Variational regularization is a well-established technique to tackle instability in inverse problems, and it requires solving a minimization problem in which a mismatch functional is endowed with a suitable regularization term. The choice of such a functional is a crucial task, and it usually relies on theoretical suggestions as well as a priori information on the desired solution. A promising approach to this task is provided by data-driven strategies, based on statistical learning: supposing that the exact solution and the measurements are distributed according to a joint probability distribution, which is partially known thanks to a training sample, we can take advantage of this statistical model to design regularization operators. In this talk, I will present a hybrid approach, which first assumes that the desired regularizer belongs to a class of operators (suitably described by a set of parameters) and then learns the optimal one within the class. In the context of linear inverse problems, I will first briefly recap the main results obtained for the family of generalized Tikhonov regularizers: a characterization of the optimal regularizer, and two learning-based techniques to approximate it, with guaranteed error estimates. Then, I will focus on a class of sparsity-promotion regularizers, which essentially leads to the task of learning a sparsifying transform for the considered data. Also in this case, it is possible to deduce theoretical error bounds between the optimal regularizer and its supervised-learning approximation as the size of the training sample grows.

Diffusion models have recently received significant interest in the imaging sciences. After achieving impressive results in image generation, focus has shifted towards finding ways to exploit the encoded prior knowledge in classical inverse problems. In this talk, we highlight another intriguing viewpoint: Instead of focusing on image reconstruction, we propose to use tractable diffusion models which also allow to estimate the noise in an image. In particular, we utilize a fields-of-experts-type model with Gaussian mixture experts that admits an analytic expression for a normalized density under diffusion, and can be trained with empirical Bayes. The normalized model can be used for noise estimation of a given image by maximizing it w.r.t. diffusion time, and simultaneously gives a Bayesian least-squares estimator for the clean image. We show results on denoising problems and propose possible applications to more involved inverse problems.

In this talk, we address the image denoising problem in presence of speckle degradation typically arising in ultra-sound images. Variational methods and Convolutional Neural Networks (CNNs) are considered well-established methods for specific noise types, such as Gaussian and Poisson noise. Nonetheless, the advances achieved by these two classes of strategies are limited when tackling the de-speckle problem. In fact, variational methods for speckle removal typically amounts to solve a non-convex functional with the related issues from the convergence viewpoint; on the other hand, the lack of large datasets of noise-free ultra-sound images has not allowed the extension of the state-of-the-art CNN denoiser methods to the case of speckle degradation. Here, we aim at combining the classical variational methods with the predictive properties of CNNs by considering a weighted total variation regularized model; the local weights are obtained as the output of a statistically inspired neural network that is trained on a small and composite dataset of natural and synthetic images. The resulting non-convex variational model, which is minimized by means of the Alternating Direction Method of Multipliers (ADMM) is proven to converge to a stationary point. Numerical tests show the effectiveness of our approach for the denoising of natural and satellite images.

Dictionary learning methods have been used in recent years to solve inverse problems without using the forward model in the traditional optimization algorithms. When the dictionaries are large, and a sparse representation of the data in terms of the atoms is desired, computationally efficient sparse optimization algorithms are needed. Furthermore, reduced dictionaries can represent the data only up to a model reduction error, and Bayesian methods for estimating modeling errors turn out to be useful in this context. In this talk, the ideas of using Bayesian hierarchical models and modeling error methods are discussed.