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Estimating the Granularity Coefficient of a PottsMarkov Random Field within a Markov Chain Monte Carlo Algorithm
, 2013
"... This paper addresses the problem of estimating the Potts parameter β jointly with the unknown parameters of a Bayesian model within a Markov chain Monte Carlo (MCMC) algorithm. Standard MCMC methods cannot be applied to this problem because performing inference on β requires computing the intracta ..."
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Cited by 7 (2 self)
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This paper addresses the problem of estimating the Potts parameter β jointly with the unknown parameters of a Bayesian model within a Markov chain Monte Carlo (MCMC) algorithm. Standard MCMC methods cannot be applied to this problem because performing inference on β requires computing the intractable normalizing constant of the Potts model. In the proposed MCMC method, the estimation of β is conducted using a likelihoodfree Metropolis–Hastings algorithm. Experimental results obtained for synthetic data show that estimating β jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of β. On the other hand, choosing an incorrect value of β can degrade estimation performance significantly. To illustrate the interest of this method, the proposed algorithm is successfully applied to real bidimensional SAR and tridimensional ultrasound images.
Sampling highdimensional Gaussian distributions for general linear inverse problems
, 2013
"... Abstract—This paper is devoted to the problem of sampling Gaussian distributions in high dimension. Solutions exist for two specific structures of inverse covariance: sparse and circulant. The proposed algorithm is valid in a more general case especially as it emerges in linear inverse problems as w ..."
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Abstract—This paper is devoted to the problem of sampling Gaussian distributions in high dimension. Solutions exist for two specific structures of inverse covariance: sparse and circulant. The proposed algorithm is valid in a more general case especially as it emerges in linear inverse problems as well as in some hierarchical or latent Gaussian models. It relies on a perturbationoptimization principle: adequate stochastic perturbation of a criterion and optimization of the perturbed criterion. It is proved that the criterion optimizer is a sample of the target distribution. The main motivation is in inverse problems related to general (nonconvolutive) linear observation models and their solution in a Bayesian framework implemented through sampling algorithms when existing samplers are infeasible. It finds a direct application in myopic / unsupervised inversion methods as well as in some nonGaussian inversion methods. An illustration focused on hyperparameter estimation for superresolution method shows the interest and the feasibility of the proposed algorithm. Index Terms—Stochastic sampling, highdimensional sampling, inverse problem, Bayesian strategy, unsupervised, myopic I.
Article Entropy, Information Theory, Information Geometry and Bayesian Inference in Data, Signal and Image Processing and Inverse Problems †
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BAYESIAN INFERENCE WITH HIERARCHICAL PRIOR MODELS FOR INVERSE PROBLEMS IN IMAGING SYSTEMS
"... Bayesian approach is nowadays commonly used for inverse problems. Simple prior laws (Gaussian, Generalized Gaussian, GaussMarkov and more general Markovian priors) are common in modeling and in their use in Bayesian inference methods. But, we need still more appropriate prior models which can accou ..."
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Bayesian approach is nowadays commonly used for inverse problems. Simple prior laws (Gaussian, Generalized Gaussian, GaussMarkov and more general Markovian priors) are common in modeling and in their use in Bayesian inference methods. But, we need still more appropriate prior models which can account for non stationnarities in signals and for the presence of the contours and homogeneous regions in images. Recently, we proposed a family of hierarchical prior models, called GaussMarkovPotts, which seems to be more appropriate for many applications in Imaging systems such as X ray Computed Tomography (CT) or Microwave imaging in Non Destructive Testing (NDT). In this tutorial paper, first some bacgrounds on the Bayesian inference and the tools for assignment of priors and doing efficiently the Bayesian computation is presented. Then, more specifically hiearachical models and particularly the GaussMarkovPotts family of prior models are presented. Finally, their real applications in image restoration, in different practical Computed Tomography (CT) or other imaging systems are presented. 1.
REVIEW Open Access
"... Bayesian approach with prior models which enforce sparsity in signal and image processing Ali MohammadDjafari In this review article, we propose to use the Bayesian inference approach for inverse problems in signal and image processing, where we want to infer on sparse signals or images. The sparsi ..."
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Bayesian approach with prior models which enforce sparsity in signal and image processing Ali MohammadDjafari In this review article, we propose to use the Bayesian inference approach for inverse problems in signal and image processing, where we want to infer on sparse signals or images. The sparsity may be directly on the original space or in a transformed space. Here, we consider it directly on the original space (impulsive signals). To enforce the sparsity, we consider the probabilistic models and try to give an exhaustive list of such prior models and try to classify them. These models are either heavy tailed (generalized Gaussian, symmetric Weibull, Studentt or Cauchy, elastic net, generalized hyperbolic and Dirichlet) or mixture models (mixture of Gaussians, BernoulliGaussian, BernoulliGamma, mixture of translated Gaussians, mixture of multinomial, etc.). Depending on the prior model selected, the Bayesian computations (optimization for the joint maximum a posteriori (MAP) estimate or MCMC or variational Bayes approximations (VBA) for posterior means (PM) or complete density estimation) may become more complex. We propose these models, discuss on different possible Bayesian estimators, drive the corresponding appropriate algorithms, and discuss on their corresponding relative complexities and performances.
TOMOGRAPHIC IMAGE RECONSTRUCTIONWITH A SPATIALLY VARYING
"... A spatially varying Gaussian mixture model (SVGMM) prior is employed to ensure the preservation of region boundaries in penalized likelihood tomographic image reconstruction. Spatially varying Gaussian mixture models are characterized by the dependence of their mixing proportions on location (contex ..."
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A spatially varying Gaussian mixture model (SVGMM) prior is employed to ensure the preservation of region boundaries in penalized likelihood tomographic image reconstruction. Spatially varying Gaussian mixture models are characterized by the dependence of their mixing proportions on location (contextual mixing proportions) and they have been successfully used in image segmentation. The proposed model imposes a Student’s tdistribution on the local differences of the contextual mixing proportions and its parameters are automatically estimated by a variational ExpectationMaximization (EM) algorithm. The tomographic reconstruction algorithm is an iterative process consisting of alternating between an optimization of the SVGMM parameters and an optimization for updating the unknown image using also the EM algorithm. Numerical experiments on various photon limited image scenarios show that the proposed model is more accurate than the widely used Gibbs prior. Index Terms — Emission tomography, iterative image reconstruction, expectationmaximization (EM) algorithm, spatially varying Gaussian mixture models (GMM), Student’s tdistribution, edge preservation. 1.
Efficient sampling of highdimensional Gaussian fields: the nonstationary / nonsparse case
, 2013
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1Images as Occlusions of Textures: A Framework for Segmentation
"... Abstract—We propose a new mathematical and algorithmic framework for unsupervised image segmentation, which is a critical step in a wide variety of image processing applications. We have found that most existing segmentation methods are not successful on histopathology images, which prompted us to i ..."
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Abstract—We propose a new mathematical and algorithmic framework for unsupervised image segmentation, which is a critical step in a wide variety of image processing applications. We have found that most existing segmentation methods are not successful on histopathology images, which prompted us to investigate segmentation of a broader class of images, namely those without clear edges between the regions to be segmented. We model these images as occlusions of random images, which we call textures, and show that local histograms are a useful tool for segmenting them. Based on our theoretical results, we describe a flexible segmentation framework that draws on existing work on nonnegative matrix factorization and image deconvolution. Results on synthetic texture mosaics and real histology images show the promise of the method. Index Terms—image segmentation, occlusion models, texture, local histograms, deconvolution, nonnegative matrix factorization I.