pypose.Adj

class pypose.Adj(input, p)

The dot product between the Adjoint matrix at the point given by an input (Lie Group) and the second point (Lie Algebra).

\[\mathrm{Adj}: (\mathcal{G}, \mathcal{g}) \mapsto \mathcal{g} \]

Parameters

• input (LieTensor) – the input LieTensor (Lie Group)

• p (LieTensor) – the second LieTensor (Lie Algebra)

Returns

the output LieTensor (Lie Algebra)

Return type

LieTensor

List of supported \(\mathrm{Adj}\) map

input ltype	\((\mathcal{G}, \mathcal{g})\) (Lie Group, Lie Algebra)	\(\mapsto\)	\(\mathcal{g}\) (Lie Algebra)	output ltype
(SO3_type, so3_type)	\((\mathcal{G}\in\mathbb{R}^{*\times4}, \mathcal{g}\in\mathbb{R}^{*\times3})\)	\(\mapsto\)	\(\mathcal{g}\in\mathbb{R}^{*\times3}\)	so3_type
(SE3_type, se3_type)	\((\mathcal{G}\in\mathbb{R}^{*\times7}, \mathcal{g}\in\mathbb{R}^{*\times6})\)	\(\mapsto\)	\(\mathcal{g}\in\mathbb{R}^{*\times6}\)	se3_type
(Sim3_type, sim3_type)	\((\mathcal{G}\in\mathbb{R}^{*\times8}, \mathcal{g}\in\mathbb{R}^{*\times7})\)	\(\mapsto\)	\(\mathcal{g}\in\mathbb{R}^{*\times7}\)	sim3_type
(RxSO3_type, rxso3_type)	\((\mathcal{G}\in\mathbb{R}^{*\times5}, \mathcal{g}\in\mathbb{R}^{*\times4})\)	\(\mapsto\)	\(\mathcal{g}\in\mathbb{R}^{*\times4}\)	rxso3_type

Let the input be (\(\mathbf{x}\), \(\mathbf{p}\)), \(\mathbf{y}\) be the output.

\[\mathbf{y}_i = \mathrm{Adj}(\mathbf{x}_i)\mathbf{p}_i, \]

where, \(\mathrm{Adj}(\mathbf{x}_i)\) is the adjoint matrix of the Lie group of \(\mathbf{x}_i\).

• If input (\(\mathbf{x}\), \(\mathbf{p}\))’s ltype are SO3_type and so3_type (input \(\mathbf{x}\) is an instance of SO3(), \(\mathbf{p}\) is an instance of so3()). Given \(\mathbf{x}_i \in\) \(\textrm{SO(3)}\). The adjoint transformation is given by:

  \[\mathrm{Adj}(\mathbf{x}_i) = \mathbf{x}_i \]

  In the case of \(\textrm{SO3}\), the adjoint transformation for an element is the same rotation matrix used to represent the element. Rotating a tangent vector by an element “moves” it from the tangent space on the right side of the element to the tangent space on the left.

• If input (\(\mathbf{x}\), \(\mathbf{p}\))’s ltype are SE3_type and se3_type (input \(\mathbf{x}\) is an instance of SE3(), \(\mathbf{p}\) is an instance of se3()). Let \(\mathbf{R}_i \in\) \(\textrm{SO(3)}\) and \(\mathbf{t}_i \in \mathbb{R}^{3\times3}\) represent the rotation and translation part of the group. Let \(\mathbf{t}_{i\times}\) be the skew matrix (pypose.vec2skew()) of \(\mathbf{t}_i\). The adjoint transformation is given by:

  \[\mathrm{Adj}(\mathbf{x}_i) = \left[ \begin{array}{cc} \mathbf{R}_i & \mathbf{t}_{i\times}\mathbf{R}_i \\ 0 & \mathbf{R}_i \end{array} \right] \in \mathbb{R}^{6\times6} \]

  where,

  \[\mathbf{x}_i = \left[ \begin{array}{cc} \mathbf{R}_i& \mathbf{t}_i \\ 0 & 1 \end{array} \right] \in \mathrm{SE(3)} \]

• If input (\(\mathbf{x}\), \(\mathbf{p}\))’s ltype are Sim3_type and sim3_type (input \(\mathbf{x}\) is an instance of Sim3(), \(\mathbf{p}\) is an instance of sim3()). Let \(\mathbf{R}_i\in\) \(\textrm{SO(3)}\), \(\mathbf{t}_i \in \mathbb{R}^{3\times3}\), and \(s_i \in \mathbb{R}^+\) represent the rotation, translation, and scale parts of the group. Let \(\mathbf{t}_{i\times}\) be the skew matrix (pypose.vec2skew()) of \(\mathbf{t}_i\). The adjoint transformation is given by:

  \[\mathrm{Adj}(\mathbf{x}_i) = \left[ \begin{array}{cc} s_i\mathbf{R}_i& \mathbf{t}_{i\times}\mathbf{R}_i& -\mathbf{t}_i \\ 0 & \mathbf{R}_i& 0 \\ 0 & 0 & 1 \end{array} \right] \in \mathbb{R}^{7\times7} \]

  where,

  \[\mathbf{x}_i = \left[ \begin{array}{cc} s_i\mathbf{R}_i & \mathbf{t}_i \\ 0 & 1 \end{array} \right] \in \textrm{Sim(3)} \]

• If input (\(\mathbf{x}\), \(\mathbf{p}\))’s ltype are RxSO3_type and rxso3_type (input \(\mathbf{x}\) is an instance of RxSO3(), \(\mathbf{p}\) is an instance of rxso3()). Let \(\mathbf{R}_i \in\) \(\textrm{SO(3)}\), and \(s_i \in \mathbb{R}^+\) represent the rotation and scale parts of the group. The adjoint transformation is given by:

  \[\mathrm{Adj}(\mathbf{x}_i) = \left[ \begin{array}{cc} \mathbf{R}_i & 0 \\ 0 & 1 \end{array} \right] \in \mathbb{R}^{4\times4} \]

  where,

  \[\mathbf{x}_i = \left[ \begin{array}{cc} s_i\mathbf{R}_i & 0 \\ 0 & 1 \end{array} \right] \in \mathrm{RxSO(3)} \]

  In the case of \(\textrm{RxSO3}\) group, the adjoint transformation is the same as the rotation matrix of the group i.e. the \(\textrm{SO3}\) part of the group.

Note

The adjoint operator is a linear map which moves an element \(\mathbf{p} \in \mathcal{g}\) in the right tangent space of \(\mathbf{x} \in \mathcal{G}\) to the left tangent space.

\[\mathrm{Exp}(\mathrm{Adj}(\mathbf{x}) \mathbf{p}) * \mathbf{x} = \mathbf{x} * \mathrm{Exp}(\mathbf{p}) \]

It can be easily verified:

>>> x, p = pp.randn_SO3(), pp.randn_so3()
>>> torch.allclose(x*p.Exp(), x.Adj(p).Exp()*x)
True

One can refer to Eq. (8) of the following paper:

• Zachary Teed et al., Tangent Space Backpropagation for 3D Transformation Groups, IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2021.

Note

\(\mathrm{Adj}\) is generally used to transform a tangent vector from the tangent space around one element to the tangent space of another. One can refer to this paper for more details:

• J. Sola et al., A micro Lie theory for state estimation in robotics, arXiv preprint arXiv:1812.01537 (2018).

The following thesis and the tutorial serve as a good reading material to learn more about deriving the adjoint matrices for different transformation groups.

• Strasdat, H., 2012. Local accuracy and global consistency for efficient visual SLAM, (Doctoral dissertation, Department of Computing, Imperial College London).

• Lie Groups for 2D and 3D Transformations., by Ethan Eade.

Example

• \(\mathrm{Adj}\): (SO3, so3) \(\mapsto\) so3

  >>> x = pp.randn_SO3(2)
  >>> p = pp.randn_so3(2)
  >>> x.Adj(p) # equivalent to: pp.Adj(x, p)
  so3Type LieTensor:
  tensor([[-0.4171,  2.1218,  0.9951],
          [ 1.8415, -1.2185, -0.4082]])
  

• \(\mathrm{Adj}\): (SE3, se3) \(\mapsto\) se3

  >>> x = pp.randn_SE3(2)
  >>> p = pp.randn_se3(2)
  >>> x.Adj(p) # equivalent to: pp.Adj(x, p)
  se3Type LieTensor:
  tensor([[-0.8536, -0.1984, -0.4554, -0.4868,  0.3231,  0.8535],
          [ 0.1577, -1.7625,  1.7997, -1.5085, -0.2098,  0.3538]])
  

• \(\mathrm{Adj}\): (Sim3, sim3) \(\mapsto\) sim3

  >>> x = pp.randn_Sim3(2)
  >>> p = pp.randn_sim3(2)
  >>> x.Adj(p) # equivalent to: pp.Adj(x, p)
  sim3Type LieTensor:
  tensor([[ 0.1455, -0.5653, -0.1845,  0.0502,  1.3125,  1.5217, -0.8964],
          [-4.8724, -0.5254,  3.9559,  1.5170,  1.7610,  0.4375,  0.4248]])
  

• \(\mathrm{Adj}\): (RxSO3, rxso3) \(\mapsto\) rxso3

  >>> x = pp.randn_RxSO3(2)
  >>> p = pp.randn_rxso3(2)
  >>> x.Adj(p) # equivalent to: pp.Adj(x, p)
  rxso3Type LieTensor:
  tensor([[-1.3590, -0.4314, -0.0297,  1.0166],
          [-0.3378, -0.4942, -2.0083, -0.4321]])