On the Continuity of Rotation Representations in Neural Networks
Open accessAbstract
In neural networks, it is often desirable to work with various representations of the same space. For example, 3D rotations can be represented with quaternions or Euler angles. In this paper, we advance a definition of a continuous representation, which can be helpful for training deep neural networks. We relate this to topological concepts such as homeomorphism and embedding. We then investigate what are continuous and discontinuous representations for 2D, 3D, and n-dimensional rotations. We demonstrate that for 3D rotations, all representations are discontinuous in the real Euclidean spaces of four or fewer dimensions. Thus, widely used representations such as quaternions and Euler angles are discontinuous and difficult for neural networks to learn. We show that the 3D rotations have continuous representations in 5D and 6D, which are more suitable for learning. We also present continuous representations for the general case of the n-dimensional rotation group SO(n). While our main focus is on rotations, we also show that our constructions apply to other groups such as the orthogonal group and similarity transforms. We finally present empirical results, which show that our continuous rotation representations outperform discontinuous ones for several practical problems in graphics and vision, including a simple autoencoder sanity test, a rotation estimator for 3D point clouds, and an inverse kinematics solver for 3D human poses.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Euler angles and quaternions contain discontinuities that hinder rotation estimation with deep learning.
“While Euler angles and quaternions are common choices, both contain discontinuities that hinder rotation estimation with deep learning [32,33].”
From Rapid patient-specific neural networks for intraoperative X-ray to volume registration (2025) · cites this paper as [32]
Continuous 6D representations create a more stable learning target and show improved performance.
“To address this, Zhou et al. [38] proposed continuous 6D representations to create a more stable learning target, showing improved performance.”
From FastPose-ViT: A Vision Transformer for Real-Time Spacecraft Pose Estimation (2026) · cites this paper as [38]
Classical parameterizations such as axis-angle, Euler angles, and quaternions suffer from degeneracies that can make pose estimation difficult.
“Classical parameterizations such as the axis-angle representation (d=3), Euler angles (d=3), and quaternions (d=4) suffer from degeneracies that can make pose estimation difficult, such as Gimbal lock or discontinuities in the representation space [57].”
From Intraoperative 2D/3D Image Registration via Differentiable X-Ray Rendering (2024) · cites this paper as [57]
Higher-dimensional parameterizations of SO(3) have been proposed that do not suffer from discontinuities.
“In point cloud registration, higher-dimensional parameterizations of SO(3) have been proposed that do not suffer from this issue [32, 40, 57].”
From Intraoperative 2D/3D Image Registration via Differentiable X-Ray Rendering (2024) · cites this paper as [57]
Higher-dimensional alternatives (d>5) have theoretical advantages.
“This has motivated the development of several higher-dimensional (d>5) alternatives with theoretical advantages [32, 40, 57].”
From Intraoperative 2D/3D Image Registration via Differentiable X-Ray Rendering (2024) · cites this paper as [57]
Prior work has focused on regressing parameterizations of rotations that introduce discontinuities in representation space.
“Prior work has focused on regressing parameterizations of rotations that have been shown to introduce discontinuities in the representation space, e.g. Euler angles and quaternions [57].”
From Intraoperative 2D/3D Image Registration via Differentiable X-Ray Rendering (2024) · cites this paper as [57]
Cites
LeCun, Y., Bengio, Y., & Hinton, G. (2015) · Nature · doi:10.1038/nature14539 · Cited by 84,126 · see citing works
PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation
Charles, R., Su, H., Mo, K., & Guibas, L. J. (2017) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2017.16 · Cited by 9,767 · see citing works
Approximation capabilities of multilayer feedforward networks
Hornik, K. (1991) · Neural Networks · doi:10.1016/0893-6080(91)90009-t · Cited by 6,091 · see citing works
Universal approximation bounds for superpositions of a sigmoidal function
Barron, A. R. (1993) · IEEE Transactions on Information Theory · doi:10.1109/18.256500 · Cited by 2,861 · see citing works
PoseNet: A Convolutional Network for Real-Time 6-DOF Camera Relocalization
Kendall, A., Grimes, M., & Cipolla, R. (2015) · 2015 IEEE International Conference on Computer Vision (ICCV) · doi:10.1109/iccv.2015.336 · Cited by 2,489 · see citing works
ShapeNet: An Information-Rich 3D Model Repository
Chang, A. X., Funkhouser, T., Guibas, L., Hanrahan, P., Huang, Q., Li, Z., Savarese, S., Savva, M., Song, S., Su, H., et al. (2015) · arXiv · doi:10.48550/arxiv.1512.03012 · Cited by 2,330 · see citing works
PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes
Xiang, Y., Schmidt, T., Narayanan, V., & Fox, D. (2018) · Robotics: Science and Systems XIV · doi:10.15607/rss.2018.xiv.019 · Cited by 2,139 · see citing works
End-to-End Recovery of Human Shape and Pose
Kanazawa, A., Black, M. J., Jacobs, D. W., & Malik, J. (2018) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2018.00744 · Cited by 1,899 · see citing works
Geometric Loss Functions for Camera Pose Regression with Deep Learning
Kendall, A., & Cipolla, R. (2017) · 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2017.694 · Cited by 803 · see citing works
DeMoN: Depth and Motion Network for Learning Monocular Stereo
Ummenhofer, B., Zhou, H., Uhrig, J., Mayer, N., Ilg, E., Dosovitskiy, A., & Brox, T. (2016) · Computer Vision and Pattern Recognition · doi:10.1109/cvpr.2017.596 · Cited by 777 · see citing works
Practical Parameterization of Rotations Using the Exponential Map
Grassia, F. S. (1998) · Journal of Graphics Tools · doi:10.1080/10867651.1998.10487493 · Cited by 490 · see citing works
Deep Kinematic Pose Regression
Zhou, X., Sun, X. N., Zhang, W., Liang, S., & Wei, Y. (2016) · Lecture notes in computer science · doi:10.1007/978-3-319-49409-8_17 · Cited by 263 · see citing works
Neural Kinematic Networks for Unsupervised Motion Retargetting
Villegas, R., Yang, J., Ceylan, D., & Lee, H. (2018) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2018.00901 · Cited by 240 · see citing works
Self-citationQuatNet: Quaternion-Based Head Pose Estimation With Multiregression Loss
Hsu, H.-W., Wu, T.-Y., Wan, S., et al. (2018) · IEEE Transactions on Multimedia · doi:10.1109/tmm.2018.2866770 · Cited by 179 · see citing works
A First Course in Algebraic Topology
Kosniowski, C. (1980) · Cambridge University Press eBooks · doi:10.1017/cbo9780511569296 · Cited by 159 · see citing works
Explorations in Homeomorphic Variational Auto-EncodingPreprint
Falorsi, L., de Haan, P., Davidson, T. R., Cao, N. D., Weiler, M., Forré, P., & Cohen, T. (2018) · arXiv.org · doi:10.48550/arxiv.1807.04689 · Cited by 134 · see citing works
Erdős, P. L., & Túrán, P. (1937) · Annals of Mathematics · doi:10.2307/1968516 · Cited by 118 · see citing works
Deep-6DPose: Recovering 6D Object Pose from a Single RGB ImagePreprint
Do, T.-T., Cai, M., Pham, T. T., & Reid, I. D. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1802.10367 · Cited by 118 · see citing works
(2013) · Choice Reviews Online · doi:10.5860/choice.50-4490 · Cited by 110 · see citing works
Learning 3-D object orientation from images
Saxena, A., Driemeyer, J., & Ng, A. (2009) · IEEE International Conference on Robotics and Automation · doi:10.1109/robot.2009.5152855 · Cited by 102 · see citing works
On solving the inverse kinematics problem using neural networks
Csiszar, A., Eilers, J., & Verl, A. W. (2017) · International Conference on Mechatronics and Machine Vision in Practice (M2VIP) · doi:10.1109/m2vip.2017.8211457 · Cited by 73 · see citing works
The essential order of approximation for neural networks
Xu, Z. (2004) · Science in China Series F Information Sciences · doi:10.1360/02yf0221 · Cited by 71 · see citing works
Simultaneous Lp-approximation order for neural networks
Xu, Z., & Cao, F. (2005) · Neural Networks · doi:10.1016/j.neunet.2005.03.013 · Cited by 66 · see citing works
(1990) · Choice Reviews Online · doi:10.5860/choice.27-5166 · Cited by 50 · see citing works
Constructive Approximation of Discontinuous Functions by Neural Networks
Llanas, B., Sánchez, S. L., & Sáinz, F. J. (2008) · Neural Processing Letters · doi:10.1007/s11063-007-9070-9 · Cited by 49 · see citing works
Occlusion Resistant Object Rotation Regression from Point Cloud Segments
Gao, G., Lauri, M., Zhang, J., & Frintrop, S. (2019) · Lecture notes in computer science · doi:10.1007/978-3-030-11009-3_44 · Cited by 20 · see citing works
Uniform Sampling of Rotations for Discrete and Continuous Learning of 2D Shape Models
Perez-Sala, X., Igual, L., Escalera, S., et al. (2012) · IGI Global eBooks · doi:10.4018/978-1-4666-2672-0.ch002 · Cited by 15 · see citing works
The construction and approximation of neural networks operators with Gaussian activation function
Chen, Z., & Cao, F. (2013) · Hrčak Portal of scientific journals of Croatia (University Computing Centre) · Cited by 10 · see citing works
Embeddings of real projective spaces
Davis, D. M. (1998) · Boletín de la Sociedad Matemática Mexicana · Cited by 8 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
Diffusion policy: Visuomotor policy learning via action diffusion
Chi, C., Xu, Z., Feng, S., Cousineau, E., Du, Y., Burchfiel, B., Tedrake, R., & Song, S. (2025) · The International Journal of Robotics Research · doi:10.1177/02783649241273668 · Cited by 613 · see citing works
“We used the 6D rotation representation proposed in Zhou et al. (2019) for all environments (real-world and simulation) with positional control action space.”
Diffusion Policy: Visuomotor Policy Learning via Action Diffusion
Chi, C., Feng, S., Du, Y., Xu, Z., Cousineau, E., Burchfiel, B., & Song, S. (2023) · Robotics: Science and Systems XIX · doi:10.15607/rss.2023.xix.026 · Cited by 530 · see citing works
MotionDiffuse: Text-Driven Human Motion Generation with Diffusion Model
Zhang, M., Cai, Z., Pan, L., Hong, F., Guo, X., Yang, L., & Liu, Z. (2022) · arXiv · doi:10.48550/arxiv.2208.15001 · Cited by 437 · see citing works
FoundationPose: Unified 6D Pose Estimation and Tracking of Novel Objects
Wen, B., Yang, W., Kautz, J., & Birchfield, S. (2024) · 2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr52733.2024.01692 · Cited by 353 · see citing works
Where2Act: From Pixels to Actions for Articulated 3D Objects
Mo, K., Guibas, L., Mukadam, M., Gupta, A., & Tulsiani, S. (2021) · arXiv · doi:10.48550/arxiv.2101.02692 · Cited by 132 · see citing works
Robust multi-task learning and online refinement for spacecraft pose estimation across domain gap
Park, T. H., & D’Amico, S. (2024) · Advances in Space Research · doi:10.1016/j.asr.2023.03.036 · Cited by 125 · see citing works
“…cientPose implementation regresses the axis-angle representation of the orientation, the rotation subnet in this work instead regresses the 6D representation studied by Zhou et al. (2019), whose continuous property has shown to outperform other parametrization methods in rotation regression tasks.”
MethodCAPTRA: CAtegory-level Pose Tracking for Rigid and Articulated Objects from Point Clouds
Weng, Y., Wang, H., Zhou, Q., Qin, Y., Duan, Y., Fan, Q., Chen, B., Su, H., & Guibas, L. J. (2021) · 2021 IEEE/CVF International Conference on Computer Vision (ICCV) · doi:10.1109/iccv48922.2021.01296 · Cited by 103 · see citing works
LEGO-Net: Learning Regular Rearrangements of Objects in Rooms
Wei, Q. A., Ding, S., Park, J. J., Sajnani, R., Poulenard, A., Sridhar, S., & Guibas, L. (2023) · 2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr52729.2023.01825 · Cited by 60 · see citing works
Intraoperative 2D/3D Image Registration via Differentiable X-Ray Rendering
Gopalakrishnan, V., Dey, N., & Golland, P. (2024) · 2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr52733.2024.01108 · Cited by 48 · see citing works
“This has motivated the development of several higher-dimensional ( d > 5 ) alternatives with theoretical advantages [32, 40, 57].”
BackgroundMethodInfluentialA Smooth Representation of Belief over SO(3) for Deep Rotation Learning with Uncertainty
Peretroukhin, V., Giamou, M., Nicholas Greene, W., Rosen, D., Kelly, J., & Roy, N. (2020) · Robotics: Science and Systems XVI · doi:10.15607/rss.2020.xvi.007 · Cited by 41 · see citing works
PVSPE: A pyramid vision multitask transformer network for spacecraft pose estimation
Yang, H., Xiao, X., Yao, M., Xiong, Y., Cui, H., & Fu, Y. (2024) · Advances in Space Research · doi:10.1016/j.asr.2024.05.011 · Cited by 41 · see citing works
ChiNet: Deep recurrent convolutional learning for multimodal spacecraft pose estimation
Rondao, D., Aouf, N., & Richardson, M. A. (2022) · IEEE Transactions on Aerospace and Electronic Systems · doi:10.1109/taes.2022.3193085 · Cited by 24 · see citing works
Deep learning-based plane pose regression in obstetric ultrasound
Di Vece, C., Dromey, B., Vasconcelos, F., David, A. L., Peebles, D., & Stoyanov, D. (2022) · International Journal of Computer Assisted Radiology and Surgery · doi:10.1007/s11548-022-02609-z · Cited by 17 · see citing works
Diffusion Policy: Visuomotor Policy Learning via Action Diffusion
Chi, C., Xu, Z., Feng, S., Cousineau, E., Du, Y., Burchfiel, B., Tedrake, R., & Song, S. (2023) · arXiv · doi:10.48550/arxiv.2303.04137 · Cited by 12 · see citing works
“We used the 6D rotation representation proposed in Zhou et al. (2019) for all environments (real-world and simulation) with positional control action space.”
Domain Generalization for In-Orbit 6D Pose Estimation
Legrand, A., Detry, R., & De Vleeschouwer, C. (2025) · Journal of Aerospace Information Systems · doi:10.2514/1.i011498 · Cited by 7 · see citing works
Wang, Z., Liu, Y., & Zhang, E. (2024) · Aerospace · doi:10.3390/aerospace11110948 · Cited by 5 · see citing works
Rapid patient-specific neural networks for intraoperative X-ray to volume registration
Gopalakrishnan, V., Chlorogiannis, D.-D., Abumoussa, A., Larson, A. M., Haouchine, N., Orbach, D. B., Frisken, S., Dey, N., & Golland, P. (2025) · arXiv · doi:10.48550/arxiv.2503.16309 · Cited by 2 · see citing works
FastPose-ViT: A Vision Transformer for Real-Time Spacecraft Pose Estimation
Ancey, P., Price, A., Javed, S., & Salzmann, M. (2026) · 2026 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV) · doi:10.1109/wacv61042.2026.00760
“To address this, Zhou et al. [38] proposed continuous 6D representations to create a more stable learning target, showing improved performance.”
- Authors
- Zhou, Y., Barnes, C., Lu, J., Yang, J., & Li, H.
- Affiliations
- University of Southern California (US)
- Adobe Systems (United States) (US)
- Venue
- 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)
- Publisher
- IEEE
- Published
- 2019-06
- Pages
- 5738-5746
- ISSN
- 1063-6919 (ISSN-L)
- OpenAlex
- W2949924544
- Cited by
- 1,255
- References
- 29
- Citations by year
- 2026: 168 · 2025: 211 · 2024: 264 · 2023: 254 · 2022: 174 · 2021: 117 · 2020: 55 · 2019: 11 · 2018: 1
- FWCI
- 103.66
- Type
- conference paper
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Semiconductor Research Corporation
- Defense Advanced Research Projects Agency
- Office of Naval Research (N00014, N00014-17)
- Army Research Laboratory
- Topics
- Physical Sciences › Engineering › Computational Mechanics › 3D Shape Modeling and Analysis
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Physical Sciences › Mathematics › Applied Mathematics › Algebraic and Geometric Analysis
- Keywords
- rotation representation, continuous representations, SO(n), quaternions, Euler angles, homeomorphism, topological embeddings, rotation group, orthogonal group, similarity transformation, inverse kinematics, 3D human pose, autoencoder
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
