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

可靠的数据驱动的决策过程取决于用于解释数据的方法的稳健性。对于许多应用，从医疗保健到天文学、国防和金融，数据解释包括通过从降级测量（例如，从核磁共振扫描获得的脑图像）估计未知对象来解决反问题，这对于高维数据来说变得更具挑战性。一个经典的方法是将未知对象定义为最小化问题的解决方案。这些问题可以使用优化算法有效地解决，其中大多数都有完善的理论保证。他们的理论分析往往是复杂的，涉及工具如凸，非凸，随机优化理论，和单调算子理论。最近，人们对涉及神经网络的优化方法越来越感兴趣。可以区分两大类：在迭代算法中注入神经网络的PnP算法，以及展开有限次迭代算法的展开神经网络。虽然这些方法已经被证明产生了高质量的结果，但它们的理论行为仍然没有被完全理解。该项目将提供涉及神经网络的新的混合优化方法，并提供理论结果，以准确解决高维逆问题。为此，将研究未展开神经网络的平均特性，并将得到的神经网络插入到导致收敛的PnP方法的近端算法中。将调查所得到的方法输出的特性。新的算法将用于计算成像。我们将特别关注光子成像在医学中的两种应用。

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.