# Unfolded proximal neural networks for robust image Gaussian denoising

arXiv: https://arxiv.org/abs/2308.03139 · HTML: https://mlsanity.com/paper/2308.03139/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 51 references that are in the map.
- First posted on arXiv 6 Aug 2023.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Robust Learning with Jacobian Regularization](https://mlsanity.com/paper/1908.02729/) (2019)
- [Deep Equilibrium Architectures for Inverse Problems in Imaging](https://mlsanity.com/paper/2102.07944/) (2021)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [Improving DNN Robustness to Adversarial Attacks using Jacobian Regularization](https://mlsanity.com/paper/1803.08680/) (2018)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators](https://mlsanity.com/paper/1903.01014/) (2019)
- [Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems](https://mlsanity.com/paper/1406.5429/) (2014)
- [Deep Equilibrium Learning of Explicit Regularizers for Imaging Inverse Problems](https://mlsanity.com/paper/2303.05386/) (2023)
- [A deep primal-dual proximal network for image restoration](https://mlsanity.com/paper/2007.00959/) (2020)
- [Deep Unfolding of the DBFB Algorithm with Application to ROI CT Imaging with Limited Angular Density](https://mlsanity.com/paper/2209.13264/) (2022)
- [Alternative design of DeepPDNet in the context of image restoration](https://mlsanity.com/paper/2202.09810/) (2022)

## Built on by (most cited first, 7 of 7)

- [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025)
- [Stability Bounds for the Unfolded Forward-Backward Algorithm](https://mlsanity.com/paper/2412.17888/) (2024)
- [Meta-Prior: Meta learning for Adaptive Inverse Problem Solvers](https://mlsanity.com/paper/2311.18710/) (2023)
- [Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms](https://mlsanity.com/paper/2411.13276/) (2024)
- [LMS-Net: A Learned Mumford-Shah Network For Few-Shot Medical Image Segmentation](https://mlsanity.com/paper/2502.05473/) (2025)
- [Deep Unfolding Network for Nonlinear Multi-Frequency Electrical Impedance Tomography](https://mlsanity.com/paper/2507.16678/) (2025)
- [LHMCF-Net: A Learned Hyperbolic Mean Curvature Flow Network for Medical Images Segmentation](https://mlsanity.com/paper/2608.20942/) (2026)

## Abstract (arXiv)

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.