Gaussian Markov Random Field Model Research Articles

Application of the sequential quadratic programming algorithm for reconstructing the distribution of optical parameters based on the time-domain radiative transfer equation.

Sequential quadratic programming (SQP) is used as an optimization algorithm to reconstruct the optical parameters based on the time-domain radiative transfer equation (TD-RTE). Numerous time-resolved measurement signals are obtained using the TD-RTE as forward model. For a high computational efficiency, the gradient of objective function is calculated using an adjoint equation technique. SQP algorithm is employed to solve the inverse problem and the regularization term based on the generalized Gaussian Markov random field (GGMRF) model is used to overcome the ill-posed problem. Simulated results show that the proposed reconstruction scheme performs efficiently and accurately.

Image Segmentation of Printed Fabrics with Hierarchical Improved Markov Random Field in the Wavelet Domain

An improved MRF algorithm–hierarchical Gauss Markov Random Field model in the wavelet domain is presented for fabric image segmentation in this paper, which obtains the relation of inter-scale dependency from the feature field modeling and label field modeling. The Gauss-Markov random field modeling is usually adopted to feature field modeling. The label field modeling employs the inter-scale causal MRF model and the intra-scale non-causal MRF model. After that, parameter estimation is the essential section in the inter-scale, enhancing modeling capabilities of the pixels partial dependency. Sequential maximum a posterior criterion is applied to achieve the results of image segmentation. Comparisons with other hybrid schemes, results are indicated that performance of the presented algorithm is effective and accurate, in terms of classification accuracy and kappa coefficient, for patterned fabric images.

A Gaussian random field model for similarity-based smoothing in Bayesian disease mapping.

Conditionally specified Gaussian Markov random field (GMRF) models with adjacency-based neighbourhood weight matrix, commonly known as neighbourhood-based GMRF models, have been the mainstream approach to spatial smoothing in Bayesian disease mapping. In the present paper, we propose a conditionally specified Gaussian random field (GRF) model with a similarity-based non-spatial weight matrix to facilitate non-spatial smoothing in Bayesian disease mapping. The model, named similarity-based GRF, is motivated for modelling disease mapping data in situations where the underlying small area relative risks and the associated determinant factors do not vary systematically in space, and the similarity is defined by "similarity" with respect to the associated disease determinant factors. The neighbourhood-based GMRF and the similarity-based GRF are compared and accessed via a simulation study and by two case studies, using new data on alcohol abuse in Portugal collected by the World Mental Health Survey Initiative and the well-known lip cancer data in Scotland. In the presence of disease data with no evidence of positive spatial correlation, the simulation study showed a consistent gain in efficiency from the similarity-based GRF, compared with the adjacency-based GMRF with the determinant risk factors as covariate. This new approach broadens the scope of the existing conditional autocorrelation models.

SU‐G‐IeP4‐03: Cone Beam X‐Ray Luminescence Computed Tomography Based On Generalized Gaussian Markov Random Field

Purpose:Cone beam X‐ray luminescence computed tomography (CB‐XLCT), which aims to achieve molecular and functional imaging by X‐rays, has recently been proposed as a new imaging modality. However, the inverse problem of CB‐XLCT is seriously ill‐conditioned, hindering us to achieve good image quality. In this work, a novel reconstruction method based on Bayesian theory is proposed to tackle this problemMethods:Bayesian theory provides a natural framework for utilizing various kinds of available prior information to improve the reconstruction image quality. A generalized Gaussian Markov random field (GGMRF) model is proposed here to construct the prior model of the Bayesian theory. The most important feature of GGMRF model is the adjustable shape parameter p, which can be continuously adjusted from 1 to 2. The reconstruction image tends to have more edge‐preserving property when p is slide to 1, while having more noise tolerance property when p is slide to 2, just like the behavior of L1 and L2 regularization methods, respectively. The proposed method provides a flexible regularization framework to adapt to a wide range of applications.Results:Numerical simulations were implemented to test the performance of the proposed method. The Digimouse atlas were employed to construct a three‐dimensional mouse model, and two small cylinders were placed inside to serve as the targets. Reconstruction results show that the proposed method tends to obtain better spatial resolution with a smaller shape parameter, while better signal‐to‐noise image with a larger shape parameter. Quantitative indexes, contrast‐to‐noise ratio (CNR) and full‐width at half‐maximum (FWHM), were used to assess the performance of the proposed method, and confirmed its effectiveness in CB‐XLCT reconstruction.Conclusion:A novel reconstruction method for CB‐XLCT is proposed based on GGMRF model, which enables an adjustable performance tradeoff between L1 and L2 regularization methods. Numerical simulations were conducted to demonstrate its performance.

Voronoi tessellation‐based regionalised segmentation for colour texture image

This study presents a region‐based algorithm for segmenting colour texture image, which uses Voronoi tessellation for partitioning the domain of the image and Markov random field (MRF) for modelling colour texture. In detail, (i) an image domain is divided into polygons (or sub‐regions) by Voronoi tessellation; (ii) two MRF models, improved Potts model and multivariate Gaussian MRF model, are used to characterise colour texture structures inter‐ and intra‐polygons, respectively; (iii) by Bayesian paradigm, a posterior distribution which characterises the segmentation and model parameters conditional on a given colour image can be obtained up to a normalising constant; (iv) a Markov chain Monte Carlo algorithm is developed to simulate from the posterior distribution; finally, (v) a maximum a posteriori scheme is employed to find an optimal segmentation and model parameters. In order to evaluate the proposed colour texture segmentation algorithm, two kinds of colour texture images are tested, including synthetic and real colour texture images. The accuracy assessments are performed qualitatively on all kinds of images and quantitatively on synthetic images. All results demonstrate that the proposed algorithm is efficiently.

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation

An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.

Fully Bayesian Field Slam Using Gaussian Markov Random Fields

AbstractThis paper presents a fully Bayesian way to solve the simultaneous localization and spatial prediction problem using a Gaussian Markov random field (GMRF) model. The objective is to simultaneously localize robotic sensors and predict a spatial field of interest using sequentially collected noisy observations by robotic sensors. The set of observations consists of the observed noisy positions of robotic sensing vehicles and noisy measurements of a spatial field. To be flexible, the spatial field of interest is modeled by a GMRF with uncertain hyperparameters. We derive an approximate Bayesian solution to the problem of computing the predictive inferences of the GMRF and the localization, taking into account observations, uncertain hyperparameters, measurement noise, kinematics of robotic sensors, and uncertain localization. The effectiveness of the proposed algorithm is illustrated by simulation results as well as by experiment results. The experiment results successfully show the flexibility and adaptability of our fully Bayesian approach in a data‐driven fashion.

On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods

A large number of statistical models are "doubly-intractable": the likelihood normalising term, which is a function of the model parameters, is intractable, as well as the marginal likelihood (model evidence). This means that standard inference techniques to sample from the posterior, such as Markov chain Monte Carlo (MCMC), cannot be used. Examples include, but are not confined to, massive Gaussian Markov random fields, autologistic models and Exponential random graph models. A number of approximate schemes based on MCMC techniques, Approximate Bayesian computation (ABC) or analytic approximations to the posterior have been suggested, and these are reviewed here. Exact MCMC schemes, which can be applied to a subset of doubly-intractable distributions, have also been developed and are described in this paper. As yet, no general method exists which can be applied to all classes of models with doubly-intractable posteriors. In addition, taking inspiration from the Physics literature, we study an alternative method based on representing the intractable likelihood as an infinite series. Unbiased estimates of the likelihood can then be obtained by finite time stochastic truncation of the series via Russian Roulette sampling, although the estimates are not necessarily positive. Results from the Quantum Chromodynamics literature are exploited to allow the use of possibly negative estimates in a pseudo-marginal MCMC scheme such that expectations with respect to the posterior distribution are preserved. The methodology is reviewed on well-known examples such as the parameters in Ising models, the posterior for Fisher-Bingham distributions on the $d$-Sphere and a large-scale Gaussian Markov Random Field model describing the Ozone Column data. This leads to a critical assessment of the strengths and weaknesses of the methodology with pointers to ongoing research.

Image retrieval based on micro-level spatial structure features and content analysis using Full Range Gaussian Markov Random Field model

This paper proposes a novel system, based on Full Range Gaussian Markov Random Field (FRGMRF) model with Bayesian approach for image retrieval. The color image is segmented into various regions according to its structure and nature. The segmented image is modeled to HSV color space, where V attributes to pixel values, and it ranges from 0 to 1. On HSV color space, the texture information and spatial orientation of the pixels are extracted. On each region, the model parameters, autocorrelation coefficient (ACC), and unique texture numbers are computed using the FRGMRF model. The model parameters and ACCs are formed as feature vectors (FVs) of the image. The Mahalanobis distance is applied to measure the distance between the query and target images. Moreover, associated probabilities are computed on the texture numbers on each region, and are used to compute the divergence between the query and target images using cosine function. The obtained results are compared to those of the existing systems. The comparative study reveals that the proposed system outperforms the existing systems.

Image Reconstruction of Two-Dimensional Highly Scattering Inhomogeneous Medium Using MAP-Based Estimation

A maximum a posteriori (MAP) estimation based on Bayesian framework is applied to image reconstruction of two-dimensional highly scattering inhomogeneous medium. The finite difference method (FDM) and conjugate gradient (CG) algorithm serve as the forward and inverse solving models, respectively. The generalized Gaussian Markov random field model (GGMRF) is treated as the regularization, and finally the influence of the measurement errors and initial distributions is investigated. Through the test cases, the MAP estimate algorithm is demonstrated to greatly improve the reconstruction results of the optical coefficients.

A Method for Calligraphy Writer Identification by Integrating Gabor Filter and Gaussian Markov Random Field

Extracting efiective features to describe texture is always a key problem in writer identiflcation. This paper proposes a novel method for texture feature extraction by integrating Gabor fllter and Gauss Markov Random Field (GMRF). That is to say, the handwriting images are flrstly flltered by a bank of Gabor fllters in which global features such as directional information can be detected; then GMRF models are developed for every flltered image that is ∞exible enough to capture the local spatial structure and the model parameters of all GMRFs are concatenated as texture features for writer identiflcation, and flnally, Support Vector Machine (SVM) is applied to evaluate the performance of calligraphic artist identiflcation. The experimental results show that this method can achieve a classiflcation rate of 98%, which has outperformed the traditional Gabor fllter method.

Spatio-temporal regression on compositional covariates: modeling vegetation in a gypsum outcrop

Investigating the relationship between vegetation cover and substrate typologies is important for habitat conservation. To study these relationships, common practice in modern ecological surveys is to collect information regarding vegetation cover and substrate typology over fine regular lattices, as derived from digital ground photos. Information on substrate typologies is often available as compositional measures, e.g., the area proportion occupied by a certain substrate. Two primary issues are of interest for ecologists: first, how much substrate typologies differ in terms of relative suitability for vegetation cover and, second, whether suitability varies over time. This paper develops a procedure for managing compositional covariates within a Bayesian hierarchical framework to effectively address the aforementioned issues. A spatio-temporal model is adopted to estimate the temporal pattern characterizing substrate relative suitability for vegetation cover and, at the same time, to account for spatio-temporal correlation. Relative suitability is modeled by time-varying regression coefficients, and spatial, temporal and spatio-temporal random effects are modeled using Gaussian Markov Random Field models.

SAR Image Categorization Using Parametric and Nonparametric Approaches Within a Dual Tree CWT

This letter presents synthetic aperture radar (SAR) image classification based on feature descriptors within the discrete wavelet transform (DWT) domain using parametric and nonparametric features. The DWT enables an efficient multiresolution description of SAR images due to its geometric and stochastic features. A 2-D DWT, a real 2-D oriented dual tree wavelet transform (2-D RODTWT) and an oriented dual tree complex wavelet transform (2-D ODTCWT) were used for the estimation of subband features. First and second moments, entropy, coding gain, and fractal dimension were used for the nonparametric approach. A parametric approach considers a Gauss Markov Random Field model for feature extraction. A database with 2000 images representing 20 different classes with 100 images per class was used for classification efficiency assessment. Several SAR scenes were divided into small patches with dimension of 200 × 200 pixels. 10% and 20% of the test images per class were used during the learning stage. Supervised learning using a support vector machine was used for all experiments. The experimental results showed that the proposed methods had superior performances compared with (GLCM) and log comulants of Fourier transform. Amongst the proposed methods, the nonparametric features within oriented dual tree complex wavelet transform gave the best results for classes when categorizing SAR images.

Bayesian hierarchical models for analysing spatial point-based data at a grid level: a comparison of approaches

Spatial data are now prevalent in a wide range of fields including environmental and health science. This has led to the development of a range of approaches for analysing patterns in these data. In this paper, we compare several Bayesian hierarchical models for analysing point-based data based on the discretization of the study region, resulting in grid-based spatial data. The approaches considered include two parametric models and a semiparametric model. We highlight the methodology and computation for each approach. Two simulation studies are undertaken to compare the performance of these models for various structures of simulated point-based data which resemble environmental data. A case study of a real dataset is also conducted to demonstrate a practical application of the modelling approaches. Goodness-of-fit statistics are computed to compare estimates of the intensity functions. The deviance information criterion is also considered as an alternative model evaluation criterion. The results suggest that the adaptive Gaussian Markov random field model performs well for highly sparse point-based data where there are large variations or clustering across the space; whereas the discretized log Gaussian Cox process produces good fit in dense and clustered point-based data. One should generally consider the nature and structure of the point-based data in order to choose the appropriate method in modelling a discretized spatial point-based data.

Despeckling and Information Extraction From SLC SAR Images

This paper presents an information extraction and image enhancement technique using single-look complex (SLC) synthetic aperture radar data. The novelty of this method is the proposed complex-domain despeckling stage. Tikhonov-like optimization is used for minimizing the cost function, which consists of a Gauss-Markov random field (GMRF) prior. The GMRF model is used for texture modeling. The texture parameters of the GMRF are estimated using the evidence maximization framework. The experimental results showed that despeckled SLC images have well-preserved textural features, structures, and point scatterers. The phase of the reconstructed image is well preserved and provides good-quality interferograms of high-resolution spotlight images.

Diffusion Adaptation Strategies for Distributed Estimation Over Gaussian Markov Random Fields

The aim of this paper is to propose diffusion strategies for distributed estimation over adaptive networks, assuming the presence of spatially correlated measurements distributed according to a Gaussian Markov random field (GMRF) model. The proposed methods incorporate prior information about the statistical dependency among observations, while at the same time processing data in real time and in a fully decentralized manner. A detailed mean-square analysis is carried out in order to prove stability and evaluate the steady-state performance of the proposed strategies. Finally, we also illustrate how the proposed techniques can be easily extended in order to incorporate thresholding operators for sparsity recovery applications. Numerical results show the potential advantages of using such techniques for distributed learning in adaptive networks deployed over GMRF.

Controlling for localised spatio-temporal autocorrelation in long-term air pollution and health studies

Estimating the long-term health impact of air pollution using an ecological spatio-temporal study design is a challenging task, due to the presence of residual spatio-temporal autocorrelation in the health counts after adjusting for the covariate effects. This autocorrelation is commonly modelled by a set of random effects represented by a Gaussian Markov random field (GMRF) prior distribution, as part of a hierarchical Bayesian model. However, GMRF models typically assume the random effects are globally smooth in space and time, and thus are likely to be collinear to any spatially and temporally smooth covariates such as air pollution. Such collinearity leads to poor estimation performance of the estimated fixed effects, and motivated by this epidemiological problem, this paper proposes new GMRF methodology to allow for localised spatio-temporal smoothing. This means random effects that are either geographically or temporally adjacent are allowed to be autocorrelated or conditionally independent, which allows more flexible autocorrelation structures to be represented. This increased flexibility results in improved fixed effects estimation compared with global smoothing models, which is evidenced by our simulation study. The methodology is then applied to the motivating study investigating the long-term effects of air pollution on respiratory ill health in Greater Glasgow, Scotland between 2007 and 2011.

Texture characterization, representation, description, and classification based on full range Gaussian Markov random field model with Bayesian approach

A statistical approach, based on full range Gaussian Markov random field model, is proposed for texture analysis such as texture characterization, unique representation, description, and classification. The parameters of the model are estimated based on the Bayesian approach. The estimated parameters are utilized to compute autocorrelation coefficients. The computed autocorrelation coefficients fall in between –1 and +1. The coefficients are converted into decimal numbers using a simple transformation. Based on the decimal numbers, two texture descriptors are proposed: (i) texnum, the local descriptor; (ii) texspectrum, the global descriptor. The decimal numbers are proposed to represent the textures present in a small image region. These numbers uniquely represent the texture primitives. The textured image under analysis is represented globally by observing the frequency of occurrences of the texnums called texspectrum. The textures are identified and are distinguished from untextured regions with edges. The classification analyses such as supervised and unsupervised are performed on the local descriptors.

A unified Bayesian hierarchical model for MRI tissue classification.

Various works have used magnetic resonance imaging (MRI) tissue classification extensively to study a number of neurological and psychiatric disorders. Various noise characteristics and other artifacts make this classification a challenging task. Instead of splitting the procedure into different steps, we extend a previous work to develop a unified Bayesian hierarchical model, which addresses both the partial volume effect and intensity non-uniformity, the two major acquisition artifacts, simultaneously. We adopted a normal mixture model with the means and variances depending on the tissue types of voxels to model the observed intensity values. We modeled the relationship among the components of the index vector of tissue types by a hidden Markov model, which captures the spatial similarity of voxels. Furthermore, we addressed the partial volume effect by construction of a higher resolution image in which each voxel is divided into subvoxels. Finally, We achieved the bias field correction by using a Gaussian Markov random field model with a band precision matrix designed in light of image filtering. Sparse matrix methods and parallel computations based on conditional independence are exploited to improve the speed of the Markov chain Monte Carlo simulation. The unified model provides more accurate tissue classification results for both simulated and real data sets.

Bayesian Spatial Transformation Models with Applications in Neuroimaging Data

The aim of this article is to develop a class of spatial transformation models (STM) to spatially model the varying association between imaging measures in a three-dimensional (3D) volume (or 2D surface) and a set of covariates. The proposed STM include a varying Box-Cox transformation model for dealing with the issue of non-Gaussian distributed imaging data and a Gaussian Markov random field model for incorporating spatial smoothness of the imaging data. Posterior computation proceeds via an efficient Markov chain Monte Carlo algorithm. Simulations and real data analysis demonstrate that the STM significantly outperforms the voxel-wise linear model with Gaussian noise in recovering meaningful geometric patterns. Our STM is able to reveal important brain regions with morphological changes in children with attention deficit hyperactivity disorder.