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Rotation equivariant vector field networks

arXiv 1612.09346 ↗First posted 29 Dec 2016cs.CVcited by 99

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Abstract (arXiv)

In many computer vision tasks, we expect a particular behavior of the output with respect to rotations of the input image. If this relationship is explicitly encoded, instead of treated as any other variation, the complexity of the problem is decreased, leading to a reduction in the size of the required model. In this paper, we propose the Rotation Equivariant Vector Field Networks (RotEqNet), a Convolutional Neural Network (CNN) architecture encoding rotation equivariance, invariance and covariance. Each convolutional filter is applied at multiple orientations and returns a vector field representing magnitude and angle of the highest scoring orientation at every spatial location. We develop a modified convolution operator relying on this representation to obtain deep architectures. We test RotEqNet on several problems requiring different responses with respect to the inputs' rotation: image classification, biomedical image segmentation, orientation estimation and patch matching. In all cases, we show that RotEqNet offers extremely compact models in terms of number of parameters and provides results in line to those of networks orders of magnitude larger.

Key facts. Cited by 99 papers in the map (data through 14 Oct 2026). First posted on arXiv 29 Dec 2016.

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Citing papers per year: 2021: 23; 2022: 23; 2023: 21; 2024: 14; 2025: 11; 2026: 7.

99 papers.

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  1. 2021ReDet: A Rotation-equivariant Detector for Aerial Object Detectioncited by 203
  2. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 187
  3. 2023EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoningcited by 97
  4. 2025EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modelingcited by 89
  5. 2021Equivariant Point Network for 3D Point Cloud Analysiscited by 81
  6. 2021SE(3)-equivariant prediction of molecular wavefunctions and electronic densitiescited by 67
  7. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 65
  8. 2022The Lie Derivative for Measuring Learned Equivariancecited by 54
  9. 2022DEVIANT: Depth EquiVarIAnt NeTwork for Monocular 3D Object Detectioncited by 48
  10. 2021Constrained Learning with Non-Convex Lossescited by 42
  11. 2023Segment anything, from space?cited by 42
  12. 2022Sample Efficiency of Data Augmentation Consistency Regularizationcited by 28
38 more
  1. 2022VN-Transformer: Rotation-Equivariant Attention for Vector Neuronscited by 27
  2. 2023E(2)-Equivariant Vision Transformercited by 27
  3. 2021Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutionscited by 26
  4. 2022Self-Supervised Equivariant Learning for Oriented Keypoint Detectioncited by 26
  5. 2023Learning Rotation-Equivariant Features for Visual Correspondencecited by 23
  6. 2021Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groupscited by 22
  7. 2022RIC-CNN: Rotation-Invariant Coordinate Convolutional Neural Networkcited by 21
  8. 2021Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scalescited by 17
  9. 2023Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restorationcited by 17
  10. 2021Equivariant Networks for Pixelized Spherescited by 16
  11. 2022Learning Invariant Weights in Neural Networkscited by 16
  12. 2022Learning Instance-Specific Augmentations by Capturing Local Invariancescited by 15
  13. 2022Reflection and Rotation Symmetry Detection via Equivariant Learningcited by 13
  14. 2024RiEMann: Near Real-Time SE(3)-Equivariant Robot Manipulation without Point Cloud Segmentationcited by 12
  15. 2021Rotation Equivariant Feature Image Pyramid Network for Object Detection in Optical Remote Sensing Imagerycited by 11
  16. 2023Normalization-Equivariant Neural Networks with Application to Image Denoisingcited by 11
  17. 2023Approximately Equivariant Quantum Neural Network for p4m Group Symmetries in Imagescited by 10
  18. 2021Training or Architecture? How to Incorporate Invariance in Neural Networkscited by 9
  19. 2022In What Ways Are Deep Neural Networks Invariant and How Should We Measure This?cited by 9
  20. 2024A Characterization Theorem for Equivariant Networks with Point-wise Activationscited by 9
  21. 2022A Principled Design of Image Representation: Towards Forensic Taskscited by 8
  22. 2023Oriented object detection in optical remote sensing images using deep learning: a surveycited by 8
  23. 2024Steerable Transformers for Volumetric Datacited by 8
  24. 2021Deformation Robust Roto-Scale-Translation Equivariant CNNscited by 7
  25. 2022Improving the Sample-Complexity of Deep Classification Networks with Invariant Integrationcited by 7
  26. 2024Tilt your Head: Activating the Hidden Spatial-Invariance of Classifierscited by 7
  27. 2021Mapping industrial poultry operations at scale with deep learning and aerial imagerycited by 6
  28. 2023Equivariant Map and Agent Geometry for Autonomous Driving Motion Predictioncited by 6
  29. 2025Rotation-Equivariant Self-Supervised Method in Image Denoisingcited by 6
  30. 2021Generating Unrestricted Adversarial Examples via Three Parameterscited by 5
  31. 2023Transformation-Invariant Network for Few-Shot Object Detection in Remote Sensing Imagescited by 5
  32. 2025A Regularization-Guided Equivariant Approach for Image Restorationcited by 5
  33. 2025Rotation Equivariant Arbitrary-scale Image Super-Resolutioncited by 5
  34. 2022Learning Continuous Rotation Canonicalization with Radial Beam Samplingcited by 4
  35. 2022Beyond CNNs: Exploiting Further Inherent Symmetries in Medical Image Segmentationcited by 4
  36. 2022PDO-s3DCNNs: Partial Differential Operator Based Steerable 3D CNNscited by 4
  37. 2024Pioneering SE(2)-Equivariant Trajectory Planning for Automated Drivingcited by 4
  38. 2024R2Det: Exploring Relaxed Rotation Equivariance in 2D object detectioncited by 4

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