# Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration

arXiv: https://arxiv.org/abs/2312.15701 · HTML: https://mlsanity.com/paper/2312.15701/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 99 references that are in the map.
- First posted on arXiv 25 Dec 2023.

## Builds on (most cited first, 30 of 50)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Image Super-Resolution Using Very Deep Residual Channel Attention Networks](https://mlsanity.com/paper/1807.02758/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Progressive Image Deraining Networks: A Better and Simpler Baseline](https://mlsanity.com/paper/1901.09221/) (2019)
- [Deep Unfolding Network for Image Super-Resolution](https://mlsanity.com/paper/2003.10428/) (2020)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Recurrent Squeeze-and-Excitation Context Aggregation Net for Single Image Deraining](https://mlsanity.com/paper/1807.05698/) (2018)
- [Blind Super-Resolution With Iterative Kernel Correction](https://mlsanity.com/paper/1904.03377/) (2019)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Spatial Attentive Single-Image Deraining with a High Quality Real Rain Dataset](https://mlsanity.com/paper/1904.01538/) (2019)
- [Unsupervised Degradation Representation Learning for Blind Super-Resolution](https://mlsanity.com/paper/2104.00416/) (2021)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Clearing the Skies: A deep network architecture for single-image rain removal](https://mlsanity.com/paper/1609.02087/) (2016)
- [Unfolding the Alternating Optimization for Blind Super Resolution](https://mlsanity.com/paper/2010.02631/) (2020)
- [A Model-driven Deep Neural Network for Single Image Rain Removal](https://mlsanity.com/paper/2005.01333/) (2020)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Rotation equivariant vector field networks](https://mlsanity.com/paper/1612.09346/) (2016)
- [Denoising Prior Driven Deep Neural Network for Image Restoration](https://mlsanity.com/paper/1801.06756/) (2018)
- [Oriented Response Networks](https://mlsanity.com/paper/1701.01833/) (2017)
- [DuDoNet: Dual Domain Network for CT Metal Artifact Reduction](https://mlsanity.com/paper/1907.00273/) (2019)

## Built on by (most cited first, 17 of 17)

- [RPCANet++: Deep Interpretable Robust PCA for Sparse Object Segmentation](https://mlsanity.com/paper/2508.04190/) (2025)
- [Rotation-Equivariant Self-Supervised Method in Image Denoising](https://mlsanity.com/paper/2505.19618/) (2025)
- [A Regularization-Guided Equivariant Approach for Image Restoration](https://mlsanity.com/paper/2505.19799/) (2025)
- [Rotation Equivariant Arbitrary-scale Image Super-Resolution](https://mlsanity.com/paper/2508.05160/) (2025)
- [Lightweight Deep Unfolding Networks with Enhanced Robustness for Infrared Small Target Detection](https://mlsanity.com/paper/2509.08205/) (2025)
- [It's Not the Target, It's the Background: Rethinking Infrared Small Target Detection via Deep Patch-Free Low-Rank Representations](https://mlsanity.com/paper/2506.10425/) (2025)
- [PreCM: The Padding-based Rotation Equivariant Convolution Mode for Semantic Segmentation](https://mlsanity.com/paper/2411.01624/) (2024)
- [Equivariant Denoisers for Image Restoration](https://mlsanity.com/paper/2412.05343/) (2024)
- [FIND-Net -- Fourier-Integrated Network with Dictionary Kernels for Metal Artifact Reduction](https://mlsanity.com/paper/2508.10617/) (2025)
- [Image-to-Image Translation Framework Embedded with Rotation Symmetry Priors](https://mlsanity.com/paper/2604.12805/) (2026)
- [Feature Alignment with Equivariant Convolutions for Burst Image Super-Resolution](https://mlsanity.com/paper/2503.08300/) (2025)
- [DUN-SRE: Deep Unrolling Network with Spatiotemporal Rotation Equivariance for Dynamic MRI Reconstruction](https://mlsanity.com/paper/2506.10309/) (2025)
- [Physically Interpretable Multi-Degradation Image Restoration via Deep Unfolding and Explainable Convolution](https://mlsanity.com/paper/2511.10166/) (2025)
- [ReDiffuse: Rotation Equivariant Diffusion Model for Multi-focus Image Fusion](https://mlsanity.com/paper/2603.21129/) (2026)
- [Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration](https://mlsanity.com/paper/2605.13744/) (2026)
- [Geometric Causal Models](https://mlsanity.com/paper/2607.05153/) (2026)
- [PE-CSNet: An equivariant network architecture with learnable patch-based sparse representation](https://mlsanity.com/paper/2608.14708/) (2026)

## Abstract (arXiv)

The deep unfolding approach has attracted significant attention in computer vision tasks, which well connects conventional image processing modeling manners with more recent deep learning techniques. Specifically, by establishing a direct correspondence between algorithm operators at each implementation step and network modules within each layer, one can rationally construct an almost ``white box'' network architecture with high interpretability. In this architecture, only the predefined component of the proximal operator, known as a proximal network, needs manual configuration, enabling the network to automatically extract intrinsic image priors in a data-driven manner. In current deep unfolding methods, such a proximal network is generally designed as a CNN architecture, whose necessity has been proven by a recent theory. That is, CNN structure substantially delivers the translational invariant image prior, which is the most universally possessed structural prior across various types of images. However, standard CNN-based proximal networks have essential limitations in capturing the rotation symmetry prior, another universal structural prior underlying general images. This leaves a large room for further performance improvement in deep unfolding approaches. To address this issue, this study makes efforts to suggest a high-accuracy rotation equivariant proximal network that effectively embeds rotation symmetry priors into the deep unfolding framework. Especially, we deduce, for the first time, the theoretical equivariant error for such a designed proximal network with arbitrary layers under arbitrary rotation degrees. This analysis should be the most refined theoretical conclusion for such error evaluation to date and is also indispensable for supporting the rationale behind such networks with intrinsic interpretability requirements.