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# pypose.mul¶

class pypose.mul(input, other)[source]

Multiply input LieTensor by other.

$\bm{y}_i = \bm{x}_i \ast \bm{a}_i$

where $$\bm{x}$$ is the input LieTensor, $$\bm{a}$$ is the other Tensor or LieTensor, and $$\bm{y}$$ is the output.

Parameters
Returns

the product of input and other.

Return type

Tensor/LieTensor

List of pypose.mul cases

input LieTensor

other

output

Lie Algebra

Number

Lie Algebra

Lie Group

Tensor $$\in \mathbb{R^{*\times3}}$$

Tensor

Lie Group

Tensor $$\in \mathbb{R^{*\times4}}$$

Tensor

Lie Group

Lie Group

Lie Group

When multiplying a Lie Group by another Lie Group, they must have the same Lie type.

Note

• When other is a Tensor, this operator is equivalent to Act().

• When other is a number and input is a Lie Algebra, this operator performs simple element-wise multiplication.

• When input is a Lie Group, more details are shown below.

• Input $$\bm{x}$$’s ltype is SO3_type (input $$\bm{x}$$ is an instance of SO3()):

${\displaystyle \bm{y}_i={\begin{bmatrix} q_i^wq_i^{w'} - q_i^xq_i^{x'} -q_i^yq_i^{y'} - q_i^zq_i^{z'}\\ q_i^wq_i^{x'} + q_i^xq_i^{w'} +q_i^yq_i^{z'} - q_i^zq_i^{y'}\\ q_i^wq_i^{y'} - q_i^xq_i^{z'} +q_i^yq_i^{w'} + q_i^zq_i^{x'}\\ q_i^wq_i^{z'} + q_i^xq_i^{y'} -q_i^yq_i^{x'} + q_i^zq_i^{w'} \end{bmatrix} }^T},$

where $$\bm{x}_i = [q_i^x, q_i^y, q_i^z, q_i^w]$$ and $$\bm{a}_i = [q_i^{x'}, q_i^{y'}, q_i^{z'}, q_i^{w'}]$$ are the input and other LieTensor, respectively.

Note

$$\mathbf{y}_i$$ can be simply derived by taking the complex number multiplication.

$\bm{y}_i = (q_i^x\mathbf{i} + q_i^y\mathbf{j} + q_i^z\mathbf{k} + q_i^w) \ast (q_i^{x'}\mathbf{i} + q_i^{y'}\mathbf{j} + q_i^{z'}\mathbf{k} + q_i^{w'}),$

where and $$\mathbf{i}$$ $$\mathbf{j}$$, and $$\mathbf{k}$$ are the imaginary units.

$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{ijk} = -1$
• Input $$\bm{x}$$’s ltype is SE3_type (input $$\bm{x}$$ is an instance of SE3()):

$\bm{y}_i = [\mathbf{q}_i * \mathbf{t}_i' + \mathbf{t}_i, \mathbf{q}_i * \mathbf{q}_i']$

where $$\bm{x}_i = [\mathbf{t}_i, \mathbf{q}_i]$$ and $$\bm{a}_i = [\mathbf{t}_i', \mathbf{q}_i']$$ are the input and other LieTensor, respectively; $$\mathbf{t}_i$$, $$\mathbf{t}_i'$$ and $$\mathbf{q}_i$$, $$\mathbf{q}_i'$$ are their translation and SO3 parts, respectively; the operator $$\ast$$ denotes the obj:SO3_type multiplication introduced above.

• Input $$\bm{x}$$’s ltype is RxSO3_type (input $$\bm{x}$$ is an instance of RxSO3())

$\bm{y}_i = [\mathbf{q}_i * \mathbf{q}_i', s_is_i']$

where $$s_i$$ and $$s_i'$$ are the scale parts of the input and other LieTensor, respectively.

• Input $$\bm{x}$$’s ltype is Sim3_type (input $$\bm{x}$$ is an instance of Sim3()):

$\bm{y}_i = [\mathbf{q}_i * \mathbf{t}_i' + \mathbf{t}_i, \mathbf{q}_i * \mathbf{q}_i', s_is_i']$

where $$\bm{x}_i = [\mathbf{t}_i, \mathbf{q}_i, s_i]$$ and $$\bm{a}_i = [\mathbf{t}_i', \mathbf{q}_i', s_i']$$ are the input and other LieTensor, respectively.

Examples

• Lie Algebra $$*$$ Number $$\mapsto$$ Lie Algebra

>>> x = pp.randn_so3()
>>> x
so3Type LieTensor:
LieTensor([ 0.3018, -1.0246,  0.7784])
>>> a = 5
>>> # The following two operations are equivalent.
>>> x * a
so3Type LieTensor:
LieTensor([ 1.5090, -5.1231,  3.8919])
>>> pp.mul(x, 5)
so3Type LieTensor:
LieTensor([ 1.5090, -5.1231,  3.8919])

• Lie Group $$*$$ Tensor $$\mapsto$$ Tensor

>>> x = pp.randn_SO3()
>>> a = torch.randn(3)
>>> x, a
(SO3Type LieTensor:
LieTensor([ 0.6047, -0.2129, -0.1781,  0.7465]),
tensor([-0.1811, -0.2278, -1.9956]))
>>> x * a
tensor([ 0.9089,  1.6984, -0.5969])
>>> a = torch.randn(4)
>>> a
tensor([ 1.5236, -1.2757, -0.7140,  0.2467])
>>> x * a
tensor([ 1.6588, -0.4687, -1.2196,  0.2467]

• SO3_type $$*$$ SO3_type $$\mapsto$$ SO3_type

>>> a = pp.randn_SO3()
>>> a
SO3Type LieTensor:
LieTensor([ 0.0118, -0.7042, -0.4516,  0.5478])
>>> x * a
SO3Type LieTensor:
LieTensor([ 0.3108, -0.3714, -0.8579,  0.1715])

• SE3_type $$*$$ SE3_type $$\mapsto$$ SE3_type

>>> x = pp.randn_SE3()
>>> a = pp.randn_SE3()
>>> x, a
(SE3Type LieTensor:
LieTensor([ 0.7819,  1.8541, -0.2857, -0.1970,  0.4742,  0.1109,  0.8509]),
SE3Type LieTensor:
LieTensor([ 0.6039, -1.4076,  0.3496,  0.7297,  0.3971,  0.2849,  0.4783]))
>>> x * a
SE3Type LieTensor:
LieTensor([ 1.8949,  0.7456, -0.3104,  0.6177,  0.7017, -0.1287,  0.3308])

• RxSO3_type $$*$$ RxSO3_type $$\mapsto$$ RxSO3_type

>>> x = pp.randn_RxSO3()
>>> a = pp.randn_RxSO3()
>>> x, a
(RxSO3Type LieTensor:
LieTensor([-0.7518, -0.6481,  0.0933, -0.0775,  1.5791]),
RxSO3Type LieTensor:
LieTensor([ 0.2757,  0.3102, -0.4086,  0.8129,  0.6593]))
>>> x * a
RxSO3Type LieTensor:
LieTensor([-0.3967, -0.8323,  0.0530,  0.3835,  1.0411])

• Sim3_type $$*$$ Sim3_type $$\mapsto$$ Sim3_type

>>> x = pp.randn_Sim3()
>>> a = pp.randn_Sim3()
>>> x, a
(Sim3Type LieTensor:
LieTensor([-0.3439, -0.2309, -0.6571,  0.3170, -0.6594, -0.1100,  0.6728, 0.6296]),
Sim3Type LieTensor:
LieTensor([-0.7434,  1.8613, -2.1315,  0.7688, -0.0268,  0.0520,  0.6367, 1.7745]))
>>> x * a
Sim3Type LieTensor:
LieTensor([ 0.5740,  1.3197, -0.2752,  0.6819, -0.5389,  0.4634,  0.1727, 1.1172])


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