Second-order Spatial Statistics Research Articles

Spatial pattern analysis of encroaching tree species (Vachellia karrooandVachellia nilotica) after fire suppression in a semi-arid savanna

Abstract:Bush encroachment has serious consequences on ecosystem functioning through alteration of species composition and ecosystem productivity. However, little is known regarding the spatial patterning of invading shrubs in semi-arid savannas. Cartesian coordinates of two invading woody species (Vachellia karrooandV. nilotica), were recorded in a 20 × 20-m plot on a grassland at Matopos research station, south-west Zimbabwe. We recorded a total of 308 plants including both saplings and shrubs from the two study plant species. Second-order spatial statistics was applied in order to understand the spatial pattern of encroaching plants. We predicted that the encroaching plants would be spatially aggregated because of facilitation that occurs in harsh environmental conditions. Consistent with our predictions, the two species were aggregated, with no evidence of inter- and intra specific species competition. This study demonstrates that encroaching trees in semi-arid savanna generally do not show self organization during early growth stages.

Statistical characterization of flow field structure in evolving braided gravel beds

Braided rivers quickly vary their geometry, modifying their boundaries and floodplains. Therefore, flooding of a braided stream is typically associated with high hydrologic risk in the surrounding areas or alluvial valleys. A forecast of these morpho-hydrodynamic processes is therefore essential when managing the environmental and economic impact of flood events. A few studies in the past demonstrated that open channels spontaneously tend to establish and maintain an equilibrium state that corresponds to a specific level of cross-sectional velocity entropy. The present study analyses for the first time the relationship between bank-full flow (described by measures of velocity entropy and second-order spatial statistics) and bed topography (described in terms of bed elevation second-order spatial statistics) in gravel-bed braided rivers at cross-sectional scale, in both equilibrium and pre-equilibrium conditions. Based on the outcome of suitable laboratory experiments, we found that, in gravel bed braided rivers, longitudinal equilibrium during floods is associated with a periodic-like spatial minimization of normalized cross-sectional velocity entropy. This periodic-like minimization of cross-sectional velocity entropy (and statistical variance) corresponds to a periodic-like maximization of normalized bed elevation variance, according to general theories on longitudinal river profile evolution. Additionally, bed evolution occurs by both entropy minima/maxima appearance and growth and entropy minima/maxima longitudinal migration.

Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics

Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial connectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes.

Pocket Model for Aluminum Agglomeration Based on Propellant Microstructure

The propellant microstructure is addressed for the interpretation and the prediction of agglomerate size distribution in aluminized composite solid rocket propellants. Although the mixing process of a propellant is intrinsically random, repetitive fuel-rich local structures (pockets) are generated in the bulk. Pockets are privileged locations for agglomerate generation. In the present work, second-order spatial statistics are applied to model propellants for the characterization of the microstructure and for the definition of an agglomeration model. The model propellants used in this work are generated by a packing code on the basis of real formulations, which are experimentally characterized for validation purposes. The average size and the metal content of the pockets are derived from the interpretation of the radial distribution function. The model is capable of predicting the size distribution of the incipient agglomerates for given propellant microstructures, using one free parameter for the tuning. The fitting of experimental agglomeration data from four different industrial propellants suggests that the free parameter can be expressed as a power function of the combustion pressure and that the microstructure agglomeration model produces particle distributions that reasonably match with the experimental data.

Texture Feature Extraction of Infrared River Ice Images using Second-Order Spatial Statistics

Texture Feature Extraction of Infrared River Ice Images using Second-Order Spatial Statistics

Spatial pattern analysis and competition between Acacia karroo trees in humid savannas

It is unclear whether inter-tree competition plays a role in determining the woody plant cover of humid savannas. Spatial point-pattern analysis can give insights to the underlying processes affecting the individuals in a population. We assessed the spatial patterns of Acacia karroo trees from savannas in KwaZulu-Natal, using conventional nearest neighbour analysis and second-order spatial statistics such as Ripley’s K- and L-functions, and the univariate and bivariate O-ring statistics. We predicted that juvenile trees would be spatially aggregated, because of facilitation between shrubs when zones of overlap are relatively small, while adult trees would be regularly spaced because of the effects of density-dependent mortality (i.e. consistent with the honeycomb rippling model). We found that juvenile trees were more aggregated than expected by chance, and the overall spatial distribution of all trees was also found to be aggregated, with no evidence of regularity among large individuals. Nearest neighbour analysis, however, revealed significant positive correlations between the sum of the distances to the four nearest neighbours and the sum of the canopy diameters of the target tree and its four nearest neighbours, indicating the presence of competition. In sum, these findings suggest that competitive interactions between A. karroo trees at these sites are relatively weak, and result in decreased performance (smaller canopy diameters) rather than mortality, thus preventing a regular pattern of tree distribution. We advocate the use of both methods of detecting competitive interactions in the field, especially if the effects of competition are too subtle to result in differential mortality.

Non-local energetics of random heterogeneous lattices

In this paper, we study the mechanics of statistically non-uniform two-phase elastic discrete structures. In particular, following the methodology proposed in Luciano and Willis (2005) [ Journal of the Mechanics and Physics of Solids 53, 1505–1522], energetic bounds and estimates of the Hashin–Shtrikman–Willis type are developed for discrete systems with a heterogeneity distribution quantified by second-order spatial statistics. As illustrated by three numerical case studies, the resulting expressions for the ensemble average of the potential energy are fully explicit, computationally feasible and free of adjustable parameters. Moreover, the comparison with reference Monte-Carlo simulations confirms a notable improvement in accuracy with respect to approaches based solely on the first-order statistics.

Folk perception of sexual dimorphism, sex ratio, and spatial repartition: implications for population dynamics of Sclerocarya birrea [(A. Rich) Hochst] populations in Benin, West Africa

In Sub-Saharan Africa, indigenous fruit trees play vital roles in nutrition and food security particularly, in food shortage times. Sclerocarya birrea subsp. birrea, an indigenous dioecious fruit tree is such a resource with strong multipurpose use characteristics in semi-arid zones of West Africa. We assessed sex ratio, spatial distribution among male and female adult trees using second-order spatial statistics and assessed folk perception of dioecism among the natural populations in protected areas and surrounding agroforestry systems. A field survey showed that 55% of interviewees were aware of sex separation in the species. Some used bark appearance to make distinction between sexes, but this morphological criterion was not consistent with statistical results. The sex ratio did not deviate significantly from 0.5 in any of the districts or land use types. Bivariate spatial analysis with pair correlation function revealed no spatial association between male and female individuals. Moreover, a strict spatial segregation of sexes was not observed even though some individuals of the same sex could sometimes be found together. Results confirmed the functional dioecy of the species and showed that the species did not display any apparent sex-specific dimorphism outside the reproduction period or any apparent sex-specific requirement for environment conditions.

Clonal and spatial genetic structure within populations of a coastal plant, Carex kobomugi (Cyperaceae)

Clarification of clonal growth pattern is critical for understanding the population dynamics and reproductive system evolution of clonal plant species. The contribution of clonality to the spatial genetic structure (SGS) within populations is also an important issue. I examined the spatial distribution of genetic variability within two populations of the coastal plant Carex kobomugi using seven microsatellite loci. Genotyping of 226 and 140 ramets within 14 × 40 m and 14 × 34 m plots on two populations revealed 36 and 33 multilocus genotypes, respectively. To quantify the extent of intermingling among clones, for each genet, I calculated the dominance of ramets belonging to a particular genet within a spatial range of the genet. Furthermore, I analyzed spatial distribution of genotypes within 2 × 2 m and 1 × 2 m quadrats using second-order spatial statistics. These analyses indicated that clones are highly intermingled, suggesting a low level of spatial interaction among clones. Spatial autocorrelation analysis of kinship coefficient including all pairs of ramets showed significantly stronger SGS than analysis considering only pairs between different genets. I conclude that clonal propagation largely contributes to SGS at a fine scale.

Ecological information from spatial patterns of plants: insights from point process theory

Summary 1. This article reviews the application of some summary statistics from current theory of spatial point processes for extracting information from spatial patterns of plants. Theoretical measures and issues connected with their estimation are described. Results are illustrated in the context of specific ecological questions about spatial patterns of trees in two forests. 2. The pair correlation function, related to Ripley’s K function, provides a formal measure of the density of neighbouring plants and makes precise the general notion of a ‘plant’s-eye’ view of a community. The pair correlation function can also be used to describe spatial relationships of neighbouring plants with different qualitative properties, such as species identity and size class. 3. The mark correlation function can be used to describe the spatial relationships of quantitative measures (e.g. biomass). We discuss two types of correlation function for quantitative marks. Applying these functions to the distribution of biomass in a temperate forest, it is shown that the spatial pattern of biomass is uncoupled from the spatial pattern of plant locations. 4. The inhomogeneous pair correlation function enables first-order heterogeneity in the environment to be removed from second-order spatial statistics. We illustrate this for a tree species in a forest of high topographic heterogeneity and show that spatial aggregation remains after allowing for spatial variation in density. An alternative method, the master function, takes a weighted average of homogeneous pair correlation functions computed in subareas; when applied to the same data and compared with the former method, the spatial aggregations are smaller in size. 5. Synthesis. These spatial statistics, especially those derived from pair densities, will help ecologists to extract important ecological information from intricate spatially correlated plants in populations and communities.

On the Sensitivity of Time-Reversal Imaging Techniques to Model Perturbations

Time-reversal (TR) techniques involve physical or synthetic retransmission of signals acquired by a set of transceivers in a time-reversed fashion, and can be used for a host of applications, including ultrawideband imaging of obscured targets. This work investigates the performance of some TR-based techniques under non-ideal conditions, and is divided into three main parts. In the first part, we investigate the robustness of TR-based imaging techniques via the decomposition of TR operator (DORT) algorithm and the multiple signal classification (MUSIC) algorithm under clutter and (additive) noise. In the second part, we examine the effect of TR-invariance breaking (due to losses in the intervening media) on the performance of both DORT and (TR-based) MUSIC. In the third part, effects of translational perturbations of the TR array are examined, as a class of model perturbations. These ensuing effects are then exploited, in a controlled manner, as a means to extract second-order spatial statistics of the probed media.

Microstructure of Composite Propellants Using Simulated Packings and X-Ray Tomography

C OMPOSITE solid propellants are heterogeneous media in which particles are embedded in a rubber binder. For industrial applications, such particles are commonly ammonium perchlorate (AP) and aluminum (Al), the sizes of which are typically below 100 m. As a general rule, simple mixture laws are often inappropriate for granular systems because they cannot capture the inherent microstructural features. This conclusion also stands for composite propellants, because various macroscale phenomena are strongly linked with the structure at microscale. They may be encountered in the field of detonation (e.g., hot-spot formation during deflagration– detonation transition), combustion (e.g., aluminum agglomeration and fine AP/binder premixed flames), mechanics (e.g., filler/binder adhesion), etc. This obviously motivates an improved knowledge of propellant microstructure and also the ability to “build” representative numericalmodels of structures as input data for detailedmultiphysics modeling (namely, computational fluid dynamics or finite element method codes). Those numerical microstructures are classically randompackings of spheres, which have been studied formany years because they serve as useful models for a variety of physical systems (granular, porous, anamorphous materials, etc.). Yet, random packings should faithfully reflect the features of the actual propellant spatial structure, not only global properties (as volume fraction, for instance), but also second-order spatial statistics. This can only be checked with some in situ experimental characterization of the structure. This work aims at comparing simulated random packings on a typical industrial AP/Al propellant together with experimental microstructure data to evaluate to which extent such simulated packing describe the actual microstructure. Experimental data are obtained through x-ray tomography of a propellant sample. This approach leads to genuine 3D statistical data in a nonintrusive way and allows a relative automatization of data sampling and processing. II. Random Packings

A completely automated CAD system for mass detection in a large mammographic database

Mass localization plays a crucial role in computer-aided detection (CAD) systems for the classification of suspicious regions in mammograms. In this article we present a completely automated classification system for the detection of masses in digitized mammographic images. The tool system we discuss consists in three processing levels: (a) Image segmentation for the localization of regions of interest (ROIs). This step relies on an iterative dynamical threshold algorithm able to select iso-intensity closed contours around gray level maxima of the mammogram. (b) ROI characterization by means of textural features computed from the gray tone spatial dependence matrix (GTSDM), containing second-order spatial statistics information on the pixel gray level intensity. As the images under study were recorded in different centers and with different machine settings, eight GTSDM features were selected so as to be invariant under monotonic transformation. In this way, the images do not need to be normalized, as the adopted features depend on the texture only, rather than on the gray tone levels, too. (c) ROI classification by means of a neural network, with supervision provided by the radiologist's diagnosis. The CAD system was evaluated on a large database of 3369 mammographic images [2307 negative, 1062 pathological (or positive), containing at least one confirmed mass, as diagnosed by an expert radiologist]. To assess the performance of the system, receiver operating characteristic (ROC) and free-response ROC analysis were employed. The area under the ROC curve was found to be Az = 0.783 +/- 0.008 for the ROI-based classification. When evaluating the accuracy of the CAD against the radiologist-drawn boundaries, 4.23 false positives per image are found at 80% of mass sensitivity.

Spatial constraint of peatland butterfly occurrences within a heterogeneous landscape.

The occurrence of ten butterfly taxa (Clossiana eunomia dawsonii, Clossiana freija, Clossiana frigga, Clossiana titania, Coenonympha inornata, Erebia discoidalis, Incisalia augustinus, Lycaena dorcas, Lycaena epixanthe, Oeneis jutta) was analyzed within three acid peatland habitat types from the Lake Superior drainage basin of northwestern Wisconsin. Both first- (nearest-neighbor spatial analysis) and second-order (Ripley's K) spatial point process statistics were used to identify the extents over which each distribution pattern significantly deviated from random expectations. Versions of these tests were used that identified significant spatial pattern uncorrelated to habitat location and habitat preference. These analyses documented non-random occurrence patterns in seven species. Deviations from random were largely confined to two extents: <50km and 70-100+km. The majority of non-random patterns at <50km extents were examples of aggregation, while the majority of non-random patterns noted at the 70-100+km scale were examples of segregation. These results demonstrate that even for winged animals inside a limited landscape, spatially constrained processes can be important determinants of distribution. It is likely that metapopulation dynamics and dispersal limitation help explain why aggregation is dominant at small scales. The mechanisms underlying the predominance of segregation at large scales are less clear, but may be related to migration history and/or weak environmental gradients.

The human visual system is optimised for processing the spatial information in natural visual images.

A fundamental tenet of visual science is that the detailed properties of visual systems are not capricious accidents, but are closely matched by evolution and neonatal experience to the environments and lifestyles in which those visual systems must work. This has been shown most convincingly for fish and insects. For mammalian vision, however, this tenet is based more upon theoretical arguments than upon direct observations. Here, we describe experiments that require human observers to discriminate between pictures of slightly different faces or objects. These are produced by a morphing technique that allows small, quantifiable changes to be made in the stimulus images. The independent variable is designed to give increasing deviation from natural visual scenes, and is a measure of the Fourier composition of the image (its second-order statistics). Performance in these tests was best when the pictures had natural second-order spatial statistics, and degraded when the images were made less natural. Furthermore, performance can be explained with a simple model of contrast coding, based upon the properties of simple cells in the mammalian visual cortex. The findings thus provide direct empirical support for the notion that human spatial vision is optimised to the second-order statistics of the optical environment.

Texture processing of synthetic aperture radar data using second-order spatial statistics

A method is presented for generating second-order spatial statistics (texture) from digital image data using the neighboring gray-level dependence matrix (NGLDM). Texture processing of a high-resolution airborne synthetic aperture radar image in an agricultural environment improved crop classification over the raw data. Classification of the large-number emphasis feature, based on the NGLDM (window size = 11 × 11, α = 24 and β = 1), showed a 30.5% improvement in validation accuracy over the tonal data.

Radio Brightness Measurement of Incoherent Sources: Second-Order Brightness Spatial Statistics

Radio brightness measurements of an incoherent source, in front of a constant-temperature background, depart from the true source brightness because of finite antenna beamwidth, internal receiver noise, and background noise. Noisy measured brightness samples may be used to estimate the measured and the true brightness, at and between the measured points. Such estimates are called and restoration, respectively. Optimum linear estimation - interpolation or restoration - requires only the second-order brightness statistics of the true and measured brightness samples, and of the corresponding noise samples. We derive these correlation functions in terms of the antenna parameters, subject to the assumptions that the source brightness varies rapidly compared to the antenna beamwidth, and that the background temperature is constant. We apply these results to calculating the interpolation and restoration errors for optimum linear processing of a finite number of noisy measured brightness samples...

Fractal modelling in image texture analysis

A fractal analysis procedure for the processing of visual textures is investigated. The method uses second-order spatial statistics to calculate the rate of change of a local estimate of the fractal dimension over different scales and orientations. Possible deficiencies caused by aliasing within a known technique are examined, together with the performance of a modified version which incorporates appropriate filtering. Results with the modified technique show that it is able to test the validity of the fractal model for real visual textures.

Statistical contrast enhancement of subtraction images for radiographic caries diagnosis

The effects of the nonlinear contrast-enhancement technique are examined in terms of diagnostic performance obtainable from subtracted dental radiographs for a variety of simulated exposures. Conventional bitewing radiographs of patients known to be free of caries were digitized by means of a computer interfaced with a conventional television camera. The resulting images were duplicated and stored in the computer. Radiolucencies similar in appearance to interproximal caries were simulated analytically in one set of the images. Reference images were superimposed spatially and subtracted from their counterparts containing the induced interproximal lesions after simulating the effects of quantum limited exposure on both sets. This was done for each separate image element independently by replacing original gray levels in each image with levels determined by a Poisson random deviate. The resulting difference images were contrast enhanced by a method which first smooths out local variations in gray level and then reassigns gray-level values in a way determined by the observed second-order spatial statistics. To aid in localization, these images were then again subtracted from the original noise-degraded pictures without lesions, rendering images similar to conventional radiographs but contrast enhanced. Observer performance by means of these enhanced images was compared with that produced from unenhanced lesion-containing controls. The results suggest that enhancement increases the certainty with which diagnosis can be made and, further, that diagnostic accuracy can be improved in severely degraded images which simulate the effects of reduced levels of exposure.