In this paper, a novel algorithm of weighted k-means clustering with geodesic criteria is presented to generate a uniform database for a star sensor. For this purpose, selecting the appropriate star catalogue and desirable minimum magnitude and eliminating double stars are among the steps of the uniformity process. Further, Delaunay triangulation and determining the scattered data density by using a Voronoi diagram were used to solve the problems of the proposed clustering method. Thus, by running a Monte Carlo simulation to count the number of stars observed in different fields of view, it was found that the uniformity leads to a significant reduction of the probability of observing a large number of stars in all fields of view. In contrast, the uniformity slightly increased the field of view needed to observe the minimum number of required stars for an identification algorithm.

Let X := X n ∪ {(0, 0), (1, 0)}, where X n is a planar Poisson point process of intensity n. We provide a first nontrivial lower bound for the distance between the expected length of the shortest path between (0, 0) and (1, 0) in the Delaunay triangulation associated with X when the intensity of X n goes to ∞. Simulations indicate that the correct value is about 1.04. We also prove that the expected length of the so-called upper path converges to 35 / 3π2, yielding an upper bound for the expected length of the smallest path.

Machine vision-based monitoring of pig lying behaviour is a fast and non-intrusive approach that could be used to improve animal health and welfare. Four pens with 22 pigs in each were selected at a commercial pig farm and monitored for 15 days using top view cameras. Three thermal categories were selected relative to room setpoint temperature. An image processing technique based on Delaunay triangulation (DT) was utilized. Different lying patterns (close, normal and far) were defined regarding the perimeter of each DT triangle and the percentages of each lying pattern were obtained in each thermal category. A method using a multilayer perceptron (MLP) neural network, to automatically classify group lying behaviour of pigs into three thermal categories, was developed and tested for its feasibility. The DT features (mean value of perimeters, maximum and minimum length of sides of triangles) were calculated as inputs for the MLP classifier. The network was trained, validated and tested and the results revealed that MLP could classify lying features into the three thermal categories with high overall accuracy (95.6%). The technique indicates that a combination of image processing, MLP classification and mathematical modelling can be used as a precise method for quantifying pig lying behaviour in welfare investigations.

A Delaunay-type mesh condition is developed for a linear finite element approximation of two-dimensional anisotropic diffusion problems to satisfy a discrete maximum principle. The condition is weaker than the existing anisotropic non-obtuse angle condition and reduces to the well known Delaunay condition for the special case with the identity diffusion matrix. Numerical results are presented to verify the theoretical findings.

In this article a novel model framework to simulate cells and their internal structure is described. The model is agent-based and suitable to simulate single cells with a detailed internal structure as well as multi-cellular compounds. Cells are simulated as a set of many interacting particles, with neighborhood relations defined via a Delaunay triangulation. The interacting sub-particles of a cell can assume specific roles – i.e., membrane sub-particle, internal sub-particle, organelles, etc –, distinguished by specific interaction potentials and, eventually, also by the use of modified interaction criteria. For example, membrane sub-particles may interact only on a two-dimensional surface embedded on three-dimensional space, described via a restricted Delaunay triangulation. The model can be used not only to study cell shape and movement, but also has the potential to investigate the coupling between internal space-resolved movement of molecules and determined cell behaviors.

The horizontal cells are known to form a mono-layered mosaic in the adult retina, but are scattered at different retinal depths in early development. To help clarifying when and which spatial constraints appear in the relative positioning of these cells, we have performed a quantitative analysis of the three-dimensional (3D) organization of the horizontal cell mosaic at different developmental stages in the postnatal rat retina. We first analyzed the two-dimensional (2D) distribution of the horizontal cell projections onto a plane parallel to the upper retinal surface in retinal flat-mounts, and thus to the future mature horizontal cell mosaic. We found that this 2D distribution was non random since postnatal day 1 (P1), and had a subsequent stepwise improvement in regularity. This preceded the alignment of cells in a single monolayer, which was observed on P6. We then computed true horizontal cell spacing in 3D, finding non-random 3D positioning already on P1. Simulation studies showed that this order might simply derive from the 2D order observed in the projections of the cells in flat-mount, combined with their limited spread in retinal depth. Throughout the period analyzed, the relative positions of horizontal cells are in good agreement with a minimal spacing rule in which the exclusion zone corresponds to the average size of the inner core of the cell dendritic tree estimated from P1 samples. These data indicate the existence of different phases in the process of horizontal cell 3D spatial ordering, supporting the view that multiple mechanisms are involved in the development of the horizontal cell mosaic.

A known gamma-type result for the Poisson process states that certain domains defined through configuration of the points (or ‘particles’) of the process have volumes which are gamma distributed. By proving the corresponding sequential gamma-type result, we show that in some cases such a domain allows for decomposition into subdomains each having independent exponentially distributed volumes. We consider other examples—based on the Voronoi and Delaunay tessellations—where a natural decomposition does not produce subdomains with exponentially distributed volumes. A simple algorithm for the construction of a typical Voronoi flower arises in this work. In our theoretical development, we generalize the classical theorem of Slivnyak, relating it to the strong Markov property of the Poisson process and to a result of Mecke and Muche (1995). This new theorem has interest beyond the specific problems being considered here.

This paper describes an algorithm for surface reconstruction from a set of scattered three-dimensional points extracted from an image sequence. In this process, additional information (such as location of the viewpoints and the points visible from a particular viewpoint) is available and can be exploited for accurately recovering the shape of the objects portrayed in the images. Initially, the set of points is subjected to a Delaunay triangulation that fills the convex hull of the set of points with disjoint tetrahedra. The key idea of the shape recovery algorithm is to eliminate triangles that obstruct the visibility of points from certain viewpoints, whose locations are known from the image acquisition process. The major contribution of this paper is that we have been able to design an algorithm for surface reconstruction that handles a wide variety of shapes, as opposed to currently existing techniques.

Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path.

We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.

There were three objectives to the work undertaken for this paper: (1) to provide a comprehensive characterization of the statistical properties of arrays of β-cell somata; (2) to develop a model that simulates cellular arrays with the same properties; and (3) to use this model to examine whether the array of β-cells should be viewed as one array or as two arrays, one each for its OFF- and ON-center cells. β-cells are morphological correlates of the electrophysiological X-cells and those β-cells whose dendrites stratify within the outer and inner sublamina of the retina's inner plexiform layer correspond, respectively, to OFF- and ON-center X-cells. Arrays of peripheral β-cell somata from two retinas were studied. A Delaunay triangulation and a Voronoi tessellation were generated for each array and measures derived from these constructs used to analyze the arrays' spatial organization. As others have shown previously with a less complete statistical characterization, we found that the arrays of OFF- and ON-center β-cells have similar spatial properties and are more regular than the array of all β-cells. We developed a model to simulate cellular arrays with spatial properties like those of arrays of β-cells. A good fit between model and real arrays was found when the model assumed an explicit spatial dependence between the placement of OFF- and ON-center cells. We propose therefore that a single array of β-cells formed of both OFF- and ON-center cells is consistent with the data currently available for β-cell somatic arrays.

The present study deals with the existence of ‘nearest-neighbour’ type Gibbs models, introduced by Baddeley and Møller in 1989. In such models, the neighbourhood relation depends on the realization of the process. After giving new sufficient conditions to prove the existence of stationary Gibbs states, we deal with the first-nearest-neighbour model, the triplets Delaunay model, Ord's model and Markov connected component type models.