This paper considers the unsupervised filtering problem for large-dimensional linear and Gaussian systems, a setup in which the optimal Kalman filter (KF) might not be usable due to the exorbitant computational cost and storage requirements. For this problem, we propose two efficient algorithms based on the variational Bayesian (VB) approach. The first algorithm is an extension of a recent VBKF algorithm to the unsupervised framework, whereas the second algorithm is an accelerated version of the first one, derived based on the adaptation of subspace optimization methods in Hilbert spaces into the space of probability density functions. Furthermore, both algorithms account for the sparsity in the state and observations through heavy-tailed Student-t priors. Results of numerical experiments conducted on a dynamical tomography problem to assess the performances of the proposed schemes are presented.

In many applications, such as Eddy-Current Testing (ECT), we are often interested in the joint model choice and parameter estimation. Nested Sampling (NS) is one of the possible methods. The key step that reflects the efficiency of the NS algorithm is how to get samples with hard constraint on the likelihood value. This contribution is based on the classical idea where the new sample is drawn within a hyper-ellipsoid, the latter being located from Gaussian approximation. This sampling strategy can automatically guarantee the hard constraint on the likelihood. Meanwhile, it shows the best sampling efficiency for models which have Gaussian-like likelihood distributions. We apply this method in ECT. The simulation results show that this method has high model choice ability and good parameter estimation accuracy, and low computational cost meanwhile.

Variational Bayesian approximations have been widely used in fully Bayesian inference for approximating an intractable posterior distribution by a separable one. Nevertheless, the classical variational Bayesian approximation (VBA) method suffers from slow convergence to the approximate solution when tackling large dimensional problems. To address this problem, we propose in this paper a more efficient VBA method. Actually, variational Bayesian issue can be seen as a functional optimization problem. The proposed method is based on the adaptation of subspace optimization methods in Hilbert spaces to the involved function space, in order to solve this optimization problem in an iterative way. The aim is to determine an optimal direction at each iteration in order to get a more efficient method. We highlight the efficiency of our new VBA method and demonstrate its application to image processing by considering an ill-posed linear inverse problem using a total variation prior. Comparisons with state of the art variational Bayesian methods through a numerical example show a notable improvement in computation time.

In this paper we provide an algorithm adapted to variational Bayesian approximation. The main contribution is to transpose a classical iterative algorithm of optimization in the metric space of measures involved in Bayesian methodology. Once given the convergence properties of this algorithm, we consider its application to large dimensional inverse problems, especially for unsupervised reconstruction. The interest of our method is enhanced by its application to large dimensional linear inverse problems involving sparse objects. Finally, we provide simulation results. First we show the good numerical performances of our method compared to classical ones on a small example. Then we deal with a large dimensional dictionary learning problem.

In this work, a new unsupervised Bayesian method for joint image super-resolution and component separation is introduced. More precisely, we are interested in super-resolution for astrophysical map-making and separation between extended and point emissions. This is tackled as an inverse problem in a Bayesian framework, where a Markovian model is used as a prior for the extended emission and a student's t-distribution is attributed for the point sources component. All model and noise parameters are unknown, therefore we chose to estimate them jointly with the images. Nevertheless, both Joint Maximum A Posteriori (JMAP) and Posterior Mean (PM) estimators are intractable. Hence, we propose to approximate the true posterior by free-form separable distribution using a gradient-like variational Bayesian approach, which allows a joint update of the shape parameters of the approximating marginals. Applications on simulated and real datasets, obtained from Herschel space observatory, show a good quality of reconstruction. Furthermore, compared to conventional methods, our method gives a higher resolution while conserving photometery and reducing noise.

This paper is devoted to an improved variational Bayesian method. Actually, variational Bayesian issue can be seen as a convex functional optimization problem. Our main contribution is the adaptation of subspace optimization methods into the functional space involved in this problem. We highlight the efficiency of our methodology on a linear inverse problem with a sparse prior. Comparisons with classical Bayesian methods through a numerical example show the notable improved computation time.

This paper introduces two unsupervised approaches for large dimensional ill-posed inverse problems. These approaches are based on improved variational Bayesian (VB) methodologies, where a functional optimization problem is involved. We propose to solve this problem by adapting the subspace optimization methods into the functional space. The application of these approaches to image processing problems is considered thanks to a TV prior. We highlight the efficiency of our approaches through comparisons with a classical VB based one on a super-resolution problem.

This paper focuses on the performance of a Wyner-Ziv coding scheme for which the correlation between the source and the side information is modeled by a hidden Markov model with Gaussian emission. Such a signal model takes the memory of the correlation into account and is hence able to describe the bursty nature of the correlation between sources in applications such as sensor networks, video coding etc. This paper provides bounds on the rate-distortion performance of a Wyner-Ziv coding scheme for such model. It proposes a practical coding scheme able to exploit the memory in the correlation. Finally, the contribution to each part of the coding and decoding scheme is analysed.

The problem of dynamical tomography consists in reconstructing a temporal sequence of images from their noisy projections. For this purpose, a recursive algorithm is usually used, like for instance the Kalman Filter (KF), due to the dynamical structure of the problem. However, since it needs the inverse of innovation matrix, KF may suffer from a huge computational cost in cases of images with very high dimensions. To solve this issue, we develop a new suboptimal version of the KF based on a Variational Bayesian (VB) approach. The proposed Variational Bayesian KF (VBKF) algorithm is compared to the KF with simulations in a small problem (images 32×32). As expected, the image quality of VBKF is as good as the KF and the VBKF algorithm is four times faster than the KF. Furthermore, in realistic high dimensional problems in which the practical implementation of KF becomes impossible, the VBKF gives an attractive alternative since it can be millions times faster than the KF

The main contribution presented here is an adaptive/unsupervised iterative thresholding algorithm for sparse representation of signals which can be modeled as the sum of two components. Such Hybrid or Morphological representations are known to be well adapted for applications in image or audio signal processing. The proposed algorithm uses a Bernoulli-Gaussian prior on the synthesis coefficients of the signal, with morphological depending parameters. Using an EM-framework introduced by Figueiredo and Nowak in the case of the convex ℓ1prior, we derive an unsupervised algorithm in the spirit of ISTA, with iteratively adapted thresholding/shrinkage.

Metal and beam-hardening artifacts are tough issues in Computed Tomography (CT) images. This paper proposes an iterative Maximum A Posteriori (MAP) reconstruction algorithm aiming to reduce both of them. This algorithm is based on a multi-energy acquisition system, a Gaussian noised measuring model and a basis material decomposition formula. In the Multi-Energy Computed Tomography (MECT) system, an energy discriminant detector which can measure the flux of photons at different energies is supposed to be employed. Our method can reconstruct two separate base material density maps where the metal and beam-hardening artifacts are highly reduced.

In this paper, we first introduce a Multi-Energy Computed Tomography (MECT) forward projection model based on the base material decomposition method. In this method, an object is considered into a linear combination of the fractions of several base materials weighted by X-ray beam energy functions. Then, three different data fusion inversion approaches are proposed to reconstruct the base material fractions. For the first pre-separation reconstruction approach, the base material decomposition is carried on in the projection space while in the second post-separation approach, the base material decomposition is carried on the attenuation coefficients. The third approach is a joint Bayesian inversion method. Finally, the reconstruction performances of the three reconstruction methods are compared on the simulated data.

In this article, we propose to reconstruct the 3D structure of the solar atmosphere, taking into account the temporal evolution of the coronal structures. To do this we develop a new method, which considers that the object of interest consists of several sources each one having different evolution from the other at different time. The data are the Radon transform of an instantaneous mixture of sources. So, we must solve a separation source problem with a linear mixture of sources. To our best knowledge, this problem is still open. To solve this problem of source separation, we have introduced a sparsity information on the sources, and we have chosen to use the variational Bayesian approach for reduce the computation time.

This paper proposes a Bayesian approach for unsupervised image deconvolution when the parameter of the gaussian PSF is unknown. The parameters of the regularization parameters are also unknown and jointly estimated with the other parameters. The solution is found by inferring on a global a posteriori law for unknown object and parameters. The estimate is chosen in the sense of the posterior mean, numerically calculated by means of a Monte-Carlo Markov chain algorithm. The computation is efficiently done in Fourier space and the practicability of the method is shown on simulated examples. Results show high-frequencies restoration in the estimated image with correct estimation of the hyperparameters and instrument parameters.

This paper proposes a Bayesian approach for estimation of instrument parameter in convex image deconvolution. The parameters of the instrument response (PSF) are jointly estimated with the image leading to a myopic deconvolution approach. In addition a special convex field allowing efficient hyperparameter estimation is used. The solution is based on a global a posteriori law for unknown parameters and object. The estimate is chosen in the sense of the posterior mean, numerically calculated by means of a Monte-Carlo Markov chain algorithm. The computation is efficient with a partial implementation in Fourier space. Simulation results are provided to assess the effectiveness of the proposed approach.

Super-resolution methods aimed to restore the spectrum of an original image above the half sampling frequency. The restoration problem is generally viewed as an inverse problem and a lot of research focus on the inversion approach. In this paper the question of super-resolution is addressed with only the forward model point of view. With the addition of the time dimension in an specific case of continuous shift scanning we obtain explicit expression for the forward model and the data spectrum. We show with a new, simple and rigorous formalism that super-resolution can be done because the whole set of data is less aliased than a single image of the scene. Moreover, limits, conditions and performance questions are also addressed as well as perspectives on sampling conditions.