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

可靠的数据驱动决策过程取决于用于解释数据的方法的稳健性。对于从医疗保健到天文学、国防和金融的许多应用，数据解释包括通过从退化测量估计未知对象来解决逆问题（例如，来自MR扫描的大脑图像），这对于高维数据变得更具挑战性。一个经典的方法是将未知对象定义为最小化问题的解。这些问题可以有效地解决使用优化算法，其中大多数具有良好的理论保证。他们的理论分析往往是复杂的，涉及的工具，如凸，非凸，随机优化理论，单调算子理论。最近，越来越多的兴趣已经得到了优化方法，涉及神经网络。可以区分两个主要类别：在迭代算法中注入NN的PNN算法，以及展开NN展开算法的有限次迭代。虽然这些方法已被证明产生高质量的结果，其理论行为仍然没有完全understood.This project will provide new hybrid optimization methods involving NNs，with theoretical results，to accurately solve high-dimensional inverse problems.为了这个目的，平均性能展开NN将进行调查，并将由此产生的NN将被插入到近端算法，导致收敛的Pennsylvania方法。将研究所得方法输出的表征。新算法将用于计算成像。我们将特别关注双光子成像在医学中的应用。

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.