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Provably Convergent Plug & Play Linearized ADMM, applied to Deblurring Spatially Varying Kernels

Charles Laroche, Andrés Almansa, Eva Coupeté, Matias Tassano

TL;DR

This paper tackles inverse image restoration under complex degradations by formulating MAP estimation with $E(x)=h(Hx)+\lambda f(x)$ and introducing a Provably Convergent Plug & Play framework based on Linearized-ADMM (PnP-LADMM). By linearizing the $x$-update with a curvature parameter $L_x$, it bypasses the costly proximal $\operatorname{prox}_{\alpha h(H\cdot)}$ while preserving convergence guarantees that hold under mild conditions, and allowing the denoiser to play the role of the regularizer without requiring Lipschitz constraints. The approach is validated on deblurring with spatially varying blur, showing that PnP-LADMM delivers competitive PSNR/SSIM/LPIPS scores with substantially faster runtimes than gradient- or CG-based PnP methods, and with convergence guarantees under weaker assumptions than prior work. The key contribution is a practical, convergent framework that extends Plug & Play to challenging degradations, enabling robust, efficient restoration using off-the-shelf denoisers such as DRUNet.

Abstract

Plug & Play methods combine proximal algorithms with denoiser priors to solve inverse problems. These methods rely on the computability of the proximal operator of the data fidelity term. In this paper, we propose a Plug & Play framework based on linearized ADMM that allows us to bypass the computation of intractable proximal operators. We demonstrate the convergence of the algorithm and provide results on restoration tasks such as super-resolution and deblurring with non-uniform blur.

Provably Convergent Plug & Play Linearized ADMM, applied to Deblurring Spatially Varying Kernels

TL;DR

This paper tackles inverse image restoration under complex degradations by formulating MAP estimation with and introducing a Provably Convergent Plug & Play framework based on Linearized-ADMM (PnP-LADMM). By linearizing the -update with a curvature parameter , it bypasses the costly proximal while preserving convergence guarantees that hold under mild conditions, and allowing the denoiser to play the role of the regularizer without requiring Lipschitz constraints. The approach is validated on deblurring with spatially varying blur, showing that PnP-LADMM delivers competitive PSNR/SSIM/LPIPS scores with substantially faster runtimes than gradient- or CG-based PnP methods, and with convergence guarantees under weaker assumptions than prior work. The key contribution is a practical, convergent framework that extends Plug & Play to challenging degradations, enabling robust, efficient restoration using off-the-shelf denoisers such as DRUNet.

Abstract

Plug & Play methods combine proximal algorithms with denoiser priors to solve inverse problems. These methods rely on the computability of the proximal operator of the data fidelity term. In this paper, we propose a Plug & Play framework based on linearized ADMM that allows us to bypass the computation of intractable proximal operators. We demonstrate the convergence of the algorithm and provide results on restoration tasks such as super-resolution and deblurring with non-uniform blur.
Paper Structure (19 sections, 9 theorems, 60 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 9 theorems, 60 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model
  3. Linearized-ADMM (LADMM)
  4. Convergence
  5. Plug & Play linearized-ADMM (PnP-LADMM)
  6. Experiments
  7. Datasets
  8. Compared methods
  9. Results
  10. Conclusion
  11. Convergence of linearized-ADMM
  12. Application to PnP-LADMM
  13. Proof of Proposition \ref{['prop:pnp-denoisers']}
  14. Proximal Gradient Step Denoiser.
  15. MMSE Denoiser.
  16. ...and 4 more sections

Key Result

Theorem 1

Under Assumption assump:1, for linearized-ADMM with hyper parameters such that: then the sequence $\{\mathcal{L}_\beta(x_k,z_k,w_k)\}$ is convergent and the primal residues $\|x_{k+1}-x_k\|$, $\|z_{k+1}-z_k\|$ and the dual residue $\|w_{k+1}-w_k\|$ converge to 0 as k approaches infinity. We also have that the sequence ${(x_k,z_k,w_k)}$ satisfies and that there exists If in addition $f$ is diffe

Figures (5)

  • Figure 1: Example of sample from the testset with restoration result.
  • Figure 2: Convergence speed of the different methods, we use 40 images with spatially-varying blur and Gaussian noise with $\sigma=10/255$.
  • Figure 3: Parameters influence
  • Figure 4: Visual results for deblurring spatially-varying blur.
  • Figure 5: Convergence of the splitting variable $\|x_{k+1}-x_k\|$.

Theorems & Definitions (19)

  • Theorem 1
  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 9 more