Yuling Zheng

Efficient Variational Bayesian Approximation Method Based on Subspace Optimization

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.

Wavelet-based Image Deconvolution and Reconstruction

Image deconvolution and reconstruction are inverse problems which are encountered in a wide array of applications. Due to the ill-posedness of such problems, their resolution generally relies on the incorporation of prior information through regularizations, which may be formulated in the original data space or through a suitable linear representation. In this article, we show the benefits which can be drawn from frame representations, such as wavelet transforms. We present an overview of recovery methods based on these representations: (i) variational formulations and non-smooth convex optimization strategies, (ii) Bayesian approaches, especially Monte Carlo Markov Chain methods and variational Bayesian approximation techniques, and (iii) Stein-based approaches. A brief introduction to blind deconvolution is also provided.

Fast variational Bayesian signal recovery in the presence of Poisson-Gaussian noise

This paper presents a new method for solving linear inverse problems where the observations are corrupted with a mixed Poisson-Gaussian noise. To generate a reliable solution, a regularized approach is often adopted in the literature. In this context, the optimal selection of the regularization parameters is of crucial importance in terms of estimation performance. The variational Bayesian-based approach we propose in this work allows us to automatically estimate the original signal and the associated regularization parameter from the observed data. A majorization-minimization technique is employed to circumvent the difficulties raised by the intricate form of the Poisson-Gaussian likelihood. Experimental results show that the proposed method is fast and achieves state-of-the art performance in comparison with approaches where the regularization parameters are manually adjusted.

A Variational Bayesian Approach for Image Restoration—Application to Image Deblurring With Poisson–Gaussian Noise

In this paper, a methodology is investigated for signal recovery in the presence of non-Gaussian noise. In contrast with regularized minimization approaches often adopted in the literature, in our algorithm the regularization parameter is reliably estimated from the observations. As the posterior density of the unknown parameters is analytically intractable, the estimation problem is derived in a variational Bayesian framework where the goal is to provide a good approximation to the posterior distribution in order to compute posterior mean estimates. Moreover, a majorization technique is employed to circumvent the difficulties raised by the intricate forms of the non-Gaussian likelihood and of the prior density. We demonstrate the potential of the proposed approach through comparisons with state-of-the-art techniques that are specifically tailored to signal recovery in the presence of mixed Poisson–Gaussian noise. Results show that the proposed approach is efficient and achieves performance comparable with other methods where the regularization parameter is manually tuned from the ground truth.

Wavelet based unsupervised variational Bayesian image reconstruction approach

In this paper, we present a variational Bayesian approach in the wavelet domain for linear image reconstruction problems. This approach is based on a Gaussian Scale Mixture prior and an improved variational Bayesian approximation method. Its main advantages are that it is unsupervised and can be used to solve various linear inverse problems. We show the good performance of our approach through comparisons with state of the art approaches on a deconvolution problem.

Fast variational Bayesian approaches applied to large dimensional problems

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.

A subspace-based variational Bayesian method

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.