Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (Normalizing Flow-based Unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pre-trained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and non-asymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography (CT) reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.

翻译：贝叶斯方法用于求解逆问题是经典方法的有力替代方案，因其能够量化解的不确定性。近年来，基于数据的逆问题求解技术凭借其优越的表征能力取得了显著成功。本研究将数据驱动模型融入一类基于朗之万采样的算法中，用于成像逆问题的贝叶斯推断。具体而言，我们提出NF-ULA（基于归一化流的非调整朗之万算法），该方法通过学习归一化流（NF）作为图像先验。采用NF学习先验的原因在于：对数先验的可处理闭式表达式可通过自动求导库实现微分运算。本算法仅需一个基于归一化流的生成网络，该网络可独立于待求解的逆问题和前向算子进行预训练。我们通过分析NF-ULA算法的适定性和非渐近收敛性开展理论验证。在图像去模糊、图像修复及有限角度X射线计算机断层扫描（CT）重建等多项图像恢复问题中，所提NF-ULA算法的有效性得到验证。结果表明，对于严重病态逆问题，NF-ULA的性能优于对比方法。