Consistent estimation of the basic neighborhood of Markov random fields

For Markov random fields on $\mathbb{Z}^d$ with finite state space, we address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the Bayesian Information Criterion, replacing likelihood by pseudo-likelihood, is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions: the estimated basic neighborhood equals the true one eventually almost surely, not assuming any prior bound on the size of the latter. Stationarity of the Markov field is not required, and phase transition does not affect the results.Comment: Published at http://dx.doi.org/10.1214/009053605000000912 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Object Segmentation in Images using EEG Signals

This paper explores the potential of brain-computer interfaces in segmenting objects from images. Our approach is centered around designing an effective method for displaying the image parts to the users such that they generate measurable brain reactions. When an image region, specifically a block of pixels, is displayed we estimate the probability of the block containing the object of interest using a score based on EEG activity. After several such blocks are displayed, the resulting probability map is binarized and combined with the GrabCut algorithm to segment the image into object and background regions. This study shows that BCI and simple EEG analysis are useful in locating object boundaries in images.Comment: This is a preprint version prior to submission for peer-review of the paper accepted to the 22nd ACM International Conference on Multimedia (November 3-7, 2014, Orlando, Florida, USA) for the High Risk High Reward session. 10 page

Data augmentation in Rician noise model and Bayesian Diffusion Tensor Imaging

Mapping white matter tracts is an essential step towards understanding brain function. Diffusion Magnetic Resonance Imaging (dMRI) is the only noninvasive technique which can detect in vivo anisotropies in the 3-dimensional diffusion of water molecules, which correspond to nervous fibers in the living brain. In this process, spectral data from the displacement distribution of water molecules is collected by a magnetic resonance scanner. From the statistical point of view, inverting the Fourier transform from such sparse and noisy spectral measurements leads to a non-linear regression problem. Diffusion tensor imaging (DTI) is the simplest modeling approach postulating a Gaussian displacement distribution at each volume element (voxel). Typically the inference is based on a linearized log-normal regression model that can fit the spectral data at low frequencies. However such approximation fails to fit the high frequency measurements which contain information about the details of the displacement distribution but have a low signal to noise ratio. In this paper, we directly work with the Rice noise model and cover the full range of $b$-values. Using data augmentation to represent the likelihood, we reduce the non-linear regression problem to the framework of generalized linear models. Then we construct a Bayesian hierarchical model in order to perform simultaneously estimation and regularization of the tensor field. Finally the Bayesian paradigm is implemented by using Markov chain Monte Carlo.Comment: 37 pages, 3 figure

Model-based tests for simplification of lattice processes

Separable processes represent a convenient class of models for data collected on a regular rectangular lattice. Three model-based tests, for testing separability and testing axial symmetry and separability together, are presented. These are shown to be much more powerful than existing model-free tests using the sample periodogram, provided the model assumptions hold. A simulation study also suggests that these tests are not very sensitive to small departures from the assumed process

Consistent estimation of the basic neighborhood of Markov random fields

This is the published version, also available here: http://dx.doi.org/10.1214/009053605000000912.For Markov random fields on ℤd with finite state space, we address the statistical estimation of the basic neighborhood, the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given. A modification of the Bayesian Information Criterion, replacing likelihood by pseudo-likelihood, is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions: the estimated basic neighborhood equals the true one eventually almost surely, not assuming any prior bound on the size of the latter. Stationarity of the Markov field is not required, and phase transition does not affect the results

Statistical Modelling and Inference in Image Analysis

The aim of the thesis is to investigate classes of model-based approaches to statistical image analysis. We explored the properties of models and examined the problem of parameter estimation from the original image data and, in particular, from noisy versions of the the scene. We concentrated on Markov random field (MRF) models, Markov mesh random field (MMRF) models and Multi-dimensional Markov chain (MDMC) models. In Chapter 2, for the one-dimensional version of Markov random fields, we developed a recursive technique which enables us to achieve maximum likelihood estimation for the underlying parameter and to carry out the EM algorithm for parameter estimation when only noisy data are available. This technique also enables us, in just a single pass, to generate a sample from a one-dimensional Markov random field. Although, unfortunately, this technique cannot be extended to two- or multi-dimensional models, it was applied to many cases in this thesis. Since, for two-dimensional Markov random fields, the density of each row (column), conditionally on all other rows (columns) is of the form of a one-dimensional Markov random field, and since the distribution of the original image, conditionally on the noisy version of data, is still a Markov random field, the technique can be used on different forms of conditional density of one row (column). In Chapter 3, therefore, we developed the line-relaxation method for simulating MRFs and maximum line pseudo-likelihood estimation of parameter(s), and in Chapter 5, we developed a simultaneous procedure of parameter estimation and restoration, in which line pseudo-likelihood and a modified EM algorithm were used. The first part of Chapter 3 and Chapter 4 concentrate on inference for two-dimensional MRFs. We obtained a matrix expression for partition functins for general models, and a more explicit form for a multi-colour Ising model, and thus located the positions of critical points of this multi-colour model. We examined the asymptotic properties of an asymmetric, two-colour Ising model. For general models, in Chapter 4, we explored asymptotic properties under an "independence" or a "near independence" condition, and then developed the approach of maximum approximate-likelihood estimation. For three-dimensional MMRF models, in chapter 6, a generalization of Devijver's F-G-H algorithm is developed for restoration. In Chapter 7, the recursive technique was again used to introduce MDMC models, which form a natural extension of a Markov chain. By suitable choice of model parameters, textures can be generated that are similar to those simulated from MRFs, but the simulation procedure is computationally much more economical. The recursive technique also enables us to maximize the likelihood function of the model. For all three sorts of prior random field models considered in this thesis, we developed a simultaneous procedure for parameter estimation and image restoration, when only noisy data are available. The currently restored image was used, together with noisy data, in modified versions of the EM algorithm. In simulation studies, quite good results were obtained, in terms of estimation of parameters in both the original model and, particularly, in the noise model, and in terms of restoration

Unsupervised image segmentation using a telegraph parameterization of Pickard random fields

This communication presents a nonsupervised three-dimensional segmentation method based upon a discrete-level unilateral Markov field model for the labels and conditionally Gaussian densities for the observed voxels. Such models have been shown to yield numerically efficient algorithms, for segmentation and for estimation of the model parameters as well. Our contribution is twofold. First, we deal with the degeneracy of the likelihood function with respect to the parameters of the Gaussian densities, which is a well-known problem for such mixture models. We introduce a bounded penalized likelihood function that has been recently shown to provide a consistent estimator in the simpler cases of independent Gaussian mixtures. On the other hand, implementation with EM reestimation formulas remains possible with only limited changes with respect to the standard case. Second, we propose a telegraphic parametrization of the unilateral Markov field. On a theoretical level, this parametrization ensures that some important properties of the field (e.g., stationarity) do hold. On a practical level, it reduces the computational complexity of the algorithm used in the segmentation and parameter estimation stages of the procedure. In addition, it decreases the number of model parameters that must be estimated, thereby improving convergence speed and accuracy of the corresponding estimation method