The relative neighbourhood graph of a finite planar set

The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n²) time and the other runs in 0(n³) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined.     

Hamiltonian triangulations for fast rendering

High-performance rendering engines are often pipelined; their speed is bounded by the rate at which triangulation data can be sent into the machine. An ordering such that consecutive triangles share a face, which reduces the data rate, exists if and only if the dual graph of the triangulation contains a Hamiltonian path. We (1) show thatany set ofn points in the plane has a Hamiltonian triangulation; (2) prove that certain nondegenerate point sets do not admit asequential triangulation; (3) test whether a polygonP has a Hamiltonian triangulation in time linear in the size of its visibility graph; and (4) show how to add Steiner points to a triangulation to create Hamiltonian triangulations that avoid narrow angles.     

Computing relative neighbourhood graphs in the plane

The relative neighbourhood graph (RNG) of a set of N points connects the relative neighbours, i.e. a pair of points is connected by an edge if those points are at least as close to each other as to any other point. The paper presents two new algorithms for constructing RNG in two-dimensional Euclidean space. The method is to determine a supergraph for RNG which can then be thinned efficiently from the extra edges. The first algorithm is simple, and works in O(N²) time. The worst case running time of the second algorithm is also O(N²), but its average case running time is O(N) for points from a homogeneous planar Poisson point process. Experimental tests have shown the usefulness of the approach.     

Graham triangulations and triangulations with a center are hamiltonean

Let P be a point set with n elements in general position. A triangulation T of P is a set of triangles with disjoint interiors such that their union is the convex hull of P, no triangle contains an element of P in its interior, and the vertices of the triangles of T are points of P. Given T we define a graph G(T) whose vertices are the triangles of T, two of which are adjacent if they share an edge. We say that T is hamiltonean if G(T) has a hamiltonean path. We prove that the triangulations produced by Graham's Scan are hamiltonean. Furthermore we prove that any triangulation T of a point set which has a point adjacent to all the points in P (a center of T) is hamiltonean.     

A linear time algorithm for max-min length triangulation of a convex polygon

We consider the following planar max-min length triangulation problem: given a set of n points in the Euclidean plane, find a triangulation such that the length of the shortest edge in the triangulation is maximized. In this paper, a linear time algorithm is proposed for computing the max-min length triangulation of a set of points in convex position. In addition, an O(nlogn) time algorithm is proposed for computing the max-min length k-set triangulation of a set of points in convex position, where we are to compute a set of k vertices such that the max-min length triangulation on them is minimized over all possible k-set. We further show that the graph version of max-min length triangulation is NP-complete, and some common heuristics such as greedy algorithm are in general not able to give a bounded-ratio approximation to the max-min length triangulation.     

New results for the minimum weight triangulation problem

Given a finite set of points in a plane, a triangulation is a maximal set of nonintersecting line segments connecting the points. The weight of a triangulation is the sum of the Euclidean lengths of its line segments. No polynomial-time algorithm is known to find a triangulation of minimum weight, nor is the minimum weight triangulation problem known to be NP-hard. This paper proposes a new heuristic algorithm that triangulates a set ofn points inO(n ³⁾ time and that never produces a triangulation whose weight is greater than that of a greedy triangulation. The algorithm produces an optimal triangulation if the points are the vertices of a convex polygon. Experimental results indicate that this algorithm rarely produces a nonoptimal triangulation and performs much better than a seemingly similar heuristic of Lingas. In the direction of showing the minimum weight triangulation problem is NP-hard, two generalizations that are quite close to the minimum weight triangulation problem are shown to be NP-hard.This research was done while the second author was with the Department of Computer Science, Virginia Polytechnic Institute and State University.     

A linear expected-time algorithm for computing planar relative neighbourhood graphs

A new algorithm for computing the relative neighbourhood graph (RNG) of a planar point set is given. The expected running time of the algorithm is linear for a point set in a unit square when the points have been generated by a homogeneous planar Poisson point process. The worst-case running time is quadratic on the number of the points. The algorithm proceeds in two steps. First, a supergraph of the RNG is constructed with the aid of a cell organization of the points. Here, a point is connected by an edge to some of its nearest neighbours in eight regions around the point. The nearest region neighbours are chosen in a special way to minimize the costs. Second, extra edges are pruned from the graph by a simple scan.     

Localized Delaunay triangulation with application in ad hoc wireless networks

Several localized routing protocols guarantee the delivery of the packets when the underlying network topology is a planar graph. Typically, relative neighborhood graph (RING) or Gabriel graph (GG) is used as such planar structure. However, it is well-known that the spanning ratios of these two graphs are not bounded by any constant (even for uniform randomly distributed points). Bose et al. (1999) recently developed a localized routing protocol that guarantees that the distance traveled by the packets is within a constant factor of the minimum if Delaunay triangulation of all wireless nodes is used, in addition, to guarantee the delivery of the packets. However, it is expensive to construct the Delaunay triangulation in a distributed manner. Given a set of wireless nodes, we model the network as a unit-disk graph (UDG), in which a link uv exists only if the distance /spl par/uv/spl par/ is at most the maximum transmission range. In this paper, we present a novel localized networking protocol that constructs a planar 2 5-spanner of UDG, called the localized Delaunay triangulation (LDEL), as network topology. It contains all edges that are both in the unit-disk graph and the Delaunay triangulation of all nodes. The total communication cost of our networking protocol is O(n log n) bits, which is within a constant factor of the optimum to construct any structure in a distributed manner. Our experiments show that the delivery rates of some of the existing localized routing protocols are increased when localized Delaunay triangulation is used instead of several previously proposed topologies. Our simulations also show that the traveled distance of the packets is significantly less when the FACE routing algorithm is applied on LDEL, rather than applied on GG.     

Relative neighborhood graphs in the Li-metric

The relative neighborhood graph of a set of n points in the plane under the L₁-metric is considered. An algorithm that runs in O(nlog n) time for constructing the relative neighborhood graph based on the Delaunay triangulation is presented, improving a previously known algorithm that runs in O(n²log n) time.     

The drawability problem for minimum weight triangulations

A graph is minimum weight drawable if it admits a straight-line drawing that is a minimum weight triangulation of the set of points representing the vertices of the graph. We study the problem of characterizing those graphs that are minimum weight drawable. Our contribution is twofold: We show that there exist infinitely many triangulations that are not minimum weight drawable. Furthermore, we present non-trivial classes of triangulations that are minimum weight drawable, along with corresponding linear time algorithms that take as input any graph from one of these classes and produce as output such a drawing. One consequence of our work is the construction of triangulations that are minimum weight drawable but not Delaunay drawable – that is, not drawable as a Delaunay triangulation.     

Efficient minimum spanning tree construction without Delaunay triangulation

Given n points in a plane, a minimum spanning tree is a set of edges which connects all the points and has a minimum total length. A naive approach enumerates edges on all pairs of points and takes at least Ω(n2) time. More efficient approaches find a minimum spanning tree only among edges in the Delaunay triangulation of the points. However, Delaunay triangulation is not well defined in rectilinear distance. In this paper, we first establish a framework for minimum spanning tree construction which is based on a general concept of spanning graphs. A spanning graph is a natural definition and not necessarily a Delaunay triangulation. Based on this framework, we then design an O(nlogn) sweep-line algorithm to construct a rectilinear minimum spanning tree without using Delaunay triangulation.     

Power图的性质及构造算法研究

点集的Power图是点集Voronoi图的推广,特别适用用来解决涉及球（圆)的几何问题,文中首先对Power图的基本性质进行了几何化的证明;之后,研究了权为负数时对Power图的影响,指出在Power图的理论中允许权为负数,从而Power图可以应用到具有负权性质的领域;最后,给出了平面点集的Power图的构造算法,该算法到用Power图与正则三角化互为对偶的原理,在点集的正则三角化的基础上构造Power图,同时给出了实例以说明算法的有效性。     

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     

平面散乱点线集三角剖分的算法

利用平面扫描的思想，即利用从右到左移动的y-轴扫描点线集．当扫描线达到某个给定点或给定线段端点时，将该点或端点与其上下相邻线段端点连接．新连线与已三角剖分的边只能在其端点处相交．该算法的时间复杂性为O(N log N)，其中N是点线集中点的数目与线段端点数之和．     

AnO(n logn) plane-sweep algorithm forL 1 andL ∞ Delaunay triangulations

TheDelaunay diagram on a set of points in the plane, calledsites, is the straight-line dual graph of the Voronoi diagram. When no degeneracies are present, the Delaunay diagram is a triangulation of the sites, called theDelaunay triangulation. When degeneracies are present, edges must be added to the Delaunay diagram to obtain a Delaunay triangulation. In this paper we describe an optimalO(n logn) plane-sweep algorithm for computing a Delaunay triangulation on a possibly degenerate set of sites in the plane under theL ₁ metric or theL _∞ metric.     

Triangulating input-constrained planar point sets

We present a linear-time algorithm for computing a triangulation of n points in 2D whose positions are constrained to n disjoint disks of uniform size, after O(nlogn) preprocessing applied to these disks. Our algorithm can be extended to any collection of convex sets of bounded areas and aspect ratios, assuming no point lies in more than some constant number of sets (bounded depth of overlap), and each set contains only a constant number of query points.     

Capacity-Constrained Delaunay Triangulation for point distributions

Sample point distributions possessing blue noise spectral characteristics play a central role in computer graphics, but are notoriously difficult to generate. We describe an algorithm to very efficiently generate these distributions. The core idea behind our method is to compute a Capacity-Constrained Delaunay Triangulation (CCDT), namely, given a simple polygon P in the plane, and the desired number of points n, compute a Delaunay triangulation of the interior of P with n Steiner points, whose triangles have areas which are as uniform as possible. This is computed iteratively by alternating update of the point geometry and triangulation connectivity. The vertex set of the CCDT is shown to have good blue noise characteristics, comparable in quality to those of state-of-the-art methods, achieved at a fraction of the runtime. Our CCDT method may be applied also to an arbitrary density function to produce non-uniform point distributions. These may be used to half-tone grayscale images.     

Trajectory triangulation: 3D reconstruction of moving points from amonocular image sequence

We consider the problem of reconstructing the 3D coordinates of a moving point seen from a monocular moving camera, i.e., to reconstruct moving objects from line-of-sight measurements only. The task is feasible only when some constraints are placed on the shape of the trajectory of the moving point. We coin the family of such tasks as “trajectory triangulation.” We investigate the solutions for points moving along a straight-line and along conic-section trajectories, We show that if the point is moving along a straight line, then the parameters of the line (and, hence, the 3D position of the point at each time instant) can be uniquely recovered, and by linear methods, from at least five views. For the case of conic-shaped trajectory, we show that generally nine views are sufficient for a unique reconstruction of the moving point and fewer views when the conic is of a known type (like a circle in 3D Euclidean space for which seven views are sufficient). The paradigm of trajectory triangulation, in general, pushes the envelope of processing dynamic scenes forward. Thus static scenes become a particular case of a more general task of reconstructing scenes rich with moving objects (where an object could be a single point)     

A Generalization of AT-Free Graphs and a Generic Algorithm for Solving Triangulation Problems

A subset A of the vertices of a graph G is an asteroidal set if for each vertex a ∈ A a connected component of G-N[a] exists containing A\backslash{a} . An asteroidal set of cardinality three is called asteriodal triple and graphs without an asteriodal triple are called AT-free . The maximum cardinality of an asteroidal set of G , denoted by \an(G) , is said to be the asteriodal number of G . We present a scheme for designing algorithms for triangulation problems on graphs. As a consequence, we obtain algorithms to compute graph parameters such as treewidth, minimum fill-in and vertex ranking number. The running time of these algorithms is a polynomial (of degree asteriodal number plus a small constant) in the number of vertices and the number of minimal separators of the input graph.     

PREPROCESSING RULES FOR TRIANGULATION OF PROBABILISTIC NETWORKS*

Currently, the most efficient algorithm for inference with a probabilistic network builds upon a triangulation of a network's graph. In this paper, we show that pre‐processing can help in finding good triangulations for probabilistic networks, that is, triangulations with a maximum clique size as small as possible. We provide a set of rules for stepwise reducing a graph, without losing optimality. This reduction allows us to solve the triangulation problem on a smaller graph. From the smaller graph's triangulation, a triangulation of the original graph is obtained by reversing the reduction steps. Our experimental results show that the graphs of some well‐known real‐life probabilistic networks can be triangulated optimally just by preprocessing; for other networks, huge reductions in their graph's size are obtained.