# Source code for pytorch3d.transforms.transform3d

```
# Copyright (c) Meta Platforms, Inc. and affiliates.
# All rights reserved.
#
# This source code is licensed under the BSD-style license found in the
# LICENSE file in the root directory of this source tree.
import math
import warnings
from typing import List, Optional, Union
import torch
from ..common.datatypes import Device, get_device, make_device
from ..common.workaround import _safe_det_3x3
from .rotation_conversions import _axis_angle_rotation
[docs]class Transform3d:
"""
A Transform3d object encapsulates a batch of N 3D transformations, and knows
how to transform points and normal vectors. Suppose that t is a Transform3d;
then we can do the following:
.. code-block:: python
N = len(t)
points = torch.randn(N, P, 3)
normals = torch.randn(N, P, 3)
points_transformed = t.transform_points(points) # => (N, P, 3)
normals_transformed = t.transform_normals(normals) # => (N, P, 3)
BROADCASTING
Transform3d objects supports broadcasting. Suppose that t1 and tN are
Transform3d objects with len(t1) == 1 and len(tN) == N respectively. Then we
can broadcast transforms like this:
.. code-block:: python
t1.transform_points(torch.randn(P, 3)) # => (P, 3)
t1.transform_points(torch.randn(1, P, 3)) # => (1, P, 3)
t1.transform_points(torch.randn(M, P, 3)) # => (M, P, 3)
tN.transform_points(torch.randn(P, 3)) # => (N, P, 3)
tN.transform_points(torch.randn(1, P, 3)) # => (N, P, 3)
COMBINING TRANSFORMS
Transform3d objects can be combined in two ways: composing and stacking.
Composing is function composition. Given Transform3d objects t1, t2, t3,
the following all compute the same thing:
.. code-block:: python
y1 = t3.transform_points(t2.transform_points(t1.transform_points(x)))
y2 = t1.compose(t2).compose(t3).transform_points(x)
y3 = t1.compose(t2, t3).transform_points(x)
Composing transforms should broadcast.
.. code-block:: python
if len(t1) == 1 and len(t2) == N, then len(t1.compose(t2)) == N.
We can also stack a sequence of Transform3d objects, which represents
composition along the batch dimension; then the following should compute the
same thing.
.. code-block:: python
N, M = len(tN), len(tM)
xN = torch.randn(N, P, 3)
xM = torch.randn(M, P, 3)
y1 = torch.cat([tN.transform_points(xN), tM.transform_points(xM)], dim=0)
y2 = tN.stack(tM).transform_points(torch.cat([xN, xM], dim=0))
BUILDING TRANSFORMS
We provide convenience methods for easily building Transform3d objects
as compositions of basic transforms.
.. code-block:: python
# Scale by 0.5, then translate by (1, 2, 3)
t1 = Transform3d().scale(0.5).translate(1, 2, 3)
# Scale each axis by a different amount, then translate, then scale
t2 = Transform3d().scale(1, 3, 3).translate(2, 3, 1).scale(2.0)
t3 = t1.compose(t2)
tN = t1.stack(t3, t3)
BACKPROP THROUGH TRANSFORMS
When building transforms, we can also parameterize them by Torch tensors;
in this case we can backprop through the construction and application of
Transform objects, so they could be learned via gradient descent or
predicted by a neural network.
.. code-block:: python
s1_params = torch.randn(N, requires_grad=True)
t_params = torch.randn(N, 3, requires_grad=True)
s2_params = torch.randn(N, 3, requires_grad=True)
t = Transform3d().scale(s1_params).translate(t_params).scale(s2_params)
x = torch.randn(N, 3)
y = t.transform_points(x)
loss = compute_loss(y)
loss.backward()
with torch.no_grad():
s1_params -= lr * s1_params.grad
t_params -= lr * t_params.grad
s2_params -= lr * s2_params.grad
CONVENTIONS
We adopt a right-hand coordinate system, meaning that rotation about an axis
with a positive angle results in a counter clockwise rotation.
This class assumes that transformations are applied on inputs which
are row vectors. The internal representation of the Nx4x4 transformation
matrix is of the form:
.. code-block:: python
M = [
[Rxx, Ryx, Rzx, 0],
[Rxy, Ryy, Rzy, 0],
[Rxz, Ryz, Rzz, 0],
[Tx, Ty, Tz, 1],
]
To apply the transformation to points which are row vectors, the M matrix
can be pre multiplied by the points:
.. code-block:: python
points = [[0, 1, 2]] # (1 x 3) xyz coordinates of a point
transformed_points = points * M
"""
[docs] def __init__(
self,
dtype: torch.dtype = torch.float32,
device: Device = "cpu",
matrix: Optional[torch.Tensor] = None,
) -> None:
"""
Args:
dtype: The data type of the transformation matrix.
to be used if `matrix = None`.
device: The device for storing the implemented transformation.
If `matrix != None`, uses the device of input `matrix`.
matrix: A tensor of shape (4, 4) or of shape (minibatch, 4, 4)
representing the 4x4 3D transformation matrix.
If `None`, initializes with identity using
the specified `device` and `dtype`.
"""
if matrix is None:
self._matrix = torch.eye(4, dtype=dtype, device=device).view(1, 4, 4)
else:
if matrix.ndim not in (2, 3):
raise ValueError('"matrix" has to be a 2- or a 3-dimensional tensor.')
if matrix.shape[-2] != 4 or matrix.shape[-1] != 4:
raise ValueError(
'"matrix" has to be a tensor of shape (minibatch, 4, 4)'
)
# set dtype and device from matrix
dtype = matrix.dtype
device = matrix.device
self._matrix = matrix.view(-1, 4, 4)
self._transforms = [] # store transforms to compose
self._lu = None
self.device = make_device(device)
self.dtype = dtype
def __len__(self) -> int:
return self.get_matrix().shape[0]
[docs] def __getitem__(
self, index: Union[int, List[int], slice, torch.Tensor]
) -> "Transform3d":
"""
Args:
index: Specifying the index of the transform to retrieve.
Can be an int, slice, list of ints, boolean, long tensor.
Supports negative indices.
Returns:
Transform3d object with selected transforms. The tensors are not cloned.
"""
if isinstance(index, int):
index = [index]
return self.__class__(matrix=self.get_matrix()[index])
[docs] def compose(self, *others: "Transform3d") -> "Transform3d":
"""
Return a new Transform3d representing the composition of self with the
given other transforms, which will be stored as an internal list.
Args:
*others: Any number of Transform3d objects
Returns:
A new Transform3d with the stored transforms
"""
out = Transform3d(dtype=self.dtype, device=self.device)
out._matrix = self._matrix.clone()
for other in others:
if not isinstance(other, Transform3d):
msg = "Only possible to compose Transform3d objects; got %s"
raise ValueError(msg % type(other))
out._transforms = self._transforms + list(others)
return out
[docs] def get_matrix(self) -> torch.Tensor:
"""
Return a matrix which is the result of composing this transform
with others stored in self.transforms. Where necessary transforms
are broadcast against each other.
For example, if self.transforms contains transforms t1, t2, and t3, and
given a set of points x, the following should be true:
.. code-block:: python
y1 = t1.compose(t2, t3).transform(x)
y2 = t3.transform(t2.transform(t1.transform(x)))
y1.get_matrix() == y2.get_matrix()
Returns:
A transformation matrix representing the composed inputs.
"""
composed_matrix = self._matrix.clone()
if len(self._transforms) > 0:
for other in self._transforms:
other_matrix = other.get_matrix()
composed_matrix = _broadcast_bmm(composed_matrix, other_matrix)
return composed_matrix
def _get_matrix_inverse(self) -> torch.Tensor:
"""
Return the inverse of self._matrix.
"""
return torch.inverse(self._matrix)
[docs] def inverse(self, invert_composed: bool = False) -> "Transform3d":
"""
Returns a new Transform3d object that represents an inverse of the
current transformation.
Args:
invert_composed:
- True: First compose the list of stored transformations
and then apply inverse to the result. This is
potentially slower for classes of transformations
with inverses that can be computed efficiently
(e.g. rotations and translations).
- False: Invert the individual stored transformations
independently without composing them.
Returns:
A new Transform3d object containing the inverse of the original
transformation.
"""
tinv = Transform3d(dtype=self.dtype, device=self.device)
if invert_composed:
# first compose then invert
tinv._matrix = torch.inverse(self.get_matrix())
else:
# self._get_matrix_inverse() implements efficient inverse
# of self._matrix
i_matrix = self._get_matrix_inverse()
# 2 cases:
if len(self._transforms) > 0:
# a) Either we have a non-empty list of transforms:
# Here we take self._matrix and append its inverse at the
# end of the reverted _transforms list. After composing
# the transformations with get_matrix(), this correctly
# right-multiplies by the inverse of self._matrix
# at the end of the composition.
tinv._transforms = [t.inverse() for t in reversed(self._transforms)]
last = Transform3d(dtype=self.dtype, device=self.device)
last._matrix = i_matrix
tinv._transforms.append(last)
else:
# b) Or there are no stored transformations
# we just set inverted matrix
tinv._matrix = i_matrix
return tinv
[docs] def stack(self, *others: "Transform3d") -> "Transform3d":
"""
Return a new batched Transform3d representing the batch elements from
self and all the given other transforms all batched together.
Args:
*others: Any number of Transform3d objects
Returns:
A new Transform3d.
"""
transforms = [self] + list(others)
matrix = torch.cat([t.get_matrix() for t in transforms], dim=0)
out = Transform3d(dtype=self.dtype, device=self.device)
out._matrix = matrix
return out
[docs] def transform_points(self, points, eps: Optional[float] = None) -> torch.Tensor:
"""
Use this transform to transform a set of 3D points. Assumes row major
ordering of the input points.
Args:
points: Tensor of shape (P, 3) or (N, P, 3)
eps: If eps!=None, the argument is used to clamp the
last coordinate before performing the final division.
The clamping corresponds to:
last_coord := (last_coord.sign() + (last_coord==0)) *
torch.clamp(last_coord.abs(), eps),
i.e. the last coordinates that are exactly 0 will
be clamped to +eps.
Returns:
points_out: points of shape (N, P, 3) or (P, 3) depending
on the dimensions of the transform
"""
points_batch = points.clone()
if points_batch.dim() == 2:
points_batch = points_batch[None] # (P, 3) -> (1, P, 3)
if points_batch.dim() != 3:
msg = "Expected points to have dim = 2 or dim = 3: got shape %r"
raise ValueError(msg % repr(points.shape))
N, P, _3 = points_batch.shape
ones = torch.ones(N, P, 1, dtype=points.dtype, device=points.device)
points_batch = torch.cat([points_batch, ones], dim=2)
composed_matrix = self.get_matrix()
points_out = _broadcast_bmm(points_batch, composed_matrix)
denom = points_out[..., 3:] # denominator
if eps is not None:
denom_sign = denom.sign() + (denom == 0.0).type_as(denom)
denom = denom_sign * torch.clamp(denom.abs(), eps)
points_out = points_out[..., :3] / denom
# When transform is (1, 4, 4) and points is (P, 3) return
# points_out of shape (P, 3)
if points_out.shape[0] == 1 and points.dim() == 2:
points_out = points_out.reshape(points.shape)
return points_out
[docs] def transform_normals(self, normals) -> torch.Tensor:
"""
Use this transform to transform a set of normal vectors.
Args:
normals: Tensor of shape (P, 3) or (N, P, 3)
Returns:
normals_out: Tensor of shape (P, 3) or (N, P, 3) depending
on the dimensions of the transform
"""
if normals.dim() not in [2, 3]:
msg = "Expected normals to have dim = 2 or dim = 3: got shape %r"
raise ValueError(msg % (normals.shape,))
composed_matrix = self.get_matrix()
# TODO: inverse is bad! Solve a linear system instead
mat = composed_matrix[:, :3, :3]
normals_out = _broadcast_bmm(normals, mat.transpose(1, 2).inverse())
# This doesn't pass unit tests. TODO investigate further
# if self._lu is None:
# self._lu = self._matrix[:, :3, :3].transpose(1, 2).lu()
# normals_out = normals.lu_solve(*self._lu)
# When transform is (1, 4, 4) and normals is (P, 3) return
# normals_out of shape (P, 3)
if normals_out.shape[0] == 1 and normals.dim() == 2:
normals_out = normals_out.reshape(normals.shape)
return normals_out
[docs] def translate(self, *args, **kwargs) -> "Transform3d":
return self.compose(Translate(device=self.device, *args, **kwargs))
[docs] def scale(self, *args, **kwargs) -> "Transform3d":
return self.compose(Scale(device=self.device, *args, **kwargs))
[docs] def rotate(self, *args, **kwargs) -> "Transform3d":
return self.compose(Rotate(device=self.device, *args, **kwargs))
[docs] def rotate_axis_angle(self, *args, **kwargs) -> "Transform3d":
return self.compose(RotateAxisAngle(device=self.device, *args, **kwargs))
[docs] def clone(self) -> "Transform3d":
"""
Deep copy of Transforms object. All internal tensors are cloned
individually.
Returns:
new Transforms object.
"""
other = Transform3d(dtype=self.dtype, device=self.device)
if self._lu is not None:
other._lu = [elem.clone() for elem in self._lu]
other._matrix = self._matrix.clone()
other._transforms = [t.clone() for t in self._transforms]
return other
[docs] def to(
self,
device: Device,
copy: bool = False,
dtype: Optional[torch.dtype] = None,
) -> "Transform3d":
"""
Match functionality of torch.Tensor.to()
If copy = True or the self Tensor is on a different device, the
returned tensor is a copy of self with the desired torch.device.
If copy = False and the self Tensor already has the correct torch.device,
then self is returned.
Args:
device: Device (as str or torch.device) for the new tensor.
copy: Boolean indicator whether or not to clone self. Default False.
dtype: If not None, casts the internal tensor variables
to a given torch.dtype.
Returns:
Transform3d object.
"""
device_ = make_device(device)
dtype_ = self.dtype if dtype is None else dtype
skip_to = self.device == device_ and self.dtype == dtype_
if not copy and skip_to:
return self
other = self.clone()
if skip_to:
return other
other.device = device_
other.dtype = dtype_
other._matrix = other._matrix.to(device=device_, dtype=dtype_)
other._transforms = [
t.to(device_, copy=copy, dtype=dtype_) for t in other._transforms
]
return other
[docs]class Translate(Transform3d):
[docs] def __init__(
self,
x,
y=None,
z=None,
dtype: torch.dtype = torch.float32,
device: Optional[Device] = None,
) -> None:
"""
Create a new Transform3d representing 3D translations.
Option I: Translate(xyz, dtype=torch.float32, device='cpu')
xyz should be a tensor of shape (N, 3)
Option II: Translate(x, y, z, dtype=torch.float32, device='cpu')
Here x, y, and z will be broadcast against each other and
concatenated to form the translation. Each can be:
- A python scalar
- A torch scalar
- A 1D torch tensor
"""
xyz = _handle_input(x, y, z, dtype, device, "Translate")
super().__init__(device=xyz.device)
N = xyz.shape[0]
mat = torch.eye(4, dtype=dtype, device=self.device)
mat = mat.view(1, 4, 4).repeat(N, 1, 1)
mat[:, 3, :3] = xyz
self._matrix = mat
def _get_matrix_inverse(self) -> torch.Tensor:
"""
Return the inverse of self._matrix.
"""
inv_mask = self._matrix.new_ones([1, 4, 4])
inv_mask[0, 3, :3] = -1.0
i_matrix = self._matrix * inv_mask
return i_matrix
[docs]class Scale(Transform3d):
[docs] def __init__(
self,
x,
y=None,
z=None,
dtype: torch.dtype = torch.float32,
device: Optional[Device] = None,
) -> None:
"""
A Transform3d representing a scaling operation, with different scale
factors along each coordinate axis.
Option I: Scale(s, dtype=torch.float32, device='cpu')
s can be one of
- Python scalar or torch scalar: Single uniform scale
- 1D torch tensor of shape (N,): A batch of uniform scale
- 2D torch tensor of shape (N, 3): Scale differently along each axis
Option II: Scale(x, y, z, dtype=torch.float32, device='cpu')
Each of x, y, and z can be one of
- python scalar
- torch scalar
- 1D torch tensor
"""
xyz = _handle_input(x, y, z, dtype, device, "scale", allow_singleton=True)
super().__init__(device=xyz.device)
N = xyz.shape[0]
# TODO: Can we do this all in one go somehow?
mat = torch.eye(4, dtype=dtype, device=self.device)
mat = mat.view(1, 4, 4).repeat(N, 1, 1)
mat[:, 0, 0] = xyz[:, 0]
mat[:, 1, 1] = xyz[:, 1]
mat[:, 2, 2] = xyz[:, 2]
self._matrix = mat
def _get_matrix_inverse(self) -> torch.Tensor:
"""
Return the inverse of self._matrix.
"""
xyz = torch.stack([self._matrix[:, i, i] for i in range(4)], dim=1)
ixyz = 1.0 / xyz
imat = torch.diag_embed(ixyz, dim1=1, dim2=2)
return imat
[docs]class Rotate(Transform3d):
[docs] def __init__(
self,
R: torch.Tensor,
dtype: torch.dtype = torch.float32,
device: Optional[Device] = None,
orthogonal_tol: float = 1e-5,
) -> None:
"""
Create a new Transform3d representing 3D rotation using a rotation
matrix as the input.
Args:
R: a tensor of shape (3, 3) or (N, 3, 3)
orthogonal_tol: tolerance for the test of the orthogonality of R
"""
device_ = get_device(R, device)
super().__init__(device=device_)
if R.dim() == 2:
R = R[None]
if R.shape[-2:] != (3, 3):
msg = "R must have shape (3, 3) or (N, 3, 3); got %s"
raise ValueError(msg % repr(R.shape))
R = R.to(device=device_, dtype=dtype)
_check_valid_rotation_matrix(R, tol=orthogonal_tol)
N = R.shape[0]
mat = torch.eye(4, dtype=dtype, device=device_)
mat = mat.view(1, 4, 4).repeat(N, 1, 1)
mat[:, :3, :3] = R
self._matrix = mat
def _get_matrix_inverse(self) -> torch.Tensor:
"""
Return the inverse of self._matrix.
"""
return self._matrix.permute(0, 2, 1).contiguous()
[docs]class RotateAxisAngle(Rotate):
[docs] def __init__(
self,
angle,
axis: str = "X",
degrees: bool = True,
dtype: torch.dtype = torch.float64,
device: Optional[Device] = None,
) -> None:
"""
Create a new Transform3d representing 3D rotation about an axis
by an angle.
Assuming a right-hand coordinate system, positive rotation angles result
in a counter clockwise rotation.
Args:
angle:
- A torch tensor of shape (N,)
- A python scalar
- A torch scalar
axis:
string: one of ["X", "Y", "Z"] indicating the axis about which
to rotate.
NOTE: All batch elements are rotated about the same axis.
"""
axis = axis.upper()
if axis not in ["X", "Y", "Z"]:
msg = "Expected axis to be one of ['X', 'Y', 'Z']; got %s"
raise ValueError(msg % axis)
angle = _handle_angle_input(angle, dtype, device, "RotateAxisAngle")
angle = (angle / 180.0 * math.pi) if degrees else angle
# We assume the points on which this transformation will be applied
# are row vectors. The rotation matrix returned from _axis_angle_rotation
# is for transforming column vectors. Therefore we transpose this matrix.
# R will always be of shape (N, 3, 3)
R = _axis_angle_rotation(axis, angle).transpose(1, 2)
super().__init__(device=angle.device, R=R)
def _handle_coord(c, dtype: torch.dtype, device: torch.device) -> torch.Tensor:
"""
Helper function for _handle_input.
Args:
c: Python scalar, torch scalar, or 1D torch tensor
Returns:
c_vec: 1D torch tensor
"""
if not torch.is_tensor(c):
c = torch.tensor(c, dtype=dtype, device=device)
if c.dim() == 0:
c = c.view(1)
if c.device != device:
c = c.to(device=device)
return c
def _handle_input(
x,
y,
z,
dtype: torch.dtype,
device: Optional[Device],
name: str,
allow_singleton: bool = False,
) -> torch.Tensor:
"""
Helper function to handle parsing logic for building transforms. The output
is always a tensor of shape (N, 3), but there are several types of allowed
input.
Case I: Single Matrix
In this case x is a tensor of shape (N, 3), and y and z are None. Here just
return x.
Case II: Vectors and Scalars
In this case each of x, y, and z can be one of the following
- Python scalar
- Torch scalar
- Torch tensor of shape (N, 1) or (1, 1)
In this case x, y and z are broadcast to tensors of shape (N, 1)
and concatenated to a tensor of shape (N, 3)
Case III: Singleton (only if allow_singleton=True)
In this case y and z are None, and x can be one of the following:
- Python scalar
- Torch scalar
- Torch tensor of shape (N, 1) or (1, 1)
Here x will be duplicated 3 times, and we return a tensor of shape (N, 3)
Returns:
xyz: Tensor of shape (N, 3)
"""
device_ = get_device(x, device)
# If x is actually a tensor of shape (N, 3) then just return it
if torch.is_tensor(x) and x.dim() == 2:
if x.shape[1] != 3:
msg = "Expected tensor of shape (N, 3); got %r (in %s)"
raise ValueError(msg % (x.shape, name))
if y is not None or z is not None:
msg = "Expected y and z to be None (in %s)" % name
raise ValueError(msg)
return x.to(device=device_)
if allow_singleton and y is None and z is None:
y = x
z = x
# Convert all to 1D tensors
xyz = [_handle_coord(c, dtype, device_) for c in [x, y, z]]
# Broadcast and concatenate
sizes = [c.shape[0] for c in xyz]
N = max(sizes)
for c in xyz:
if c.shape[0] != 1 and c.shape[0] != N:
msg = "Got non-broadcastable sizes %r (in %s)" % (sizes, name)
raise ValueError(msg)
xyz = [c.expand(N) for c in xyz]
xyz = torch.stack(xyz, dim=1)
return xyz
def _handle_angle_input(
x, dtype: torch.dtype, device: Optional[Device], name: str
) -> torch.Tensor:
"""
Helper function for building a rotation function using angles.
The output is always of shape (N,).
The input can be one of:
- Torch tensor of shape (N,)
- Python scalar
- Torch scalar
"""
device_ = get_device(x, device)
if torch.is_tensor(x) and x.dim() > 1:
msg = "Expected tensor of shape (N,); got %r (in %s)"
raise ValueError(msg % (x.shape, name))
else:
return _handle_coord(x, dtype, device_)
def _broadcast_bmm(a, b) -> torch.Tensor:
"""
Batch multiply two matrices and broadcast if necessary.
Args:
a: torch tensor of shape (P, K) or (M, P, K)
b: torch tensor of shape (N, K, K)
Returns:
a and b broadcast multiplied. The output batch dimension is max(N, M).
To broadcast transforms across a batch dimension if M != N then
expect that either M = 1 or N = 1. The tensor with batch dimension 1 is
expanded to have shape N or M.
"""
if a.dim() == 2:
a = a[None]
if len(a) != len(b):
if not ((len(a) == 1) or (len(b) == 1)):
msg = "Expected batch dim for bmm to be equal or 1; got %r, %r"
raise ValueError(msg % (a.shape, b.shape))
if len(a) == 1:
a = a.expand(len(b), -1, -1)
if len(b) == 1:
b = b.expand(len(a), -1, -1)
return a.bmm(b)
@torch.no_grad()
def _check_valid_rotation_matrix(R, tol: float = 1e-7) -> None:
"""
Determine if R is a valid rotation matrix by checking it satisfies the
following conditions:
``RR^T = I and det(R) = 1``
Args:
R: an (N, 3, 3) matrix
Returns:
None
Emits a warning if R is an invalid rotation matrix.
"""
N = R.shape[0]
eye = torch.eye(3, dtype=R.dtype, device=R.device)
eye = eye.view(1, 3, 3).expand(N, -1, -1)
orthogonal = torch.allclose(R.bmm(R.transpose(1, 2)), eye, atol=tol)
det_R = _safe_det_3x3(R)
no_distortion = torch.allclose(det_R, torch.ones_like(det_R))
if not (orthogonal and no_distortion):
msg = "R is not a valid rotation matrix"
warnings.warn(msg)
return
```