Robust and Interpretable Learned Regularization for Solving Inverse Problems in Medical Imaging

Imaging-based sciences often involve solving inverse problems, where a hidden image, such as the inside of the human body, is reconstructed from noisy measurement data. In medicine, examples include X-ray computed tomography (CT) and magnetic resonance imaging (MRI), both essential in clinical practice. However, reconstructing images can be difficult due to noise or missing data, creating a demand for improved computational methods. For years, variational reconstruction methods, which solve an optimization problem using hand-crafted regularizers based on prior knowledge, were the gold standard due to their interpretability. Recently, deep learning methods have produced superior results but often lack interpretability and theoretical guarantees, raising concerns about their use in critical settings. This project aims to bridge the gap by developing new learnable regularizers that preserve interpretability and provide theoretical assurances. Inspired by deep learning, we identified two concepts missing in traditional regularizers: local adaptivity and long-range dependencies. Local adaptivity allows regularizers to adjust to local structures, preserving sharp image details, while long-range dependencies capture global properties like symmetries and patterns. We propose to integrate these principles using conditioning and multi-scale modeling. Preliminary experiments with conditional regularizers have shown improved performance, and we aim to retain interpretability and ensure theoretical guarantees. This new approach introduces a dependency of the regularizer on the data, requiring novel theoretical analysis. We will test our findings on real-world inverse problems, such as low-field MRI reconstruction.