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.

# pyre-unsafe

import math
import os
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
from .se3 import se3_log_map

[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 =[tN.transform_points(xN), tM.transform_points(xM)], dim=0) y2 = tN.stack(tM).transform_points([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 latter are converted to homogeneous (4D) coordinates and right-multiplied by the M matrix: .. code-block:: python points = [[0, 1, 2]] # (1 x 3) xyz coordinates of a point [transformed_points, 1] ∝ [points, 1] @ 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) or (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.BoolTensor, torch.LongTensor] ) -> "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: """ Returns a 4×4 matrix corresponding to each transform in the batch. If the transform was composed from others, the matrix for the composite transform will be returned. 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() Where necessary, those transforms are broadcast against each other. Returns: A (N, 4, 4) batch of transformation matrices representing the stored transforms. See the class documentation for the conventions. """ 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
[docs] def get_se3_log(self, eps: float = 1e-4, cos_bound: float = 1e-4) -> torch.Tensor: """ Returns a 6D SE(3) log vector corresponding to each transform in the batch. 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.get_matrix()) 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 [2]. See the docstring for `se3.se3_log_map` and [1], Sec 9.4.2. for more detailed description. Args: 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: A (N, 6) tensor, rows of which represent the individual transforms stored in the object as SE(3) logarithms. Raises: ValueError if the stored transform is not Euclidean (e.g. R is not a rotation matrix or the last column has non-zeros in the first three places). [1] [2] """ return se3_log_map(self.get_matrix(), eps, cos_bound)
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 =[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 =[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(*args, device=self.device, dtype=self.dtype, **kwargs) )
[docs] def scale(self, *args, **kwargs) -> "Transform3d": return self.compose( Scale(*args, device=self.device, dtype=self.dtype, **kwargs) )
[docs] def rotate(self, *args, **kwargs) -> "Transform3d": return self.compose( Rotate(*args, device=self.device, dtype=self.dtype, **kwargs) )
[docs] def rotate_axis_angle(self, *args, **kwargs) -> "Transform3d": return self.compose( RotateAxisAngle(*args, device=self.device, dtype=self.dtype, **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 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 =, dtype=dtype_) other._transforms = [, copy=copy, dtype=dtype_) for t in other._transforms ] return other
[docs] def cpu(self) -> "Transform3d": return"cpu")
[docs] def cuda(self) -> "Transform3d": return"cuda")
[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, dtype=dtype) 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, dtype=dtype) 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) # pyre-fixme[58]: `/` is not supported for operand types `float` and `Tensor`. ixyz = 1.0 / xyz # pyre-fixme[6]: For 1st param expected `Tensor` but got `float`. 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_, dtype=dtype) 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 =, dtype=dtype) if os.environ.get("PYTORCH3D_CHECK_ROTATION_MATRICES", "0") == "1": # Note: aten::all_close in the check is computationally slow, so we # only run the check when PYTORCH3D_CHECK_ROTATION_MATRICES is on. _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.float32, 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, dtype=dtype)
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 or c.dtype != dtype: c =, dtype=dtype) 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, dtype=dtype) 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