src/utils/transforms.py

import functools
import torch
import torch.nn.functional as F

########################Implementations of the functions in the PyTorch3D########################
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)


import numpy as np
def rotation_6d_to_matrix_np(d6: np.ndarray) -> np.ndarray:
    a1, a2 = d6[..., :3], d6[..., 3:]
    b1 = a1 / np.linalg.norm(a1, axis=-1, keepdims=True)
    b2 = a2 - np.sum(b1 * a2, axis=-1, keepdims=True) * b1
    b2 = b2 / np.linalg.norm(b2, axis=-1, keepdims=True)
    b3 = np.cross(b1, b2, axis=-1)
    return np.stack((b1, b2, b3), axis=-2)

def matrix_to_rotation_6d_np(matrix: np.ndarray) -> np.ndarray:
    return matrix[..., :2, :].reshape(*matrix.shape[:-2], 6)

########################Implementations of the functions in the PyTorch3D########################



from einops import rearrange
def transform_points(x, mat):
    shape = x.shape
    x = rearrange(x, 'b t (j c) -> b (t j) c', c=3)  # B x N x 3
    x = torch.einsum('bpc,bck->bpk', mat[:, :3, :3], x.permute(0, 2, 1))  # B x 3 x N   N x B x 3
    x = x.permute(2, 0, 1) + mat[:, :3, 3]
    x = x.permute(1, 0, 2)
    x = x.reshape(shape)
    return x


def transform_points_numpy(x, mat):
    shape = x.shape
    x = x.reshape(shape[0], -1, 3)  # b x (t*j) x c
    x = np.einsum('bpc,bck->bpk', mat[:, :3, :3], np.transpose(x, (0, 2, 1)))
    x = np.transpose(x, (2, 0, 1)) + mat[:, :3, 3]
    x = np.transpose(x, (1, 0, 2))
    x = x.reshape(shape)
    return x


def zup_to_yup(coord):
    if len(coord.shape) > 1:
        coord = coord[..., [0, 2, 1]]
        coord[..., 2] *= -1
    else:
        coord = coord[[0, 2, 1]]
        coord[2] *= -1
    return coord


def rigid_transform_3D(A, B, scale=False):
    assert len(A) == len(B)
    N = A.shape[0]  # total points
    centroid_A = np.mean(A, axis=0)
    centroid_B = np.mean(B, axis=0)

    # center the points
    AA = A - np.tile(centroid_A, (N, 1))
    BB = B - np.tile(centroid_B, (N, 1))
    if scale:
        H = np.transpose(BB) * AA / N
    else:
        H = np.transpose(BB) * AA

    U, S, Vt = np.linalg.svd(H)
    R = Vt.T * U.T
    # special reflection case
    if np.linalg.det(R) < 0:
        Vt[2, :] *= -1
        R = Vt.T * U.T

    if scale:
        varA = np.var(A, axis=0).sum()
        c = 1 / (1 / varA * np.sum(S))  # scale factor
        t = -R * (centroid_B.T * c) + centroid_A.T
    else:
        c = 1
        t = -R * centroid_B.T + centroid_A.T

    return c, R, t




##################joints blending######################
@torch.jit.script
def slerp(q0: torch.Tensor, q1: torch.Tensor, t: torch.Tensor) -> torch.Tensor:
    """
    Spherical linear interpolation between two quaternions.

    Args:
        q0: (..., 4) tensor of quaternions
        q1: (..., 4) tensor of quaternions
        t: (..., 1) tensor of interpolation coefficients

    Returns:
        (..., 4) tensor of quaternions
    """
    cos_half_theta = torch.sum(q0 * q1, dim=-1)

    neg_mask = cos_half_theta < 0
    q1 = q1.clone()
    q1[neg_mask] = -q1[neg_mask]
    cos_half_theta = torch.abs(cos_half_theta)
    cos_half_theta = torch.unsqueeze(cos_half_theta, dim=-1)

    half_theta = torch.acos(cos_half_theta)
    sin_half_theta = torch.sqrt(1.0 - cos_half_theta * cos_half_theta)
    
    ratioA = torch.sin((1 - t) * half_theta) / sin_half_theta
    ratioB = torch.sin(t * half_theta) / sin_half_theta

    new_q = ratioA * q0 + ratioB * q1

    new_q = torch.where(torch.abs(sin_half_theta) < 0.001, 0.5 * q0 + 0.5 * q1, new_q)
    new_q = torch.where(torch.abs(cos_half_theta) >= 1, q0, new_q)

    return new_q


def blend_joint_rot_batch(body_pose_1, body_pose_2, t):
    """
    Blend two batches of joint rotations using spherical linear interpolation.

    Args:
        body_pose_1: (batch_size, sequence_length, num_joints, 3) tensor of axis-angle rotations
        body_pose_2: (batch_size, sequence_length, num_joints, 3) tensor of axis-angle rotations
        t: (batch_size, 1, num_joints, 1) tensor of interpolation coefficients

    Returns:
        (batch_size, sequence_length, num_joints, 3) tensor of axis-angle rotations
    """
    shape = body_pose_1.shape
    if len(shape) == 3:
        body_pose_1 = body_pose_1.reshape(shape[0], shape[1], -1, 3)
        body_pose_2 = body_pose_2.reshape(shape[0], shape[1], -1, 3)
    ret = quaternion_to_axis_angle(
        slerp(axis_angle_to_quaternion(body_pose_1), axis_angle_to_quaternion(body_pose_2), t)
    )
    if len(shape) == 3:
        ret = ret.reshape(shape)
        
    return ret