Hence, in practice, tomographic reconstruction is performed on computers using discretizations, i.e., reduced models of finite dimensions.
This naturally raises the question of whether the discrete problems are related to our continuous understanding of computed tomography. One cannot hope for discrete inverse problems to fully describe the infinite-dimensional setting; however, one would expect that when refining discretizations, e.g., by increasing resolutions, the discretizations are more and more representative of the infinite-dimensional situation.
The investigations move in multiple directions that benefit each other. Quite naturally the discrete forward operator should be as representative of the true operator as possible, a goal that leads to development of better discretization schemes. But that alone is not enough to guarantee that the solutions we obtained for the discrete problems, in fact, converge to solutions of the continuous inverse problem. Investigating for which methods and which senses this is the case is another key aspect of this project.Thus, preprocessing the data into a suitable form is a key step of tomographic reconstruction. To that end, detecting and correcting said corruptions prior to reconstruction is crucial. Doing so requires differentiation between plausible and implausible data.
One way to describe the plausibility of data is by checking whether they are in the range of tomographic projection operators describing the measurement process. The level of deviation from said range can indicate the level of corruption. However, how can we check whether data is in the range without doing reconstruction? Data in the range inherently possesses information overlaps between different projections that systematic corruptions will perturb.
Hence, the goal of this research is the characterization of tomographic projection operators' ranges and their related inherent information overlap. We develop general methods for characterizing the range of tomographic projection operators and describe how they can be employed in tomographic preprocessing.The field of learning PDE-based models from data is growing rapidly. It is driven by the need to bridge physical principles with data-driven insights and enables successful advances in learning unknown parameters in partially specified PDEs, discovering entirely new PDE structures, and approximating solution operators with remarkable accuracy. Alongside practical applications, there is increasing attention to theoretical analysis addressing consistency, convergence, and generalization. These are essential for ensuring the reliability and robustness of learned models in scientific and engineering contexts.
Our research group focuses on the framework of "structured model learning." Structured model learning builds on approximate physical knowledge and addresses the limitations of overly coarse abstractions by incorporating fine-scale physics learned from data. This approach improves interpretability, accuracy, and generality, enabling the model to handle complex scenarios and external factors beyond the scope of simplified physical principles. Augmenting known physics with data-driven components enables the model to effectively describe phenomena, even in non-ideal or challenging contexts. However, it is crucial to learn only what is necessary to ensure that the augmentation remains efficient and grounded in physical understanding.
Our research group is actively engaged in structured model learning, focusing on both the theoretical foundations and the practical applications. We have investigated the unique identifiability of learned fine-scale physics and established a convergence result with practical relevance (see [HM2024]). Currently, a major focus of our work is learning multi-pool dynamics in magnetic resonance imaging (MRI).
Variational methods are a state-of-the-art approach for solving inverse problems in image processing and beyond. By employing suitable regularization functionals, variational methods enable provably stable and consistent reconstructions for a wide range of inverse imaging problems and effectively compensate for missing data through appropriate mathematical modeling.
Our research group has made several fundamental contributions to the field of variational methods for inverse problems. These include:
