New proximal algorithms for computational imaging: From optimisation theory to enhanced deep learning

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Reliable data-driven decision-making processes depend on the robustness of the methods used to interpret the data. For many applications, ranging from healthcare to astronomy, defence and finance, data interpretation consists of solving an inverse problem, by estimating an unknown object from degraded measurements (e.g., a brain image from an MR scan), that becomes even more challenging for high dimensional data. A classical approach is to define the unknown object as a solution to a minimisation problem. Such problems can be solved efficiently using optimisation algorithms, most of them having well established theoretical guarantees. Their theoretical analysis are often complex, involving tools as convex, nonconvex, stochastic optimisation theories, and monotone operator theory. Recently, growing interest has been given to optimisation methods involving NNs. Two main classes can be distinguished: PnP algorithms injecting NNs in iterative algorithms, and unfolded NNs unrolling finite number of iterations of an algorithm. Although these approaches have been shown to produce high quality results, their theoretical behavior is still not fully understood.This project will provide new hybrid optimisation methods involving NNs, with theoretical results, to accurately solve high dimensional inverse problems. To this aim, averaging properties of unfolded NNs will be investigated, and the resulting NNs will be plugged into proximal algorithms leading to convergent PnP methods. Characterisation of the resulting method outputs will be investigated. The new algorithms will be used for computational imaging. We will particularly focus on two photon imaging applications in medicine.