We have studied two efficient sampling methods, Langevin and Hessian adapted Metropolis Hastings (MH), applied to a parameter estimation problem of the mathematical model (Lorentzian, Laplacian, Gaussian) that describes the Power Spectral Density (PSD) of a texture. The novelty brought by this paper consists in the exploration of textured images modeled by centered, stationary Gaussian fields using directional stochastic sampling methods. Our main contribution is the study of the behavior of the previously mentioned two samplers and the improvement of the Hessian MH method by using the Fisher information matrix instead of the Hessian to increase the stability of the algorithm and the computational speed.The directional methods yield superior performances as compared to the more popular Independent and standard Random Walk MH for the PSD described by the three models, but can easily be adapted to any target law respecting the differentiability constraint. The Fisher MH produces the best results as it combines the advantages of the Hessian, i.e., approaches the most probable regions of the target in a single iteration, and of the Langevin MH, as it requires only first order derivative computations.
Piecewise constant denoising can be solved either by deterministic optimization approaches, based on the Potts model, or by stochastic Bayesian procedures. The former lead to low computational time but require the selection of a regularization parameter, whose value significantly impacts the achieved solution, and whose automated selection remains an involved and challenging problem. Conversely, fully Bayesian formalisms encapsulate the regularization parameter selection into hierarchical models, at the price of high computational costs. This contribution proposes an operational strategy that combines hierarchical Bayesian and Potts model formulations, with the double aim of automatically tuning the regularization parameter and of maintaining computational efficiency. The proposed procedure relies on formally connecting a Bayesian framework to a 2-Potts functional. Behaviors and performance for the proposed piecewise constant denoising and regularization parameter tuning techniques are studied qualitatively and assessed quantitatively, and shown to compare favorably against those of a fully Bayesian hierarchical procedure, both in accuracy and in computational load. * J. Frecon, N. Pustelnik (Corresponding author) and P. Abry are with
Bayesian method for texture model choice from blurred and noisy (i.e., indirect) observations is presented. The textures are modeled by stationary Random Fields, with various distribution laws, either Gaussian or Scale Mixtures of Gaussians. The power spectral densities of the fields are modeled by parametric functions and the aim is to select the most appropriate model among a set of candidates. This is achieved by computing the a posteriori model probabilities through parameter marginalization. The marginalization is done by sampling and harmonic mean approach, considering separately each model, in a within-model sampling strategy. The highly nonlinear dependency with respect to the parameters imposes the use of the Metropolis-Hastings sampler. Moreover, to achieve efficient sampling, the paper proposes a new fast algorithm based on the Fisher information matrix, the Fisher Metropolis-Hastings.
The paper tackles the problem of joint deconvolution and segmentation of textured images. The images are composed of regions containing a patch of texture that belongs to a set of K possible classes. Each class is described by a Gaussian random field with parametric power spectral density whose parameters are unknown. The class labels are modelled by a Potts field driven by a granularity coefficient that is also unknown. The method relies on a hierarchical model and a Bayesian strategy to jointly estimate the labels, the K textured images in addition to hyperparameters: the signal and the noise levels as well as the texture parameters and the granularity coefficient. The capability to estimate the latter is an important feature of the paper. The estimates are designed in an optimal manner as a risk minimizer that yields the marginal posterior maximizer for the labels and the posterior mean for the rest of the unknowns. They are computed based on a convergent procedure from samples of the posterior obtained through an advanced MCMC algorithm: Perturbation-Optimization step and Fisher Metropolis-Hastings step within a Gibbs loop. Various numerical evaluations provide encouraging results despite the strong difficulty of the problem.
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