src/models/utils/rotation.py

# Modified from PyTorch3D, https://github.com/facebookresearch/pytorch3d

import torch
import numpy as np
import torch.nn.functional as F


def quat_to_rotmat(quaternions: torch.Tensor) -> torch.Tensor:
    """
    Quaternion Order: XYZW or say ijkr, scalar-last

    Convert rotations given as quaternions to rotation matrices.
    Args:
        quaternions: quaternions with real part last,
            as tensor of shape (..., 4).

    Returns:
        Rotation matrices as tensor of shape (..., 3, 3).
    """
    i, j, k, r = 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 rotmat_to_quat(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 last, as tensor of shape (..., 4).
        Quaternion Order: XYZW or say ijkr, scalar-last
    """
    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 = torch.tensor(0.1).to(dtype=q_abs.dtype, device=q_abs.device)
    quat_candidates = quat_by_rijk / (2.0 * q_abs[..., None].max(flr))

    # if not for numerical problems, quat_candidates[i] should be same (up to a sign),
    # forall i; we pick the best-conditioned one (with the largest denominator)
    out = quat_candidates[F.one_hot(q_abs.argmax(dim=-1), num_classes=4) > 0.5, :].reshape(batch_dim + (4,))

    # Convert from rijk to ijkr
    out = out[..., [1, 2, 3, 0]]

    out = standardize_quaternion(out)

    return out


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
    if torch.is_grad_enabled():
        ret[positive_mask] = torch.sqrt(x[positive_mask])
    else:
        ret = torch.where(positive_mask, torch.sqrt(x), ret)
    return ret


def standardize_quaternion(quaternions: torch.Tensor) -> torch.Tensor:
    """
    Convert a unit quaternion to a standard form: one in which the real
    part is non negative.

    Args:
        quaternions: Quaternions with real part last,
            as tensor of shape (..., 4).

    Returns:
        Standardized quaternions as tensor of shape (..., 4).
    """
    return torch.where(quaternions[..., 3:4] < 0, -quaternions, quaternions)