A variational Bayesian approach for image restoration with Poisson-Gaussian noise

In this talk, a methodology is investigated for signal recovery in the presence of mixed Poisson-Gaussian noise [1]. Existing strategies for solving inverse problems often define the estimate as a minimizer of an appropriate cost function. Several algorithms have been proposed to tackle the problem of restoration for signals corrupted with non-Gaussian noise by using minimization approaches (see [2] and the references therein). In these approaches, the regularization parameter allows a tradeoff to be performed between fidelity to the observations and the prior information. The problem of selecting the regularization parameter remains an open issue especially in situations where the images are acquired under poor conditions i.e. when the noise level is very high. To address the shortcomings of these methods, one can adopt the Bayesian framework. In particular, Bayesian estimation methods based on Markov Chain Monte Carlo (MCMC) sampling algorithms have been recently extended to inverse problems involving non-Gaussian noise. However, despite good estimation performance that has been obtained, such methods remain computationally expensive for large scale problems. Another alternative approach which is explored here is to rely on variational Bayesian approximation (VBA). Instead of simulating from the true posterior distribution, VBA approaches aim at approximating the intractable true posterior distribution with a tractable one from which the posterior mean can be easily computed. These methods can lead generally to a relatively low computational complexity when compared with sampling based algorithms. In this work, we propose such a VBA estimation approach for signals degraded by an arbitrary linear operator and corrupted with non-Gaussian noise. One of the main advantages of the proposed method is that it allows us to jointly estimate the original signal and the required regularization parameter from the observed data by providing good approximations of the Minimum Mean Square Estimator (MMSE) for the problem of interest. While using VBA, the main difficulty arising in the non-Gaussian case is that the involved likelihood and the prior density may have a complicated form and are not necessarily conjugate. To address this problem, a majorization technique is adopted providing a tractable VBA solution for non-conjugate distributions. Our approach allows us to employ a wide class of a priori distributions accounting for the possible sparsity of the target signal after some appropriate linear transformation. It can be easily applied to several non Gaussian likelihoods that have been widely used. In particular, experiments in the case of images corrupted by Poisson Gaussian noise showcase the good performance of our approach compared with methods using the discrepancy principle for estimating the regularization parameter. Moreover, we propose variants of our method leading to a significant reduction of the computational cost while maintaining a satisfactory restoration quality.

References
[1] Y. Marnissi, Y. Zheng, E. Chouzenoux and J.-C. Pesquet, A variational Bayesian approach for image restoration. Application to image deblurring with Poisson- Gaussian noise, IEEE Transactions on Computational Imaging, vol. 3, no. 4, pages 722-737, 2017.
[2] A. Jezierska, E. Chouzenoux, J.-C. Pesquet and H. Talbot, A convex approach for image restoration with exact Poisson-Gaussian likelihood, SIAM Journal on Imaging Sciences, vol. 8, no. 4, pages 2662-2682, 2015.