pose

Representations and conversions of SE(3)

Rigid transformations

We represent rigid transforms as \(4 \times 4\) matrices

\[\begin{equation} \begin{bmatrix} \mathbf R & \mathbf R \mathbf t \\ \mathbf 0 & 1 \end{bmatrix} \in \mathbf{SE}(3) \,, \end{equation}\]

where \(\mathbf R \in \mathbf{SO}(3)\) is a rotation matrix and \(\mathbf t\in \mathbb R^3\) represents a translation.

Tip

In this parameterization, \(\mathbf R \mathbf t\) represents the position of the C-arm source in world coordinates and \(\mathbf R\) represents the orientation of the C-arm.

Note that since rotation matrices are orthogonal (\(\mathbf R \mathbf R^T = \mathbf R^T \mathbf R = \mathbf I\)), we have a simple closed-form equation for the inverse: \[\begin{equation} \begin{bmatrix} \mathbf R & \mathbf R \mathbf t \\ \mathbf 0 & 1 \end{bmatrix}^{-1} = \begin{bmatrix} \mathbf R^T & -\mathbf t \\ \mathbf 0 & 1 \end{bmatrix} \,. \end{equation}\]

source

RigidTransform

 RigidTransform (matrix)

Applies rigid transforms in SE(3) to point clouds. Can handle batched rigid transforms, composition of transforms, closed-form inversion, and conversions to various representations of SE(3).

SE(3) Conversions

source

convert

 convert (*args, parameterization, convention=None)

9D rotation parameterization

SVDO+ (Levinson et al., 2020) use the SVD to symetmetrically orthogonalize a matrix.

source

matrix_to_rotation_9d

 matrix_to_rotation_9d (matrix:torch.Tensor)

source

rotation_9d_to_matrix

 rotation_9d_to_matrix (rotation:torch.Tensor)

Convert a 9-vector to a symmetrically orthogonalized rotation matrix via SVD.

10D rotation parameterizations

Implementations to convert rotation_10d (Peretroukhin et al., 2021) and quaternion_adjugate (Hanson and Hanson, 2022) parameterizations of SO(3) to quaternions.

source

quaternion_to_rotation_10d

 quaternion_to_rotation_10d (q:torch.Tensor)

source

rotation_10d_to_quaternion

 rotation_10d_to_quaternion (rotation:torch.Tensor)

*Convert a 10-vector into a symmetric matrix, whose eigenvector corresponding to the eigenvalue of minimum modulus is the resulting quaternion.

Source: https://arxiv.org/abs/2006.01031*

source

quaternion_to_quaternion_adjugate

 quaternion_to_quaternion_adjugate (q:torch.Tensor)

source

quaternion_adjugate_to_quaternion

 quaternion_adjugate_to_quaternion (rotation:torch.Tensor)

*Convert a 10-vector in the quaternion adjugate, a symmetric matrix whose eigenvector corresponding to the eigenvalue of maximum modulus is the (unnormalized) quaternion. Uses a fast method to solve for the eigenvector without explicity computing the eigendecomposition.

Source: https://arxiv.org/abs/2205.09116*

PyTorch3D conversions port

PyTorch3D has many useful conversion functions for transforming between multiple parameterizations of \(\mathbf{SO}(3)\) and \(\mathbf{SE}(3)\). However, installing PyTorch3D can be annoying for users not on Linux. We include the required conversion functions for PyTorch3D below. The original LICENSE from PyTorch3D is also included:

BSD License

For PyTorch3D software

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