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

Rafael Flock, Shuigen Liu, Yiqiu Dong, Xin T. Tong

TL;DR

This work addresses uncertainty quantification for Bayesian image deblurring with a total variation prior by developing a local MALA-within-Gibbs sampler that exploits sparse, neighborhood-based conditional structure. By smoothing the TV prior and proving that the resulting smoothing error distributes uniformly across blocks, the authors obtain dimension-independent block acceptance and convergence properties, while enabling efficient local and parallel updates. Numerical experiments on Cameraman and House demonstrate that MLwG outperforms global MALA in effective sample size and wall-clock time, particularly as image resolution grows. The method offers scalable, uncertainty-aware image recovery and points to GPU-accelerated extensions and broader priors for future work.

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.

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

TL;DR

This work addresses uncertainty quantification for Bayesian image deblurring with a total variation prior by developing a local MALA-within-Gibbs sampler that exploits sparse, neighborhood-based conditional structure. By smoothing the TV prior and proving that the resulting smoothing error distributes uniformly across blocks, the authors obtain dimension-independent block acceptance and convergence properties, while enabling efficient local and parallel updates. Numerical experiments on Cameraman and House demonstrate that MLwG outperforms global MALA in effective sample size and wall-clock time, particularly as image resolution grows. The method offers scalable, uncertainty-aware image recovery and points to GPU-accelerated extensions and broader priors for future work.

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.
Paper Structure (28 sections, 6 theorems, 84 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 28 sections, 6 theorems, 84 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Outline
  5. Preliminaries
  6. Problem setting
  7. MALA-within-Gibbs (MLwG)
  8. Posterior smoothing with dimension-independent error
  9. Dimension-independent block acceptance and convergence rate
  10. Local and parallel MLwG algorithm
  11. Local block likelihood
  12. Local block prior
  13. Local & parallel MLwG algorithm
  14. Numerical examples
  15. Cameraman
  16. ...and 13 more sections

Key Result

Theorem 3.2

\newlabelthm:TV_ApproxErr0 Consider the target distribution $\pi$ defined in eq:pos and its smooth approximation $\pi_\varepsilon$ in eq:smooth_pos_expl. Assume that ${\boldsymbol{A}}^{\rm T} {\boldsymbol{A}}$ is $c$-diagonal block dominant as in def:local. Suppose $\frac{\lambda}{\delta} \geq \fr Here, $\pi_i$ and $\pi_{\varepsilon,i}$ denote the marginal distributions of $\pi$ and $\pi_\varepsi

Figures (7)

  • Figure 1: Example of a partition with 16 blocks. When updating a block, the neighboring blocks must be fixed. This is exemplified here for block $6$, where the hatched blocks must be fixed. A possible parallel updated scheme for MLwG is indicated by the index sets $\mathcal{U}_l,\,l=1\dots4$, where for a given $l$, all blocks associated to $\mathcal{U}_l$ can be updated in parallel.
  • Figure 1: Examples of extended blocks for an interior block (left) and a corner block (right) in order to compute the local block likelihood.
  • Figure 1: Left: True “cameraman” image and partition into deblurring problems of different sizes (black frames). All sections are again partitioned into blocks of equal size $64\times 64$ (white frames). Right: Data obtained via Gaussian blur and additive Gaussian noise.
  • Figure 2: MAP estimates, sample means, and widths of the 90% sample CIs for $\varepsilon\in\{10^{-3}, 10^{-5}, 10^{-7}\}$. The sampling results are obtained with the local & parallel MLwG given in \ref{['alg:loc_TV']}.
  • Figure 3: Block acceptance rates of MLwG for the different problem sizes in %. The acceptance rates are listed according to the problem sizes in the order shown on the right.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1
  • Remark 3.1
  • Theorem 3.2
  • Proof 1
  • Proposition 4.1
  • Remark 4.2
  • Proposition 4.3
  • Remark 4.4
  • Lemma 5.1
  • Proof 2
  • ...and 5 more