Local MALA-within-Gibbs for Bayesian image deblurring with total variation prior

TL;DR

This paper introduces a dimension-independent MALA-within-Gibbs sampling method for Bayesian image deblurring with TV prior, enabling efficient high-resolution image reconstruction by exploiting local structures.

Contribution

It develops a novel MALA-within-Gibbs algorithm that leverages local image structures and approximates TV to achieve dimension-independent convergence in high-dimensional Bayesian inference.

Findings

The proposed MLwG algorithm has dimension-independent acceptance rates.

MLwG demonstrates superior performance over standard MALA in high-resolution tests.

The method efficiently handles local dependencies in the posterior distribution.

Abstract

We consider Bayesian inference for image deblurring with total variation (TV) prior. Since the posterior is analytically intractable, we resort to Markov chain Monte Carlo (MCMC) methods. However, since most MCMC methods significantly deteriorate in high dimensions, they are not suitable to handle high resolution imaging problems. In this paper, we show how low-dimensional sampling can still be facilitated by exploiting the sparse conditional structure of the posterior. To this end, we make use of the local structures of the blurring operator and the TV prior by partitioning the image into rectangular blocks and employing a blocked Gibbs sampler with proposals stemming from the Metropolis-Hastings adjusted Langevin Algorithm (MALA). We prove that this MALA-within-Gibbs (MLwG) sampling algorithm has dimension-independent block acceptance rates and dimension-independent convergence rate. In order to apply the MALA proposals, we approximate the TV by a smoothed version, and show that the introduced approximation error is evenly distributed and dimension-independent. Since the posterior is a Gibbs density, we can use the Hammersley-Clifford Theorem to identify the posterior conditionals which only depend locally on the neighboring blocks. We outline computational strategies to evaluate the conditionals, which are the target densities in the Gibbs updates, locally and in parallel. In two numerical experiments, we validate the dimension-independent properties of the MLwG algorithm and demonstrate its superior performance over MALA.