Ill-posed linear inverse problems that combine knowledge of the forward measurement model with prior models arise frequently in various applications, from computational photography to medical imaging. Recent research has focused on solving these problems with score-based generative models (SGMs) that produce perceptually plausible images, especially in inpainting problems. In this study, we exploit the particular structure of the prior defined in the SGM to formulate recovery in a Bayesian framework as a Feynman--Kac model adapted from the forward diffusion model used to construct score-based diffusion. To solve this Feynman--Kac problem, we propose the use of Sequential Monte Carlo methods. The proposed algorithm, MCGdiff, is shown to be theoretically grounded and we provide numerical simulations showing that it outperforms competing baselines when dealing with ill-posed inverse problems.

翻译：结合正向测量模型知识与先验模型的病态线性逆问题频繁出现在从计算摄影到医学成像的各种应用中。近期研究聚焦于利用基于分数的生成模型（SGMs）来解决此类问题，尤其是图像修复任务中生成感知上合理的图像。在本研究中，我们利用SGM中定义的先验的特定结构，将贝叶斯框架下的恢复问题建模为一种Feynman-Kac模型，该模型改编自用于构建基于分数的扩散的正向扩散模型。为解决这一Feynman-Kac问题，我们提出使用序列蒙特卡洛方法。理论分析表明，所提出的算法MCGdiff具有理论基础，而数值仿真结果显示，在处理病态逆问题时，该算法优于现有基准方法。