pytorch3d.py

import functools

import torch
import torch.nn.functional as F


def quaternion_to_matrix(quaternions):
    r, i, j, k = torch.unbind(quaternions, -1)
    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 _copysign(a, b):
    signs_differ = (a < 0) != (b < 0)
    return torch.where(signs_differ, -a, a)


def _sqrt_positive_part(x: torch.Tensor) -> torch.Tensor:
    ret = torch.zeros_like(x)
    positive_mask = x > 0
    ret[positive_mask] = torch.sqrt(x[positive_mask])
    return ret


def matrix_to_quaternion(matrix: torch.Tensor) -> torch.Tensor:
    if matrix.size(-1) != 3 or matrix.size(-2) != 3:
        raise ValueError(f"Invalid rotation matrix  shape f{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,
        )
    )

    quat_by_rijk = torch.stack(
        [
            torch.stack([q_abs[..., 0] ** 2, m21 - m12, m02 - m20, m10 - m01], dim=-1),
            torch.stack([m21 - m12, q_abs[..., 1] ** 2, m10 + m01, m02 + m20], dim=-1),
            torch.stack([m02 - m20, m10 + m01, q_abs[..., 2] ** 2, m12 + m21], dim=-1),
            torch.stack([m10 - m01, m20 + m02, m21 + m12, q_abs[..., 3] ** 2], dim=-1),
        ],
        dim=-2,
    )

    quat_candidates = quat_by_rijk / (2.0 * q_abs[..., None].max(q_abs.new_tensor(0.1)))

    return quat_candidates[
        F.one_hot(q_abs.argmax(dim=-1), num_classes=4) > 0.5, :
    ].reshape(*batch_dim, 4)


def _axis_angle_rotation(axis: str, angle):
    cos = torch.cos(angle)
    sin = torch.sin(angle)
    one = torch.ones_like(angle)
    zero = torch.zeros_like(angle)

    if axis == "X":
        R_flat = (one, zero, zero, zero, cos, -sin, zero, sin, cos)
    if axis == "Y":
        R_flat = (cos, zero, sin, zero, one, zero, -sin, zero, cos)
    if axis == "Z":
        R_flat = (cos, -sin, zero, sin, cos, zero, zero, zero, one)

    return torch.stack(R_flat, -1).reshape(angle.shape + (3, 3))


def euler_angles_to_matrix(euler_angles, convention: str):
    if euler_angles.dim() == 0 or euler_angles.shape[-1] != 3:
        raise ValueError("Invalid input euler angles.")
    if len(convention) != 3:
        raise ValueError("Convention must have 3 letters.")
    if convention[1] in (convention[0], convention[2]):
        raise ValueError(f"Invalid convention {convention}.")
    for letter in convention:
        if letter not in ("X", "Y", "Z"):
            raise ValueError(f"Invalid letter {letter} in convention string.")
    matrices = map(_axis_angle_rotation, convention, torch.unbind(euler_angles, -1))
    return functools.reduce(torch.matmul, matrices)


def _angle_from_tan(
    axis: str, other_axis: str, data, horizontal: bool, tait_bryan: bool
):
    i1, i2 = {"X": (2, 1), "Y": (0, 2), "Z": (1, 0)}[axis]
    if horizontal:
        i2, i1 = i1, i2
    even = (axis + other_axis) in ["XY", "YZ", "ZX"]
    if horizontal == even:
        return torch.atan2(data[..., i1], data[..., i2])
    if tait_bryan:
        return torch.atan2(-data[..., i2], data[..., i1])
    return torch.atan2(data[..., i2], -data[..., i1])


def _index_from_letter(letter: str):
    if letter == "X":
        return 0
    if letter == "Y":
        return 1
    if letter == "Z":
        return 2


def matrix_to_euler_angles(matrix, convention: str):
    if len(convention) != 3:
        raise ValueError("Convention must have 3 letters.")
    if convention[1] in (convention[0], convention[2]):
        raise ValueError(f"Invalid convention {convention}.")
    for letter in convention:
        if letter not in ("X", "Y", "Z"):
            raise ValueError(f"Invalid letter {letter} in convention string.")
    if matrix.size(-1) != 3 or matrix.size(-2) != 3:
        raise ValueError(f"Invalid rotation matrix  shape f{matrix.shape}.")
    i0 = _index_from_letter(convention[0])
    i2 = _index_from_letter(convention[2])
    tait_bryan = i0 != i2
    if tait_bryan:
        central_angle = torch.asin(
            matrix[..., i0, i2] * (-1.0 if i0 - i2 in [-1, 2] else 1.0)
        )
    else:
        central_angle = torch.acos(matrix[..., i0, i0])

    o = (
        _angle_from_tan(
            convention[0], convention[1], matrix[..., i2], False, tait_bryan
        ),
        central_angle,
        _angle_from_tan(
            convention[2], convention[1], matrix[..., i0, :], True, tait_bryan
        ),
    )
    return torch.stack(o, -1)


def standardize_quaternion(quaternions):
    return torch.where(quaternions[..., 0:1] < 0, -quaternions, quaternions)


def quaternion_raw_multiply(a, b):
    aw, ax, ay, az = torch.unbind(a, -1)
    bw, bx, by, bz = torch.unbind(b, -1)
    ow = aw * bw - ax * bx - ay * by - az * bz
    ox = aw * bx + ax * bw + ay * bz - az * by
    oy = aw * by - ax * bz + ay * bw + az * bx
    oz = aw * bz + ax * by - ay * bx + az * bw
    return torch.stack((ow, ox, oy, oz), -1)


def quaternion_multiply(a, b):
    ab = quaternion_raw_multiply(a, b)
    return standardize_quaternion(ab)


def quaternion_invert(quaternion):
    return quaternion * quaternion.new_tensor([1, -1, -1, -1])


def quaternion_apply(quaternion, point):
    if point.size(-1) != 3:
        raise ValueError(f"Points are not in 3D, f{point.shape}.")
    real_parts = point.new_zeros(point.shape[:-1] + (1,))
    point_as_quaternion = torch.cat((real_parts, point), -1)
    out = quaternion_raw_multiply(
        quaternion_raw_multiply(quaternion, point_as_quaternion),
        quaternion_invert(quaternion),
    )
    return out[..., 1:]


def axis_angle_to_matrix(axis_angle):
    return quaternion_to_matrix(axis_angle_to_quaternion(axis_angle))


def matrix_to_axis_angle(matrix):
    return quaternion_to_axis_angle(matrix_to_quaternion(matrix))


def axis_angle_to_quaternion(axis_angle):
    angles = torch.norm(axis_angle, p=2, dim=-1, keepdim=True)
    half_angles = 0.5 * angles
    eps = 1e-6
    small_angles = angles.abs() < eps
    sin_half_angles_over_angles = torch.empty_like(angles)
    sin_half_angles_over_angles[~small_angles] = (
        torch.sin(half_angles[~small_angles]) / angles[~small_angles]
    )
    # for x small, sin(x/2) is about x/2 - (x/2)^3/6
    # so sin(x/2)/x is about 1/2 - (x*x)/48
    sin_half_angles_over_angles[small_angles] = (
        0.5 - (angles[small_angles] * angles[small_angles]) / 48
    )
    quaternions = torch.cat(
        [torch.cos(half_angles), axis_angle * sin_half_angles_over_angles], dim=-1
    )
    return quaternions


def quaternion_to_axis_angle(quaternions):
    norms = torch.norm(quaternions[..., 1:], p=2, dim=-1, keepdim=True)
    half_angles = torch.atan2(norms, quaternions[..., :1])
    angles = 2 * half_angles
    eps = 1e-6
    small_angles = angles.abs() < eps
    sin_half_angles_over_angles = torch.empty_like(angles)
    sin_half_angles_over_angles[~small_angles] = (
        torch.sin(half_angles[~small_angles]) / angles[~small_angles]
    )
    # for x small, sin(x/2) is about x/2 - (x/2)^3/6
    # so sin(x/2)/x is about 1/2 - (x*x)/48
    sin_half_angles_over_angles[small_angles] = (
        0.5 - (angles[small_angles] * angles[small_angles]) / 48
    )
    return quaternions[..., 1:] / sin_half_angles_over_angles


def rotation_6d_to_matrix(d6: torch.Tensor) -> torch.Tensor:
    a1, a2 = d6[..., :3], d6[..., 3:]
    b1 = F.normalize(a1, dim=-1)
    b2 = a2 - (b1 * a2).sum(-1, keepdim=True) * b1
    b2 = F.normalize(b2, dim=-1)
    b3 = torch.cross(b1, b2, dim=-1)
    return torch.stack((b1, b2, b3), dim=-2)


def matrix_to_rotation_6d(matrix: torch.Tensor) -> torch.Tensor:
    return matrix[..., :2, :].clone().reshape(*matrix.size()[:-2], 6)