Blind image deconvolution refers to the problem of simultaneously estimating the blur kernel and the true image from a set of observations when both the blur kernel and the true image are unknown. Sometimes, additional image and/or blur information is available and the term semi-blind deconvolution (SBD) is used. We consider a recently introduced Bayesian conjugate hierarchical model for SBD, formulated on an extended cyclic lattice to allow a computationally scalable Gibbs sampler. In this article, we extend this model to the general SBD problem, rewrite the previously proposed Gibbs sampler so that operations are performed in the Fourier domain whenever possible, and introduce a new marginal Hamiltonian Monte Carlo (HMC) blur update, obtained by analytically integrating the blur-image joint conditional over the image. The cyclic formulation combined with non-trivial linear algebra manipulations allows a Fourier-based, scalable HMC update, otherwise complicated by the rigid constraints of the SBD problem. Having determined the padding size in the cyclic embedding through a numerical experiment, we compare the mixing and exploration behaviour of the Gibbs and HMC blur updates on simulated data and on a real geophysical seismic imaging problem where we invert a grid with $300\times50$ nodes, corresponding to a posterior with approximately $80,000$ parameters.

翻译：盲图像反卷积是指在模糊核与真实图像均未知的情况下，从一组观测数据中同时估计模糊核与真实图像的问题。当存在额外的图像和/或模糊信息时，该问题通常被称为半盲反卷积。本文研究一种近期提出的用于半盲反卷积的贝叶斯共轭分层模型，该模型构建于扩展的循环格点上，以实现计算可扩展的吉布斯采样器。本文中，我们将该模型推广至一般半盲反卷积问题，重构了先前提出的吉布斯采样器以尽可能在傅里叶域中执行运算，并引入了一种新的边缘哈密顿蒙特卡洛模糊核更新方法——该方法通过对图像与模糊核的联合条件分布进行解析积分得到。循环格点构造结合非平凡的线性代数操作，实现了基于傅里叶变换的可扩展哈密顿蒙特卡洛更新，而这一更新过程原本因半盲反卷积问题的刚性约束而难以实现。通过数值实验确定了循环嵌入所需的填充尺寸后，我们在模拟数据及真实地球物理地震成像问题上比较了吉布斯采样与哈密顿蒙特卡洛模糊核更新的混合与探索性能，其中反演网格包含 $300\times50$ 个节点，对应约 $80,000$ 个参数的后验分布。