# Equivariant plug-and-play image reconstruction

arXiv: https://arxiv.org/abs/2312.01831 · HTML: https://mlsanity.com/paper/2312.01831/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 39 of its 54 references that are in the map.
- First posted on arXiv 4 Dec 2023.

## Builds on (most cited first, 30 of 39)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [SwinIR: Image Restoration Using Swin Transformer](https://mlsanity.com/paper/2108.10257/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [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)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Denoising Diffusion Models for Plug-and-Play Image Restoration](https://mlsanity.com/paper/2305.08995/) (2023)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Learning Proximal Operators: Using Denoising Networks for Regularizing Inverse Imaging Problems](https://mlsanity.com/paper/1704.03488/) (2017)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imaging](https://mlsanity.com/paper/2203.17061/) (2022)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [One Network to Solve Them All --- Solving Linear Inverse Problems using Deep Projection Models](https://mlsanity.com/paper/1703.09912/) (2017)
- [Learning Maximally Monotone Operators for Image Recovery](https://mlsanity.com/paper/2012.13247/) (2020)
- [Equivariant Q Learning in Spatial Action Spaces](https://mlsanity.com/paper/2110.15443/) (2021)
- [Regularization by Denoising via Fixed-Point Projection (RED-PRO)](https://mlsanity.com/paper/2008.00226/) (2020)
- [Seven ways to improve example-based single image super resolution](https://mlsanity.com/paper/1511.02228/) (2015)
- [Recovery of Continuous 3D Refractive Index Maps from Discrete Intensity-Only Measurements using Neural Fields](https://mlsanity.com/paper/2112.00002/) (2021)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)

## Built on by (most cited first, 26 of 26)

- [Analysis of Deep Image Prior and Exploiting Self-Guidance for Image Reconstruction](https://mlsanity.com/paper/2402.04097/) (2024)
- [Constrained Diffusion with Trust Sampling](https://mlsanity.com/paper/2411.10932/) (2024)
- [MRI Super-Resolution with Deep Learning: A Comprehensive Survey](https://mlsanity.com/paper/2511.16854/) (2025)
- [Empirical Bayesian image restoration by Langevin sampling with a denoising diffusion implicit prior](https://mlsanity.com/paper/2409.04384/) (2024)
- [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025)
- [Rotation Equivariant Arbitrary-scale Image Super-Resolution](https://mlsanity.com/paper/2508.05160/) (2025)
- [NPN: Non-Linear Projections of the Null-Space for Imaging Inverse Problems](https://mlsanity.com/paper/2510.01608/) (2025)
- [EquiReg: Equivariance Regularized Diffusion for Inverse Problems](https://mlsanity.com/paper/2505.22973/) (2025)
- [From Learning to Optimize to Learning Optimization Algorithms](https://mlsanity.com/paper/2405.18222/) (2024)
- [Robust plug-and-play methods for highly accelerated non-Cartesian MRI reconstruction](https://mlsanity.com/paper/2411.01955/) (2024)
- [A Tunable Despeckling Neural Network Stabilized via Diffusion Equation](https://mlsanity.com/paper/2411.15921/) (2024)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)
- [Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problems](https://mlsanity.com/paper/2503.16222/) (2025)
- [Axis-level Symmetry Detection with Group-Equivariant Representation](https://mlsanity.com/paper/2508.10740/) (2025)
- [An analytic theory of convolutional neural network inverse problems solvers](https://mlsanity.com/paper/2601.10334/) (2026)
- [Equivariant Denoisers for Image Restoration](https://mlsanity.com/paper/2412.05343/) (2024)
- [Equivariant Splitting: Self-supervised learning from incomplete data](https://mlsanity.com/paper/2510.00929/) (2025)
- [Equivariant Sampling for Improving Diffusion Model-based Image Restoration](https://mlsanity.com/paper/2511.09965/) (2025)
- [GSNR: Graph Smooth Null-Space Representation for Inverse Problems](https://mlsanity.com/paper/2602.20328/) (2026)
- [Fast Equivariant Imaging: Accelerating Unsupervised Learning and Model Adaptation via Inexact Splitting](https://mlsanity.com/paper/2507.06764/) (2025)
- [prNet: Data-Driven Phase Retrieval via Stochastic Refinement](https://mlsanity.com/paper/2507.09608/) (2025)
- [Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problems](https://mlsanity.com/paper/2511.12257/) (2025)
- [Multilevel Stochastic Plug-and-Play for Sparse-View CT Reconstruction](https://mlsanity.com/paper/2606.24567/) (2026)
- [Stabilizing Deep Reconstruction Operators with Contractive Anchoring](https://mlsanity.com/paper/2607.23341/) (2026)
- [Embedding Rotation Invariance for Provable Multi-Oriented Scene Text Recognition](https://mlsanity.com/paper/2608.10684/) (2026)
- [Convergent Plug-and-Play Image Restoration with Annealed Noise Levels](https://mlsanity.com/paper/2609.32393/) (2026)

## Abstract (arXiv)

Plug-and-play algorithms constitute a popular framework for solving inverse imaging problems that rely on the implicit definition of an image prior via a denoiser. These algorithms can leverage powerful pre-trained denoisers to solve a wide range of imaging tasks, circumventing the necessity to train models on a per-task basis. Unfortunately, plug-and-play methods often show unstable behaviors, hampering their promise of versatility and leading to suboptimal quality of reconstructed images. In this work, we show that enforcing equivariance to certain groups of transformations (rotations, reflections, and/or translations) on the denoiser strongly improves the stability of the algorithm as well as its reconstruction quality. We provide a theoretical analysis that illustrates the role of equivariance on better performance and stability. We present a simple algorithm that enforces equivariance on any existing denoiser by simply applying a random transformation to the input of the denoiser and the inverse transformation to the output at each iteration of the algorithm. Experiments on multiple imaging modalities and denoising networks show that the equivariant plug-and-play algorithm improves both the reconstruction performance and the stability compared to their non-equivariant counterparts.