Data-driven regularisation techniques for inverse problems

In inverse problems, such as computed tomography (CT), the goal is to reconstruct a quantity of interest from indirect
and often noisy measurements. These problems are challenging due to their ill-posed nature and the complexity of
the data. Traditional approaches, like variational regularisation have been effective, but are often limited by their
handcrafted design (e.g. promoting smoothness in the reconstruction), which can be restrictive when capturing
diverse features of the quantity of interest.
Recently, data-driven regularisation methods, such as plug-and-play (PnP), have been introduced to integrate
learned priors from training data. These methods are designed to address issues of traditional approaches and are
more flexible to a wider range of inverse problems. This project will explore the application of such data-driven
approaches to medical imaging, focusing on their theoretical foundations, practical implementation, and mathematical
guarantees.
To help facilitate the project, mathematical methods in numerical analysis and optimisation will be used alongside
numerical experiments with the use of coding languages.
Overall, this PhD aims to understand the inner workings of these data-driven methods and identify the most suitable
regularisation techniques for specific inverse problem settings for imaging.