Dynamic PET is one of the few possibilities when trying to understand neurochemical processes in-vivo. However, most analysis focuses on individual time activity curve modelling rather than a full spatio-temporal analysis, primarily due to the high dimensional nature of the full spatio-temporal problem. However, recent advances in computational statistics, in particular sequential Monte Carlo (SMC) methods for the analysis of PET models [1,2] mean that it is now possible to consider more complex spatial dependencies between the time activity curves. This project will be to investigate the theoretical guarantees of using spatial models (such as Potts models) within PET imaging, and facilitate their implementation on both test and realistic examples. Additional, spatial-temporal priors in term of appropriate sparsity and geometry promoting regularisation functionals will be crucial in this analysis, picking up spatial and temporal changes as intrinsically given in the data. In this context, we are especially interested in ideas related to bilevel learning which learns the right decomposition into spatial and temporal components from the image data itself [3,4].
The project could also examine other modalities, such as dynamic MRI, where similar models also are appropriate. This project could appeal to a candidate with an interest in computational statistics, applied analysis, inverse problems and optimisation.
 Y Zhou, JAD Aston and AM Johansen. Bayesian Model Comparison for Compartmental Models with Applications in Positron Emission Tomography (2013), Journal of Applied Statistics, 40:993-1016.
 Y Zhou, AM Johansen and JAD Aston. Towards Automatic Model Comparison: An Adaptive Sequential Monte Carlo Approach, Journal of Computational and Graphical Statistics, in press.
 J. C. de Los Reyes, C.-B. Schönlieb and T. Valkonen, The structure of optimal parameters for image restoration problems, Journal of Mathematical Analysis and Applications 434 (2016), 464–500.
 L. Calatroni, C. Cao, J. C. De Los Reyes, C.-B. Schönlieb, and T. Valkonen, Bilevel approaches for learning of variational imaging models, to appear in Radon book series 2015.