Scientific imaging problems are often severely ill-posed, and hence have significant intrinsic uncertainty. Accurately quantifying the uncertainty in the solutions to such problems is therefore critical for the rigorous interpretation of experimental results as well as for reliably using the reconstructed images as scientific evidence. Unfortunately, existing imaging methods are unable to quantify the uncertainty in the reconstructed images in a manner that is robust to experiment replications. This paper presents a new uncertainty quantification methodology based on an equivariant formulation of the parametric bootstrap algorithm that leverages symmetries and invariance properties commonly encountered in imaging problems. Additionally, the proposed methodology is general and can be easily applied with any image reconstruction technique, including unsupervised training strategies that can be trained from observed data alone, thus enabling uncertainty quantification in situations where there is no ground truth data available. We demonstrate the proposed approach with a series of numerical experiments and through comparisons with alternative uncertainty quantification strategies from the state-of-the-art, such as Bayesian strategies involving score-based diffusion models and Langevin samplers. In all our experiments, the proposed method delivers remarkably accurate high-dimensional confidence regions and outperforms the competing approaches in terms of estimation accuracy, uncertainty quantification accuracy, and computing time.

翻译：科学成像问题通常严重病态，因此具有显著的内在不确定性。准确量化此类问题解的不确定性对于严格解释实验结果以及可靠地将重建图像用作科学证据至关重要。遗憾的是，现有成像方法无法以对实验复制鲁棒的方式量化重建图像中的不确定性。本文提出了一种新的不确定性量化方法，其基于参数化自举算法的等变公式，利用了成像问题中常见的对称性和不变性。此外，所提出的方法具有通用性，可轻松应用于任何图像重建技术，包括仅从观测数据训练的无监督训练策略，从而在无真实数据的情况下实现不确定性量化。我们通过一系列数值实验以及与现有最先进的不确定性量化策略（如基于分数的扩散模型和Langevin采样器的贝叶斯策略）的比较来展示所提出的方法。在所有实验中，所提方法均能提供显著准确的高维置信区域，并在估计精度、不确定性量化精度和计算时间方面优于竞争方法。