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Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling

Elhadji C. Faye, Mame Diarra Fall, Nicolas Dobigeon

TL;DR

This paper addresses ill-posed image inversion by placing Regularization by Denoising (RED) into a Bayesian framework and deriving a posterior that couples a data-fidelity term with a RED-based prior. It introduces Langevin-within-split Gibbs sampling (LwSGS) using asymptotically exact data augmentation (AXDA) to sample from the RED posterior, decoupling the likelihood and the RED prior via an auxiliary variable. The authors provide convergence guarantees and a bound on the bias due to the Langevin step, and they draw connections between AXDA and PnP-ULA, showing equivalences under MMSE denoisers. Empirical results on deblurring, inpainting, and super-resolution demonstrate competitive performance and, crucially, provide uncertainty quantification through posterior samples, reinforcing the practical value of data-driven priors in Bayesian imaging.

Abstract

This paper introduces a Bayesian framework for image inversion by deriving a probabilistic counterpart to the regularization-by-denoising (RED) paradigm. It additionally implements a Monte Carlo algorithm specifically tailored for sampling from the resulting posterior distribution, based on an asymptotically exact data augmentation (AXDA). The proposed algorithm is an approximate instance of split Gibbs sampling (SGS) which embeds one Langevin Monte Carlo step. The proposed method is applied to common imaging tasks such as deblurring, inpainting and super-resolution, demonstrating its efficacy through extensive numerical experiments. These contributions advance Bayesian inference in imaging by leveraging data-driven regularization strategies within a probabilistic framework.

Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling

TL;DR

This paper addresses ill-posed image inversion by placing Regularization by Denoising (RED) into a Bayesian framework and deriving a posterior that couples a data-fidelity term with a RED-based prior. It introduces Langevin-within-split Gibbs sampling (LwSGS) using asymptotically exact data augmentation (AXDA) to sample from the RED posterior, decoupling the likelihood and the RED prior via an auxiliary variable. The authors provide convergence guarantees and a bound on the bias due to the Langevin step, and they draw connections between AXDA and PnP-ULA, showing equivalences under MMSE denoisers. Empirical results on deblurring, inpainting, and super-resolution demonstrate competitive performance and, crucially, provide uncertainty quantification through posterior samples, reinforcing the practical value of data-driven priors in Bayesian imaging.

Abstract

This paper introduces a Bayesian framework for image inversion by deriving a probabilistic counterpart to the regularization-by-denoising (RED) paradigm. It additionally implements a Monte Carlo algorithm specifically tailored for sampling from the resulting posterior distribution, based on an asymptotically exact data augmentation (AXDA). The proposed algorithm is an approximate instance of split Gibbs sampling (SGS) which embeds one Langevin Monte Carlo step. The proposed method is applied to common imaging tasks such as deblurring, inpainting and super-resolution, demonstrating its efficacy through extensive numerical experiments. These contributions advance Bayesian inference in imaging by leveraging data-driven regularization strategies within a probabilistic framework.
Paper Structure (26 sections, 10 theorems, 78 equations, 3 figures, 4 tables, 1 algorithm)

This paper contains 26 sections, 10 theorems, 78 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Bayesian formulation of RED inversion
  3. Regularization by denoising (RED)
  4. Probabilistic counterpart of RED
  5. RED posterior distribution
  6. Proposed algorithm
  7. Langevin-within-split Gibbs sampler
  8. Related Monte Carlo algorithms
  9. Theoretical analysis
  10. Revisiting PnP-ULA and AXDA from the RED paradigm
  11. Experiments
  12. Experimental setup
  13. Compared methods
  14. Figures-of-merit
  15. Experimental results
  16. ...and 11 more sections

Key Result

Proposition 1

If Assumption ass:well_defined_density and Conditions RedCond:C3--RedCond:C4 hold, then and $p_{\textrm{red}}(\cdot)$ in prior defines a proper pdf.

Figures (3)

  • Figure 1: FFHQ data set: images recovered by the compared methods for deblurring (top), inpainting (middle) and super-resolution (bottom).
  • Figure 2: ImageNet data set: images recovered by the compared methods for deblurring (top), inpainting (middle) and super-resolution (bottom).
  • Figure 3: FFHQ data set: (absolute) autocorrelation function (ACF) of the samples generated by the compared algorithms for deblurring (left), inpainting (middle) and super-resolution (right).

Theorems & Definitions (22)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 1
  • proof
  • proof
  • Lemma 2
  • ...and 12 more