# Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements

arXiv: https://arxiv.org/abs/2111.12855 · HTML: https://mlsanity.com/paper/2111.12855/

## Key facts

- Cited by 56 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 42 references that are in the map.
- First posted on arXiv 25 Nov 2021.

## Builds on (most cited first, 18 of 18)

- [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)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Learning a Variational Network for Reconstruction of Accelerated MRI Data](https://mlsanity.com/paper/1704.00447/) (2017)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Robust and interpretable blind image denoising via bias-free convolutional neural networks](https://mlsanity.com/paper/1906.05478/) (2019)
- [Equivariant Imaging: Learning Beyond the Range Space](https://mlsanity.com/paper/2103.14756/) (2021)
- [From Denoising to Compressed Sensing](https://mlsanity.com/paper/1406.4175/) (2014)
- [Learned D-AMP: Principled Neural Network based Compressive Image Recovery](https://mlsanity.com/paper/1704.06625/) (2017)
- [Unsupervised Learning with Stein's Unbiased Risk Estimator](https://mlsanity.com/paper/1805.10531/) (2018)
- [Training Deep Learning Based Denoisers without Ground Truth Data](https://mlsanity.com/paper/1803.01314/) (2018)
- [Equivariant neural networks for inverse problems](https://mlsanity.com/paper/2102.11504/) (2021)
- [Training deep learning based image denoisers from undersampled measurements without ground truth and without image prior](https://mlsanity.com/paper/1806.00961/) (2018)
- [Deep Image Reconstruction using Unregistered Measurements without Groundtruth](https://mlsanity.com/paper/2009.13986/) (2020)

## Built on by (most cited first, 30 of 56)

- [Equivariant Multi-Modality Image Fusion](https://mlsanity.com/paper/2305.11443/) (2023)
- [GSURE-Based Diffusion Model Training with Corrupted Data](https://mlsanity.com/paper/2305.13128/) (2023)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise](https://mlsanity.com/paper/2205.10278/) (2022)
- [Sensing Theorems for Unsupervised Learning in Linear Inverse Problems](https://mlsanity.com/paper/2203.12513/) (2022)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Unsupervised Learning From Incomplete Measurements for Inverse Problems](https://mlsanity.com/paper/2201.12151/) (2022)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [Scale-Equivariant Imaging: Self-Supervised Learning for Image Super-Resolution and Deblurring](https://mlsanity.com/paper/2312.11232/) (2023)
- [Generalized Recorrupted-to-Recorrupted: Self-Supervised Learning Beyond Gaussian Noise](https://mlsanity.com/paper/2412.04648/) (2024)
- [Invertible Diffusion Models for Compressed Sensing](https://mlsanity.com/paper/2403.17006/) (2024)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Measurement Score-Based Diffusion Model](https://mlsanity.com/paper/2505.11853/) (2025)
- [Learning to Reconstruct Signals From Binary Measurements](https://mlsanity.com/paper/2303.08691/) (2023)
- [Unsupervised Spectral Demosaicing with Lightweight Spectral Attention Networks](https://mlsanity.com/paper/2307.01990/) (2023)
- [Perspective-Equivariance for Unsupervised Imaging with Camera Geometry](https://mlsanity.com/paper/2403.09327/) (2024)
- [Equivariant Imaging for Self-supervised Hyperspectral Image Inpainting](https://mlsanity.com/paper/2404.13159/) (2024)
- [A Regularization-Guided Equivariant Approach for Image Restoration](https://mlsanity.com/paper/2505.19799/) (2025)
- [Self-Supervised Learning from Noisy and Incomplete Data](https://mlsanity.com/paper/2601.03244/) (2026)
- [3D helical CT Reconstruction with a Memory Efficient Learned Primal-Dual Architecture](https://mlsanity.com/paper/2205.11952/) (2022)
- [Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging](https://mlsanity.com/paper/2311.08972/) (2023)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [Rotationally Equivariant Super-Resolution of Velocity Fields in Two-Dimensional Fluids Using Convolutional Neural Networks](https://mlsanity.com/paper/2202.11099/) (2022)
- [Nonlinear Equivariant Imaging: Learning Multi-Parametric Tissue Mapping without Ground Truth for Compressive Quantitative MRI](https://mlsanity.com/paper/2211.12786/) (2022)
- [Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimation](https://mlsanity.com/paper/2404.05445/) (2024)
- [Fully Unsupervised Dynamic MRI Reconstruction via Diffeo-Temporal Equivariance](https://mlsanity.com/paper/2410.08646/) (2024)
- [Unsupervised Accelerated MRI Reconstruction via Ground-Truth-Free Flow Matching](https://mlsanity.com/paper/2502.17174/) (2025)
- [Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurements](https://mlsanity.com/paper/2510.11964/) (2025)
- [Efficient Model-Based Deep Learning via Network Pruning and Fine-Tuning](https://mlsanity.com/paper/2311.02003/) (2023)

## Abstract (arXiv)

Deep networks provide state-of-the-art performance in multiple imaging inverse problems ranging from medical imaging to computational photography. However, most existing networks are trained with clean signals which are often hard or impossible to obtain. Equivariant imaging (EI) is a recent self-supervised learning framework that exploits the group invariance present in signal distributions to learn a reconstruction function from partial measurement data alone. While EI results are impressive, its performance degrades with increasing noise. In this paper, we propose a Robust Equivariant Imaging (REI) framework which can learn to image from noisy partial measurements alone. The proposed method uses Stein's Unbiased Risk Estimator (SURE) to obtain a fully unsupervised training loss that is robust to noise. We show that REI leads to considerable performance gains on linear and nonlinear inverse problems, thereby paving the way for robust unsupervised imaging with deep networks. Code is available at: https://github.com/edongdongchen/REI.