On the Continuity of Rotation Representations in Neural Networks

By Connelly Barnes, Jingwan Lu, Jimei Yang, Hao Li. 2018

1,859 citations recorded by the source corpus.

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.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition

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Citations in the corpus

Visual Imitation Made Easy (2020, methodology): “We train on a 6D rotation representation [47] because it is continuous in the real Euclidean space and thus more suitable for learning as opposed to more commonly used axis-angle and quaternion based representations.”
Perpetual Humanoid Control for Real-time Simulated Avatars (2023, neutral): “We define the reference pose as q̂t ≜ (θ̂t, p̂t), consisting of 3D joint rotation θ̂t ∈ RJ×6 and position p̂t ∈ RJ×3 of all J links on the humanoid (we use the 6 DoF rotation representation [54]).”
MotionCLIP: Exposing Human Motion Generation to CLIP Space (2022, background): “A sequence of length T denoted p1:T such that pi ∈ R24×6 defines orientations in 6D representation[50] for global body orientation and 23 SMPL joints, at the i frame.”
Deep Global Registration (2020, methodology): “[52], rather than Euler angles or quaternions.”
Contact-GraspNet: Efficient 6-DoF Grasp Generation in Cluttered Scenes (2021, background): “representation and facilitates the regression of 3D rotations [36].”
Towards Unconstrained Joint Hand-Object Reconstruction From RGB Videos (2021, methodology): “1 for the PCA components of the MANO [53] articulated hand model and the continuous rotation parameters [66].”
UniSim: A Neural Closed-Loop Sensor Simulator (2023, methodology): “We parameterize the rotation component using a 6D representation [104].”
RDT-1B: a Diffusion Foundation Model for Bimanual Manipulation (2024, positive): “We also employ the 6D representation (Zhou et al., 2019) for the EEF rotation to overcome the gimbal lock issue.”
MotionDiffuse: Text-Driven Human Motion Generation With Diffusion Model (2022, background): “j ∈ RJ×6 is the 6D rotation (Zhou et al., 2019) of each joint.”
CosyPose: Consistent multi-view multi-object 6D pose estimation (2020, methodology): “Second, we exploit the rotation parametrization recently introduced in [41], which has been shown to lead to more stable CNN training than quaternions.”
EDGE: Editable Dance Generation From Music (2022, methodology): “Pose Representation We represent dances as sequences of poses in the 24-joint SMPL format [35], using the 6DOF rotation representation [64] for every joint and a single root translation: w ∈ R24·6+3=147.”
3D Diffuser Actor: Policy Diffusion with 3D Scene Representations (2024, positive): “We represent rotations using the 6D rotation representation of [85] for all environments in all our experiments, to avoid the discontinuities of the quaternion representation.”
Humans in 4D: Reconstructing and Tracking Humans with Transformers (2023, methodology): “We follow [35] and regress 3D rotations using the representation of [91].”
NASA: Neural Articulated Shape Approximation (2019, background): “…diﬀerential as: resulting in the optimization: where we parameterize the rotational portion of elements in the collection { ∆ B − 1 b } by two (initially orthogonal) vectors [59], and re-orthogonalize them before inversion within each optimization update are the quantities the solver optimizes for.”
GRAB: A Dataset of Whole-Body Human Grasping of Objects (2020, methodology): “with the latent code as the input (condition) to the decoder. The outputs of the decoder are MANO translation ( 2 R 3 ) and joint angles ( 2 R 96 ) in the continuous 6-dimensional representation of [15]. Using our validation set, we found out that 16 dimensions for the latent space results in gener”
Perceiving 3D Human-Object Spatial Arrangements from a Single Image in the Wild (2020, methodology): “1 Implementation details We represent rotations for the object poses using the 6-DoF rotation representation introduced in [6].”
Monocular, One-stage, Regression of Multiple 3D People (2020, background): “The pose parameters θ ∈ R6×22 contain the 3D rotation of the 22 body joints in a 6D representation [59].”