Inverse problems naturally arise in many scientific settings, and the study of these problems has been crucial in, for example, the development of medical imaging technology. In an inverse problem, the goal is to reconstruct an image from its measurements, using knowledge of the measurement process. The inversion of the measurement process is usually ill-posed and needs to be stably approximated to ensure high-quality reconstructions if the measurements are noisy. A well-established paradigm for doing so is the variational regularisation approach, in which images are estimated by minimising a functional that trades off the fit to the measurements and the fit to a model of the prior knowledge about the true image.

Although this has proven to be a very successful approach, it generally requires a lot of manual intervention to tune the parameters that are involved and select a reasonable hand-crafted model of the image prior. Recently there has been a considerable amount of interest in using data to learn reconstruction methods (inspired by) the variational regularisation approach.

This project explores various aspects of such data-driven approaches to image reconstruction. On the one hand, it is possible to learn various parts of a variational regularisation problem (e.g. the regularisation parameter and the measurement strategy) using a bilevel optimisation framework. This results in a method that has the desirable guarantees of the variational regularisation approach, but the learning problem requires a large amount of computational effort. On the other hand, the variational regularisation approach has inspired unrolled iterative neural network methods, which are computationally more efficient and produce empirically excellent results, but come with few guarantees. This project studies how the shortcomings of these two methods can be overcome.