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Plug and Play Splitting Techniques for Poisson Image Restoration

Alessandro Benfenati

TL;DR

The paper tackles Poisson image restoration by formulating a variational objective with a KL fidelity term $KL(Hx+b,g)$ and a regularizer $R(x)$ over $x\in\mathbb{R}^n_+$, a setting where Gaussian-based theory fails due to non-Lipschitz Poisson dynamics. It extends PIDSplit+ by embedding the Plug and Play ADMM framework and substituting the regularization proximal step with a firmly nonexpansive Gaussian denoiser, yielding a closed-form deblurring update and guaranteed convergence. Convergence is established within the ADMM core, relying on a denoiser whose resolvent is maximally monotone, and the method demonstrates robust performance under heavy blur and noise with reduced computational cost relative to inner-solver baselines. Extensive experiments on Set5 and BSD500 across multiple blur operators and noise levels show competitive PSNR/SSIM, controlled runtimes, and resilience to iteration budgets, highlighting practical impact for Poisson image restoration tasks.

Abstract

Plug and Play (PnP) methods achieve remarkable results in the framework of image restoration problems for Gaussian data. Nonetheless, the theory available for the Gaussian case cannot be extended to the Poisson case, due to the non-Lipschitz gradient of the fidelity function, the Kullback-Leibler functional, or the absence of closed-form solution for the proximal operator of such term, leading to employ iterative solvers for the inner subproblem. In this work we extend the idea of PIDSplit+ algorithm, exploiting the Alternating Direction Method of Multipliers, to PnP scheme: this allows to provide a closed form solution for the deblurring step, with no need for iterative solvers. The convergence of the method is assured by employing a firmly non expansive denoiser. The proposed method, namely PnPSplit+, is tested on different Poisson image restoration problems, showing remarkable performance even in presence of high noise level and severe blurring conditions.

Plug and Play Splitting Techniques for Poisson Image Restoration

TL;DR

The paper tackles Poisson image restoration by formulating a variational objective with a KL fidelity term and a regularizer over , a setting where Gaussian-based theory fails due to non-Lipschitz Poisson dynamics. It extends PIDSplit+ by embedding the Plug and Play ADMM framework and substituting the regularization proximal step with a firmly nonexpansive Gaussian denoiser, yielding a closed-form deblurring update and guaranteed convergence. Convergence is established within the ADMM core, relying on a denoiser whose resolvent is maximally monotone, and the method demonstrates robust performance under heavy blur and noise with reduced computational cost relative to inner-solver baselines. Extensive experiments on Set5 and BSD500 across multiple blur operators and noise levels show competitive PSNR/SSIM, controlled runtimes, and resilience to iteration budgets, highlighting practical impact for Poisson image restoration tasks.

Abstract

Plug and Play (PnP) methods achieve remarkable results in the framework of image restoration problems for Gaussian data. Nonetheless, the theory available for the Gaussian case cannot be extended to the Poisson case, due to the non-Lipschitz gradient of the fidelity function, the Kullback-Leibler functional, or the absence of closed-form solution for the proximal operator of such term, leading to employ iterative solvers for the inner subproblem. In this work we extend the idea of PIDSplit+ algorithm, exploiting the Alternating Direction Method of Multipliers, to PnP scheme: this allows to provide a closed form solution for the deblurring step, with no need for iterative solvers. The convergence of the method is assured by employing a firmly non expansive denoiser. The proposed method, namely PnPSplit+, is tested on different Poisson image restoration problems, showing remarkable performance even in presence of high noise level and severe blurring conditions.

Paper Structure

This paper contains 15 sections, 2 theorems, 26 equations, 7 figures, 5 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation.
  3. Plug and Play Methods
  4. Proposed Method
  5. Previous Work: PIDSPLIT+
  6. PnPSplit$^{+}$
  7. Numerical Experiments
  8. On the choice of $\beta\gamma$
  9. Comparison with State-of-the-Art Algorithms
  10. Severely Corrupted Images
  11. Other Blurring Operators
  12. Adaptive strategy for $\gamma$
  13. Performance without Convergence Guarantees
  14. Conclusion
  15. Acknowledgments.

Key Result

Proposition 1

For any starting point and for any $\gamma\in\mathbb{R}^+$ the sequences $\{\bm{\lambda}^{k}\}_k$ and $\{{\mathbf{w}}^{k}\}_k$ generated by al:ADMM converge. The sequence $\{{\mathbf{x}}^{k}\}_k$ calculated by al:ADMM converges to a solution of the primal problem eq:varProb if one of the following c

Figures (7)

  • Figure 1: Sketch of the network used in the experiments. The network consists of 20 blocks of convolutional and LeakyReLU layers, with a skip connection between the input and the output. The number of channels is denoted by $C$: $C=3$ for RGB images, $C=1$ for black&white images.
  • Figure 2: Visual inspection of the recovered images provided by PnPSplit$^{+}$ and B-PnP algorithms. First column: ground truth images. Second column: simulated recorded data, perturbed with a Gaussian PSF and Poisson noise at level 20. Third column: B-PnP reconstruction. Fourth column: PnPSplit$^{+}$ reconstruction. Both algorithms have run for 2500 iterations. The B-PnP reconstructions suffer from the presence of some artifacts, while PnPSplit$^{+}$ ones presents more smooth results.
  • Figure 3: PSNR assessment on 20 images of the BSDS500 dataset. The panels shows the results for PnPSplit$^{+}$ and B-PnP when the maximum number of iteration is set to 1000 (left) and to 400 (right). The dots represent the average PSNR of the recovered images for each method. PnPSplit$^{+}$ reveals to be quite robust with respect to the maximum number of iterations, and even when B-PnP runs with the optimal number of iterations PnPSplit$^{+}$ is competitive. The curved behaviour is due to the cubic spline used for plotting the results.
  • Figure 4: Comparinson on the reconstruction achieved by PnPSplit$^{+}$, QAB-PNP and P$^4$IP, respectively on the second, third and fourth column. The first column shows the corrupted data ${\mathbf{g}}$. The results of P$^4$IP are shown in a different scale: while PnPSplit$^{+}$ and QAB provide reconstructions in [0,1], P$^4$IP failed to recover images with values higher than 0.04 in all cases.
  • Figure 5: image results when the perturbation on the recorded data is particularly strong, in terms of noise level or blurring. First row: reconstructions obtained for a PSF with $\sigma=1$ and noise level set to 5. Second row: reconstructions obtained for a PSF with $\sigma=2.5$ and noise level set to 20.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Remark 1
  • Proposition 1: Setzer2012
  • Remark 2
  • Proposition 2
  • proof
  • Remark 3
  • Remark 4