Random spanning tree Markov random field priors for Bayesian inverse problems in imaging

TL;DR

This paper introduces a novel Bayesian prior for imaging inverse problems using random spanning trees to better preserve edges and reduce contrast loss, combined with an efficient Gibbs sampling method.

Contribution

It proposes a hyperprior based on random spanning trees for Markov random fields, enhancing edge preservation in Bayesian imaging.

Findings

Sparse regularization of edges improves image quality.

Random spanning tree priors reduce contrast loss compared to standard methods.

The Gibbs sampler effectively explores the posterior distribution.

Abstract

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.