Integrating variational and learning models for imaging inverse problems

Imaging inverse problems are fundamental in various fields like diagnostic medicine and manufacturing engineering. Current methods for reconstruction can be divided into variational and learning-based models. Variational techniques use knowledge of the acquisition model to to reformulate the inverse problem as an optimization problem. The performances of these approach is often limited by regularization choices. Learning-based methods learn reconstruction maps directly from the data but lack consistent theoretical understanding. This thesis explores different integrated frameworks, specifically developed to overcome some limitations of both approaches, demonstrating improvements without sacrificing the performances. Variational models incorporating data-driven techniques are improved, considering a bilevel framework for Total Variation regularization or a Plug-and-Play convergent schemes for iterative reconstruction. Several deep learning architectures for image reconstruction are presented, including models for super resolution microscopy and few-view CT, as well as regularization strategies within the Deep Image Prior framework. These approaches highlight the importance of architecture choice and the potential for improvement by incorporating handcrafted regularization terms in deep learning framework. The proposed approaches show how the choice of the architecture in learning-based models is crucial. In addition, their general performances can be improved by employing handcrafted regularization terms, as in the variational framework. In conclusion, the models presented in this thesis confirm that the tools, developed by regularization theory, represent an important component to analyze and control the theoretical guarantees and properties of learning-based techniques, when applied to imaging inverse problems.