# Equivariant Imaging: Learning Beyond the Range Space

arXiv: https://arxiv.org/abs/2103.14756 · HTML: https://mlsanity.com/paper/2103.14756/

## Key facts

- Cited by 83 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 40 references that are in the map.
- First posted on arXiv 26 Mar 2021.

## Builds on (most cited first, 25 of 25)

- [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)
- [ESRGAN: Enhanced Super-Resolution Generative Adversarial Networks](https://mlsanity.com/paper/1809.00219/) (2018)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Least Squares Generative Adversarial Networks](https://mlsanity.com/paper/1611.04076/) (2016)
- [Free-Form Image Inpainting with Gated Convolution](https://mlsanity.com/paper/1806.03589/) (2018)
- [Generative Image Inpainting with Contextual Attention](https://mlsanity.com/paper/1801.07892/) (2018)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [Image Inpainting for Irregular Holes Using Partial Convolutions](https://mlsanity.com/paper/1804.07723/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [SRFlow: Learning the Super-Resolution Space with Normalizing Flow](https://mlsanity.com/paper/2006.14200/) (2020)
- [Low Dose CT Image Denoising Using a Generative Adversarial Network with Wasserstein Distance and Perceptual Loss](https://mlsanity.com/paper/1708.00961/) (2017)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Neural Proximal Gradient Descent for Compressive Imaging](https://mlsanity.com/paper/1806.03963/) (2018)
- [Generalization Error of Invariant Classifiers](https://mlsanity.com/paper/1610.04574/) (2016)
- [Deep Decomposition Learning for Inverse Imaging Problems](https://mlsanity.com/paper/1911.11028/) (2019)
- [Improving Transformation Invariance in Contrastive Representation Learning](https://mlsanity.com/paper/2010.09515/) (2020)
- [Compressive MR Fingerprinting reconstruction with Neural Proximal Gradient iterations](https://mlsanity.com/paper/2006.15271/) (2020)
- [Noise2Inverse: Self-supervised deep convolutional denoising for tomography](https://mlsanity.com/paper/2001.11801/) (2020)

## Built on by (most cited first, 30 of 83)

- [Equivariant Multi-Modality Image Fusion](https://mlsanity.com/paper/2305.11443/) (2023)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Sensing Theorems for Unsupervised Learning in Linear Inverse Problems](https://mlsanity.com/paper/2203.12513/) (2022)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Online Deep Equilibrium Learning for Regularization by Denoising](https://mlsanity.com/paper/2205.13051/) (2022)
- [Transformer Meets Boundary Value Inverse Problems](https://mlsanity.com/paper/2209.14977/) (2022)
- [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)
- [Diffusion Posterior Proximal Sampling for Image Restoration](https://mlsanity.com/paper/2402.16907/) (2024)
- [Generalized Recorrupted-to-Recorrupted: Self-Supervised Learning Beyond Gaussian Noise](https://mlsanity.com/paper/2412.04648/) (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)
- [Deep Physics-Guided Unrolling Generalization for Compressed Sensing](https://mlsanity.com/paper/2307.08950/) (2023)
- [Progressive Content-aware Coded Hyperspectral Compressive Imaging](https://mlsanity.com/paper/2303.09773/) (2023)
- [Auto-Linear Phenomenon in Subsurface Imaging](https://mlsanity.com/paper/2305.13314/) (2023)
- [MEPNet: A Model-Driven Equivariant Proximal Network for Joint Sparse-View Reconstruction and Metal Artifact Reduction in CT Images](https://mlsanity.com/paper/2306.14274/) (2023)
- [Glimpse: Generalized Locality for Scalable and Robust CT](https://mlsanity.com/paper/2401.00816/) (2024)
- [QN-Mixer: A Quasi-Newton MLP-Mixer Model for Sparse-View CT Reconstruction](https://mlsanity.com/paper/2402.17951/) (2024)
- [Scale-Equivariant Unrolled Neural Networks for Data-Efficient Accelerated MRI Reconstruction](https://mlsanity.com/paper/2204.10436/) (2022)
- [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)
- [Stochastic Primal-Dual Deep Unrolling](https://mlsanity.com/paper/2110.10093/) (2021)
- [D3C2-Net: Dual-Domain Deep Convolutional Coding Network for Compressive Sensing](https://mlsanity.com/paper/2207.13560/) (2022)

## Abstract (arXiv)

In various imaging problems, we only have access to compressed measurements of the underlying signals, hindering most learning-based strategies which usually require pairs of signals and associated measurements for training. Learning only from compressed measurements is impossible in general, as the compressed observations do not contain information outside the range of the forward sensing operator. We propose a new end-to-end self-supervised framework that overcomes this limitation by exploiting the equivariances present in natural signals. Our proposed learning strategy performs as well as fully supervised methods. Experiments demonstrate the potential of this framework on inverse problems including sparse-view X-ray computed tomography on real clinical data and image inpainting on natural images. Code has been made available at: https://github.com/edongdongchen/EI.