Data driven regularization models of non-linear ill-posed inverse problems in imaging

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.

Data surveillance for infectious diseases: models, complex networks and machine learning

In this thesis, we investigate the role of applied physics in epidemiological surveillance through the application of mathematical models, network science and machine learning. The spread of a communicable disease depends on many biological, social, and health factors. The large masses of data available make it possible, on the one hand, to monitor the evolution and spread of pathogenic organisms; on the other hand, to study the behavior of people, their opinions and habits. Presented here are three lines of research in which an attempt was made to solve real epidemiological problems through data analysis and the use of statistical and mathematical models. In Chapter 1, we applied language-inspired Deep Learning models to transform influenza protein sequences into vectors encoding their information content. We then attempted to reconstruct the antigenic properties of different viral strains using regression models and to identify the mutations responsible for vaccine escape. In Chapter 2, we constructed a compartmental model to describe the spread of a bacterium within a hospital ward. The model was informed and validated on time series of clinical measurements, and a sensitivity analysis was used to assess the impact of different control measures. Finally (Chapter 3) we reconstructed the network of retweets among COVID-19 themed Twitter users in the early months of the SARS-CoV-2 pandemic. By means of community detection algorithms and centrality measures, we characterized users’ attention shifts in the network, showing that scientific communities, initially the most retweeted, lost influence over time to national political communities. In the Conclusion, we highlighted the importance of the work done in light of the main contemporary challenges for epidemiological surveillance. In particular, we present reflections on the importance of nowcasting and forecasting, the relationship between data and scientific research, and the need to unite the different scales of epidemiological surveillance.