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Unfolded proximal neural networks for robust image Gaussian denoising

Hoang Trieu Vy Le, Audrey Repetti, Nelly Pustelnik

TL;DR

The paper addresses Gaussian image denoising by reframing MAP estimation with a proximal objective $F(\mathbf{x})=\tfrac{1}{2}\|\mathbf{x}-\mathbf{z}\|_2^2+\nu g(\mathbf{D}\mathbf{x})+\iota_{C}(\mathbf{x})$ and solves it via proximal schemes. It introduces a unified unfolded neural-network framework (PNNs) based on dual-FB and primal–dual CP algorithms, augmented with inertial variants to enable skip connections, and two learning strategies (LNO, LFO) to balance convergence guarantees with expressive power. The authors instantiate four unfolded schemes (DDFB, DDiFB, DCP, DScCP) and their LNO/LFO variants, and demonstrate that they achieve competitive denoising performance with dramatically lighter architectures than DRUnet, while offering enhanced robustness as measured by Lipschitz-like properties and stability in plug-and-play restoration. Extensive experiments on Gaussian and non-Gaussian noise, plus PnP deblurring, show that inertia and convergence-aware learning yield stronger robustness and practical performance, with DDFB-LNO and DScCP-LNO often providing the best trade-offs. The work highlights the practical impact of optimization-informed architecture design, offering scalable, robust denoisers suitable for broader image restoration pipelines.

Abstract

A common approach to solve inverse imaging problems relies on finding a maximum a posteriori (MAP) estimate of the original unknown image, by solving a minimization problem. In thiscontext, iterative proximal algorithms are widely used, enabling to handle non-smooth functions and linear operators. Recently, these algorithms have been paired with deep learning strategies, to further improve the estimate quality. In particular, proximal neural networks (PNNs) have been introduced, obtained by unrolling a proximal algorithm as for finding a MAP estimate, but over a fixed number of iterations, with learned linear operators and parameters. As PNNs are based on optimization theory, they are very flexible, and can be adapted to any image restoration task, as soon as a proximal algorithm can solve it. They further have much lighter architectures than traditional networks. In this article we propose a unified framework to build PNNs for the Gaussian denoising task, based on both the dual-FB and the primal-dual Chambolle-Pock algorithms. We further show that accelerated inertial versions of these algorithms enable skip connections in the associated NN layers. We propose different learning strategies for our PNN framework, and investigate their robustness (Lipschitz property) and denoising efficiency. Finally, we assess the robustness of our PNNs when plugged in a forward-backward algorithm for an image deblurring problem.

Unfolded proximal neural networks for robust image Gaussian denoising

TL;DR

The paper addresses Gaussian image denoising by reframing MAP estimation with a proximal objective and solves it via proximal schemes. It introduces a unified unfolded neural-network framework (PNNs) based on dual-FB and primal–dual CP algorithms, augmented with inertial variants to enable skip connections, and two learning strategies (LNO, LFO) to balance convergence guarantees with expressive power. The authors instantiate four unfolded schemes (DDFB, DDiFB, DCP, DScCP) and their LNO/LFO variants, and demonstrate that they achieve competitive denoising performance with dramatically lighter architectures than DRUnet, while offering enhanced robustness as measured by Lipschitz-like properties and stability in plug-and-play restoration. Extensive experiments on Gaussian and non-Gaussian noise, plus PnP deblurring, show that inertia and convergence-aware learning yield stronger robustness and practical performance, with DDFB-LNO and DScCP-LNO often providing the best trade-offs. The work highlights the practical impact of optimization-informed architecture design, offering scalable, robust denoisers suitable for broader image restoration pipelines.

Abstract

A common approach to solve inverse imaging problems relies on finding a maximum a posteriori (MAP) estimate of the original unknown image, by solving a minimization problem. In thiscontext, iterative proximal algorithms are widely used, enabling to handle non-smooth functions and linear operators. Recently, these algorithms have been paired with deep learning strategies, to further improve the estimate quality. In particular, proximal neural networks (PNNs) have been introduced, obtained by unrolling a proximal algorithm as for finding a MAP estimate, but over a fixed number of iterations, with learned linear operators and parameters. As PNNs are based on optimization theory, they are very flexible, and can be adapted to any image restoration task, as soon as a proximal algorithm can solve it. They further have much lighter architectures than traditional networks. In this article we propose a unified framework to build PNNs for the Gaussian denoising task, based on both the dual-FB and the primal-dual Chambolle-Pock algorithms. We further show that accelerated inertial versions of these algorithms enable skip connections in the associated NN layers. We propose different learning strategies for our PNN framework, and investigate their robustness (Lipschitz property) and denoising efficiency. Finally, we assess the robustness of our PNNs when plugged in a forward-backward algorithm for an image deblurring problem.
Paper Structure (16 sections, 6 theorems, 31 equations, 13 figures, 4 tables)

This paper contains 16 sections, 6 theorems, 31 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Denoising (accelerated) proximal schemes
  3. Proposed unfolded denoising NNs
  4. Primal-dual building block iteration
  5. Arrow-Hurwicz unfolded building block
  6. Proposed unfolded strategies
  7. Proposed learning strategies
  8. Experiments
  9. Denoising training setting
  10. Architecture comparison
  11. PNN denoising performance comparison
  12. PNN robustness comparison
  13. PNN robustness comparison on non-Gaussian noise
  14. Robustness comparison: Image deblurring with PnP
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $(\mathrm{u}_k, \mathrm{v}_k)_{k\in \mathbb{N}}$ be generated by eq:fista. Assume that one of the following conditions is satisfied. Then we have where $\widehat{\mathrm{x}}_{\mathrm{MAP}}$ is defined in prob:2-terms-minimization-eq:prox-function.

Figures (13)

  • Figure 1: Top: Architecture of the proposed DAH-Unified block for the $k$-th layer. Linearities, biases, and activation functions are shown in blue, green and red, respectively. Bottom: Inertial step for ScCP (top) and DiFB (bottom), for the $k$-th layer.
  • Figure 2: PNN denoising performance on Gaussian noise (Training Setting 1). Average PSNR obtained with the proposed PNNs, on $100$ images from BSD500 degraded with noise level $\delta=0.08$. Results are shown for $J\in \{8,16,32,64\}$ and $K\in \{5,10,15,20,25\}$. Top row: LNO settings. Bottom row: LFO settings.
  • Figure 3: PNN denoising performance on Gaussian noise (Training Setting 2). PSNR values obtained with the proposed PNNs (with $(K,J) = (20, 64)$), for 20 images of BSDS500 validation set, degraded with noise level $\delta=0.05$.
  • Figure 4: Denoising performance on Gaussian noise (Training Setting 2). Example of denoised images (and PSNR values) for Gaussian noise $\delta=0.05$ obtained with BM3D, DRUnet, and the proposed DDFB-LNO and DScCP-LNO, with $(K,J)=(20,64)$.
  • Figure 5: PNN robustness comparison (Training Setting 1). Values $\chi = \max_{s \in \mathbb{J}} \|\operatorname{J} f_\Theta(\mathrm{z}_s) \|_S$ ($\log_2$ scale) for the proposed PNNs, with $J\in \{8,16,32,64\}$ and $K\in \{5,10,15,20,25\}$. Top row: LNO settings. Bottom row: LFO settings.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Theorem 2.1: Combettes_Vu_2010chambolle2015convergence
  • Theorem 2.2: chambolle2011first
  • Proposition 3.1
  • proof
  • Corollary 1: Limit case for deep unfolded NNs
  • Proposition 4.1
  • Theorem 4.1