Mathematics

I am interested in the analysis and the development of optimisation methods for inverse imaging problems. My research touches such areas of mathematics as optimisation, geometric measure theory, variational analysis, and partial differential equations. A lot of my research provides new insight into functions of bounded variation and deformation.

In inverse problems research, lacking unique solutions, one usually models prior assumptions of what a good solution is, through a regulariser. One can then attempt to reconstruct an image – such as a photograph or a medical, biological, or an astronomical image – by minimising an objecive consisting of a fidelity term, involving data f, and the regulariser. A popular and prototypical regulariser in image processing is total variation, or the (distributional) one-norm of the gradient. This is mainly due to its edge-preserving properties, although it has several drawbacks, such as the stair-casing effect. Recently, therefore, higher-order and curvature-based regularisers have, such as total generalised variation have received increased attention due to their better visual qualities. Very little is known about them analytically. Yet this would be desirable from the point of view of reliability of the techniques in practise, to show that no new artefacts are introduced, and that desired features are restored correctly. I do mathematical analysis on this kind of questions.

A photograph taken with current state-of-the-art digital cameras has between 10 and 20 million pixels. Some cameras, such as the semi-prototypical Nokia 808 PureView have up to 41 million sensor pixels. Some medical imaging data sets, such as full three-dimensional diffusion tensor (DTI) or HARDI images also contain tens of millions of data points. The resulting optimisation problems for the improvment of such images, or for reconstructing them from partial data, are huge, and computationally very intensive. Moreover, the aforementioned state-of-the-art regularisers are generally non-smooth, which causes difficulties in the application of conventional optimisation methods. State-of-the-art image processing techniques based on mathematical principles are only up to processing tiny images in real time. A question that interests me, is whether we can design optimisation algorithms that would make the processing of real high-resolution images fast?

If you wish to learn more about these topics, please see my publications.