# Source code for pytorch3d.transforms.se3

```# Copyright (c) Meta Platforms, Inc. and affiliates.
#
# LICENSE file in the root directory of this source tree.

import torch
from pytorch3d.common.compat import solve

from .so3 import _so3_exp_map, hat, so3_log_map

[docs]def se3_exp_map(log_transform: torch.Tensor, eps: float = 1e-4) -> torch.Tensor:
"""
Convert a batch of logarithmic representations of SE(3) matrices `log_transform`
to a batch of 4x4 SE(3) matrices using the exponential map.
See e.g. , Sec 9.4.2. for more detailed description.

A SE(3) matrix has the following form:
```
[ R 0 ]
[ T 1 ] ,
```
where `R` is a 3x3 rotation matrix and `T` is a 3-D translation vector.
SE(3) matrices are commonly used to represent rigid motions or camera extrinsics.

In the SE(3) logarithmic representation SE(3) matrices are
represented as 6-dimensional vectors `[log_translation | log_rotation]`,
i.e. a concatenation of two 3D vectors `log_translation` and `log_rotation`.

The conversion from the 6D representation to a 4x4 SE(3) matrix `transform`
is done as follows:
```
transform = exp( [ hat(log_rotation) 0 ]
[   log_translation 1 ] ) ,
```
where `exp` is the matrix exponential and `hat` is the Hat operator .

Note that for any `log_transform` with `0 <= ||log_rotation|| < 2pi`
(i.e. the rotation angle is between 0 and 2pi), the following identity holds:
```
se3_log_map(se3_exponential_map(log_transform)) == log_transform
```

The conversion has a singularity around `||log(transform)|| = 0`
which is handled by clamping controlled with the `eps` argument.

Args:
log_transform: Batch of vectors of shape `(minibatch, 6)`.
eps: A threshold for clipping the squared norm of the rotation logarithm
to avoid unstable gradients in the singular case.

Returns:
Batch of transformation matrices of shape `(minibatch, 4, 4)`.

Raises:
ValueError if `log_transform` is of incorrect shape.

 https://en.wikipedia.org/wiki/Hat_operator
"""

if log_transform.ndim != 2 or log_transform.shape != 6:
raise ValueError("Expected input to be of shape (N, 6).")

N, _ = log_transform.shape

log_translation = log_transform[..., :3]
log_rotation = log_transform[..., 3:]

# rotation is an exponential map of log_rotation
(
R,
rotation_angles,
log_rotation_hat,
log_rotation_hat_square,
) = _so3_exp_map(log_rotation, eps=eps)

# translation is V @ T
V = _se3_V_matrix(
log_rotation,
log_rotation_hat,
log_rotation_hat_square,
rotation_angles,
eps=eps,
)
T = torch.bmm(V, log_translation[:, :, None])[:, :, 0]

transform = torch.zeros(
N, 4, 4, dtype=log_transform.dtype, device=log_transform.device
)

transform[:, :3, :3] = R
transform[:, :3, 3] = T
transform[:, 3, 3] = 1.0

return transform.permute(0, 2, 1)

[docs]def se3_log_map(
transform: torch.Tensor, eps: float = 1e-4, cos_bound: float = 1e-4
) -> torch.Tensor:
"""
Convert a batch of 4x4 transformation matrices `transform`
to a batch of 6-dimensional SE(3) logarithms of the SE(3) matrices.
See e.g. , Sec 9.4.2. for more detailed description.

A SE(3) matrix has the following form:
```
[ R 0 ]
[ T 1 ] ,
```
where `R` is an orthonormal 3x3 rotation matrix and `T` is a 3-D translation vector.
SE(3) matrices are commonly used to represent rigid motions or camera extrinsics.

In the SE(3) logarithmic representation SE(3) matrices are
represented as 6-dimensional vectors `[log_translation | log_rotation]`,
i.e. a concatenation of two 3D vectors `log_translation` and `log_rotation`.

The conversion from the 4x4 SE(3) matrix `transform` to the
6D representation `log_transform = [log_translation | log_rotation]`
is done as follows:
```
log_transform = log(transform)
log_translation = log_transform[3, :3]
log_rotation = inv_hat(log_transform[:3, :3])
```
where `log` is the matrix logarithm
and `inv_hat` is the inverse of the Hat operator .

Note that for any valid 4x4 `transform` matrix, the following identity holds:
```
se3_exp_map(se3_log_map(transform)) == transform
```

The conversion has a singularity around `(transform=I)` which is handled
by clamping controlled with the `eps` and `cos_bound` arguments.

Args:
transform: batch of SE(3) matrices of shape `(minibatch, 4, 4)`.
eps: A threshold for clipping the squared norm of the rotation logarithm
to avoid division by zero in the singular case.
cos_bound: Clamps the cosine of the rotation angle to
[-1 + cos_bound, 3 - cos_bound] to avoid non-finite outputs.
The non-finite outputs can be caused by passing small rotation angles
to the `acos` function in `so3_rotation_angle` of `so3_log_map`.

Returns:
Batch of logarithms of input SE(3) matrices
of shape `(minibatch, 6)`.

Raises:
ValueError if `transform` is of incorrect shape.
ValueError if `R` has an unexpected trace.

 https://en.wikipedia.org/wiki/Hat_operator
"""

if transform.ndim != 3:
raise ValueError("Input tensor shape has to be (N, 4, 4).")

N, dim1, dim2 = transform.shape
if dim1 != 4 or dim2 != 4:
raise ValueError("Input tensor shape has to be (N, 4, 4).")

if not torch.allclose(transform[:, :3, 3], torch.zeros_like(transform[:, :3, 3])):
raise ValueError("All elements of `transform[:, :3, 3]` should be 0.")

# log_rot is just so3_log_map of the upper left 3x3 block
R = transform[:, :3, :3].permute(0, 2, 1)
log_rotation = so3_log_map(R, eps=eps, cos_bound=cos_bound)

# log_translation is V^-1 @ T
T = transform[:, 3, :3]
V = _se3_V_matrix(*_get_se3_V_input(log_rotation), eps=eps)
log_translation = solve(V, T[:, :, None])[:, :, 0]

def _se3_V_matrix(
log_rotation: torch.Tensor,
log_rotation_hat: torch.Tensor,
log_rotation_hat_square: torch.Tensor,
rotation_angles: torch.Tensor,
eps: float = 1e-4,
) -> torch.Tensor:
"""
A helper function that computes the "V" matrix from , Sec 9.4.2.
"""

V = (
torch.eye(3, dtype=log_rotation.dtype, device=log_rotation.device)[None]
+ log_rotation_hat
* ((1 - torch.cos(rotation_angles)) / (rotation_angles ** 2))[:, None, None]
+ (
log_rotation_hat_square
* ((rotation_angles - torch.sin(rotation_angles)) / (rotation_angles ** 3))[
:, None, None
]
)
)

return V

def _get_se3_V_input(log_rotation: torch.Tensor, eps: float = 1e-4):
"""
A helper function that computes the input variables to the `_se3_V_matrix`
function.
"""
nrms = (log_rotation ** 2).sum(-1)
rotation_angles = torch.clamp(nrms, eps).sqrt()
log_rotation_hat = hat(log_rotation)
log_rotation_hat_square = torch.bmm(log_rotation_hat, log_rotation_hat)
return log_rotation, log_rotation_hat, log_rotation_hat_square, rotation_angles
```