Types Of Random Fields Research Articles

Machine Learned Texture Prior From Full-Dose CT Database via Multi-Modality Feature Selection for Bayesian Reconstruction of Low-Dose CT.

In our earlier study, we proposed a regional Markov random field type tissue-specific texture prior from previous full-dose computed tomography (FdCT) scan for current low-dose CT (LdCT) imaging, which showed clinical benefits through task-based evaluation. Nevertheless, two assumptions were made for early study. One assumption is that the center pixel has a linear relationship with its nearby neighbors and the other is previous FdCT scans of the same subject are available. To eliminate the two assumptions, we proposed a database assisted end-to-end LdCT reconstruction framework which includes a deep learning texture prior model and a multi-modality feature based candidate selection model. A convolutional neural network-based texture prior is proposed to eliminate the linear relationship assumption. And for scenarios in which the concerned subject has no previous FdCT scans, we propose to select one proper prior candidate from the FdCT database using multi-modality features. Features from three modalities are used including the subjects' physiological factors, the CT scan protocol, and a novel feature named Lung Mark which is deliberately proposed to reflect the z-axial property of human anatomy. Moreover, a majority vote strategy is designed to overcome the noise effect from LdCT scans. Experimental results showed the effectiveness of Lung Mark. The selection model has accuracy of 84% testing on 1,470 images from 49 subjects. The learned texture prior from FdCT database provided reconstruction comparable to the subjects having corresponding FdCT. This study demonstrated the feasibility of bringing clinically relevant textures from available FdCT database to perform Bayesian reconstruction of any current LdCT scan.

Stochastic seepage analysis in embankment dams using different types of random fields

Random field theory has become a very popular approach for probabilistic analysis of earth dams. This study performs a stochastic analysis of seepage through an embankment dam to account for the spatial variability of hydraulic parameters. The originality of this study consists in using and comparing three types of Random Fields RFs for the seepage analysis: classical stationary RFs, conditional RFs (to consider the position of available measurements), and non-stationary RFs (to consider the mean variation with depth). The same sample of hydraulic conductivity values is considered to generate the three types of random fields. The hydraulic conductivity RFs are integrated into a Finite Element seepage analysis model to evaluate the flow velocity, hydraulic gradient, pore water pressure, and flow rate. The process is repeated by Monte Carlo simulations to characterize the spatial variability of the seepage results in terms of mean and standard deviation at each point of the studied dam and its foundation. The results show a higher dispersion for the stationary RF case and a lower dispersion for the conditional RF case. For the case study, the results also show that the use of different RFs for modelling the spatial variability of hydraulic conductivity induces differences that can be considered significant for flow velocity, moderate for hydraulic gradient and minor for pore pressure.

Theoretical Probabilistic Nonlinear Analysis of Post-Tensioned Bridge Cross-Sections with the Application of Random Fields

Abstract The paper presents a proprietary procedure for the analysis of normal stress distributions in post-tensioned cross-sections. It has a significant advantage over conventional commonly used approaches based solely on the envelope analysis as it provides stress levels in all components of the cross-section. The procedure was used in a series of probabilistic analyses with the adoption of random fields. These fields represented uncertainties in strain-stress relationship in concrete. The analysis covered several types of cross-sections and several types of random fields. Key observations from the conducted simulations are as follows: (I) the widest ranges of the probable maximum stresses (i.e. the lowest indexes of reliability) were obtained for sections with relatively low heights of the compressive zone. (II) The highest probabilistic sensitivity to the type of random field used was found in tall sections with a relatively large compressive zone. (III) The greatest sensitivity to batch uncertainties was evident in all cross-sections when using squared exponential random fields. (IV) The greatest relative sensitivity to the batch uncertainties in the form of the random field compliant with the guidelines of the Joint Comity of Structural Safety (JCSS) was evident in the analyses of the tallest cross-section corresponding to the incrementally launched bridges.

Driven particle dispersion in narrow disordered racetracks

We study the disorder-induced deterministic dispersion of particles uniformly driven in an array of narrow tracks. For different toy models with quenched disorder we obtain exact analytical expressions for the steady-state mean velocity $v$ and the dispersion constant $D$ for any driving force $f$ above a putative depinning threshold. For short-range correlated pinning forces we find that at large drives $D\sim 1/v$ for random-field type of disorder while $D \sim 1/v^3$ for the random-bond type. We show numerically that these results are robust: the same scaling holds for models of massive damped particles, soft particles, particles in quasi-one dimensional or two dimensional tracks, and for a model of a magnetic domain wall with two degrees of freedom driven either by electrical current or magnetic field. Crossover and finite temperature effects are discussed. The universal features we identify may be relevant for describing the fluctuating dynamics of stable localized objects such solitons, superconducting vortices, magnetic domain walls and skyrmions, and colloids driven in quasi one-dimensional track arrays. In particular, the drive dependence of $D$ appears as a sensitive tool for characterizing and assessing the nature of disorder in the host materials.

History-dependent phase transition character

We consider history-dependent behavior in domain-type configurations in orientational order that are formed in configurations reached via continuous symmetry-breaking phase transitions. In equilibrium, these systems exhibit in absence of impurities a spatially homogeneous order. We focus on cases where domains are formed via (i) Kibble-Zurek mechanism in fast enough quenches or by (ii) Kibble mechanism in strongly supercooled phases. In both cases, domains could be arrested due to pinned topological defects that are formed at domain walls. In systems exhibiting polar or quadrupolar order, point and line defects (disclinations) dominate, respectively. In particular, the disclinations could form complex entangled structures and are more efficient in stabilizing domains. Domain patterns formed by fast quenches could be arrested by impurities imposing a strong enough random-field type disorder, as suggested by the Imry-Ma theorem. On the other hand, domains formed in supercooled systems could be also formed if large enough energy barriers arresting domains are established due to large enough systems’ stiffness. The resulting effective interactions in established domain-type patterns could be described by random matrices. The resulting eigenvectors reveal expected structural excitations formed in such structures. The most important role is commonly played by the random matrix largest eigenvector. Qualitatively different behavior is expected if this eigenvector exhibits a localized or extended character. In the former case, one expects a gradual, non-critical-type transition into a glass-type structure. However, in the latter case, a critical-like phase behavior could be observed.Graphical abstract

Stripe order, impurities, and symmetry breaking in a diluted frustrated magnet

We investigate the behavior of the frustrated $J_1$-$J_2$ Ising model on a square lattice under the influence of random dilution and spatial anisotropies. Spinless impurities generate a random-field type disorder for the spin-density wave (stripe) order parameter. These random fields destroy the long-range stripe order in the case of spatially isotropic interactions. Combining symmetry arguments, percolation theory and large-scale Monte Carlo simulations, we demonstrate that arbitrarily weak spatial interaction anisotropies restore the stripe phase. More specifically, the transition temperature $T_c$ into the stripe phase depends on the interaction anisotropy $\Delta J$ via $T_c \sim 1/|\ln (\Delta J)|$ for small $\Delta J$. This logarithmic dependence implies that very weak anisotropies are sufficient to restore the transition temperature to values comparable to that of the undiluted system. We analyze the critical behavior of the emerging transition and find it to belong to the disordered two-dimensional Ising universality class, which features the clean Ising critical exponents and universal logarithmic corrections. We also discuss the generality of our results and their consequences for experiments.

Random-field critical scaling in a model of dipolar glass and relaxor ferroelectric behavior.

Heat-bath Monte Carlo simulations have been used to study a 12-state discretized Heisenberg model with a type of random field on simple cubic lattices of size 128×128×128. The 12 states correspond to the [110] directions of a cube. The model has the standard nonrandom two-spin exchange term with coupling energy J and a random field which consists of adding an energy h_{R} to two of the 12 spin states, chosen randomly and independently at each site. We report on the case h_{R}/J=3, which has a sharp phase transition at about T_{c}/J=1.40625. Below T_{c}, the model has long-range ferroelectric order oriented along one of the eight [111] directions. At T_{c}, the behavior of the peak in the structure factor, S(k), at small |k| is a straight line on a log-log plot, which gives the result η[over ¯]=1.214±0.014. The onset of orientational order below T_{c} is very rapid for this value of h_{R}. There are peaks in the specific heat and longitudinal susceptibility at T_{c}. Below T_{c} there is a strong correction to ordinary scaling, which is probably caused by the cubic anisotropy, which is a dangerous irrelevant variable.

Phase transitions of the variety of random-field Potts models

The phase transitions of random-field q-state Potts models in d=3 dimensions are studied by renormalization-group theory by exact solution of a hierarchical lattice and, equivalently, approximate Migdal–Kadanoff solutions of a cubic lattice. The recursion, under rescaling, of coupled random-field and random-bond (induced under rescaling by random fields) coupled probability distributions is followed to obtain phase diagrams. Unlike the Ising model (q=2), several types of random fields can be defined for q≥3 Potts models, including random-axis favored, random-axis disfavored, random-axis randomly favored or disfavored cases, all of which are studied. Quantitatively very similar phase diagrams are obtained, for a given q for the three types of field randomness, with the low-temperature ordered phase persisting, increasingly as temperature is lowered, up to random-field threshold in d=3, which is calculated for all temperatures below the zero-field critical temperature. Phase diagrams thus obtained are compared as a function of q. The ordered phase in the low-q models reaches higher temperatures, while in the high-q models it reaches higher random fields. This renormalization-group calculation result is physically explained.

Modelling and comparison of different types of random fields: case of a real earth dam

The random field (RF) theory is widely used for describing the soil spatial variability in geotechnical engineering. This article presents soil variability modellings by using different types of RF-based on the available measurements of an earth dam. The effects of these RFs on dam reliability are investigated as well. The studied dam is well-documented, and there are many geo-localized measurements for the dry density. These measurements are firstly used to estimate the basic parameters of unconditional-stationary RFs and are then explored to define two more complex RFs (one is conditional RF and the other considers the mean variation with depth). The three mentioned types of RFs are all implemented in the same reliability analysis procedure for comparison. The results demonstrate that using different RFs for soil spatial modelling would induce insignificant differences in terms of reliability results if the dam construction was well controlled (careful selection for the construction material and controlled compaction). Therefore, a simple RF (unconditional-stationary) is enough to obtain satisfactory results in the case of carefully controlled dams during their construction. Otherwise, conditional RFs are recommended if more accurate results are needed, given that this type of RF is conditioned on the available data and can consider the non-stationarity of a soil property.

Random Field Ising Model and Parisi-Sourlas supersymmetry. Part I. Supersymmetric CFT

Quenched disorder is very important but notoriously hard. In 1979, Parisi and Sourlas proposed an interesting and powerful conjecture about the infrared fixed points with random field type of disorder: such fixed points should possess an unusual supersymmetry, by which they reduce in two less spatial dimensions to usual non-supersymmetric non- disordered fixed points. This conjecture however is known to fail in some simple cases, but there is no consensus on why this happens. In this paper we give new non-perturbative arguments for dimensional reduction. We recast the problem in the language of Conformal Field Theory (CFT). We then exhibit a map of operators and correlation functions from Parisi-Sourlas supersymmetric CFT in d dimensions to a (d − 2)-dimensional ordinary CFT. The reduced theory is local, i.e. it has a local conserved stress tensor operator. As required by reduction, we show a perfect match between superconformal blocks and the usual conformal blocks in two dimensions lower. This also leads to a new relation between conformal blocks across dimensions. This paper concerns the second half of the Parisi-Sourlas conjecture, while the first half (existence of a supersymmetric fixed point) will be examined in a companion work.

Fracture of beams with random field properties: Fractal and Hurst effects

The classical problem of peeling a beam off a substrate is studied through a re-examination of Griffith's fracture criterion in the presence of multiscale random properties. Four types of wide-sense homogeneous Gaussian random fields of the vector {Young's modulus E, surface energy density γ}, parametrized by the beam axis, are considered: Ornstein–Uhlenbeck, Matérn, Cauchy, and Dagum. The latter two are multiscale and allow decoupling of the fractal dimension and Hurst effects. Also calculated is the variance of the crack driving force G with any given type of random field in terms of the covariances of E and γ, under either fixed-grip or dead-load conditions. This investigation is complemented by a study of the stochastic crack stability which involves a stochastic competition between potential and surface energies. Overall, we find that, for Cauchy and Dagum models, the introduction of fractal-and-Hurst effects strongly influences the fracture mechanics results. Notably, while the Cauchy and Dagum models represent a more realistic scenario of random fields, given the same covariance on input, the response on output is strongest for Matérn, then Ornstein–Uhlenbeck, then Cauchy and, finally, Dagum model.

A Domain Decomposition Model Reduction Method for Linear Convection-Diffusion Equations with Random Coefficients

We developed a domain decomposition model reduction method for linear steady- state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equation with random diffusivity and the convection-dominated transport equation with random velocity. We investigated two types of random fields, i.e., colored noises and discrete white noises, both of which can lead to high-dimensional parametric dependence in practice. The motivation is to exploit domain decomposition to reduce the parametric dimension of local problems in subdomains, such that an entire parametric map can be approximated with a small number of expensive partial differential equation (PDE) simulations. The new method combines domain decomposition with model reduction and sparse polynomial approximation, so as to simultaneously handle the high dimensionality and irregular behavior of the PDEs under consideration. The advantages of our method lie in three aspects: (i) online-offline decomposition, i.e., the online cost is independent of the size of the triangle mesh; (ii) sparse approximation of operators involving nonaffine high-dimensional random fields; (iii) an effective strategy to capture irregular behaviors, e.g., sharp transitions of the PDE solutions. Two numerical examples are provided to demonstrate the advantageous performance of our method.

Comparison of statistical methods for testing the hypothesis of constant global mean in spatial statistics

In spatial statistics in general, and in geostatistics in particular, the choice between a spatial model with drift and a model with constant global mean is often critical, especially when only a small number of samples are available. A statistical test provides an objective means of making this choice. Among the many available statistical tests, a variance-ratio test has been widely used for making this choice because of its good statistical properties but, in addition to a semi-variogram model, it also requires an alternative drift model hypothesis. Another test statistic is the global D-statistic, which is a complementary test in the sense that it does not require an alternative hypothesis model. In this paper, we use sparse data from simulated random fields to evaluate and compare the performances of these two methods for testing the hypothesis of constant global mean in spatial statistics. We do so by considering the influence of four factors: the amount of data, the type of random field, the amount of spatial or temporal correlation and parametric drifts. In addition, we evaluate their performances in time series analysis, in which testing the hypothesis of constant global mean is also of significant interest. The two test statistics are compared in terms of their achieved confidence level and achieved power. The better method is the one that achieves the nominal confidence level and has higher power. We discuss departures from the nominal values and the results are used to highlight the importance of this problem in spatial statistics.

Emergence of Quantum Mechanics from Theory of Random Fields

Our aim in this paper is to enlighten the possibility to treat quantum mechanics as emergent from a kind of classical physical model, in spite of recent remarkable experiments demonstrating a violation of the Bell inequality. To proceed in a rigorous way, we use the methodology of ontic–epistemic modeling of physical phenomena. This methodology is rooted in the old Bild conception about theoretical and observational models in physics. This conception was elaborated in the fundamental works of Hertz, Boltzmann, and Schrodinger. Our ontic model (generating the quantum model) is of the random field type, prequantum classical statistical field theory (PCSFT). We present a brief review of its basic features without overloading the presentation by mathematical details. Then we show that the Bell inequality can be violated not only at the epistemic level, i.e., for observed correlations, but even at the ontic level, for classical random fields. We devote the important part of the paper to an analysis of the internal energy structure of prequantum random fields and their coupling with the background field of subquantum fluctuations. Finally, we present a unified picture of the microworld based on the composition of prequantum random fields from elementary fluctuations. Since quantum systems are treated as the symbolic representation of prequantum fields, this picture leads to a unifying treatment of all quantum systems as special blocks of elementary fluctuations carrying negligibly small energies.

Including fringe fields from a nearby ferromagnet in a percolation theory of organic magnetoresistance

Random hyperfine fields are essential to mechanisms of low-field magnetoresistance in organic semiconductors. Recent experiments have shown that another type of random field --- fringe fields due to a nearby ferromagnet --- can also dramatically affect the magnetoresistance. A theoretical analysis of the effect of these fringe fields is challenging, as the fringe field magnitudes and their correlation lengths are orders of magnitude larger than that of the hyperfine couplings. We extend a recent theory of organic magnetoresistance to calculate the magnetoresistance with both hyperfine and fringe fields present. This theory describes several key features of the experimental fringe-field magnetoresistance, including the applied fields where the magnetoresistance reaches extrema, the applied field range of large magnetoresistance effects from the fringe fields, and the sign of the effect.

Random pinning limits the size of membrane adhesion domains

Theoretical models describing specific adhesion of membranes predict (for certain parameters) a macroscopic phase separation of bonds into adhesion domains. We show that this behavior is fundamentally altered if the membrane is pinned randomly due to, e.g., proteins that anchor the membrane to the cytoskeleton. Perturbations which locally restrict membrane height fluctuations induce quenched disorder of the random-field type. This rigorously prevents the formation of macroscopic adhesion domains following the Imry-Ma argument [Imry and Ma, Phys. Rev. Lett. 35, 1399 (1975)]. Our prediction of random-field disorder follows from analytical calculations and is strikingly confirmed in large-scale Monte Carlo simulations. These simulations are based on an efficient composite Monte Carlo move, whereby membrane height and bond degrees of freedom are updated simultaneously in a single move. The application of this move should prove rewarding for other systems also.

Space-time models for moving fields with an application to significant wave height fields

The surface of the ocean, and so such quantities as the significant wave height, , can be thought of as a random surface that develops over time. In this paper, we explore certain types of random fields in space and time, with and without dynamics that may or may not be driven by a physical law, as models for the significant wave height. Reanalysis data is used to estimate the sea-state motion which is modeled as a hidden Markov chain in a state space framework by means of an AR(1) process or in the presence of the dispersion relation. Parametric covariance models with and without dynamics are fitted to reanalysis and satellite data and compared to the empirical covariance functions. The derived models have been validated against satellite and buoy data. Copyright © 2010 JohnWiley & Sons, Ltd.

Isotropic Random Fields with the Von Kármán–Whittle Type Correlation Structure

This article proposes a subclass of stationary Gaussian or elliptically contoured random fields whose covariance functions are isotropic and are the linear combination of the von Kármán–Whittle structure. A particular example of this type of random fields is the so-called spartan random field recently appeared in the applied literature.

Texture Based Segmentation using Statistical Properties for Mammographic Images

Segmentation is very basic and important step in computer vision and image processing. For medical images specifically accuracy is much more important than the computational complexity and thus time required by process. But as volume of data of patients goes on increasing then it becomes necessary to think about the processing time along with accuracy. Here in this paper, new algorithm is proposed for texture based segmentation using statistical properties. For that probability of each intensity value of image is calculated directly and image is formed by replacing intensity by its probability . Variance is calculated in three different ways to extract the texture features of the mammographic images. These results of proposed algorithm are compared with well known GLCM and Watershed algorithm. Segmenting mammographic images into homogeneous texture regions representing disparate tissue types is often a useful preprocessing step in the computer-assisted detection of breast cancer. With the increasing size and number of medical images, the use of computers in facilitating their processing and analysis has become necessary. Estimation of the volume of the whole organ, parts of the organ and/or objects within an organ i.e. tumors is clinically important in the analysis of medical image. The relative change in size, shape and the spatial relationships between anatomical structures obtained from intensity distributions provide important information in clinical diagnosis for monitoring disease progression. Therefore, radiologists are particularly interested to observe the size, shape and texture of the organs and/or parts of the organ. For this, organ and tissue morphometry performed in every radiological imaging centre. Texture based image segmentation is area of intense research activity in the past few years and many algorithms were published in consequence of all this effort, starting from simple thresholding method up to the most sophisticated random field type method. The repeating occurrence of homogeneous regions of images is texture. Texture image segmentation identifies image regions that have homogeneous with respect to a selected texture measure. Recent approaches to texture based segmentation are based on linear transforms and multiresolution feature extraction (1), Markov random filed models (2,3), Wavelets (4 - 6) and fractal dimension (7). Although unsupervised texture-based image segmentation is

Bayesian Nonlinear Principal Component Analysis Using Random Fields

We propose a novel model for nonlinear dimension reduction motivated by the probabilistic formulation of principal component analysis. Nonlinearity is achieved by specifying different transformation matrices at different locations of the latent space and smoothing the transformation using a Markov random field type prior. The computation is made feasible by the recent advances in sampling from von Mises-Fisher distributions. The computational properties of the algorithm are illustrated through simulations as well as an application to handwritten digits data.