Inverse problems in imaging science: from classical regularization methods to state of the art Bayesian methods

Inverses problems arise in almost all the engineering and applied sciences where we have indirect measurement. Many classical signal and image processing research subjects are directly expressed as inverse problems: signal deconvolution, image restoration, image reconstruction in many imaging systems such as X ray Tomography, Microwave and Ultrasound imaging, Synthetic aperture radar (SAR), etc. In this tutorial, first we express in a unifying approach all these applications in a common mathematical framework. Then, mentioning the ill-posed nature of these inverse problems, we describe the regularization methods which were very successful during 1960-2000. Mentioning the limitations of these methods, we see how the Bayesian approach can give tools to go beyond these difficulties. In particular, we will see how this approach can be useful to account for many different a priori knowledges: smoothness, positivity, piecewise continuity, sparsity, finite number of materials (compact homogeneous regions), etc. We also discuss the computational aspects of the Bayesian approach and the practical implementations of the proposed algorithms.