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Classical methods for inverse problems are mainly based on regularization theory, in particular those, that are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two terms and a great number of optimization algorithms have been proposed. When these two terms are distance or divergence measures, they can have a Bayesian Maximum A Posteriori (MAP) interpretation where these two terms correspond to the likelihood and prior-probability models, respectively. The Bayesian approach gives more flexibility in choosing these terms and, in particular, the prior term via hierarchical models and hidden variables. 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) and particularly convolutional NN, deep NN, physics-informed neural networks, etc. can become helpful to obtain approximate practical solutions to inverse problems. In this tutorial article, particular examples of image denoising, image restoration, and computed-tomography (CT) image reconstruction will illustrate this cooperation between ML and inversion. Full article

Signale and image processing has always been the main tools in many area and in particular in Medical and Biomedical applications. Nowadays, there are great number of toolboxes, general purpose and very specialized, in which classical techniques are implemented and can be used: all the transformation based methods (Fourier, Wavelets, ...) as well as model based and iterative regularization methods. Statistical methods have also shown their success in some area when parametric models are available. Bayesian inference based methods had great success, in particular, when the data are noisy, uncertain, incomplete (missing values) or with outliers and where there is a need to quantify uncertainties. In some applications, nowadays, we have more and more data. To use these “Big Data” to extract more knowledge, the Machine Learning and Artificial Intelligence tools have shown success and became mandatory. However, even if in many domains of Machine Learning such as classification and clustering these methods have shown success, their use in real scientific problems are limited. The main reasons are twofold: First, the users of these tools cannot explain the reasons when the are successful and when they are not. The second is that, in general, these tools can not quantify the remaining uncertainties. Model based and Bayesian inference approach have been very successful in linear inverse problems. However, adjusting the hyper parameters is complex and the cost of the computation is high. The Convolutional Neural Networks (CNN) and Deep Learning (DL) tools can be useful for pushing farther these limits. At the other side, the Model based methods can be helpful for the selection of the structure of CNN and DL which are crucial in ML success. In this work, I first provide an overview and then a survey of the aforementioned methods and explore the possible interactions between them. Full article

3D X-ray Computed Tomography (CT) is used in medicine and non-destructive testing (NDT) for industry to visualize the interior of a volume and control its healthiness. Compared to analytical reconstruction methods, model-based iterative reconstruction (MBIR) methods obtain high-quality reconstructions while reducing the dose. Nevertheless, usual Maximum-A-Posteriori (MAP) estimation does not enable to quantify the uncertainties on the reconstruction, which can be useful for the control performed afterwards. Herein, we propose to estimate these uncertainties jointly with the reconstruction by computing Posterior Mean (PM) thanks to Variational Bayesian Approach (VBA). We present our reconstruction algorithm using a Gauss-Markov-Potts prior model on the volume to reconstruct. For PM calculation in VBA, the uncertainties on the reconstruction are given by the variances of the posterior distribution of the volume. To estimate these variances in our algorithm, we need to compute diagonal coefficients of the posterior covariance matrix. Since this matrix is not available in 3D X-ray CT, we propose an efficient solution to tackle this difficulty, based on the use of a matched pair of projector and backprojector. In our simulations using the Separable Footprint (SF) pair, we compare our PM estimation with MAP estimation. Perspectives for this work are applications to real data as improvement of our GPU implementation of SF pair. Full article