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

Area of Science:

• Computational imaging
• Bayesian inference
• Image processing

Background:

• Traditional image inversion methods often struggle with ill-posed problems.
• Regularization-by-denoising (RED) offers data-driven priors but lacks a fully probabilistic formulation.
• Integrating RED into a Bayesian framework is crucial for robust uncertainty quantification.

Purpose of the Study:

• To develop a probabilistic Bayesian framework for image inversion.
• To introduce a novel Monte Carlo sampling algorithm for the derived posterior distribution.
• To demonstrate the framework's effectiveness on various image restoration tasks.

Main Methods:

• Derivation of a probabilistic counterpart to the regularization-by-denoising (RED) paradigm.
• Implementation of a Monte Carlo algorithm based on asymptotically exact data augmentation (AXDA).
• The algorithm is an approximate split Gibbs sampling (SGS) incorporating a Langevin Monte Carlo step.

Main Results:

• The proposed Bayesian framework successfully handles image inversion tasks.
• Extensive numerical experiments validate the efficacy of the method in deblurring, inpainting, and super-resolution.
• The developed Monte Carlo algorithm provides efficient posterior sampling.

Conclusions:

• The study establishes a robust Bayesian framework for image inversion by extending the RED paradigm.
• The novel sampling algorithm enables practical application of data-driven regularization within a probabilistic context.
• This work advances Bayesian inference in computational imaging, offering improved performance and uncertainty estimation.