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

Abstract

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.

Regularization, Bayesian Inference, and Machine Learning Methods for Inverse Problems

“…In fact, the two terms of the regularization methods can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond to the likelihood and prior models, respectively. Indeed, the Bayesian approach gives more flexibility in choosing these terms and particularly the prior term via hierarchical models and hidden variables [ 6 , 7 , 8 , 9 ] However, the Bayesian computations can become very heavy computationally. The machine-learning (ML) methods, such as classification, clustering, segmentation, and regression, based on neural networks (NN), such as convolutional NN and deep NN, physics-informed neural networks, etc., can become helpful to obtain approximate but good-quality and practical solutions to inverse problems [ 10 , 11 , 12 , 13 ].…”