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"""Contains linear algebra related utility functions. |
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For licensing see accompanying LICENSE file. |
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Copyright (C) 2025 Apple Inc. All Rights Reserved. |
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""" |
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from __future__ import annotations |
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import torch |
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import torch.nn.functional as F |
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from scipy.spatial.transform import Rotation |
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def rotation_matrices_from_quaternions(quaternions: torch.Tensor) -> torch.Tensor: |
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"""Convert batch of quaternions into rotations matrices. |
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Args: |
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quaternions: The quaternions convert to matrices. |
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Returns: |
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The rotations matrices corresponding to the (normalized) quaternions. |
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""" |
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device = quaternions.device |
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shape = quaternions.shape[:-1] |
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quaternions = quaternions / torch.linalg.norm(quaternions, dim=-1, keepdim=True) |
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real_part = quaternions[..., 0] |
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vector_part = quaternions[..., 1:] |
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vector_cross = get_cross_product_matrix(vector_part) |
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real_part = real_part[..., None, None] |
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matrix_outer = vector_part[..., :, None] * vector_part[..., None, :] |
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matrix_diag = real_part.square() * eyes(3, shape=shape, device=device) |
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matrix_cross_1 = 2 * real_part * vector_cross |
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matrix_cross_2 = vector_cross @ vector_cross |
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return matrix_outer + matrix_diag + matrix_cross_1 + matrix_cross_2 |
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def quaternions_from_rotation_matrices(matrices: torch.Tensor) -> torch.Tensor: |
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"""Convert batch of rotation matrices to quaternions. |
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Args: |
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matrices: The matrices to convert to quaternions. |
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Returns: |
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The quaternions corresponding to the rotation matrices. |
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Note: this operation is not differentiable and will be performed on the CPU. |
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""" |
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if not matrices.shape[-2:] == (3, 3): |
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raise ValueError(f"matrices have invalid shape {matrices.shape}") |
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matrices_np = matrices.detach().cpu().numpy() |
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quaternions_np = Rotation.from_matrix(matrices_np.reshape(-1, 3, 3)).as_quat() |
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quaternions_np = quaternions_np[:, [3, 0, 1, 2]] |
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quaternions_np = quaternions_np.reshape(matrices_np.shape[:-2] + (4,)) |
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return torch.as_tensor(quaternions_np, device=matrices.device, dtype=matrices.dtype) |
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def get_cross_product_matrix(vectors: torch.Tensor) -> torch.Tensor: |
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"""Generate cross product matrix for vector exterior product.""" |
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if not vectors.shape[-1] == 3: |
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raise ValueError("Only 3-dimensional vectors are supported") |
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device = vectors.device |
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shape = vectors.shape[:-1] |
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unit_basis = eyes(3, shape=shape, device=device) |
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return torch.cross(vectors[..., :, None], unit_basis, dim=-2) |
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def eyes( |
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dim: int, shape: tuple[int, ...], device: torch.device | str | None = None |
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) -> torch.Tensor: |
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"""Create batch of identity matrices.""" |
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return torch.eye(dim, device=device).broadcast_to(shape + (dim, dim)).clone() |
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def quaternion_product(q1, q2): |
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"""Compute dot product between two quaternions.""" |
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real_1 = q1[..., :1] |
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real_2 = q2[..., :1] |
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vector_1 = q1[..., 1:] |
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vector_2 = q2[..., 1:] |
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real_out = real_1 * real_2 - (vector_1 * vector_2).sum(dim=-1, keepdim=True) |
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vector_out = real_1 * vector_2 + real_2 * vector_1 + torch.cross(vector_1, vector_2) |
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return torch.concatenate([real_out, vector_out], dim=-1) |
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def quaternion_conj(q): |
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"""Get conjugate of a quaternion.""" |
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real = q[..., :1] |
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vector = q[..., 1:] |
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return torch.concatenate([real, -vector], dim=-1) |
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def project(u: torch.Tensor, basis: torch.Tensor) -> torch.Tensor: |
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"""Project tensor u to unit basis a.""" |
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unit_u = F.normalize(u, dim=-1) |
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inner_prod = (unit_u * basis).sum(dim=-1, keepdim=True) |
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return inner_prod * u |
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