# Imaging with Equivariant Deep Learning

arXiv: https://arxiv.org/abs/2209.01725 · HTML: https://mlsanity.com/paper/2209.01725/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 5 Sep 2022.

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

- [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)
- [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)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Understanding image representations by measuring their equivariance and equivalence](https://mlsanity.com/paper/1411.5908/) (2014)
- [End-to-End Variational Networks for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2004.06688/) (2020)
- [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)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [RARE: Image Reconstruction using Deep Priors Learned without Ground Truth](https://mlsanity.com/paper/1912.05854/) (2019)
- [Image Reconstruction: From Sparsity to Data-adaptive Methods and Machine Learning](https://mlsanity.com/paper/1904.02816/) (2019)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [Deep Residual Learning for Accelerated MRI using Magnitude and Phase Networks](https://mlsanity.com/paper/1804.00432/) (2018)
- [Improved Generalization Bounds of Group Invariant / Equivariant Deep Networks via Quotient Feature Spaces](https://mlsanity.com/paper/1910.06552/) (2019)
- [Equivariant neural networks for inverse problems](https://mlsanity.com/paper/2102.11504/) (2021)
- [Sensing Theorems for Unsupervised Learning in Linear Inverse Problems](https://mlsanity.com/paper/2203.12513/) (2022)
- [Data augmentation for deep learning based accelerated MRI reconstruction with limited data](https://mlsanity.com/paper/2106.14947/) (2021)
- [Truly shift-equivariant convolutional neural networks with adaptive polyphase upsampling](https://mlsanity.com/paper/2105.04040/) (2021)
- [FRIST - Flipping and Rotation Invariant Sparsifying Transform Learning and Applications](https://mlsanity.com/paper/1511.06359/) (2015)
- [Performance characterization of a novel deep learning-based MR image reconstruction pipeline](https://mlsanity.com/paper/2008.06559/) (2020)

## Built on by (most cited first, 30 of 35)

- [Equivariant Multi-Modality Image Fusion](https://mlsanity.com/paper/2305.11443/) (2023)
- [Equivariant Adaptation of Large Pretrained Models](https://mlsanity.com/paper/2310.01647/) (2023)
- [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)
- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [Diffusion Posterior Proximal Sampling for Image Restoration](https://mlsanity.com/paper/2402.16907/) (2024)
- [Fully Convolutional Slice-to-Volume Reconstruction for Single-Stack MRI](https://mlsanity.com/paper/2312.03102/) (2023)
- [Self-Supervised Scalable Deep Compressed Sensing](https://mlsanity.com/paper/2308.13777/) (2023)
- [Improved Canonicalization for Model Agnostic Equivariance](https://mlsanity.com/paper/2405.14089/) (2024)
- [MRI Super-Resolution with Deep Learning: A Comprehensive Survey](https://mlsanity.com/paper/2511.16854/) (2025)
- [Do Bayesian imaging methods report trustworthy probabilities?](https://mlsanity.com/paper/2405.08179/) (2024)
- [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)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [WaveMo: Learning Wavefront Modulations to See Through Scattering](https://mlsanity.com/paper/2404.07985/) (2024)
- [SRE-CNN: A Spatiotemporal Rotation-Equivariant CNN for Cardiac Cine MR Imaging](https://mlsanity.com/paper/2409.08537/) (2024)
- [EquiReg: Equivariance Regularized Diffusion for Inverse Problems](https://mlsanity.com/paper/2505.22973/) (2025)
- [Convergence analysis of equilibrium methods for inverse problems](https://mlsanity.com/paper/2306.01421/) (2023)
- [From Learning to Optimize to Learning Optimization Algorithms](https://mlsanity.com/paper/2405.18222/) (2024)
- [An analytic theory of convolutional neural network inverse problems solvers](https://mlsanity.com/paper/2601.10334/) (2026)
- [Bi-directional Self-Registration for Misaligned Infrared-Visible Image Fusion](https://mlsanity.com/paper/2505.06920/) (2025)
- [Equivariant Sampling for Improving Diffusion Model-based Image Restoration](https://mlsanity.com/paper/2511.09965/) (2025)
- [SHARE: A Fully Unsupervised Framework for Single Hyperspectral Image Restoration](https://mlsanity.com/paper/2601.13987/) (2026)
- [Riemannian AmbientFlow: Towards Simultaneous Manifold Learning and Generative Modeling from Corrupted Data](https://mlsanity.com/paper/2601.18728/) (2026)
- [PolarAPP: Beyond Polarization Demosaicking for Polarimetric Applications](https://mlsanity.com/paper/2603.23071/) (2026)
- [Hypothesis Testing in Imaging Inverse Problems](https://mlsanity.com/paper/2505.22481/) (2025)
- [DUN-SRE: Deep Unrolling Network with Spatiotemporal Rotation Equivariance for Dynamic MRI Reconstruction](https://mlsanity.com/paper/2506.10309/) (2025)
- [Bayesian model selection and misspecification testing in imaging inverse problems only from noisy and partial measurements](https://mlsanity.com/paper/2510.27663/) (2025)
- [Deep Lightweight Unrolled Network for High Dynamic Range Modulo Imaging](https://mlsanity.com/paper/2601.12526/) (2026)

## Abstract (arXiv)

From early image processing to modern computational imaging, successful models and algorithms have relied on a fundamental property of natural signals: symmetry. Here symmetry refers to the invariance property of signal sets to transformations such as translation, rotation or scaling. Symmetry can also be incorporated into deep neural networks in the form of equivariance, allowing for more data-efficient learning. While there has been important advances in the design of end-to-end equivariant networks for image classification in recent years, computational imaging introduces unique challenges for equivariant network solutions since we typically only observe the image through some noisy ill-conditioned forward operator that itself may not be equivariant. We review the emerging field of equivariant imaging and show how it can provide improved generalization and new imaging opportunities. Along the way we show the interplay between the acquisition physics and group actions and links to iterative reconstruction, blind compressed sensing and self-supervised learning.