Estimating the Granularity Coefficient of a Potts-Markov Random Field Within a Markov Chain Monte Carlo Algorithm

Abstract

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 likelihood-free 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, assuming that the value of β is known can degrade estimation performance significantly if this value is incorrect. To illustrate the interest of this method, the proposed algorithm is successfully applied to real bidimensional SAR and tridimensional ultrasound images.

Local Autoencoding for Parameter Estimation in a Hidden Potts-Markov Random Field

“…Recently, Pereyra et al [10], [27] proposed a likelihoodfree method based on the approximative Bayesian computation [28] and the likelihood-free Metropolis-Hastings sampling [17], [29]. Instead of the gradient descent rules that are derived from the likelihood or pseudo-likelihood function, it jointly estimates the label, the prior parameter and other auxiliary variables by the hybrid Gibbs sampling.…”

“…In a Potts field, each unit can be in a number of Q states (classes). Statistically, when the local correlation is assumed to be Markovian, the configuration space can be modeled by a Gibbs distribution which is characterised by a vector of parameters β (also known as interaction parameters [5], spatial cohesion strength [9], granularity coefficient [10] or prior parameters [11]). However, with the notable exception of the special case of an isotropic Ising-MRF with circular boundary conditions [12], a direct estimation of the prior parameters is not possible, because a part of the model, the partition function, is computationally intractable [10].…”

“…Instead, two types of approximation strategies are used to find optimal parameters, namely sampling methods based on Monte-Carlo sampling, such as the Markov chain Monte-Carlo (MCMC) algorithm [14], [15], MCMC Maximum Likelihood (MCMCML) [16], and likelihood-free methods [10], [17], and heuristic methods based on the conditional independence assumption, e.g. pseudo-likelihood maximization (MPL) [18], [19], coding-based MPL [20], mean-field approximation [21], [22], least-square (LSQR) algorithm [5], [23].…”

“…They may be more precise, but are computationally more expensive [10], [16]. In engineering applications, heuristic methods may appear thus more favorable.…”

Deconvolution-segmentation for textured images

“…However, based on stochastic simulation, we have precomputed it for several image sizes and class numbers [27,28]. The reader is invited to consult papers such as [29,30] for alternatives.…”