Local statistics and shuffling for dimers on a square-hexagon lattice

Local statistical analysis has long been of interest to social and environmental scientists who analyze geographic data. Research into local spatial statistics experienced a step-change in the mid-1990s, which provided a large class of local statistical methods and models. The local Moran statistic is one commonly used local indicator of spatial association, able to detect both areas of similarity and observations that are very dissimilar from their surroundings. From this, many further local statistics have been developed to characterize spatial clusters and outliers. These statistics have seen limited adoption because they do not sufficiently model the relationships involved in confounded spatial data, where the analyst seeks to understand the local spatial structure of a given outcome variable that is influenced by one or more additional factors. Recent innovations used to do joint multivariate local analysis also do not model this kind of conditional local structure in data. This article provides tools to rigorously characterize confounded local inference and a new and different class of multivariate conditional local Moran statistics that can account for confounding. To do this, we return to the Moran scatterplot as the critical tool for local Moran-style covariance statistics. Extending this concept, a new method is available directly from a “Moran-form” multiple regression. We show the empirical and theoretical properties of this statistic, show how some existing heuristic approaches arise naturally from this framework, and show how the use of conditional inference can change interpretations in an empirical analysis of rent and housing stock in a rapidly changing neighborhood.

Using local spatial statistics to explore local spatial association of geo-referenced data has attracted much attention. As is known, a local statistic is formulated at a particular sampling unit based on a prespecific proximity relationship and the observations in the neighborhood of this sampling unit. However, geostatistical data such as meteorological data and air pollution data are generally collected from meteorological or monitoring stations which are usually sparsely located or highly clustered over space. For such data, a local spatial statistic formulated at an isolate sampling point may be ineffective because of its distant neighbors, or the statistic is undefinable in the sub-regions where no observations are available, which limits the comprehensive exploration of local spatial association over the whole studied region. In order to overcome the predicament, a local-linear geographically weighted interpolation method is proposed in this paper to obtain the predictors of the underlying spatial process on a lattice spatial tessellation, on which a local spatial statistic can be well formulated at each interpolation point. Furthermore, the bootstrap test is suggested to identify the locations where local spatial association is significant using the interpolated-value-based local spatial statistics. Simulation with comparison to some existing interpolation and test methods is conducted to assess the performance of the proposed interpolation and the suggested test methods and a case study based on PM2.5 concentration data in Guangdong province, China, is used to demonstrate their applicability. The results show that the proposed interpolation method performs accurately in retrieving an underlying spatial process and the bootstrap test with the interpolated-value-based local statistics is powerful in identifying local patterns of spatial association.

Recent neurophysiological and computational studies have proposed the hypothesis that our brain automatically codes the nth-order transitional probabilities (TPs) embedded in sequential phenomena such as music and language (i.e., local statistics in nth-order level), grasps the entropy of the TP distribution (i.e., global statistics), and predicts the future state based on the internalized nth-order statistical model. This mechanism is called statistical learning (SL). SL is also believed to contribute to the creativity involved in musical improvisation. The present study examines the interactions among local statistics, global statistics, and different levels of orders (mutual information) in musical improvisation interact. Interactions among local statistics, global statistics, and hierarchy were detected in higher-order SL models of pitches, but not lower-order SL models of pitches or SL models of rhythms. These results suggest that the information-theoretical phenomena of local and global statistics in each order may be reflected in improvisational music. The present study proposes novel methodology to evaluate musical creativity associated with SL based on information theory.

Estimation of partially occluded 2D human joints with a Bayesian approach

Ultrasound (US) data suffer from speckle noise as well as intensity inhomogeneities due to underlying changes in acoustic properties of tissue structure and/or the effects of acoustic focusing and attenuation. This paper describes a 2D and 3D variational level-set method for segmenting such data. To deal with the local statistics of speckle noise, the data term of the level-set energy function is based on local gamma distributions which have shown an ability to model envelope data and gray-level pixel intensities of B-mode clinical images. Local statistics are estimated at a controllable scale using a smooth function with a compact support, a mollifyer, and the method of moments. Compared to manual segmentation, the investigated method provides a high Dice similarity coefficient (DSC) on 3D simulated data, an average DSC of 0.915 on 12 B-mode images of murine tumors acquired with a clinical US system, and average DSCs of 0.920, 0.806 and 0.975 for three media of 54 3D envelope data sets acquired with a high-frequency, focused transducer from dissected human lymph nodes. It also outperforms methods that employ local Gaussian statistics instead of local gamma statistics.

The past decade has witnessed extensive development of measures that examine characteristics of spatial subsets (local spaces) defined with respect to a complete data set (global space). Such procedures have evolved independently in fields such as geography, GIS, cartography, remote sensing, and landscape ecology. Collectively, we label these procedures as local spatial methods. We focus on those methods that share a common goal of identifying subsets whose characteristics are statistically ‘significant’ in some way. We propose the concept of local spatial statistical analysis (LoSSA) both as an integrative structure for existing methods and as a framework that facilitates the development of new local and global statistics. By formalizing what is involved when a particular local statistic is used, LoSSA helps to reveal the key features and limitations of the procedure. These include a consideration of the nature of the spatial subsets, their spatial relationship to the complete data set, and the relationship between a given global statistic and the corresponding local statistics computed for the data set.

Edges in natural scenes can result from a number of different causes. In this study, we investigated the statistical differences between edges arising from occlusions and nonocclusions (reflectance differences, surface change, and cast shadows). In the first experiment, edges in natural scenes were identified using the Canny edge detection algorithm. Observers then classified these edges as either an occlusion edge (one region of an image occluding another) or a nonocclusion edge. The nonocclusion edges were further subclassified as due to a reflectance difference, a surface change, or a cast shadow. We found that edges were equally likely to be classified as occlusion or nonocclusion edges. Of the nonocclusion edges, approximately 33% were classified as reflectance changes, 9% as cast shadows, and 58% as surface changes. We also analyzed local statistical properties like contrast, average edge profile, and slope of the edges. We found significant differences between the contrast values for each category. Based on the local contrast statistics, we developed a maximum likelihood classifier to label occlusion and nonocclusion edges. An 80%-20% cross validation demonstrated that the human classification could be predicted with 83% accuracy. Overall, our results suggest that for many edges in natural scenes, there exists local statistical information regarding the cause of the edge. We believe that this information can potentially be used by the early visual system to begin the process of segregating objects from their backgrounds.

In many image sequence compression applications, Huffman coding is used to reduce statistical redundancy in quantized transform coefficients. The Huffman codeword table is often pre-defined to reduce coding delay and table transmission overhead. Local symbol statistics, however, may be much different from the global one manifested in the pre-defined table. In this paper, we propose a dynamic Huffman coding method which can adaptively modify the given codeword and symbol association according to the local statistics. Over a certain set of blocks, local symbol statistics is observed and used to re-associate the symbols to the codewords in such a way that shorter codewords are assigned to more frequency symbols. This modified code table is used to code the next set of blocks. A parameter is set up so that the relative degree of sensitivity of the local statistics to the global one can be controlled. By performing the same modification to the code table using the decoded symbols, it is possible to keep up with the code table changes in receiving side. The code table modification information need not be transmitted to the receiver. Therefore, there is no extra transmission overhead in employing this method.© (1994) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Segmentation of ultrasound images––multiresolution 2D and 3D algorithm based on global and local statistics

Purpose:To facilitate adaptive image filtering operations, addressing spatial variations in both noise and signal. Such issues are prevalent in cone‐beam projections, where physical effects such as X‐ray scattering result in spatially variant noise, violating common assumptions of homogeneous noise and challenging conventional filtering approaches to signal extraction and noise suppression.Methods:We present a computational mechanism for probing into and quantifying the spatial variance of noise throughout an image. The mechanism builds a pyramid of local statistics at multiple spatial scales; local statistical information at each scale includes (weighted) mean, median, standard deviation, median absolute deviation, as well as histogram or dynamic range after local mean/median shifting. Based on inter‐scale differences of local statistics, the spatial scope of distinguishable noise variation is detected in a semi‐ or un‐supervised manner. Additionally, we propose and demonstrate the incorporation of such information in globally parametrized (i.e., non‐adaptive) filters, effectively transforming the latter into spatially adaptive filters. The multi‐scale mechanism is materialized by efficient algorithms and implemented in parallel CPU/GPU architectures.Results:We demonstrate the impact of local statistics for adaptive image processing and analysis using cone‐beam projections of a Catphan phantom, fitted within an annulus to increase X‐ray scattering. The effective spatial scope of local statistics calculations is shown to vary throughout the image domain, necessitating multi‐scale noise and signal structure analysis. Filtering results with and without spatial filter adaptation are compared visually, illustrating improvements in imaging signal extraction and noise suppression, and in preserving information in low‐contrast regions.Conclusion:Local image statistics can be incorporated in filtering operations to equip them with spatial adaptivity to spatial signal/noise variations. An efficient multi‐scale computational mechanism is developed to curtail processing latency. Spatially adaptive filtering may impact subsequent processing tasks such as reconstruction and numerical gradient computations for deformable registration.NIH Grant No. R01‐184173

A statistical active contour model for interactive clutter image segmentation using graph cut optimization

In a previous work we have presented an adaptive segmentation algorithm of dirty images in a Bayesian framework. The segmentation problem is formulated as a maximum a posteriori (MAP) estimation problem. The optimization is achieved using Besag's iterated conditional modes algorithm. A multiresolution implementation of the segmentation algorithm, using the discrete wavelet transform, has been used. This work focuses on the adaptive character of the algorithm and discusses how global and local statistics can be taken into account in the segmentation process. We propose an improvement on the adaptivity by introducing an enhancement to control the adaptive properties of the segmentation process. A weighting function taking into account both local and global statistics is introduced in the minimization. The new formulation of the segmentation problem allows us to control the effective contribution of each statistic. Results of segmentation carried out on synthetic images are presented.

Plants, especially terrestrial long-lived perennials such as trees, do not usually move once established. Spatial patterns of sessile organisms can suggest or reveal ecological processes affecting the population or community in the present or the past – dispersal, establishment, competition, mortality, facilitation, growth – and as such, patterns of plants motivated early developments in spatial statistics (Pielou, 1977; Diggle, 1983). Specifically, it is intuitive to treat individual plants (or other sessile organisms) as discrete events on a plane whose locations are known and generated by point pattern processes (Ripley, 1981; Diggle, 1983; Fortin and Dale, 2005). Second-order point pattern statistics are used to measure their spatial pattern. Arthur Getis (Getis and Franklin, 1987) introduced ecologists to the application of local spatial statistics, specifically neighborhood second-order point pattern analysis, to maps of organisms. As Wiegand and Moloney (2004) noted in their review paper, second-order global statistics based on the distribution of distances between pairs of points, especially Ripley’s K-function (Ripley, 1976, 1977) derived from distances between all pairs, have been widely used in plant ecology. However, their review does not mention neighborhood analysis or local measures of spatial association (Anselin, 1995) at all. This chapter revisits the impact of the Getis and Franklin paper on the practice of spatial point pattern analysis in plant ecology, and specifically aims to determine if local statistics are being used and how.

This paper addresses the problem of local/spatially-variant/adaptive image processing based on direct estimates of local autocovariance functions. In order to quantify the non-stationarity of images, and often, to implement spatially-variant processing, we require estimates or measurements of the local image statistics, specifically the autocovariance function. The simplest way to achieve this is to divide the image into N x N - pixel sub-blocks (e.g. N = 16), and calculate the usual biased or unbiased autocovariance function of each sub ock. In effect, each subblock is treated as part of a wide-sense stationary field. It is well-known, however, that reliable power spectral estimates require much larger amounts of data. Nevertheless, as our work shows, it is possible to obtain useful maps of local autoco-variance parameters if we assume simple parametric autocovariance models. Specifically, we employ popular first-order models, such as the nonseparable exponential model. We discuss a procedure for estimating local autocovariance parameters. The resulting parameters are seen to correlate with observed signal activity. We also outline techniques for spatially-variant image processing - coding, restoration, and enhancement - based on local statistics. Processed examples are given.

This paper explores a new class of measures for the detection of changes in images, specially for images acquired from different classes of sensors such as synthetic aperture radar (SAR) systems or computerized axial tomography (CAT) systems, monitoring patients. The problems become very challenging as the local statistics may be different even though the observations in the images may be similar. By exploiting this similarity new approaches are proposed for change detection. Based on the assumption that some form of dependence exists between the images, this dependence can be modeled by copulas. By using the conditional copula and the second image to simulate the distribution of first image, the dependence between the two images may be more closely modeled by the ensuing joint distribution. As a follow on, the symmetrical Kullback-Leibler distance can be used to obtain the change indicator between the distributions associated with the two images. In this paper the conditional copula is used as a change detector and applied to scenes from two distinct and different image families -SAR and CAT, and its performance compared with that of conventional change detection algorithms, based on a pixel based difference measure and on local pixel statistics.