Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie (2103.04715v6)

Abstract: Since the seminal work of Venkatakrishnan et al. in 2013, Plug & Play (PnP) methods have become ubiquitous in Bayesian imaging. These methods derive Minimum Mean Square Error (MMSE) or Maximum A Posteriori (MAP) estimators for inverse problems in imaging by combining an explicit likelihood function with a prior that is implicitly defined by an image denoising algorithm. The PnP algorithms proposed in the literature mainly differ in the iterative schemes they use for optimisation or for sampling. In the case of optimisation schemes, some recent works guarantee the convergence to a fixed point, albeit not necessarily a MAP estimate. In the case of sampling schemes, to the best of our knowledge, there is no known proof of convergence. There also remain important open questions regarding whether the underlying Bayesian models and estimators are well defined, well-posed, and have the basic regularity properties required to support these numerical schemes. To address these limitations, this paper develops theory, methods, and provably convergent algorithms for performing Bayesian inference with PnP priors. We introduce two algorithms: 1) PnP-ULA (Unadjusted Langevin Algorithm) for Monte Carlo sampling and MMSE inference; and 2) PnP-SGD (Stochastic Gradient Descent) for MAP inference. Using recent results on the quantitative convergence of Markov chains, we establish detailed convergence guarantees for these two algorithms under realistic assumptions on the denoising operators used, with special attention to denoisers based on deep neural networks. We also show that these algorithms approximately target a decision-theoretically optimal Bayesian model that is well-posed. The proposed algorithms are demonstrated on several canonical problems such as image deblurring, inpainting, and denoising, where they are used for point estimation as well as for uncertainty visualisation and quantification.

• The paper introduces the PnP-ULA method, uniting Langevin dynamics and Tweedie’s formula to achieve convergence-guaranteed Bayesian image reconstruction.
• It leverages state-of-the-art MMSE denoisers to implicitly define priors, demonstrating robust performance in deblurring and inpainting tasks.
• Empirical results reveal enhanced uncertainty quantification and reconstruction quality, setting the stage for scalable imaging research.

Bayesian Imaging Using Plug-and-Play Priors: The Intersection of Langevin Processes and Tweedie's Formula

The paper on "Bayesian Imaging Using Plug and Play (PnP) Priors: When Langevin Meets Tweedie" introduces a comprehensive framework for leveraging advanced plug-and-play priors in Bayesian imaging problems. The innovation in this research lies within the integration of stochastic computational methods, specifically Langevin dynamics, and inference strategies derived from Tweedie's formula, to address the inherent challenges of inverse imaging problems.

Inverse imaging problems, where the goal is to reconstruct an original image from degraded observations, are notably challenging due to their ill-posed nature. Traditionally, Bayesian frameworks provide a means of regularization by stipulating priors over solution spaces. The PnP paradigm is especially appealing as it implicitly defines priors through state-of-the-art image denoisers, such as those constructed using deep neural networks, thus allowing the combination of proximal optimization and deep learning strategies.

Theoretical Insights and Contributions

This paper contributes to the theoretical foundations necessary for incorporating plug-and-play priors within Bayesian imaging. The authors propose the PnP Unadjusted Langevin Algorithm (PnP-ULA), a stochastic method for Bayesian computation that accommodates MMSE (Minimum Mean Squared Error) denoisers. Importantly, the paper addresses the previously unresolved issue of algorithmic convergence, demonstrating convergence guarantees of PnP-ULA for image deblurring and inpainting, underpinned by realistic assumptions on the denoisers used.

A key theoretical insight presented is the alignment between the gradient of the prior and the denoising operator facilitated by Tweedie's identity. Through this identity, it is shown that the expectation of the denoising operation aligns with the score of the prior, making it possible to employ existing denoisers in iterative Bayesian methods without the need to directly define a complex prior distribution.

Numerical Experiments and Practical Implications

The paper's empirical section demonstrates the practical application of PnP-ULA across different imaging challenges, notably in image deblurring and inpainting tasks. The experiments underline the algorithm’s ability to provide not only effective point estimates but also rich uncertainty quantification. The denoisers' efficacy is reflected in the empirical results, which showcase convergence properties and efficacy in addressing prior-related information inadequacies in images with complex textures.

The results indicate that using state-of-the-art denoising neural networks as prior models within a PnP context offers a robust mechanism to achieve high-quality reconstructions. The uncertainty quantification capabilities further constitute a significant advancement, providing comprehensive insights for applications where the quality of information is pivotal, such as in medical imaging and remote sensing.

Future Directions and Considerations

This work opens up several avenues for future research. The robustness of PnP methods when combined with alternative state-of-the-art learning-based priors like generative adversarial networks and variational autoencoders is an area ripe for exploration. Additionally, adaptive strategies for denoiser training to ensure stability and rapid convergence across a wider array of problem domains remain an open challenge.

The integration of further regularization terms into the Bayesian framework, combining both implicit and explicit priors for enhanced flexibility and performance, is also promising. Moreover, the development of accelerated and scalable stochastic simulation methods, extending beyond current proximal approaches, could be transformative for large-scale imaging applications.

Overall, the paper provides a significant advancement in the domain of Bayesian imaging, setting a solid theoretical and practical foundation for future investigations into the applicability and efficacy of plug-and-play approaches in broader computational imaging contexts.