Lecturer: Martin Burger
This course will develop a detailed view on regularization methods for inverse problems with a view on mathematical tasks and caveats of image reconstruction problems. We introduce the theoretical framework of inverse problems and how different modalities and measurement errors can lead to unfavorable reconstructions. Furthermore, we consider regularization strategies to overcome these issues, both from the theoretical and practical side. We will investigate classical aspects of regularization methods such as basic well-posedness and convergence questions. Moreover, we will develop a theoretical understanding of typical / favoured solutions by regularization methods and aspects of bias introduced by those. Finally, we also explore the related Bayesian viewpoint, where we also consider the problem of uncertainty quantification.
The fundamental knowledge on regularization methods is to be used as a starting point for the study of learning approaches in inverse problems. We will discuss the basic issues encountered with supervised learning approaches and thus further study methods that learn a regularization method from data about favourable solutions instead. We will discuss adversarial regularizers and plug-and-play approaches and then further proceed to flow-based and diffusion models, which have recently gained a lot of attention. In addition to a practical view we discuss attempts to gain a theoretical understanding of such approaches, their potential, and their limitations.