Sphere Of Influence Graph Research Articles

On the SIG-Dimension of Trees Under the L ∞-Metric

Let \({{\mathcal P},}\) where \({|{\mathcal P}| \geq 2,}\) be a set of points in d-dimensional space with a given metric ρ. For a point \({p \in {\mathcal P},}\) let r p be the distance of p with respect to ρ from its nearest neighbor in \({{\mathcal P}.}\) Let B(p,r p ) be the open ball with respect to ρ centered at p and having the radius r p . We define the sphere-of-influence graph (SIG) of \({{\mathcal P}}\) as the intersection graph of the family of sets \({\{B(p,r_p)\ | \ p\in {\mathcal P}\}.}\) Given a graph G, a set of points \({{\mathcal P}_G}\) in d-dimensional space with the metric ρ is called a d-dimensional SIG-representation of G, if G is isomorphic to the SIG of \({{\mathcal P}_G.}\) It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have a SIG-representation under the L ∞-metric in some space of finite dimension. The SIG-dimension under the L ∞-metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG-representation under the L ∞-metric. It is denoted by SIG ∞(G). We study the SIG-dimension of trees under the L ∞-metric and almost completely answer an open problem posed by Michael and Quint (Discrete Appl Math 127:447–460, 2003). Let T be a tree with at least two vertices. For each \({v\in V(T),}\) let leaf-degree(v) denote the number of neighbors of v that are leaves. We define the maximum leaf-degree as \({\alpha(T) = \max_{x \in V(T)}}\) leaf-degree(x). Let \({ S = \{v\in V(T)\|\}}\) leaf-degree{(v) = α}. If |S| = 1, we define β(T) = α(T) − 1. Otherwise define β(T) = α(T). We show that for a tree \({T, SIG_\infty(T) = \lceil \log_2(\beta + 2)\rceil}\) where β = β (T), provided β is not of the form 2k − 1, for some positive integer k ≥ 1. If β = 2k − 1, then \({SIG_\infty (T) \in \{k, k+1\}.}\) We show that both values are possible.

Weighted Least Squares for Visualization of Scanned Point Clouds

AbstractThis paper describes several improvements to implicit surface fitting over point clouds. Noisy stamped part point cloud data gathered by both a laser digitizer scanning system and a stereo vision sensor is processed using a method based on the weighted least squares algorithm. Two kernel functions are proposed: one using an enhanced sphere-of-influence graph (SIG) method to determine nearest neighbours; and a second using an improved Gaussian kernel. Automatic bandwidth adjustment is implemented for both methods. A method of incorporating a scalar attribute at each data point, which can represent any desired quantity and relate it to the surface at that point, is introduced. In this case surface strain data was gathered and processed by the scanning system, with a resulting thickness strain measurement calculated at each point. This strain data was used to introduce a temperature plot style colour map over the surface, allowing for fast and simple analysis of the strain of the formed part.

Large deviations for functionals of spatial point processes with applications to random packing and spatial graphs

Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. We prove general Donsker–Varadhan large deviation principles (LDP) for such functionals and show that the general result can be applied to prove LDPs for various particular functionals, including those concerned with random packing, nearest neighbor graphs, and lattice versions of the Voronoi and sphere of influence graphs.

Gaussian limits for random measures in geometric probability

We establish Gaussian limits for general measures induced by binomial and Poisson point processes in d-dimensional space. The limiting Gaussian field has a covariance functional which depends on the density of the point process. The general results are used to deduce central limit theorems for measures induced by random graphs (nearest neighbor, Voronoi and sphere of influence graph), random sequential packing models (ballistic deposition and spatial birth–growth models) and statistics of germ–grain models.

On a sphere of influence graph in a one-dimensional space

On a sphere of influence graph in a one-dimensional space

Point cloud surfaces using geometric proximity graphs

We present a new definition of an implicit surface over a noisy point cloud, based on the weighted least-squares approach. It can be evaluated very fast, but artifacts are significantly reduced. We propose to use a different kernel function that approximates geodesic distances on the surface by utilizing a geometric proximity graph. From a variety of possibilities, we have examined the Delaunay graph and the sphere-of-influence graph (SIG), for which we propose several extensions. The proximity graph also allows us to estimate the local sampling density, which we utilize to automatically adapt the bandwidth of the kernel and to detect boundaries. Consequently, our method is able to handle point clouds of varying sampling density without manual tuning. Our method can be integrated into other surface definitions, such as moving least squares, so that these benefits carry over.

Weak laws of large numbers in geometric probability

Using a coupling argument, we establish a general weak law of large numbers for functionals of binomial point processes in d-dimensional space, with a limit that depends explicitly on the (possibly nonuniform) density of the point process. The general result is applied to the minimal spanning tree, the k-nearest neighbors graph, the Voronoi graph and the sphere of influence graph. Functionals of interest include total edge length with arbitrary weighting, number of vertices of specified degree and number of components. We also obtain weak laws of large numbers functionals of marked point processes, including statistics of Boolean models.

Sphere of influence graphs and the L∞-metric

We introduce sphere of influence graphs (SIGs) in the L ∞-metric and study their elementary properties. We argue that SIGs defined with the L ∞-metric are superior to Euclidean SIGs of Toussaint in capturing low-level perceptual information in certain dot patterns. Every graph without isolated vertices is a SIG in the L ∞-metric for all sufficiently high dimensions, and this allows us to define a graphical parameter, the SIG-dimension, that is akin to boxicity. We determine the SIG-dimensions for some classes of graphs and obtain inequalities for others.

Central Limit Theorems for Some Graphs in Computational Geometry

Let $(B_n)$ be an increasing sequence of regions in $d$ -dimensional space with volume $n$ and with union $\mathbb{R}^d$. We prove a general central limit theorem for functionals of point sets, obtained either by restricting a homogeneous Poisson process to $(B_n)$, or by by taking $n$ uniformly distributed points in $(B_n)$. The sets $(B_n)$ could be all cubes but a more general class of regions$(B_n)$ is considered. Using this general result we obtain central limit theorems for specific functionals suchas total edge lengthand number of components, defined in terms of graphs such as the $k$-nearest neighbors graph, the sphere of influence graph and the Voronoi graph.

On the size of a random sphere of influence graph

We approach sphere of influence graphs (SIGs) from a probabilistic perspective. Ordinary SIGs were first introduced by Toussaint as a type of proximity graph for use in pattern recognition, computer vision and other low-level vision tasks. A random sphere of influence graph (RSIG) is constructed as follows. Consider n points uniformly and independently distributed within the unit square in d dimensions. Around each point, Xi, draw an open ball (‘sphere of influence’) with radius equal to the distance to Xi's nearest neighbour. Finally, draw an edge between two points if their spheres of influence intersect. Asymptotically exact values for the expected number of edges in a RSIG are determined for all values of d; previously, just upper and lower bounds were known for this quantity. A modification of the Azuma-Hoeffding exponential inequality is employed to exhibit the sharp concentration of the number of edges around its expected value.

On the size of a random sphere of influence graph

We approach sphere of influence graphs (SIGs) from a probabilistic perspective. Ordinary SIGs were first introduced by Toussaint as a type of proximity graph for use in pattern recognition, computer vision and other low-level vision tasks. A random sphere of influence graph (RSIG) is constructed as follows. Consider n points uniformly and independently distributed within the unit square in d dimensions. Around each point, X i , draw an open ball (‘sphere of influence’) with radius equal to the distance to X i 's nearest neighbour. Finally, draw an edge between two points if their spheres of influence intersect. Asymptotically exact values for the expected number of edges in a RSIG are determined for all values of d; previously, just upper and lower bounds were known for this quantity. A modification of the Azuma-Hoeffding exponential inequality is employed to exhibit the sharp concentration of the number of edges around its expected value.

Sphere of influence graphs in general metric spaces

Let χ = { X 1…, X n } be a set of points in a metric space ( n ≥ 2). Let r i , denote the minimum distance between X i and the other points in χ. The sphere of influence at X i is the open ball with center X i and radius r i . The sphere of influence graph has vertex set χ with an edge joining a pair of distinct vertices provided the corresponding spheres of influence intersect. This proximity graph was introduced by Toussaint to model computer vision and pattern recognition problems in the Euclidean plane. We discuss the abstract properties of sphere of influence graphs in general metric spaces with emphasis on normed linear spaces. Our work generalizes and simplifies some theorems in the literature on sphere of influence graphs and lays the groundwork for future work.

Sphere of influence graphs in general metric spaces

Let @g = {X1..., Xn} be a set of points in a metric space (n >= 2). Let ri, denote the minimum distance between Xi and the other points in @g. The sphere of influence at Xi is the open ball wi...