Inverse problems in signal and image processing and Bayesian inference framework- from basic to advanced Bayesian computation

Ali Mohammad-Djafari (CNRS, L2S, FR)
March 27, 2015 — 10:30 — "None"

Abstract

In signal and image processing community, we can distinguish two categories: - Those who start from the observed signals and images and do classical processing: filtering for denoising, change detection, contour detection, segmentation, compression, … - The second category called “model based”, before doing any processing try first to understand from where those signals and images come from and why they are here . So, first defining what quantity has been at the origin of those observations, then modeling their link by “forward modeling” and finally doing inversion. This approach is often called “Inverse problem approach”. Then, noting the “ill-posedness” of the inverse problems, many “Regularization methods” have been proposed and applied successfully. However, deterministic regularization has a few limitations and recently the Bayesian inference approach has become the main approach for proposing unsupervised methods and effective solutions in many real applications. Interestingly, even many classical methods have found better understanding when re-stated as inverse problem. The Bayesian approach with simple prior models such as Gaussian, Generalized Gaussian, Sparsity enforcing priors or more sophisticated Hierarchical models such as Mixture models, Gaussian Scale Mixture or Gauss-Markov-Potts models have been proposed in different applications of imaging systems with great success. However, Bayesian computation still is too costly and need more practical algorithms than MCMC. Variational Bayesian Approximation (VBA) methods have recently became a standard for computing the posterior means in unsupervized methods. Interestingly, we show that VBA includes Joint Maximum A Posteriori (JMAP) and Expectation-Maximization (EM) as special cases. VBA is much faster than MCMC methods, but, it gives only access to the posterior means. This talk gives an overview of these methods with examples in Deconvolution (simple or blind, signal or image) and in Computed Tomography (CT).