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import torch.nn.functional as F
def normalise_quat(x: torch.Tensor):
return x / torch.clamp(x.square().sum(dim=-1).sqrt().unsqueeze(-1), min=1e-10)
def norm_tensor(tensor: torch.Tensor) -> torch.Tensor:
return tensor / torch.linalg.norm(tensor, ord=2, dim=-1, keepdim=True)
"""
Below is a continuous 6D rotation representation adapted from
On the Continuity of Rotation Representations in Neural Networks
https://arxiv.org/pdf/1812.07035.pdf
https://github.com/papagina/RotationContinuity/blob/master/sanity_test/code/tools.py
"""
def normalize_vector(v, return_mag=False):
batch = v.shape[0]
v_mag = torch.sqrt(v.pow(2).sum(1))
v_mag = torch.clamp(v_mag, 1e-8)
v_mag = v_mag.view(batch, 1).expand(batch, v.shape[1])
v = v / v_mag
if return_mag:
return v, v_mag[:, 0]
else:
return v
def cross_product(u, v):
batch = u.shape[0]
i = u[:, 1] * v[:, 2] - u[:, 2] * v[:, 1]
j = u[:, 2] * v[:, 0] - u[:, 0] * v[:, 2]
k = u[:, 0] * v[:, 1] - u[:, 1] * v[:, 0]
out = torch.cat((i.view(batch, 1), j.view(batch, 1), k.view(batch, 1)), 1)
return out # batch*3
def compute_rotation_matrix_from_ortho6d(ortho6d):
x_raw = ortho6d[:, 0:3] # batch*3
y_raw = ortho6d[:, 3:6] # batch*3
x = normalize_vector(x_raw) # batch*3
z = cross_product(x, y_raw) # batch*3
z = normalize_vector(z) # batch*3
y = cross_product(z, x) # batch*3
x = x.view(-1, 3, 1)
y = y.view(-1, 3, 1)
z = z.view(-1, 3, 1)
matrix = torch.cat((x, y, z), 2) # batch*3*3
return matrix
def get_ortho6d_from_rotation_matrix(matrix):
# The orhto6d represents the first two column vectors a1 and a2 of the
# rotation matrix: [ | , |, | ]
# [ a1, a2, a3]
# [ | , |, | ]
ortho6d = matrix[:, :, :2].permute(0, 2, 1).flatten(-2)
return ortho6d
def orthonormalize_by_gram_schmidt(matrix):
"""Post-processing a 9D matrix with Gram-Schmidt orthogonalization.
Args:
matrix: A tensor of shape (..., 3, 3)
Returns:
A tensor of shape (..., 3, 3) with orthogonal rows.
"""
a1, a2, a3 = matrix[..., :, 0], matrix[..., :, 1], matrix[..., :, 2]
b1 = F.normalize(a1, dim=-1)
b2 = a2 - (b1 * a2).sum(-1, keepdim=True) * b1
b2 = F.normalize(b2, dim=-1)
b3 = a3 - (b1 * a3).sum(-1, keepdim=True) * b1 - (b2 * a3).sum(-1, keepdim=True) * b2
b3 = F.normalize(b3, dim=-1)
return torch.stack([b1, b2, b3], dim=-1)
def quaternion_to_matrix(quaternions: torch.Tensor) -> torch.Tensor:
"""
Convert rotations given as quaternions to rotation matrices.
Args:
quaternions: quaternions with real part first,
as tensor of shape (..., 4).
Returns:
Rotation matrices as tensor of shape (..., 3, 3).
"""
r, i, j, k = torch.unbind(quaternions, -1)
# pyre-fixme[58]: `/` is not supported for operand types `float` and `Tensor`.
two_s = 2.0 / (quaternions * quaternions).sum(-1)
o = torch.stack(
(
1 - two_s * (j * j + k * k),
two_s * (i * j - k * r),
two_s * (i * k + j * r),
two_s * (i * j + k * r),
1 - two_s * (i * i + k * k),
two_s * (j * k - i * r),
two_s * (i * k - j * r),
two_s * (j * k + i * r),
1 - two_s * (i * i + j * j),
),
-1,
)
return o.reshape(quaternions.shape[:-1] + (3, 3))
def _sqrt_positive_part(x: torch.Tensor) -> torch.Tensor:
"""
Returns torch.sqrt(torch.max(0, x))
but with a zero subgradient where x is 0.
"""
ret = torch.zeros_like(x)
positive_mask = x > 0
# ret[positive_mask] = torch.sqrt(x[positive_mask])
ret = torch.where(positive_mask, torch.sqrt(x), ret)
return ret
def matrix_to_quaternion(matrix: torch.Tensor) -> torch.Tensor:
"""
Convert rotations given as rotation matrices to quaternions.
Args:
matrix: Rotation matrices as tensor of shape (..., 3, 3).
Returns:
quaternions with real part first, as tensor of shape (..., 4).
"""
if matrix.size(-1) != 3 or matrix.size(-2) != 3:
raise ValueError(f"Invalid rotation matrix shape {matrix.shape}.")
batch_dim = matrix.shape[:-2]
m00, m01, m02, m10, m11, m12, m20, m21, m22 = torch.unbind(
matrix.reshape(batch_dim + (9,)), dim=-1
)
q_abs = _sqrt_positive_part(
torch.stack(
[
1.0 + m00 + m11 + m22,
1.0 + m00 - m11 - m22,
1.0 - m00 + m11 - m22,
1.0 - m00 - m11 + m22,
],
dim=-1,
)
)
# we produce the desired quaternion multiplied by each of r, i, j, k
quat_by_rijk = torch.stack(
[
# pyre-fixme[58]: `**` is not supported for operand types `Tensor` and
# `int`.
torch.stack([q_abs[..., 0] ** 2, m21 - m12, m02 - m20, m10 - m01], dim=-1),
# pyre-fixme[58]: `**` is not supported for operand types `Tensor` and
# `int`.
torch.stack([m21 - m12, q_abs[..., 1] ** 2, m10 + m01, m02 + m20], dim=-1),
# pyre-fixme[58]: `**` is not supported for operand types `Tensor` and
# `int`.
torch.stack([m02 - m20, m10 + m01, q_abs[..., 2] ** 2, m12 + m21], dim=-1),
# pyre-fixme[58]: `**` is not supported for operand types `Tensor` and
# `int`.
torch.stack([m10 - m01, m20 + m02, m21 + m12, q_abs[..., 3] ** 2], dim=-1),
],
dim=-2,
)
# We floor here at 0.1 but the exact level is not important; if q_abs is small,
# the candidate won't be picked.
flr = 0.1
q_abs_safe = torch.clamp(q_abs, min=flr) # ensures stability
# shape: [..., 4, 4]
quat_candidates = quat_by_rijk / (2.0 * q_abs_safe[..., None])
# Get best-conditioned candidate per batch using argmax
best_idx = q_abs.argmax(dim=-1) # shape: [...], values in [0, 3]
# Use gather to extract the best candidate along the quaternion axis
# First, expand index shape to match quat_candidates
index = best_idx.unsqueeze(-1).unsqueeze(-1) # [..., 1, 1]
index = index.expand(*quat_candidates.shape[:-2], 1, 4) # [..., 1, 4]
best_quat = torch.gather(quat_candidates, dim=-2, index=index).squeeze(-2)
return best_quat.reshape(batch_dim + (4,)) # shape: [..., 4]
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