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.

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.

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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.

Variational Bayesian approaches have been successfully applied to image segmentation. They usually rely on a Potts model for the hidden label variables and a Gaussian assumption on pixel intensities within a given class. Such models may however be limited, especially in the case of multicomponent images. We overcome this limitation with HOGMep, a Bayesian formulation based on a higher-order graphical model (HOGM) on labels and a Multivariate Exponential Power (MEP) prior for intensities in a class. Then, we develop an efficient statistical estimation method to solve the associated problem. Its flexibility accommodates to a broad range of applications, demonstrated on multicomponent image segmentation and restoration.

Bioequivalence (BE) studies are an integral component of new drug development process, and play an important role in approval and marketing of generic drug products. However, existing design and evaluation methods are basically under the framework of frequentist theory, while few implements Bayesian ideas. Based on the bioequivalence predictive probability model and sample re-estimation strategy, we propose a new Bayesian two-stage adaptive design and explore its application in bioequivalence testing. The new design differs from existing two-stage design (such as Potvin’s method B, C) in the following aspects. First, it not only incorporates historical information and expert information, but further combines experimental data flexibly to aid decision-making. Secondly, its sample re-estimation strategy is based on the ratio of the information in interim analysis to total information, which is simpler in calculation than the Potvin’s method. Simulation results manifested that the two-stage design can be combined with various stop boundary functions, and the results are different. Moreover, the proposed method saves sample size compared to the Potvin’s method under the conditions that type I error rate is below 0.05 and statistical power reaches 80 %.