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Stochastic Optimisation and Simulation in Image Processing

Final Report Summary - SOSIP (Stochastic Optimisation and Simulation in Image Processing)

The aim of this research project has been to investigate new computational methodologies to solve challenging problems arising in mathematical image processing. Precisely, to develop new stochastic simulation and optimisation methods specifically designed for performing Bayesian inference in high-dimensional imaging inverse problems, with a focus on problems that are blind, semi-blind or unsupervised (i.e. with fully or partially known statistical models), and which are beyond the scope of modern mathematical imaging techniques. Particularly, method that, in addition to point estimates, also support uncertainty quantification tasks and inform decision-making.

The expected outcome is that these new tools will enable methodologies for solving a number of difficult problems that are not well addressed by existing approaches, such as unsupervised image segmentation and unsupervised resolution enhancement. Such challenging problems arise in many important application domains, such as medical imaging and remote sensing, and it is expected that in the future the research conducted in this project will lead to significant social and economic benefit through impact on these application domains. During the project proposed methods were applied successfully to specific medical imaging and remote sensing problems, which the fellow investigated in collaboration with partners from the Scotland, France, Portugal and Argentina.

More precisely, during this project the fellow made a number of important contributions related to new high-dimensional stochastic simulation and optimisation approaches for imaging. A main contribution was the development of proximal Markov chain Monte Carlo algorithms. These algorithms combine high-dimensional stochastic simulation with mathematical tools from convex analysis and optimisation to provide a new computation methodology that is particularly suitable for imaging inverse problems that are convex and potentially very high dimensional. From an imaging perspective, this enables a range of Bayesian inference analyses that were previously beyond the scope of statistical image processing, such as model selection and uncertainty quantification via the computation of credibility regions. The fellow already published a computational statistics journal paper on this topic, which has received significant recognition from the community on an international scale, and has submitted a second manuscript related to a new method with faster convergence properties (joint work with A. Durmus and E. Moulines, Ecole Polythecnique, France). Two additional papers are in preparation, one on applications to uncertainty quantification in imaging, and the other one on efficient implementation strategies that self-adjust algorithm parameters to accelerate convergence and use automatic stopping rules.

Another important contribution is a new methodology that combines convex optimisation with probability theory to approximate Bayesian confidence regions in any inverse problem that is convex, enabling uncertainty quantification for images. This work, which was presented at the IEEE SSP’2016 conference, is already receiving significant attention in applied communities (a long manuscript is in press in SIAM Journal on Imaging Sciences). Following on from this, the fellow is now preparing two new papers about applications to astronomical imaging (with J. McEwen, Mullard Space Science Lab.), and to medical imaging (with Y. Wiaux, Heriot-Watt University).

Moreover, the fellow has also proposed a new high-dimensional optimisation approach to perform hierarchical Bayesian estimation for large-scale unsupervised inverse problems involving unknown regularisation parameters (these problems are too high-dimensional to be addressed by using Monte Carlo techniques). Two main instances of this approach have been considered: image segmentation, and linear inverse problems. A long paper on image segmentation will appear soon in IEEE Transaction on Image Processing (co-authored with Steve McLaughlin, Heriot-Watt University). The approach for linear inverse problems is discussed in a short paper in the proceedings of the European Signal Processing Conference (EUSIPCO 2016), and a journal paper is currently in preparation (joint work with Mario Figueiredo and José Bioucas-Dias, Technical University of Lisbon).

In addition to this methodological work, the fellow has also co-authored two invited survey papers on stochastic simulation and optimisation for Bayesian computation. To stimulate synergy between the computational statistics and the imaging community on this topic, one paper was published in Statistics and Computing and described applications to image processing, and the other one in a special issue on simulation and optimisation organised by the fellow for the IEEE Signal Processing Society (note that the fellow also proposed and served as guest editor for the this special issue).

Furthermore, as mentioned previously, the fellow has also collaborated actively with application experts to apply these new methods to specific medical imaging and remote sensing problems and deliver impact. He published two journal papers related to new Bayesian methods and algorithms for hyper-spectral satellite imaging, a remote sensing modality with important applications in environmental sciences and defence (joint work with Yoann Altmann and Steve McLaughlin, Heriot-Watt University, and José Bioucas-Dias, Technical University of Lisbon). Similarly, he collaborated with medical imaging and bioengineering partners in Toulouse and Buenos Aires to develop a new statistical EEG signal processing method for epilepsy prognosis. This collaboration has led to two short communications in the proceedings of the ADNTIIC and the IEEE IVMSP conferences. A long paper is currently in preparation for the Elsevier Digital Signal Processing journal.

Finally, the fellow has also been actively involved in a joint medical imaging research project between the School of Mathematics of the University of Bristol, Bristol Clinical Imaging Research Centre, and Bristol's Southmead Hospital on the topic of magnetic resonance imaging for stroke diagnosis. This work, led by Risto Kaupinnen, has been awarded a Dunhill Medical Trust research grant, and has led to a publication in Biomedical Spectroscopy in Imaging.