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A Unified Plug-and-Play Algorithm with Projected Landweber Operator for Split Convex Feasibility Problems

Shuchang Zhang, Hongxia Wang

TL;DR

The paper addresses ill-posed inverse imaging problems by unifying Plug-and-Play priors with split convex feasibility through a learning-based fixed-point prior and the data-fidelity constraint $A x\in Q$. It introduces the PnP-PLO algorithm, which uses an extrapolated Landweber operator to avoid estimating the operator norm $\|A\|$ and to provide relaxed, adaptive step sizes via the extrapolated function $\tau(\mathbf{x})$ and Polyak stepping. Theoretical results establish convergence to $F=\mathrm{Fix}(T)\cap A^{-1}(Q)$ under mild assumptions and derive an $o(1/\sqrt{k})$ rate for the objective with Polyak steps. Empirically, PnP-PLO outperforms state-of-the-art methods (RED, RED-PRO, PnP-FBS) in image deblurring, super-resolution, and CS-MRI, demonstrating improved convergence speed and robustness across noise models. This framework offers a scalable, theory-backed plug-and-play approach applicable to a broad class of inverse imaging tasks without reliance on exact operator-norm estimates.

Abstract

In recent years Plug-and-Play (PnP) methods have achieved state-of-the-art performance in inverse imaging problems by replacing proximal operators with denoisers. Based on the proximal gradient method, some theoretical results of PnP have appeared, where appropriate step size is crucial for convergence analysis. However, in practical applications, applying PnP methods with theoretically guaranteed step sizes is difficult, and these algorithms are limited to Gaussian noise. In this paper,from a perspective of split convex feasibility problems (SCFP), an adaptive PnP algorithm with Projected Landweber Operator (PnP-PLO) is proposed to address these issues. Numerical experiments on image deblurring, super-resolution, and compressed sensing MRI experiments illustrate that PnP-PLO with theoretical guarantees outperforms state-of-the-art methods such as RED and RED-PRO.

A Unified Plug-and-Play Algorithm with Projected Landweber Operator for Split Convex Feasibility Problems

TL;DR

The paper addresses ill-posed inverse imaging problems by unifying Plug-and-Play priors with split convex feasibility through a learning-based fixed-point prior and the data-fidelity constraint . It introduces the PnP-PLO algorithm, which uses an extrapolated Landweber operator to avoid estimating the operator norm and to provide relaxed, adaptive step sizes via the extrapolated function and Polyak stepping. Theoretical results establish convergence to under mild assumptions and derive an rate for the objective with Polyak steps. Empirically, PnP-PLO outperforms state-of-the-art methods (RED, RED-PRO, PnP-FBS) in image deblurring, super-resolution, and CS-MRI, demonstrating improved convergence speed and robustness across noise models. This framework offers a scalable, theory-backed plug-and-play approach applicable to a broad class of inverse imaging tasks without reliance on exact operator-norm estimates.

Abstract

In recent years Plug-and-Play (PnP) methods have achieved state-of-the-art performance in inverse imaging problems by replacing proximal operators with denoisers. Based on the proximal gradient method, some theoretical results of PnP have appeared, where appropriate step size is crucial for convergence analysis. However, in practical applications, applying PnP methods with theoretically guaranteed step sizes is difficult, and these algorithms are limited to Gaussian noise. In this paper,from a perspective of split convex feasibility problems (SCFP), an adaptive PnP algorithm with Projected Landweber Operator (PnP-PLO) is proposed to address these issues. Numerical experiments on image deblurring, super-resolution, and compressed sensing MRI experiments illustrate that PnP-PLO with theoretical guarantees outperforms state-of-the-art methods such as RED and RED-PRO.
Paper Structure (19 sections, 5 theorems, 35 equations, 3 figures, 8 tables, 1 algorithm)

This paper contains 19 sections, 5 theorems, 35 equations, 3 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Implicit learning-based prior
  4. PnP theory
  5. SCFP
  6. PnP via Projected Landweber Operator
  7. Learning-based fixed point prior
  8. Projected Landweber Operator
  9. PnP-PLO
  10. Experiments
  11. Convergence
  12. Image deblurring and super-resolution
  13. Compressed sensing
  14. Conclusion
  15. Proof of Lemma \ref{['lemma: averaged demicontractive denoiser']}
  16. ...and 4 more sections

Key Result

Lemma 3.2

If $T$ is a $\alpha$- demicontractive denoiser, for any $w\in (0, 1-d)$, then $T_w = w T+(1-w)\mathrm{Id}$ is $\frac{w}{1-d}-$ averaged operator, i.e. there exist a nonexpansive operaotr $N$ such that $T_w = \frac{w}{1-d} N+(1-\frac{w}{1-d})\mathrm{Id}$.

Figures (3)

  • Figure 1: An illustration of the convergence comparison of Barbara degraded by a Gaussian PSF ($\sigma=\sqrt{2}$) compared with RED, RED-PRO, and PnP-FBS.
  • Figure 2: Deblurring of Starfish degraded with the indicated blur kernel and input noise level $0.01$.
  • Figure 3: Super-resolution (x2) of Parrot degraded with the indicated blur kernel and input noise level $0.01$.

Theorems & Definitions (16)

  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Definition 3.4
  • Theorem 3.5
  • Remark 3.6
  • Theorem 3.7
  • Definition 1.1
  • Proposition 1.2
  • proof
  • ...and 6 more