Markov random fields are common prior distributions used in Bayesian inverse imaging problems. In particular, difference priors assign probability distributions to differences between neighbouring pixels, such as Gaussian, Laplace, or Cauchy distributions. Depending on the chosen difference distribution, these priors have smoothing or edge-preserving properties. In this work, we propose a hyperprior on the connectivity graph of the pixel grid in the form of a random spanning tree, i.e., a random connected graph with the minimal number of edges, thereby coupling continuous and discrete random variables in the prior. By using random spanning trees, only a sparse random subset of edges is regularized, which helps preserve edges in the image with reduced contrast loss compared to standard difference-based Markov random fields. We discuss how fractal-like interfaces arise in high-resolution prior samples due to the random-tree connectivity. Finally, we propose a Gibbs sampler that alternates between the discrete tree updates and continuous pixel updates to efficiently explore the posterior distribution. We apply the method to various standard test image restoration problems, including denoising, deblurring, and inpainting, to study the impact of the proposed prior in comparison with existing Markov random fields.

翻译：马尔可夫随机场是贝叶斯成像反问题中常用的先验分布。特别地，差分先验将概率分布赋予相邻像素之间的差值，例如高斯分布、拉普拉斯分布或柯西分布。根据所选差值分布的不同，这些先验具有平滑或边缘保持特性。本文提出一种基于像素网格连通图上的超先验，形式为随机生成树（即具有最少边数的随机连通图），从而在先验中耦合连续型和离散型随机变量。通过使用随机生成树，仅对稀疏随机子集中的边进行正则化，相较于标准基于差分的马尔可夫随机场，该方法有助于在降低对比度损失的同时保留图像边缘。我们讨论了高分辨率先验样本中因随机树连通性产生的分形界面。最后，提出一种吉布斯采样器，通过交替进行离散树更新与连续像素更新来高效探索后验分布。我们将该方法应用于多种标准测试图像恢复问题（包括去噪、去模糊和图像修补），通过与现有马尔可夫随机场的比较，研究所提出先验的影响。