Regular Point Pattern Research Articles

Analytical Tractability of Hexagonal Network Model With Random User Location

Explicit derivation of interferences in hexagonal wireless networks has been widely considered intractable and requires extensive computations with system level simulations. In this paper, we fundamentally tackle this problem and explicitly evaluate the downlink Interference-to-Signal Ratio (ISR) for any mobile location $m$ in a hexagonal wireless network, whether composed of omni-directional or tri-sectorized sites. The explicit formula of ISR is a very convergent series on $m$ and involves the use of Gauss hypergeometric and Hurwitz Riemann zeta functions. Besides, we establish simple identities that well approximate this convergent series and turn out quite useful compared to other approximations in literature. The derived expression of ISR is easily extended to any frequency reuse pattern. Moreover, it is also exploited in the derivation of an explicit form of SINR distribution for any arbitrary distribution of mobile user locations, reflecting the spatial traffic density in the network. Knowing explicitly about interferences and SINR distribution is very useful information in capacity and coverage planning of wireless cellular networks and particularly for macro-cells' layer that forms almost a regular point pattern.

On tree intensity estimation for forest inventories: Some statistical issues

The effect of the plot shape, number of subplots and their spatial arrangement on the sample variance for spatially explicit point populations is analysed for a simple intensity estimator. We derive the sample variance and covariance for sampling designs involving more than one subplot. Some numerical approximations are also presented. If a clustered point pattern has to be sampled, the best strategy to reduce the sample variance is to consider as many rectangular subplots as possible, for a prescribed total sample area, distributed over a grid. In contrast, if a regular point pattern is to be sampled, then a single circular subplot should be considered. If we assume that the point configuration is Poisson, then we can consider any subplot shape and spatial distribution ensuring no overlapping between the subplots. A case study in forestry is considered to assess the validity of our results.

Cellular organization of the trophoblastic epithelium in elongating conceptuses of ruminants

We present a computational approach to analyze the cellular organization during the elongation process of the ovine conceptus. First, we selected a set of mathematical descriptors to quantify cell geometry and cell neighborhood within the external epithelial layer of the conceptus: the trophoblast. Second, we established a hybrid image segmentation framework, and analyzed the extracted features with statistical tools to describe and compare the spatio-temporal dynamics of cellular organization within this epithelium. The main results indicated that the average geometry and neighborhood of the trophoblast cells are relatively stable from a sampling position to another and from a stage of development to another. Further, their elongation axes are randomly distributed. The cellular organization fits the Poisson's process. Moreover, no clustering structures or grid-like regular point patterns are found inside the studied cell population. This suggests that the trophoblast elongation observed in ruminants is not due to the geometrical change of cell shape but might be the consequence of cell addition associated with peculiar plans of cell division or intercalation. To cite this article: J. Wang et al., C. R. Biologies 332 (2009).

Order Distance in Regular Point Patterns

This article examines the kth nearest neighbor distance for three regular point patterns: square, triangular, and hexagonal lattices. The probability density functions of the kth nearest distance and the average kth nearest distances are theoretically derived for k=1, 2, …, 7. As an application of the kth nearest distance, we consider a facility location problem with closing of facilities. The problem is to find the optimal regular pattern that minimizes the average distance to the nearest open facility. Assuming that facilities are closed independently and at random, we show that the triangular lattice is optimal if at least 68% of facilities are open by comparing the upper and lower bounds of the average distances.

Preprocessing of region of interest localization based on local surface curvature analysis for three-dimensional reconstruction with multiresolution

We present an approach to integrate a preprocessing step of the region of interest (ROI) localization into 3-D scanners (laser or stereoscopic). The definite objective is to make the 3-D scanner intelligent enough to localize rapidly in the scene, during the preprocessing phase, the regions with high surface curvature, so that precise scanning will be done only in these regions instead of in the whole scene. In this way, the scanning time can be largely reduced, and the results contain only pertinent data. To test its feasibility and efficiency, we simulated the preprocessing process under an active stereoscopic system composed of two cameras and a video projector. The ROI localization is done in an iterative way. First, the video projector projects a regular point pattern in the scene, and then the pattern is modified iteratively according to the local surface curvature of each reconstructed 3-D point. Finally, the last pattern is used to determine the ROI. Our experiments showed that with this approach, the system is capable to localize all types of objects, including small objects with small depth.

On a model of sequential point patterns

A finite point process motivated by the cooperative sequential adsorption model is proposed. Analytical properties of the point process are considered in details. It is shown that the introduced point process is useful for modeling both aggregated and regular point patterns. A possible scheme of maximum likelihood estimation of the process parameters is briefly discussed.

Comparison of point pattern analysis methods for classifying the spatial distributions of spruce-fir stands in the north-east USA

The spatial distributions of tree locations in spruce-fir forest stands in the north-east USA were explored by various methods of spatial point pattern analysis. The results indicated that the 13 nearest neighbour statistics were not reliable because of the assumption violations for independent distance measures and large sample size requirements. Three other methods (i.e. refined nearest neighbour functions, Ripley's K-function and pair correlation function) seemed to capture the different aspects of the spatial patterns of these spruce-fir stands. The edge effect correction was very important to obtain unbiased results for the spatial point pattern analysis. The toroidal edge correction proved to be simple and satisfactory in this study compared with other edge correction methods. The results indicated that 24 plots (48 per cent) were classified as complete spatial random (CSR) point pattern, 17 (34 per cent) regular point pattern and 9 (18 per cent) clustered point pattern among the 50 plots. It was evident that the clustered plots were younger in age with much higher density and much smaller tree sizes than the CSR or regular plots. This classification scheme can be used as the basis for other spatial studies such as spatial point process modelling.

The K-function for nearly regular point processes.

We propose modeling a nearly regular point pattern by a generalized Neyman-Scott process in which the offspring are Gaussian perturbations from a regular mean configuration. The mean configuration of interest is an equilateral grid, but our results can be used for any stationary regular grid. The case of uniformly distributed points is first studied as a benchmark. By considering the square of the interpoint distances, we can evaluate the first two moments of the K-function. These results can be used for parameter estimation, and simulations are used to both verify the theory and to assess the accuracy of the estimators. The methodology is applied to an investigation of regularity in plumes observed from swimming microorganisms.

Size analysis of nearly regular Delaunay triangulations

Consider a nearly regular point pattern in which a Delaunay triangulation is comprised of nearly equilateral triangles of the same size. We propose to model this set of points with Gaussian perturbations about a regular mean configuration. By investigating triangle subsets in detail we obtain various distributions of statistics based on size, or squared size of the triangles which is closely related to the mean (squared) distance to the six nearest neighbors. A scaleless test statistic, corresponding to a coefficient of variation for squared sizes, is proposed and its asymptotic properties described. The methodology is applied to an investigation of regularity in human muscle fiber cross-sections. We compare the approach with an alternative technique in a power study.

Likelihood estimation of soft-core interaction potentials for Gibbsian point patterns

The likelihood method is developed for the analysis of socalled regular point patterns. Approximating the normalizing factor of Gibbs canonical distribution, we simultaneously estimate two parameters, one for the scale and the other which measures the softness (or hardness), of repulsive interactions between points. The approximations are useful up to a considerably high density. Some real data are analyzed to illustrate the utility of the parameters for characterizing the regular point pattern.