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Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling (2402.12292v1)

Published 19 Feb 2024 in stat.ML, cs.CV, and cs.LG

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.

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Citations (4)
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Summary

  • The paper proposes a novel Bayesian framework that integrates RED via a Langevin-within-split Gibbs sampler for effective inverse problem solving.
  • It employs an asymptotically exact data augmentation scheme to decouple fidelity and regularization, ensuring robust convergence in image restoration tasks.
  • Empirical results demonstrate competitive performance in deblurring, inpainting, and super-resolution while providing valuable uncertainty quantification.

Exploring the Intersection of Bayesian Inference and RED in Imaging Inverse Problems

In the constantly evolving field of computational imaging, the quest for robust and versatile methodologies for image restoration and reconstruction remains paramount. The recent contribution titled "Regularization by denoising: Bayesian model and Langevin-within-split Gibbs sampling" offers an intriguing intersection of Bayesian inference principles with the regularization-by-denoising (RED) strategy, providing a comprehensive data-driven approach for tackling various imaging inverse problems.

The Evolution of RED within Bayesian Frameworks

The concept of RED has garnered significant attention due to its ability to leverage advanced denoising engines for the regularization of inverse problems. Traditionally applied in a deterministic optimization context, RED's ability to incorporate complex image priors without explicitly defining them has made it a popular choice for various imaging tasks. The leap to a probabilistic setting involves crafting a Bayesian model that encapsulates the essence of RED by defining an appropriate prior distribution from the RED potential. This novel integration facilitates the application of data-driven regularization within a probabilistic framework, enhancing both theoretical and practical aspects of image restoration.

A Tailored Monte Carlo Algorithm: Langevin-within-Split Gibbs Sampling

The derivation of a specific Monte Carlo algorithm is a testament to the innovative fusion of RED within Bayesian inference. Given the unique characteristics of the RED posterior distribution, standard sampling methods fall short of efficiently exploring the solution space. The proposed Langevin-within-split Gibbs (LwSGS) algorithm, rooted in an asymptotically exact data augmentation (AXDA) scheme, emerges as a sophisticated solution. By decoupling the data fidelity and regularization terms through a split Gibbs sampler framework and integrating a Langevin Monte Carlo step, LwSGS exhibits a remarkable capability to traverse the posterior distribution effectively.

Theoretical Insights and Empirical Validation

The comprehensive theoretical analysis accompanying the proposed method establishes strong convergence guarantees, ensuring the generated samples approximate the target distribution closely. Through extensive numerical experiments across common imaging tasks such as deblurring, inpainting, and super-resolution, the method demonstrates competitive performance against both variational and Monte Carlo benchmarks. The inclusion of a deep network-based denoiser, specifically DRUNet, without further fine-tuning, highlights the method's adaptability and robustness in handling real-world data.

Towards Comprehensive Solution Characterization

One of the distinct advantages of the proposed Bayesian RED framework is its ability to provide a holistic view of the solution space, including uncertainty quantification. This added layer of insight is invaluable for various applications where understanding the confidence in the reconstructed images can guide subsequent decision-making processes. The method's ability to offer detailed uncertainty maps alongside high-quality reconstructions is a notable step forward in the computational imaging domain.

Concluding Remarks

The integration of RED into a Bayesian framework through the novel LwSGS algorithm opens up new pathways for leveraging advanced denoising engines in solving inverse problems. This work not only enhances the theoretical foundations bridging deterministic and probabilistic approaches but also sets a new benchmark in data-driven image restoration and reconstruction methodologies. As we move forward, the potential for further exploration and refinement of this approach promises exciting developments in the field of computational imaging.

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