"The aim of this fellowship proposal on Stochastic Optimisation and Simulation in Image Processing (SOSIP) is to investigate new computational methods to perform Bayesian inference in challenging inverse problems arising in statistical image processing. Precisely, this proposal intends to explore new stochastic approximation and Markov chain Monte Carlo methods to perform Bayesian inference in high-dimensional statistical models that are not differentiable (e.g. involving l1 or total-variation regularizations). A special focus will be given to methods that combine state-of-the-art stochastic optimisation and simulation with techniques from modern high-dimensional convex optimisation (e.g., proximal splitting, dualisation, augmented Langrangian decomposition, Moreau envelope, etc.). Two main classes of methods will be considered: (1) optimisation-empowered MCMC algorithms to simulate efficiently from high-dimensional models that are not differentiable and (2) MCMC-empowered (stochastic) optimisation schemes to maximize intractable functions related to complex Bayesian models with latent variables (e.g. marginal likelihoods and marginal posterior distributions). Such methods would offer the potential to advance significantly the state-of-the-art in image processing and its application domains (e.g. medical imaging, remote sensing, astronomy, etc.). The proposed methodologies will be applied to two challenging medical imaging problems that cannot be satisfactorily solved using existing simulation or optimisation techniques: (1) unsupervised blind dynamic EEG image reconstruction for low-cost functional brain imaging and (2) non-rigid multi-modal EPID + CT image fusion for 'on-line' radiotherapy-treatment-plan monitoring. The proposed work will be conducted in collaboration with researchers at Technical University of Lisbon, University of Toulouse, Buenos Aires Institute of Technology, FLENI Hospital of Buenos Aires, Edinburgh Cancer Centre and Heriot-Watt University."