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

arXiv 2111.12855 ↗First posted 25 Nov 2021cs.CV eess.IV eess.SPcited by 56

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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.

Key facts. Cited by 56 papers in the map (data through 7 Oct 2026). Builds on 18 of its 42 references that are in the map. First posted on arXiv 25 Nov 2021.

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18 of 42 references.

References · 18

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  3. 2016Group Equivariant Convolutional Networkscited by 1,154
  4. 2017Deep Image Priorcited by 1,097
  5. 2018Noise2Noise: Learning Image Restoration without Clean Datacited by 526
  6. 2018fastMRI: An Open Dataset and Benchmarks for Accelerated MRIcited by 464
  7. 2016Deep Convolutional Neural Network for Inverse Problems in Imagingcited by 431
  8. 2017Learning a Variational Network for Reconstruction of Accelerated MRI Datacited by 340
  9. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  10. 2019Robust and interpretable blind image denoising via bias-free convolutional neural networkscited by 87
  11. 2021Equivariant Imaging: Learning Beyond the Range Spacecited by 83
  12. 2014From Denoising to Compressed Sensingcited by 67
6 more
  1. 2017Learned D-AMP: Principled Neural Network based Compressive Image Recoverycited by 65
  2. 2018Unsupervised Learning with Stein's Unbiased Risk Estimatorcited by 47
  3. 2018Training Deep Learning Based Denoisers without Ground Truth Datacited by 43
  4. 2021Equivariant neural networks for inverse problemscited by 29
  5. 2018Training deep learning based image denoisers from undersampled measurements without ground truth and without image priorcited by 15
  6. 2020Deep Image Reconstruction using Unregistered Measurements without Groundtruthcited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 7; 2023: 15; 2024: 9; 2025: 13; 2026: 12.

56 papers.

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  1. 2023Equivariant Multi-Modality Image Fusioncited by 98
  2. 2023GSURE-Based Diffusion Model Training with Corrupted Datacited by 40
  3. 2022Imaging with Equivariant Deep Learningcited by 35
  4. 2024Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Datacited by 32
  5. 2022A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noisecited by 29
  6. 2022Sensing Theorems for Unsupervised Learning in Linear Inverse Problemscited by 28
  7. 2023What's in a Prior? Learned Proximal Networks for Inverse Problemscited by 24
  8. 2022Unsupervised Learning From Incomplete Measurements for Inverse Problemscited by 20
  9. 2023Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problemscited by 17
  10. 2023Scale-Equivariant Imaging: Self-Supervised Learning for Image Super-Resolution and Deblurringcited by 14
  11. 2024Generalized Recorrupted-to-Recorrupted: Self-Supervised Learning Beyond Gaussian Noisecited by 13
  12. 2024Invertible Diffusion Models for Compressed Sensingcited by 12
38 more
  1. 2023Self-Supervised Scalable Deep Compressed Sensingcited by 8
  2. 2025Measurement Score-Based Diffusion Modelcited by 7
  3. 2023Learning to Reconstruct Signals From Binary Measurementscited by 5
  4. 2023Unsupervised Spectral Demosaicing with Lightweight Spectral Attention Networkscited by 5
  5. 2024Perspective-Equivariance for Unsupervised Imaging with Camera Geometrycited by 5
  6. 2024Equivariant Imaging for Self-supervised Hyperspectral Image Inpaintingcited by 5
  7. 2025A Regularization-Guided Equivariant Approach for Image Restorationcited by 5
  8. 2026Self-Supervised Learning from Noisy and Incomplete Datacited by 5
  9. 20223D helical CT Reconstruction with a Memory Efficient Learned Primal-Dual Architecturecited by 4
  10. 2023Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imagingcited by 4
  11. 2025Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstructioncited by 4
  12. 2022Rotationally Equivariant Super-Resolution of Velocity Fields in Two-Dimensional Fluids Using Convolutional Neural Networkscited by 3
  13. 2022Nonlinear Equivariant Imaging: Learning Multi-Parametric Tissue Mapping without Ground Truth for Compressive Quantitative MRIcited by 3
  14. 2024Unsupervised Training of Convex Regularizers using Maximum Likelihood Estimationcited by 3
  15. 2024Fully Unsupervised Dynamic MRI Reconstruction via Diffeo-Temporal Equivariancecited by 3
  16. 2025Normalization-equivariant Diffusion Models: Learning Posterior Samplers From Noisy And Partial Measurementscited by 3
  17. 2023Efficient Model-Based Deep Learning via Network Pruning and Fine-Tuningcited by 2
  18. 2025Unsupervised Accelerated MRI Reconstruction via Ground-Truth-Free Flow Matchingcited by 2
  19. 2025Efficient Bayesian Computation Using Plug-and-Play Priors for Poisson Inverse Problemscited by 2
  20. 2023Single-photon Image Super-resolution via Self-supervised Learningcited by 1
  21. 2024Self-Supervised Denoiser Frameworkcited by 1
  22. 2025Self-supervised conformal prediction for uncertainty quantification in Poisson imaging problemscited by 1
  23. 2025Double Blind Imaging with Generative Modelingcited by 1
  24. 2025Equivariant Splitting: Self-supervised learning from incomplete datacited by 1
  25. 2025Equivariant Sampling for Improving Diffusion Model-based Image Restorationcited by 1
  26. 2026PolarAPP: Beyond Polarization Demosaicking for Polarimetric Applicationscited by 1
  27. 2023DULDA: Dual-domain Unsupervised Learned Descent Algorithm for PET image reconstructioncited by 0
  28. 2023Meta-Prior: Meta learning for Adaptive Inverse Problem Solverscited by 0
  29. 2023Classification of complex local environments in systems of particle shapes through shape-symmetry encoded data augmentationcited by 0
  30. 2023Rotational Augmented Noise2Inverse for Low-dose Computed Tomography Reconstructioncited by 0
  31. 2024Equivariant Test-Time Training with Operator Sketching for Imaging Inverse Problemscited by 0
  32. 2025Hypothesis Testing in Imaging Inverse Problemscited by 0
  33. 2025Fast Equivariant Imaging: Accelerating Unsupervised Learning and Model Adaptation via Inexact Splittingcited by 0
  34. 2025UPMRI: Unsupervised Parallel MRI Reconstruction via Projected Conditional Flow Matchingcited by 0
  35. 2026HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Trainingcited by 0
  36. 2026Equivariant Learning for Unsupervised Image Dehazingcited by 0
  37. 2026Perspective-Equivariant Fine-tuning for Multispectral Demosaicing without Ground Truthcited by 0
  38. 2026SPLIT: Self-supervised Partitioning for Learned Inversion in Nonlinear Tomographycited by 0

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