Estimating high-quality images while also quantifying their uncertainty are two desired features in an image reconstruction algorithm for solving ill-posed inverse problems. In this paper, we propose plug-and-play Monte Carlo (PMC) as a principled framework for characterizing the space of possible solutions to a general inverse problem. PMC is able to incorporate expressive score-based generative priors for high-quality image reconstruction while also performing uncertainty quantification via posterior sampling. In particular, we develop two PMC algorithms that can be viewed as the sampling analogues of the traditional plug-and-play priors (PnP) and regularization by denoising (RED) algorithms. To improve the sampling efficiency, we introduce weighted annealing into these PMC algorithms, further developing two additional annealed PMC algorithms (APMC). We establish a theoretical analysis for characterizing the convergence behavior of PMC algorithms. Our analysis provides non-asymptotic stationarity guarantees in terms of the Fisher information, fully compatible with the joint presence of weighted annealing, potentially non-log-concave likelihoods, and imperfect score networks. We demonstrate the performance of the PMC algorithms on multiple representative inverse problems with both linear and nonlinear forward models. Experimental results show that PMC significantly improves reconstruction quality and enables high-fidelity uncertainty quantification.

翻译：在求解病态逆问题的图像重建算法中，同时实现高质量图像估计及其不确定性量化是两个理想特性。本文提出即插即用蒙特卡洛（PMC）作为一个原则性框架，用于刻画一般逆问题可能解的空间。PMC能够结合富有表现力的分数生成先验以实现高质量图像重建，同时通过后验采样执行不确定性量化。具体而言，我们开发了两种PMC算法，可视为传统即插即用先验（PnP）与去噪正则化（RED）算法的采样对应形式。为提升采样效率，我们在这些PMC算法中引入了加权退火机制，进一步开发出两种退火PMC算法（APMC）。我们建立了理论分析以刻画PMC算法的收敛行为。该分析提供了基于费舍尔信息的非渐近平稳性保证，完全兼容加权退火、潜在非对数凹似然函数以及非完美分数网络的联合存在条件。我们在多个具有线性和非线性前向模型的典型逆问题上验证了PMC算法的性能。实验结果表明，PMC显著提升了重建质量，并实现了高保真的不确定性量化。