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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