Neuro fuzzy and punctual kriging based filter for image restoration

In this paper, we present a hybrid, image restoration approach. The proposed approach combines the geostatistical interpolation of punctual kriging, artificial neural networks (ANNs), and fuzzy logic based approaches. Images degraded with Gaussian white noise are restored by first utilizing fuzzy logic for selecting pixels that needs kriging. Three fuzzy systems are employed. Both type-I and type-II fuzzy sets in addition with neuro fuzzy classifier (NFC) have been used for the detection of noisy pixels. To avoid edge pixels, a post processing technique is used to check the edge pixel connectivity up to lag 5. If the pixel under consideration is an edge pixel, it is excluded from the fuzzy map and thus not estimated. The concept of punctual kriging is then used to estimate the intensity of a noisy pixel. ANN is employed to minimize the cost function of the kriging based pixel intensity estimation procedure. ANN, in contrast to analytical methodologies, avoids both matrix inversion failure and negative weights problems. Image restoration performance based comparison has been made against adaptive Weiner filter and existing fuzzy kriging approaches. Experimental results using 450 images are used to validate the effectiveness of the proposed approach. Different image quality measures are used to compare the efficacy of the proposed NFC and fuzzy type-II approaches for detecting noisy pixels in conjunction with ANN and kriging based estimation.     

Kriging metamodel with modified nugget-effect: The heteroscedastic variance case

Metamodels are commonly used to approximate and analyze simulation models. However, in cases where the simulation output variances are non-zero and not constant, many of the current metamodels which assume homogeneity, fail to provide satisfactory estimation. In this paper, we present a kriging model with modified nugget-effect adapted for simulations with heterogeneous variances. The new model improves the estimations of the sensitivity parameters by explicitly accounting for location dependent non-constant variances and smoothes the kriging predictor’s output accordingly. We look into the effects of stochastic noise on the parameter estimation for the classic kriging model that assumes deterministic outputs and note that the stochastic noise increases the variability of the classic parameter estimation. The nugget-effect and proposed modified nugget-effect stabilize the estimated parameters and decrease the erratic behavior of the predictor by penalizing the likelihood function affected by stochastic noise. Several numerical examples suggest that the kriging model with modified nugget-effect outperforms the kriging model with nugget-effect and the classic kriging model in heteroscedastic cases.     

High-resolution satellite image fusion using regression kriging

Image fusion is an important component of digital image processing and quantitative image analysis. Image fusion is the technique of integrating and merging information from different remote sensors to achieve refined or improved data. A number of fusion algorithms have been developed in the past two decades, and most of these methods are efficient for applications especially for same-sensor and single-date images. However, colour distortion is a common problem for multi-sensor or multi-date image fusion. In this study, a new image fusion method of regression kriging is presented. Regression kriging takes consideration of correlation between response variable (i.e., the image to be fused) and predictor variables (i.e., the image with finer spatial resolutions), spatial autocorrelation among pixels in the predictor images, and the unbiased estimation with minimized variance. Regression kriging is applied to fuse multi-temporal (e.g., Ikonos, QuickBird, and OrbView-3) images. The significant properties of image fusion using regression kriging are spectral preservation and relatively simple procedures. The qualitative assessments indicate that there is no apparent colour distortion in the fused images that coincides with the quantitative checks, which show that the fused images are highly correlated with the initial data and the per-pixel differences are too small to be considered as significant errors. Besides a basic comparison of image fusion between a wavelet based approach and regression kriging, general comparisons with other published fusion algorithms indicate that regression kriging is comparable with other sophisticated techniques for multi-sensor and multi-date image fusion.     

The application of median indicator kriging and neural network in modeling mixed population in an iron ore deposit

This paper presents the results of a study comparing median indicator kriging and an artificial neural network in the estimation of iron grades in the Jalal-Abad Zarrand iron ore deposit located in the southern Iran. The data used in this study is from 2017 composite samples with 2 m length from 32 exploration boreholes. The iron grade data is sparse, irregularly spaced and has mixed distribution, which can be problematic for the stationarity assumptions of the widely used ordinary kriging estimation method. The two estimation techniques applied in this study make no assumptions about the distribution of the sample data, and accommodate moderately mixed sample populations.     

Kriging, cokriging, radial basis functions and the role of positive definiteness

There are at least three developments for interpolators that lead to the same functional form for the interpolator; the thin plate spline, radial basis functions and the regression method known as kriging. The key to the interrelationship lies in the positive definiteness of the kernel function. Micchelli has known that a weak form of positive definiteness is sufficient to ensure a unique solution to the system of equations determining the coefficients in the interpolator. Both the positive definiteness and the interpolator can be extended to vector valued functions via the kriging approach which is also independent of the dimension of the underlying space. The kriging approach leads naturally to various methods for simulation as well.     

Comparing error estimation measures for polynomial and kriging approximation of noise-free functions

Error estimation measures are useful for assessing uncertainty in surrogate predictions. We use a suite of test problems to appraise several error estimation measures for polynomial response surfaces and kriging. In addition, we study the performance of cross-validation error measures that can be used with any surrogate. We use 1,000 experimental designs to obtain the variability of error estimates with respect to the experimental designs for each problem. We find that the (actual) errors for polynomial response surfaces are less sensitive to the choice of experimental designs than the kriging errors. This is attributed to the variability in the maximum likelihood estimates of the kriging parameters. We find that no single error measure outperforms other measures on all the problems. Computationally expensive integrated local error measures (standard error for polynomials and mean square error for kriging) estimate the actual root mean square error very well. The distribution-free cross-validation error characterized the actual errors reasonably well. While the estimated root mean square error for polynomial response surface is a good estimate of the actual errors, the process variance for kriging is not. We explore a few methods of simultaneously using multiple error measures and demonstrate that the geometric means of several combinations of error measures improve the assessment of the actual errors over individual error measures.     

Classification of remotely-sensed imagery using an indicator kriging approach: application to the problem of calcite-dolomite mineral mapping

A new image classification technique for analysis of remotely-sensed data based on geostatistical indicator kriging is introduced. Conventional classification techniques require ground truth information, use only the spectral characteristics of an unknown pixel in comparison, rely on a Gaussian probability distribution for the spectral signature of the training data, and work on a pixel support or spatial resolution without allowing classification on larger or smaller volumes. The indicator kriging classifier overcomes such problems because: (1) it relies on spectral information from laboratory studies rather than on ground truth data, (2) through the kriging estimation variances an estimate of uncertainly is derived, (3) it incorporates spatial aspects because it uses local estimation techniques, (4) it is distribution-free, (5) and may be applied on different supports if the data are corrected for support changes. Comparison of classification results applied to the problem of mapping calcite and dolomite from GER imaging spectrometry data shows that indicator kriging performs better than the conventional classification algorithms and gives insight in the accuracy of the results without prior field knowledge     

Systems that Should Have Failed: Critical Infrastructure Protection in an Institutionally Fragmented Environment

Recent years have witnessed major governmental initiatives regarding critical infrastructure protection (CIP). During that same time, critical infrastructures (CIs) have undergone massive institutional restructuring under the headings of privatization, deregulation and liberalization. Little research has gone into understanding the interactions between these two developments. In this article, we outline the consequences of institutional restructuring for the changing ways in which CIs ensure the reliability and security of their networks and services. Neither Normal Accident Theory nor High‐Reliability Theory can account for reliability under these conditions. We then investigate the implications of these findings for CIP.     

Spatially Adaptive Image Restoration Using Fuzzy Punctual Kriging

We present a general formulation based on punctual kriging and fuzzy concepts for image restoration in spatial domain. Gray-level images degraded with Gaussian white noise have been considered. Based on the pixel local neighborhood, fuzzy logic has been employed intelligently to avoid unnecessary estimation of a pixel. The intensity estimation of the selected pixels is then carried out by employing punctual kriging in conjunction with the method of Lagrange multipliers and estimates of local semi-variances. Application of such a hybrid technique performing both selection and intensity estimation of a pixel demonstrates substantial improvement in the image quality as compared to the adaptive Wiener filter and existing fuzzykriging approaches. It has been found that these filters achieve noise reduction without loss of structural detail information, as indicated by their higher structure similarity indices, peak signal to noise ratios and the new variogram based quality measures.     

Modeling uncertainties in sodium spatial dispersion using a computational intelligence-based kriging method

Classical and geostatistical methods have been used to create continuous surfaces from sampled data. A common geostatistical method is kriging, which provides an accurate estimation based on the existing spatial structure of sample points. However, kriging is sensitive to errors in the input data, the dispersion of the sample points, and the fit of the model to the variogram. The purpose of this research is to develop a new method to address the uncertainties resulting from the input data and choice of model in the kriging method. In our approach, the existing uncertainties in the input data are modeled by fuzzy computations, and the variogram variables are optimized by a genetic algorithm. To test this new hybrid method, sodium contamination values in the Zanjan aquifer were used. The results show a general improvement in accuracy compared with the ordinary kriging method. Consideration of all equations and values in fuzzy computations highlights the complexity of the computation. Herein, the integration problems experienced by other researchers when trying to use fuzzy kriging are resolved.     

Sequential sampling for contour estimation with concurrent function evaluations

We demonstrate the use of multiple surrogates and kriging believer for parallelizing surrogate-based contour estimation. For the demonstration example, we reduce wall clock time with minimal penalty in number of simulations.     

Comparison of interpolation methods for depth to groundwater and its temporal and spatial variations in the Minqin oasis of northwest China

Severe water shortages and dramatic declines in groundwater levels have resulted in environmental deterioration in the Minqin oasis, an arid region of northwest China. Understanding temporal and spatial variations in the depth to groundwater in the region is important for developing management strategies. Depth to groundwater records for 48 observation wells in the Minqin oasis were available for 22 years from 1981 to 2003, allowing us to compare three different interpolation methods based on three selected years (1981, 1990, 2002) as starting points. The three methods were inverse distance weighting (IDW), radial basis function (RBF), and kriging (including ordinary kriging (OK), simple kriging (SK), and universal kriging (UK)). Cross-validation was applied to evaluate the accuracy of the various methods, and two indices – the correlation coefficient (R²) and the root mean squared error (RMSE) – were used to compare the interpolation methods. Another two indices – deviation of estimation errors (σ) and 95% prediction interval (95 PPI) – were used to assess prediction errors. Comparison of interpolated values with observed values indicates that simple kriging is the optimal method for interpolating depth to groundwater in this region: it had the lowest standard deviation of estimation errors and smallest 95% prediction interval (95 PPI). By using the simple kriging method and an autoregressive model for depth to groundwater based on the data from 1981 to 2003, this work revealed systematic temporal and spatial variations in the depth to groundwater in the Minqin oasis. The water table has declined rapidly over the past 22 years, with the average depth to groundwater increasing from 4.95 m in 1981 to 14.07 m in 2002. We attribute the decline in the water table to excessive extraction and to decreases in irrigation channel leakage.     

Reliability-based design optimization using kriging surrogates and subset simulation

The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.     

Cokriging for cross-attribute fusion in sensor networks

Optimisation of the number of required measurement points and their location is an important research topic in sensor networks. Finding the optimal positions increases spatial coverage and reduces deployment costs. This paper presents an approach for the case that two attributes have to be measured with a different number of available sensors. The proposed cokriging method performs cross-attribute fusion in sensor networks by being based on the analysis of multi-variable spatial correlations. To the best of our knowledge, this scientific work is the first one considering kriging and cokriging interpolations as IF methods. The single-variable ordinary kriging and bi-variable methods were applied to experimental data. The combination of humidity and temperature data in a refrigerated container is used as exemplary case, humidity measurements are considered to be the expensive attribute to measure. The average estimation error for intermediate points was estimated as a function of the number of humidity sensors. When variability is high, data fusion using the bi-variable method produced results as accurate as the single-variable one, without the necessity of deploying a large number of humidity measuring points, by complementing the estimation with temperature measurements.     

异质性地表土壤冻融循环监测网络的优化采样设计-以黑河祁连山山前地区为例

在黑河祁连山山前地区建立分布式的土壤温度/水分传感器监测网络,准确获取异质性地表的遥感像元真值,用于地表冻融状态分类及土壤水分定量反演算法的发展完善和真实性检验,以及两者的降尺度研究均具有重要意义。传感器监测网络节点的空间布局直接影响观测有效性及其数据质量,一种基于异质性地表的均值估计方法被用于空间节点的优化采样设计：即以地表温度为目标变量,将研究区划分为相对均质的若干层(子区域),计算各层及总体的变异函数参数作为代价函数的输入,通过最小化目标变量的估计方差,实现传感器网络节点在各层的空间分布,准确地捕捉区域内部的异质性。结果表明,分层后各层的异质性相对于总体都有所下降,优化的节点空间布局具有较好的属性代表性,对于异质性较强的局部区域,有较高的样本密度。     

Interpolating datasets with trends: A modified median polish approach

This article investigates an alternative method to deal with data sets in the presence of trends. Median polish kriging (MPK) was introduced as an alternative solution to universal kriging or intrinsic random functions of order k (IRF-k) for estimation in the presence of trends. The maps obtained using the original MPK algorithm show banding artefacts which do not appear in the reference data set. A modified version of MPK was introduced to attempt to remove the banding artefacts. The results confirm the improvement in quality of estimate using the modified version of MPK (called MPKm), which takes into account the problems of clustered samples and boundary effect associated with the re-addition of the trend along bands. The variation introduced in the median polish algorithm proved to be satisfactory in eliminating the artefacts.     

Whole field validation of numerical and experimental results

The kriging estimator and its associated covariance model are introduced as a means of describing the verisimilitude of spatial datasets describing flow-fields in their entirety, and further as a means of interpreting and blending said datasets. In this manner a means of comparing uncertain nodal data from numerical models and experimental flow-field anemometry is developed. For spatial datasets, this activity has heretofore been considered to be a simple extension of established methodologies in validation and verification, which have been developed with the validation of scalar data – lift, drag, point velocity components or pressure, in mind. It is demonstrated that a more complex and complete comparison arises when the entire fields of data are correlated via spatial covariance functions, instead. These spatial covariance functions then inform the subsequent estimation, smoothing and blending of velocity fields; known as cokriging. In this paper, the theoretical model underlying kriging estimation is elucidated, and the techniques are demonstrated with reference to Laser Doppler anemometry and Finite Volume modelling of a subsonic flow of air around an experimental model. It is proposed that by developing spatial correlations between datasets, a more rigorous and flexible model for spatial comparison and validation of flow-fields emerges.     

Consensus of a group of mobile agents in three dimensions

This paper investigates the consensus problem of a three dimensional (3D) version of the widely studied Vicsek model, where each member moves in 3D Euclidean space with a constant speed with headings updated according to the average direction of neighbors. In comparison with the original Vicsek model, each agent’s heading is determined by two angle sequences interacting with each other, one of which evolves according to a “linear” non-homogeneous equation, which makes the theoretical analysis quite complicated. By analyzing the underlying structure of the system and relying on estimation of some characteristics concerning the initial states, we will establish consensus results for 3D Vicsek model under random framework without resorting to any a priori connectivity assumption.     

Augmented instrumental variable method for position and heading estimation with RDOA measurements

In this paper, we proposed a position and heading estimation algorithm using only range difference of arrival (RDOA) measurements. Based on RDOA measurements, an uncertain linear measurement model is derived and both position and heading are estimated with the instrumental variable (IV) method which can show unbiased estimation results for the uncertainty of the model. In addition, to remove the unknown bias included in the measurement model error, we augment the bias to the state vector of the model. Since the proposition inherits the characteristic of the IV method, it does not need the stochastic information of the RDOA measurement excepting the assumption that the RDOA measurement noise is zero mean and white, and the zero mean error performance can be guaranteed when variances of RDOA measurement noises are identical. Through simulations, the performance of the proposed algorithm is verified at various positions and headings in the sensor network and compared with the robust least squares method which shows a zero mean error performance under the assumption that the stochastic information is known exactly.     

Automatic interpolation methods for mapping piezometric surfaces

The purpose of analysis of spatial data should determine the approach to be adopted. If estimation is concerned with detailed local interpretation rather than some form of global averaging, then kriging would seem to be an inappropriate method. This is especially so for the interpolation of precise observations which consequently are better estimated by deterministic methods. Five such efficient direct automatic interpolation methods are discussed and applied to the contour mapping of a piezometric surface.