In a great number of tasks in science and engineering, the goal is to infer an unknown image from a small number of measurements collected from a known forward model describing certain sensing or imaging modality. Due to resource constraints, this task is often extremely ill-posed, which necessitates the adoption of expressive prior information to regularize the solution space. Score-based diffusion models, due to its impressive empirical success, have emerged as an appealing candidate of an expressive prior in image reconstruction. In order to accommodate diverse tasks at once, it is of great interest to develop efficient, consistent and robust algorithms that incorporate {\em unconditional} score functions of an image prior distribution in conjunction with flexible choices of forward models. This work develops an algorithmic framework for employing score-based diffusion models as an expressive data prior in general nonlinear inverse problems. Motivated by the plug-and-play framework in the imaging community, we introduce a diffusion plug-and-play method (\textsf{DPnP}) that alternatively calls two samplers, a proximal consistency sampler based solely on the likelihood function of the forward model, and a denoising diffusion sampler based solely on the score functions of the image prior. The key insight is that denoising under white Gaussian noise can be solved {\em rigorously} via both stochastic (i.e., DDPM-type) and deterministic (i.e., DDIM-type) samplers using the unconditional score functions. We establish both asymptotic and non-asymptotic performance guarantees of \textsf{DPnP}, and provide numerical experiments to illustrate its promise in solving both linear and nonlinear image reconstruction tasks. To the best of our knowledge, \textsf{DPnP} is the first provably-robust posterior sampling method for nonlinear inverse problems using unconditional diffusion priors.

翻译：在科学与工程的众多任务中，目标是从已知前向模型（描述特定传感或成像模态）采集的少量测量数据中推断未知图像。由于资源限制，该任务通常极度病态，这需要借助表达性先验信息来正则化解空间。基于分数的扩散模型凭借其令人印象深刻的实证成功，已成为图像重建中表达性先验的极具吸引力的候选方案。为同时适应多样化任务，开发高效、一致且鲁棒的算法至关重要，这类算法需将图像先验分布的无条件分数函数与灵活的前向模型选择相结合。本文提出一种算法框架，将基于分数的扩散模型作为通用非线性逆问题中的表达性数据先验。受成像领域即插即用框架启发，我们引入了一种扩散即插即用方法（DPnP），该方法交替调用两个采样器：仅基于前向模型似然函数的近端一致性采样器，以及仅基于图像先验分数函数的去噪扩散采样器。关键洞察在于：通过无条件分数函数，白高斯噪声下的去噪问题可严格地使用随机（即DDPM型）和确定性（即DDIM型）采样器求解。我们建立了DPnP的渐近与非渐近性能保证，并通过数值实验阐明其在求解线性和非线性图像重建任务中的潜力。据我们所知，DPnP是首个针对非线性逆问题使用无条件扩散先验的可证明鲁棒后验采样方法。