Imaging technologies are widely used in application fields such as natural sciences, engineering, medicine, and life sciences. A broad class of imaging problems reduces to solve ill-posed inverse problems (IPs). Traditional strategies to solve these ill-posed IPs rely on variational regularization
methods, which are based on minimization of suitable energies, and make use of knowledge about the image formation model (forward operator) and prior knowledge on the solution, but lack in incorporating knowledge directly from data. On the other hand, the more recent learned approaches can easily learn the intricate statistics of images depending on a large set of data, but do not have a
systematic method for incorporating prior knowledge about the image formation model. The main purpose of this thesis is to discuss data-driven image reconstruction methods which combine the benefits of these two different reconstruction strategies for the solution of highly nonlinear ill-posed
inverse problems. Mathematical formulation and numerical approaches for image IPs, including linear
as well as strongly nonlinear problems are described. More specifically we address the Electrical
impedance Tomography (EIT) reconstruction problem by unrolling the regularized Gauss-Newton
method and integrating the regularization learned by a data-adaptive neural network. Furthermore
we investigate the solution of non-linear ill-posed IPs introducing a deep-PnP framework that integrates
the graph convolutional denoiser into the proximal Gauss-Newton method with a practical
application to the EIT, a recently introduced promising imaging technique. Efficient algorithms are
then applied to the solution of the limited electrods problem in EIT, combining compressive sensing
techniques and deep learning strategies. Finally, a transformer-based neural network architecture
is adapted to restore the noisy solution of the Computed Tomography problem recovered using the
filtered back-projection method.