Accuracy of MAP segmentation with hidden Potts and Markov mesh prior models via Path Constrained Viterbi Training, Iterated Conditional Modes and Graph Cut based algorithms

In this paper, we study statistical classification accuracy of two different Markov field environments for pixelwise image segmentation, considering the labels of the image as hidden states and solving the estimation of such labels as a solution of the MAP equation. The emission distribution is assumed the same in all models, and the difference lays in the Markovian prior hypothesis made over the labeling random field. The a priori labeling knowledge will be modeled with a) a second order anisotropic Markov Mesh and b) a classical isotropic Potts model. Under such models, we will consider three different segmentation procedures, 2D Path Constrained Viterbi training for the Hidden Markov Mesh, a Graph Cut based segmentation for the first order isotropic Potts model, and ICM (Iterated Conditional Modes) for the second order isotropic Potts model. We provide a unified view of all three methods, and investigate goodness of fit for classification, studying the influence of parameter estimation, computational gain, and extent of automation in the statistical measures Overall Accuracy, Relative Improvement and Kappa coefficient, allowing robust and accurate statistical analysis on synthetic and real-life experimental data coming from the field of Dental Diagnostic Radiography. All algorithms, using the learned parameters, generate good segmentations with little interaction when the images have a clear multimodal histogram. Suboptimal learning proves to be frail in the case of non-distinctive modes, which limits the complexity of usable models, and hence the achievable error rate as well. All Matlab code written is provided in a toolbox available for download from our website, following the Reproducible Research Paradigm

A Bayesian Image Analysis of the Change in Tumor/Brain Contrast Uptake Induced by Radiation via Reversible Jump Markov Chain Monte Carlo

This work is motivated by a pilot study on the change in tumor/brain contrast uptake induced by radiation via quantitative Magnetic Resonance Imaging. The results inform the optimal timing of administering chemotherapy in the context of radiotherapy. A noticeable feature of the data is spatial heterogeneity. The tumor is physiologically and pathologically distinct from surrounding healthy tissue. Also, the tumor itself is usually highly heterogeneous. We employ a Gaussian Hidden Markov Random Field model that respects the above features. The model introduces a latent layer of discrete labels from an Markov Random Field (MRF) governed by a spatial regularization parameter. We further assume that conditional on the hidden labels, the observed data are independent and normally distributed, We treat the regularization parameter of the MRF, as well as the number of states of the MRF as parameters, and estimate them via the Reversible Jump Markov chain Monte Carlo algorithm. We propose a novel and nontrivial implementation of the Reversible Jump moves. The performance of the method is examined in both simulation studies and real data analysis. We report the pixel-wise posterior mean and standard deviation of the change in contrast uptake marginalized over the number of states and hidden labels. We also compare the performance with a Markov chain with fixed number of states and a parallel Expectation-Maximization approach from a frequentist perspective

Combining spatial priors and anatomical information for fMRI detection

In this paper, we analyze Markov Random Field (MRF) as a spatial regularizer in fMRI detection. The low signal-to-noise ratio (SNR) in fMRI images presents a serious challenge for detection algorithms, making regularization necessary to achieve good detection accuracy. Gaussian smoothing, traditionally employed to boost SNR, often produces over-smoothed activation maps. Recently, the use of MRF priors has been suggested as an alternative regularization approach. However, solving for an optimal configuration of the MRF is NP-hard in general. In this work, we investigate fast inference algorithms based on the Mean Field approximation in application to MRF priors for fMRI detection. Furthermore, we propose a novel way to incorporate anatomical information into the MRF-based detection framework and into the traditional smoothing methods. Intuitively speaking, the anatomical evidence increases the likelihood of activation in the gray matter and improves spatial coherency of the resulting activation maps within each tissue type. Validation using the receiver operating characteristic (ROC) analysis and the confusion matrix analysis on simulated data illustrates substantial improvement in detection accuracy using the anatomically guided MRF spatial regularizer. We further demonstrate the potential benefits of the proposed method in real fMRI signals of reduced length. The anatomically guided MRF regularizer enables significant reduction of the scan length while maintaining the quality of the resulting activation maps.National Institutes of Health (U.S.) (National Institute for Biomedical Imaging and Bioengineering (U.S.)/National Alliance for Medical Image Computing (U.S.) Grant U54-EB005149)National Science Foundation (U.S.) (Grant IIS 9610249)National Institutes of Health (U.S.) (National Center for Research Resources (U.S.)/Biomedical Informatics Research Network Grant U24-RR021382)National Institutes of Health (U.S.) (National Center for Research Resources (U.S.)/Neuroimaging Analysis Center (U.S.) Grant P41-RR13218)National Institutes of Health (U.S.) (National Institute of Neurological Disorders and Stroke (U.S.) Grant R01-NS051826)National Science Foundation (U.S.) (CAREER Grant 0642971)National Science Foundation (U.S.). Graduate Research FellowshipNational Center for Research Resources (U.S.) (FIRST-BIRN Grant)Neuroimaging Analysis Center (U.S.

Doctor of Philosophy

dissertationFunctional magnetic resonance imaging (fMRI) measures the change of oxygen consumption level in the blood vessels of the human brain, hence indirectly detecting the neuronal activity. Resting-state fMRI (rs-fMRI) is used to identify the intrinsic functional patterns of the brain when there is no external stimulus. Accurate estimation of intrinsic activity is important for understanding the functional organization and dynamics of the brain, as well as differences in the functional networks of patients with mental disorders. This dissertation aims to robustly estimate the functional connectivities and networks of the human brain using rs-fMRI data of multiple subjects. We use Markov random field (MRF), an undirected graphical model to represent the statistical dependency among the functional network variables. Graphical models describe multivariate probability distributions that can be factorized and represented by a graph. By defining the nodes and the edges along with their weights according to our assumptions, we build soft constraints into the graph structure as prior information. We explore various approximate optimization methods including variational Bayesian, graph cuts, and Markov chain Monte Carlo sampling (MCMC). We develop the random field models to solve three related problems. In the first problem, the goal is to detect the pairwise connectivity between gray matter voxels in a rs-fMRI dataset of the single subject. We define a six-dimensional graph to represent our prior information that two voxels are more likely to be connected if their spatial neighbors are connected. The posterior mean of the connectivity variables are estimated by variational inference, also known as mean field theory in statistical physics. The proposed method proves to outperform the standard spatial smoothing and is able to detect finer patterns of brain activity. Our second work aims to identify multiple functional systems. We define a Potts model, a special case of MRF, on the network label variables, and define von Mises-Fisher distribution on the normalized fMRI signal. The inference is significantly more difficult than the binary classification in the previous problem. We use MCMC to draw samples from the posterior distribution of network labels. In the third application, we extend the graphical model to the multiple subject scenario. By building a graph including the network labels of both a group map and the subject label maps, we define a hierarchical model that has richer structure than the flat single-subject model, and captures the shared patterns as well as the variation among the subjects. All three solutions are data-driven Bayesian methods, which estimate model parameters from the data. The experiments show that by the regularization of MRF, the functional network maps we estimate are more accurate and more consistent across multiple sessions

Bayesian Parameter Estimation for Latent Markov Random Fields and Social Networks

Undirected graphical models are widely used in statistics, physics and machine vision. However Bayesian parameter estimation for undirected models is extremely challenging, since evaluation of the posterior typically involves the calculation of an intractable normalising constant. This problem has received much attention, but very little of this has focussed on the important practical case where the data consists of noisy or incomplete observations of the underlying hidden structure. This paper specifically addresses this problem, comparing two alternative methodologies. In the first of these approaches particle Markov chain Monte Carlo (Andrieu et al., 2010) is used to efficiently explore the parameter space, combined with the exchange algorithm (Murray et al., 2006) for avoiding the calculation of the intractable normalising constant (a proof showing that this combination targets the correct distribution in found in a supplementary appendix online). This approach is compared with approximate Bayesian computation (Pritchard et al., 1999). Applications to estimating the parameters of Ising models and exponential random graphs from noisy data are presented. Each algorithm used in the paper targets an approximation to the true posterior due to the use of MCMC to simulate from the latent graphical model, in lieu of being able to do this exactly in general. The supplementary appendix also describes the nature of the resulting approximation.Comment: 26 pages, 2 figures, accepted in Journal of Computational and Graphical Statistics (http://www.amstat.org/publications/jcgs.cfm

Learning from Complex Systems: On the Roles of Entropy and Fisher Information in Pairwise Isotropic Gaussian Markov Random Fields

Markov Random Field models are powerful tools for the study of complex systems. However, little is known about how the interactions between the elements of such systems are encoded, especially from an information-theoretic perspective. In this paper, our goal is to enlight the connection between Fisher information, Shannon entropy, information geometry and the behavior of complex systems modeled by isotropic pairwise Gaussian Markov random fields. We propose analytical expressions to compute local and global versions of these measures using Besag's pseudo-likelihood function, characterizing the system's behavior through its \emph{Fisher curve}, a parametric trajectory accross the information space that provides a geometric representation for the study of complex systems. Computational experiments show how the proposed tools can be useful in extrating relevant information from complex patterns. The obtained results quantify and support our main conclusion, which is: in terms of information, moving towards higher entropy states (A --> B) is different from moving towards lower entropy states (B --> A), since the \emph{Fisher curves} are not the same given a natural orientation (the direction of time).Comment: 46 pages, 16 Figure

MRF Stereo Matching with Statistical Estimation of Parameters

For about the last ten years, stereo matching in computer vision has been treated as a combinatorial optimization problem. Assuming that the points in stereo images form a Markov Random Field (MRF), a variety of combinatorial optimization algorithms has been developed to optimize their underlying cost functions. In many of these algorithms, the MRF parameters of the cost functions have often been manually tuned or heuristically determined for achieving good performance results. Recently, several algorithms for statistical, hence, automatic estimation of the parameters have been published. Overall, these algorithms perform well in labeling, but they lack in performance for handling discontinuity in labeling along the surface borders. In this dissertation, we develop an algorithm for optimization of the cost function with automatic estimation of the MRF parameters – the data and smoothness parameters. Both the parameters are estimated statistically and applied in the cost function with support of adaptive neighborhood defined based on color similarity. With the proposed algorithm, discontinuity handling with higher consistency than of the existing algorithms is achieved along surface borders. The data parameters are pre-estimated from one of the stereo images by applying a hypothesis, called noise equivalence hypothesis, to eliminate interdependency between the estimations of the data and smoothness parameters. The smoothness parameters are estimated applying a combination of maximum likelihood and disparity gradient constraint, to eliminate nested inference for the estimation. The parameters for handling discontinuities in data and smoothness are defined statistically as well. We model cost functions to match the images symmetrically for improved matching performance and also to detect occlusions. Finally, we fill the occlusions in the disparity map by applying several existing and proposed algorithms and show that our best proposed segmentation based least squares algorithm performs better than the existing algorithms. We conduct experiments with the proposed algorithm on publicly available ground truth test datasets provided by the Middlebury College. Experiments show that results better than the existing algorithms’ are delivered by the proposed algorithm having the MRF parameters estimated automatically. In addition, applying the parameter estimation technique in existing stereo matching algorithm, we observe significant improvement in computational time