Project Details
Projekt
Robust and Interpretable Learned Regularization for Solving Inverse Problems in Medical Imaging
Subject Area
Mathematics
Term
since 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 543939932
Numerous tasks in imaging-based sciences can be modeled as inverse problems. Here, a hidden image, for example, the inside of the human body, must be reconstructed from previously acquired noisy measurement data. In the medical domain, prototypical examples are X-ray computed tomography (CT) and magnetic resonance imaging (MRI), which are both indispensable in today's clinical practice. For lots of inverse problems, the reconstruction process is very challenging due to high noise sensitivity or severe amounts of missing data. Hence, there is a need for ever-improving computational methods. For many years, well-analyzed variational reconstruction methods have been the gold standard. Here, the reconstruction is the solution of an optimization problem involving a regularizer, which is typically hand-crafted based on prior knowledge about the class of feasible reconstructions. Hence, they are often considered as interpretable. Nowadays, deep-learning-based methods have become the state-of-the-art in many fields. These yield much better reconstructions, but often lack a clear theoretical justification and good interpretatability. Consequently, there is an ongoing debate about their usability in clinical practice, where misdiagnoses could be fatal. In this project, we aim to combine the best of both worlds by designing novel learnable regularizers under the premise that interpretability and theoretical guarantees are preserved. Using recent deep learning approaches as a source of inspiration, we identified two essential concepts that are, up to now, hardly reflected in hand-crafted regularizers - local adaptivity and long-range dependencies. Here, adaptivity refers to the ability of a regularizer to reflect local structures in its regularization strength. In turn, sharp or accentuated image structures can be better preserved. Beyond local structures, long-range dependencies model properties of the entire image, such as symmetries, large patterns, or characteristic objects. These require an information exchange over long distances, which competes with the local nature of most of today's hand-crafted regularizers. Starting from interpretable local regularizers, we propose to incorporate both principles via conditioning and multi-scale modeling. Indeed, some preliminary numerical experiments with conditional regularizers led to significantly improved reconstruction performance. Moreover, by carefully designing the regularizer, we can maintain interpretability to a large extent and also expect to obtain theoretical guarantees. Compared to classical regularization approaches, the conditioning results in a dependency of the regularizer on the measured data. Hence, a completely new theoretical analysis becomes necessary. Finally, we will experimentally verify the gained theoretical findings on challenging real-world inverse problems such as the recently popularized low-field MRI setting.
DFG Programme
Priority Programmes
Subproject of
SPP 2298:
Theoretical Foundations of Deep Learning
