We contribute to an uncertainty quantification problem in imaging that evaluates a hypothesis test questioning the existence of local "artefacts" appearing in the maximum a posteriori (MAP) estimate (obtained from standard numerical tools). Such a method, called Bayesian uncertainty quantification by optimization (BUQO), was introduced a few years ago as an efficient and scalable alternative to sampling methods when per-pixel error-bars are not needed. BUQO formulates a hypothesis test for probing the existence of local structures in the MAP estimate as a minimization problem, that can be solved efficiently with standard optimization algorithms. In this context, BUQO requires a "mathematical" definition of the "local artefact". This definition can be interpreted as an inpainting of the structure. However, only simple hand-crafted techniques have been proposed so far due to the complexity of the problem. In this work, we propose a data-driven alternative to BUQO where the inpainting procedure in the algorithm is performed using a convolutional inpainting neural network (NN). This results in a plug-and-play algorithm, based on the primal-dual Condat-Vu iterations,where the inpainting procedure is performed with a NN. The proposed approach is assessed on two image reconstruction problems inspired by medicine. We specifically perform simulations on two Fourier undersampling problems (discrete and non-uniform) encountered in magnetic resonance imaging, as well as a computed tomography problem using the Radon measurement operator.

翻译：我们针对成像中的不确定性量化问题提出了一种方法，该方法用于评估一个假设检验，该检验质疑最大后验（MAP）估计（由标准数值工具获得）中局部“伪影”的存在性。几年前，一种名为“基于优化的贝叶斯不确定性量化（BUQO）”的方法被提出，作为不需要逐像素误差条时采样方法的高效且可扩展的替代方案。BUQO将探测MAP估计中局部结构存在性的假设检验表述为一个最小化问题，该问题可通过标准优化算法高效求解。在此背景下，BUQO需要“局部伪影”的“数学”定义，该定义可被解释为结构的图像修复。然而，由于问题的复杂性，目前仅提出了简单的手工设计技术。本文提出了一种数据驱动的BUQO替代方法，其中算法中的图像修复过程使用卷积图像修复神经网络（NN）执行。这产生了一种基于原始-对偶Condat-Vu迭代的即插即用算法，其中图像修复过程由NN完成。所提出的方法在两个受医学启发的图像重建问题上进行了评估。我们具体针对磁共振成像中遇到的两种傅里叶欠采样问题（离散和非均匀）以及使用Radon测量算子的计算机断层扫描问题进行了仿真。