of work done since the sequential time bound for this problem is Ω(n log n) . Our algorithm improves by an O(log n) factor the previously best known deterministic parallel algorithm, given by Goodrich, ó'Dúnlaing, and Yap, which runs in O( log ² n) time using O(n) processors. We obtain this result by using a new ``two-stage' random sampling technique. By choosing large samples in the first stage of the algorithm, we avoid the hurdle of problem-size ``blow-up' that is typical in recursive parallel geometric algorithms. We combine the two-stage sampling technique with efficient search and merge procedures to obtain an optimal algorithm. This technique gives an alternative optimal algorithm for the Voronoi diagram of points as well (all other optimal parallel algorithms for this problem use the transformation to three-dimensional half-space intersection).