Solving Imaging Inverse Problems Using Plug-and-Play Denoisers: Regularization and Optimization Perspectives

TL;DR

The chapter addresses ill-posed imaging inverse problems by integrating powerful learned denoisers into classical variational and proximal-splitting frameworks, yielding plug-and-play (PnP) methods that decouple the forward model from the prior. It surveys foundational theory (proximal calculus, monotone operators, ADMM/DRS/PGD), convergence guarantees under non-expansive or structured denoisers, and practical training strategies to enforce these properties, while also connecting to Regularization-by-Denoising (RED) and Tweedie’s formula. A major emphasis is placed on convergence guarantees (fixed-point, KL, objective convergence) and on extensions to posterior sampling via score-based and diffusion-driven methods, enabling uncertainty quantification in imaging. The practical impact spans medical imaging, remote sensing, and computational microscopy by enabling high-fidelity reconstructions with principled uncertainty assessments and scalable, modular priors. The synthesis highlights open theoretical questions about non-expansive nonlinear denoisers, domain-adaptation of priors, and efficient diffusion-based posterior sampling at scale.

Abstract

Inverse problems lie at the heart of modern imaging science, with broad applications in areas such as medical imaging, remote sensing, and microscopy. Recent years have witnessed a paradigm shift in solving imaging inverse problems, where data-driven regularizers are used increasingly, leading to remarkably high-fidelity reconstruction. A particularly notable approach for data-driven regularization is to use learned image denoisers as implicit priors in iterative image reconstruction algorithms. This chapter presents a comprehensive overview of this powerful and emerging class of algorithms, commonly referred to as plug-and-play (PnP) methods. We begin by providing a brief background on image denoising and inverse problems, followed by a short review of traditional regularization strategies. We then explore how proximal splitting algorithms, such as the alternating direction method of multipliers (ADMM) and proximal gradient descent (PGD), can naturally accommodate learned denoisers in place of proximal operators, and under what conditions such replacements preserve convergence. The role of Tweedie's formula in connecting optimal Gaussian denoisers and score estimation is discussed, which lays the foundation for regularization-by-denoising (RED) and more recent diffusion-based posterior sampling methods. We discuss theoretical advances regarding the convergence of PnP algorithms, both within the RED and proximal settings, emphasizing the structural assumptions that the denoiser must satisfy for convergence, such as non-expansiveness, Lipschitz continuity, and local homogeneity. We also address practical considerations in algorithm design, including choices of denoiser architecture and acceleration strategies.

Figures (3)

• Figure 1: Example reconstructions for a test image, with PSNR to ground truth in brackets. The image is blurred with a $9\times 9$ uniform blur kernel, with subsequent 3% additive Gaussian noise. Observe that PnP-PGD and PnP-DRSdiff have the same eventual PSNR, due to targeting the same underlying functional.
• Figure 2: Residual convergence for deblurring on the CBSD10 dataset, with a uniform $9\times 9$ blur kernel and 3% additive Gaussian noise. Each solid line represents one image. We observe that while DPIR has slow residual convergence, the provable PnP methods all have a convergent behavior, often reaching their stopping criteria given by the change in objective value. In particular, the quasi-Newton PnP-LBFGS method converges very quickly within 100 iterations.
• Figure 3: PSNR curves for deblurring on the CBSD10 dataset, with uniform $9\times 9$ blur kernel and 3% additive Gaussian noise. Each solid line represents one image. We observe that the non-provable DPIR method gradually decreases in PSNR at later iterations, eventually leading to instability. In contrast, the provable PnP methods all have stable convergence curves, reaching their stopping criteria.

Theorems & Definitions (17)

• Proposition 2.1: rockafellar1997convexekeland_bookrockafellar2009variational
• Definition 2.2: Monotonicity
• Theorem 2.3: Forward-backward algorithm bauschke2011convex
• Theorem 2.4: Douglas--Rachford Splitting bauschke2011convex
• Theorem 2.5: Proximal Gradient Convergence beck2017first
• Theorem 2.6: hurault2024convergent
• Theorem 2.7: he2015convergence
• Theorem 3.1: Impossibility of explicit regularization reehorst2019regularization
• Lemma 3.2: reehorst2019regularization
• Theorem 3.3: reehorst2019regularization