Regularisation theory in the data driven setting

Inverse problems deal with the reconstruction of some quantity of interest from indirectly measured data. A typical example is medical imaging, where there is no direct access to the quantity of interest (the inside of the patient's body) and imaging techniques, such X-Ray imaging and magnetic resonance imaging (MRI), are used. The classical approach to inverse problems uses models that describe the physics of the measurement. For example, in X-Ray imaging this model would describe how X-Rays pass through the body. In the era of big data, however, it becomes increasingly popular not to model the physics but to use vast amounts of data instead that relate known images with corresponding measurements.

The theory of such data driven methods, however, is not well developed yet. It is not well understood, under which conditions on the training data such methods are stable with respect to small changes in the measurement and how well they adapt to images that are different from the training images. It is important to understand this, since otherwise the reconstruction algorithm can miss important features of the image if they weren't present in the training set, such as tumours at previously unseen locations.

In this project I will extend the state-of-the-art model based theory to this data driven setting. I will study under which conditions can data driven methods achieve regularisation, i.e. when can they stably solve an otherwise unstable problem. This will make it easier to analyse stability of data driven reconstruction methods and help developing novel, stable data driven inversion methods with mathematical guarantees. I will also collaborate with the National Physical Laboratory and the Department of Chemical Engineering and Biotechnology in Cambridge on applications of my methods in imaging to reduce the time needed to acquire an image and make the reconstructions more reliable.

Inverse problems arise whenever directly accessing the quantities of interest is impossible and indirect measurements have to be used. Such problems are ubiquitous in science and technology, from microscopy to astronomy, from medical imaging to Earth exploration to non-destructive testing to airport security screening and so on.

Due to the availability of large amounts of domain-specific training data, the paradigm in many applications is shifting towards methods that rely on learning rather than careful mathematical modelling of the measurement process. However, mathematical understanding of such methods is far from being complete. The aim of this project is to reduce this gap by extending the state-of-the-art model-based inverse problems theory to the data driven setting. As a result, we will have a better understanding of data driven approaches to inverse problems and will have novel methods with improved stability properties and solid mathematical guarantees.

This will have impact on the following areas.

- Policy
Theoretical understanding of data driven methods will help shape policy on the use of such methods in sensitive applications from the societal point of view, such as medical imaging used for diagnosis or airport security screening. I will work with Cambridge based charities who help designing policy in emerging technologies in healthcare and other areas.

- Developing National Standards
In this project I will collaborate with the National Physical Laboratory, UK's National Metrology Institute responsible for developing and maintaining measurement standards. Our collaboration will help, in the long run, to design standards for the interpretation of indirect measurements using data driven approaches.

- Industry
Part of this project is applying the developed methods in image reconstruction. This type of problems occur in many areas of technology, such as material manufacturing and the energy sector. I will collaborate with the Department of Chemical Engineering and Biotechnology to speed up acquisition times in magnetic resonance imaging to enable imaging faster processes.

- Public Engagement and Outreach
An important aspect of this project is promoting mathematics to a wider audience through public engagement projects, such as the Cambridge Science Festival, and outreach, e.g., publishing articles in popular science journals.

- Academic Impact
The project will have impact on several fields of research, including imaging science and other areas that use imaging, such as chemical engineering and biomedical sciences. It will also impact data science more broadly alongside with the specialist field of inverse problems. Research will be published in high quality specialist as well as interdisciplinary journals and disseminated through international conferences and workshops. Prototype software and preprints of academic papers will be made available on public repositories.

- Teaching
The results obtained in this project will be integrated into a course taught in Cambridge. Summer research projects and other research projects related to this fellowship will be offered to students, who will benefit from participating in cutting edge research.