lib/pymafx/utils/geometry.py

import torch
from torch.nn import functional as F
import numpy as np
import numbers
from einops.einops import rearrange
"""
Useful geometric operations, e.g. Perspective projection and a differentiable Rodrigues formula
Parts of the code are taken from https://github.com/MandyMo/pytorch_HMR
"""


def batch_rodrigues(theta):
    """Convert axis-angle representation to rotation matrix.
    Args:
        theta: size = [B, 3]
    Returns:
        Rotation matrix corresponding to the quaternion -- size = [B, 3, 3]
    """
    l1norm = torch.norm(theta + 1e-8, p=2, dim=1)
    angle = torch.unsqueeze(l1norm, -1)
    normalized = torch.div(theta, angle)
    angle = angle * 0.5
    v_cos = torch.cos(angle)
    v_sin = torch.sin(angle)
    quat = torch.cat([v_cos, v_sin * normalized], dim=1)
    return quat_to_rotmat(quat)


def quat_to_rotmat(quat):
    """Convert quaternion coefficients to rotation matrix.
    Args:
        quat: size = [B, 4] 4 <===>(w, x, y, z)
    Returns:
        Rotation matrix corresponding to the quaternion -- size = [B, 3, 3]
    """
    norm_quat = quat
    norm_quat = norm_quat / norm_quat.norm(p=2, dim=1, keepdim=True)
    w, x, y, z = norm_quat[:, 0], norm_quat[:, 1], norm_quat[:, 2], norm_quat[:, 3]

    B = quat.size(0)

    w2, x2, y2, z2 = w.pow(2), x.pow(2), y.pow(2), z.pow(2)
    wx, wy, wz = w * x, w * y, w * z
    xy, xz, yz = x * y, x * z, y * z

    rotMat = torch.stack(
        [
            w2 + x2 - y2 - z2, 2 * xy - 2 * wz, 2 * wy + 2 * xz, 2 * wz + 2 * xy, w2 - x2 + y2 - z2,
            2 * yz - 2 * wx, 2 * xz - 2 * wy, 2 * wx + 2 * yz, w2 - x2 - y2 + z2
        ],
        dim=1
    ).view(B, 3, 3)
    return rotMat


def rotation_matrix_to_angle_axis(rotation_matrix):
    """
    This function is borrowed from https://github.com/kornia/kornia

    Convert 3x4 rotation matrix to Rodrigues vector

    Args:
        rotation_matrix (Tensor): rotation matrix.

    Returns:
        Tensor: Rodrigues vector transformation.

    Shape:
        - Input: :math:`(N, 3, 4)`
        - Output: :math:`(N, 3)`

    Example:
        >>> input = torch.rand(2, 3, 4)  # Nx4x4
        >>> output = tgm.rotation_matrix_to_angle_axis(input)  # Nx3
    """
    if rotation_matrix.shape[1:] == (3, 3):
        rot_mat = rotation_matrix.reshape(-1, 3, 3)
        hom = torch.tensor([0, 0, 1], dtype=torch.float32, device=rotation_matrix.device).reshape(
            1, 3, 1
        ).expand(rot_mat.shape[0], -1, -1)
        rotation_matrix = torch.cat([rot_mat, hom], dim=-1)

    quaternion = rotation_matrix_to_quaternion(rotation_matrix)
    aa = quaternion_to_angle_axis(quaternion)
    aa[torch.isnan(aa)] = 0.0
    return aa


def quaternion_to_angle_axis(quaternion: torch.Tensor) -> torch.Tensor:
    """
    This function is borrowed from https://github.com/kornia/kornia

    Convert quaternion vector to angle axis of rotation.

    Adapted from ceres C++ library: ceres-solver/include/ceres/rotation.h

    Args:
        quaternion (torch.Tensor): tensor with quaternions.

    Return:
        torch.Tensor: tensor with angle axis of rotation.

    Shape:
        - Input: :math:`(*, 4)` where `*` means, any number of dimensions
        - Output: :math:`(*, 3)`

    Example:
        >>> quaternion = torch.rand(2, 4)  # Nx4
        >>> angle_axis = tgm.quaternion_to_angle_axis(quaternion)  # Nx3
    """
    if not torch.is_tensor(quaternion):
        raise TypeError("Input type is not a torch.Tensor. Got {}".format(type(quaternion)))

    if not quaternion.shape[-1] == 4:
        raise ValueError(
            "Input must be a tensor of shape Nx4 or 4. Got {}".format(quaternion.shape)
        )
    # unpack input and compute conversion
    q1: torch.Tensor = quaternion[..., 1]
    q2: torch.Tensor = quaternion[..., 2]
    q3: torch.Tensor = quaternion[..., 3]
    sin_squared_theta: torch.Tensor = q1 * q1 + q2 * q2 + q3 * q3

    sin_theta: torch.Tensor = torch.sqrt(sin_squared_theta)
    cos_theta: torch.Tensor = quaternion[..., 0]
    two_theta: torch.Tensor = 2.0 * torch.where(
        cos_theta < 0.0, torch.atan2(-sin_theta, -cos_theta), torch.atan2(sin_theta, cos_theta)
    )

    k_pos: torch.Tensor = two_theta / sin_theta
    k_neg: torch.Tensor = 2.0 * torch.ones_like(sin_theta)
    k: torch.Tensor = torch.where(sin_squared_theta > 0.0, k_pos, k_neg)

    angle_axis: torch.Tensor = torch.zeros_like(quaternion)[..., :3]
    angle_axis[..., 0] += q1 * k
    angle_axis[..., 1] += q2 * k
    angle_axis[..., 2] += q3 * k
    return angle_axis


def quaternion_to_angle(quaternion: torch.Tensor) -> torch.Tensor:
    """
    Convert quaternion vector to angle of the rotation.

    Args:
        quaternion (torch.Tensor): tensor with quaternions.

    Return:
        torch.Tensor: tensor with angle axis of rotation.

    Shape:
        - Input: :math:`(*, 4)` where `*` means, any number of dimensions
        - Output: :math:`(*, 1)`

    Example:
        >>> quaternion = torch.rand(2, 4)  # Nx4
        >>> angle_axis = tgm.quaternion_to_angle(quaternion)  # Nx1
    """
    if not torch.is_tensor(quaternion):
        raise TypeError("Input type is not a torch.Tensor. Got {}".format(type(quaternion)))

    if not quaternion.shape[-1] == 4:
        raise ValueError(
            "Input must be a tensor of shape Nx4 or 4. Got {}".format(quaternion.shape)
        )
    # unpack input and compute conversion
    q1: torch.Tensor = quaternion[..., 1]
    q2: torch.Tensor = quaternion[..., 2]
    q3: torch.Tensor = quaternion[..., 3]
    sin_squared_theta: torch.Tensor = q1 * q1 + q2 * q2 + q3 * q3

    sin_theta: torch.Tensor = torch.sqrt(sin_squared_theta)
    cos_theta: torch.Tensor = quaternion[..., 0]
    theta: torch.Tensor = 2.0 * torch.where(
        cos_theta < 0.0, torch.atan2(-sin_theta, -cos_theta), torch.atan2(sin_theta, cos_theta)
    )

    # theta: torch.Tensor = 2.0 * torch.atan2(sin_theta, cos_theta)

    # theta2 = torch.where(sin_squared_theta > 0.0, - theta, theta)

    return theta.unsqueeze(-1)


def rotation_matrix_to_quaternion(rotation_matrix, eps=1e-6):
    """
    This function is borrowed from https://github.com/kornia/kornia

    Convert 3x4 rotation matrix to 4d quaternion vector

    This algorithm is based on algorithm described in
    https://github.com/KieranWynn/pyquaternion/blob/master/pyquaternion/quaternion.py#L201

    Args:
        rotation_matrix (Tensor): the rotation matrix to convert.

    Return:
        Tensor: the rotation in quaternion

    Shape:
        - Input: :math:`(N, 3, 4)`
        - Output: :math:`(N, 4)`

    Example:
        >>> input = torch.rand(4, 3, 4)  # Nx3x4
        >>> output = tgm.rotation_matrix_to_quaternion(input)  # Nx4
    """
    if not torch.is_tensor(rotation_matrix):
        raise TypeError("Input type is not a torch.Tensor. Got {}".format(type(rotation_matrix)))

    if len(rotation_matrix.shape) > 3:
        raise ValueError(
            "Input size must be a three dimensional tensor. Got {}".format(rotation_matrix.shape)
        )
    # if not rotation_matrix.shape[-2:] == (3, 4):
    #     raise ValueError(
    #         "Input size must be a N x 3 x 4  tensor. Got {}".format(
    #             rotation_matrix.shape))

    rmat_t = torch.transpose(rotation_matrix, 1, 2)

    mask_d2 = rmat_t[:, 2, 2] < eps

    mask_d0_d1 = rmat_t[:, 0, 0] > rmat_t[:, 1, 1]
    mask_d0_nd1 = rmat_t[:, 0, 0] < -rmat_t[:, 1, 1]

    t0 = 1 + rmat_t[:, 0, 0] - rmat_t[:, 1, 1] - rmat_t[:, 2, 2]
    q0 = torch.stack(
        [
            rmat_t[:, 1, 2] - rmat_t[:, 2, 1], t0, rmat_t[:, 0, 1] + rmat_t[:, 1, 0],
            rmat_t[:, 2, 0] + rmat_t[:, 0, 2]
        ], -1
    )
    t0_rep = t0.repeat(4, 1).t()

    t1 = 1 - rmat_t[:, 0, 0] + rmat_t[:, 1, 1] - rmat_t[:, 2, 2]
    q1 = torch.stack(
        [
            rmat_t[:, 2, 0] - rmat_t[:, 0, 2], rmat_t[:, 0, 1] + rmat_t[:, 1, 0], t1,
            rmat_t[:, 1, 2] + rmat_t[:, 2, 1]
        ], -1
    )
    t1_rep = t1.repeat(4, 1).t()

    t2 = 1 - rmat_t[:, 0, 0] - rmat_t[:, 1, 1] + rmat_t[:, 2, 2]
    q2 = torch.stack(
        [
            rmat_t[:, 0, 1] - rmat_t[:, 1, 0], rmat_t[:, 2, 0] + rmat_t[:, 0, 2],
            rmat_t[:, 1, 2] + rmat_t[:, 2, 1], t2
        ], -1
    )
    t2_rep = t2.repeat(4, 1).t()

    t3 = 1 + rmat_t[:, 0, 0] + rmat_t[:, 1, 1] + rmat_t[:, 2, 2]
    q3 = torch.stack(
        [
            t3, rmat_t[:, 1, 2] - rmat_t[:, 2, 1], rmat_t[:, 2, 0] - rmat_t[:, 0, 2],
            rmat_t[:, 0, 1] - rmat_t[:, 1, 0]
        ], -1
    )
    t3_rep = t3.repeat(4, 1).t()

    mask_c0 = mask_d2 * mask_d0_d1
    mask_c1 = mask_d2 * ~mask_d0_d1
    mask_c2 = ~mask_d2 * mask_d0_nd1
    mask_c3 = ~mask_d2 * ~mask_d0_nd1
    mask_c0 = mask_c0.view(-1, 1).type_as(q0)
    mask_c1 = mask_c1.view(-1, 1).type_as(q1)
    mask_c2 = mask_c2.view(-1, 1).type_as(q2)
    mask_c3 = mask_c3.view(-1, 1).type_as(q3)

    q = q0 * mask_c0 + q1 * mask_c1 + q2 * mask_c2 + q3 * mask_c3
    q /= torch.sqrt(
        t0_rep * mask_c0 + t1_rep * mask_c1 +    # noqa
        t2_rep * mask_c2 + t3_rep * mask_c3
    )    # noqa
    q *= 0.5
    return q


def batch_euler2matrix(r):
    return quaternion_to_rotation_matrix(euler_to_quaternion(r))


def euler_to_quaternion(r):
    x = r[..., 0]
    y = r[..., 1]
    z = r[..., 2]

    z = z / 2.0
    y = y / 2.0
    x = x / 2.0
    cz = torch.cos(z)
    sz = torch.sin(z)
    cy = torch.cos(y)
    sy = torch.sin(y)
    cx = torch.cos(x)
    sx = torch.sin(x)
    quaternion = torch.zeros_like(r.repeat(1, 2))[..., :4].to(r.device)
    quaternion[..., 0] += cx * cy * cz - sx * sy * sz
    quaternion[..., 1] += cx * sy * sz + cy * cz * sx
    quaternion[..., 2] += cx * cz * sy - sx * cy * sz
    quaternion[..., 3] += cx * cy * sz + sx * cz * sy
    return quaternion


def quaternion_to_rotation_matrix(quat):
    """Convert quaternion coefficients to rotation matrix.
    Args:
        quat: size = [B, 4] 4 <===>(w, x, y, z)
    Returns:
        Rotation matrix corresponding to the quaternion -- size = [B, 3, 3]
    """
    norm_quat = quat
    norm_quat = norm_quat / norm_quat.norm(p=2, dim=1, keepdim=True)
    w, x, y, z = norm_quat[:, 0], norm_quat[:, 1], norm_quat[:, 2], norm_quat[:, 3]

    B = quat.size(0)

    w2, x2, y2, z2 = w.pow(2), x.pow(2), y.pow(2), z.pow(2)
    wx, wy, wz = w * x, w * y, w * z
    xy, xz, yz = x * y, x * z, y * z

    rotMat = torch.stack(
        [
            w2 + x2 - y2 - z2, 2 * xy - 2 * wz, 2 * wy + 2 * xz, 2 * wz + 2 * xy, w2 - x2 + y2 - z2,
            2 * yz - 2 * wx, 2 * xz - 2 * wy, 2 * wx + 2 * yz, w2 - x2 - y2 + z2
        ],
        dim=1
    ).view(B, 3, 3)
    return rotMat


def rot6d_to_rotmat(x):
    """Convert 6D rotation representation to 3x3 rotation matrix.
    Based on Zhou et al., "On the Continuity of Rotation Representations in Neural Networks", CVPR 2019
    Input:
        (B,6) Batch of 6-D rotation representations
    Output:
        (B,3,3) Batch of corresponding rotation matrices
    """
    if x.shape[-1] == 6:
        batch_size = x.shape[0]
        if len(x.shape) == 3:
            num = x.shape[1]
            x = rearrange(x, 'b n d -> (b n) d', d=6)
        else:
            num = 1
        x = rearrange(x, 'b (k l) -> b k l', k=3, l=2)
        # x = x.view(-1,3,2)
        a1 = x[:, :, 0]
        a2 = x[:, :, 1]
        b1 = F.normalize(a1)
        b2 = F.normalize(a2 - torch.einsum('bi,bi->b', b1, a2).unsqueeze(-1) * b1)
        b3 = torch.cross(b1, b2, dim=-1)

        mat = torch.stack((b1, b2, b3), dim=-1)
        if num > 1:
            mat = rearrange(mat, '(b n) h w-> b n h w', b=batch_size, n=num, h=3, w=3)
    else:
        x = x.view(-1, 3, 2)
        a1 = x[:, :, 0]
        a2 = x[:, :, 1]
        b1 = F.normalize(a1)
        b2 = F.normalize(a2 - torch.einsum('bi,bi->b', b1, a2).unsqueeze(-1) * b1)
        b3 = torch.cross(b1, b2, dim=-1)
        mat = torch.stack((b1, b2, b3), dim=-1)
    return mat


def rotmat_to_rot6d(x):
    """Convert 3x3 rotation matrix to 6D rotation representation.
    Based on Zhou et al., "On the Continuity of Rotation Representations in Neural Networks", CVPR 2019
    Input:
        (B,3,3) Batch of corresponding rotation matrices
    Output:
        (B,6) Batch of 6-D rotation representations
    """
    batch_size = x.shape[0]
    x = x[:, :, :2]
    x = x.reshape(batch_size, 6)
    return x


def rotmat_to_angle(x):
    """Convert rotation to one-D angle.
    Based on Zhou et al., "On the Continuity of Rotation Representations in Neural Networks", CVPR 2019
    Input:
        (B,2) Batch of corresponding rotation
    Output:
        (B,1) Batch of 1-D angle
    """
    a = F.normalize(x)
    angle = torch.atan2(a[:, 0], a[:, 1]).unsqueeze(-1)

    return angle


def projection(pred_joints, pred_camera, retain_z=False, iwp_mode=True):
    """ Project 3D points on the image plane based on the given camera info, 
        Identity rotation and Weak Perspective (IWP) camera is used when iwp_mode=True, more about camera settings:
        SPEC: Seeing People in the Wild with an Estimated Camera, ICCV 2021
    """

    batch_size = pred_joints.shape[0]
    if iwp_mode:
        cam_sxy = pred_camera['cam_sxy']
        pred_cam_t = torch.stack(
            [cam_sxy[:, 1], cam_sxy[:, 2], 2 * 5000. / (224. * cam_sxy[:, 0] + 1e-9)], dim=-1
        )

        camera_center = torch.zeros(batch_size, 2)
        pred_keypoints_2d = perspective_projection(
            pred_joints,
            rotation=torch.eye(3).unsqueeze(0).expand(batch_size, -1, -1).to(pred_joints.device),
            translation=pred_cam_t,
            focal_length=5000.,
            camera_center=camera_center,
            retain_z=retain_z
        )
        # # Normalize keypoints to [-1,1]
        # pred_keypoints_2d = pred_keypoints_2d / (224. / 2.)
    else:
        assert type(pred_camera) is dict

        bbox_scale, bbox_center = pred_camera['bbox_scale'], pred_camera['bbox_center']
        img_w, img_h, crop_res = pred_camera['img_w'], pred_camera['img_h'], pred_camera['crop_res']
        cam_sxy, cam_rotmat, cam_intrinsics = pred_camera['cam_sxy'], pred_camera[
            'cam_rotmat'], pred_camera['cam_intrinsics']
        if 'cam_t' in pred_camera:
            cam_t = pred_camera['cam_t']
        else:
            cam_t = convert_to_full_img_cam(
                pare_cam=cam_sxy,
                bbox_height=bbox_scale * 200.,
                bbox_center=bbox_center,
                img_w=img_w,
                img_h=img_h,
                focal_length=cam_intrinsics[:, 0, 0],
            )

        pred_keypoints_2d = perspective_projection(
            pred_joints,
            rotation=cam_rotmat,
            translation=cam_t,
            cam_intrinsics=cam_intrinsics,
        )

    return pred_keypoints_2d


def perspective_projection(
    points,
    rotation,
    translation,
    focal_length=None,
    camera_center=None,
    cam_intrinsics=None,
    retain_z=False
):
    """
    This function computes the perspective projection of a set of points.
    Input:
        points (bs, N, 3): 3D points
        rotation (bs, 3, 3): Camera rotation
        translation (bs, 3): Camera translation
        focal_length (bs,) or scalar: Focal length
        camera_center (bs, 2): Camera center
    """
    batch_size = points.shape[0]
    if cam_intrinsics is not None:
        K = cam_intrinsics
    else:
        # raise
        K = torch.zeros([batch_size, 3, 3], device=points.device)
        K[:, 0, 0] = focal_length
        K[:, 1, 1] = focal_length
        K[:, 2, 2] = 1.
        K[:, :-1, -1] = camera_center

    # Transform points
    points = torch.einsum('bij,bkj->bki', rotation, points)
    points = points + translation.unsqueeze(1)

    # Apply perspective distortion
    projected_points = points / points[:, :, -1].unsqueeze(-1)

    # Apply camera intrinsics
    projected_points = torch.einsum('bij,bkj->bki', K, projected_points)

    if retain_z:
        return projected_points
    else:
        return projected_points[:, :, :-1]


def convert_to_full_img_cam(pare_cam, bbox_height, bbox_center, img_w, img_h, focal_length):
    # Converts weak perspective camera estimated by PARE in
    # bbox coords to perspective camera in full image coordinates
    # from https://arxiv.org/pdf/2009.06549.pdf
    s, tx, ty = pare_cam[:, 0], pare_cam[:, 1], pare_cam[:, 2]
    res = 224
    r = bbox_height / res
    tz = 2 * focal_length / (r * res * s)

    cx = 2 * (bbox_center[:, 0] - (img_w / 2.)) / (s * bbox_height)
    cy = 2 * (bbox_center[:, 1] - (img_h / 2.)) / (s * bbox_height)

    if torch.is_tensor(pare_cam):
        cam_t = torch.stack([tx + cx, ty + cy, tz], dim=-1)
    else:
        cam_t = np.stack([tx + cx, ty + cy, tz], axis=-1)

    return cam_t


def estimate_translation_np(S, joints_2d, joints_conf, focal_length=5000, img_size=(224., 224.)):
    """Find camera translation that brings 3D joints S closest to 2D the corresponding joints_2d.
    Input:
        S: (25, 3) 3D joint locations
        joints: (25, 3) 2D joint locations and confidence
    Returns:
        (3,) camera translation vector
    """

    num_joints = S.shape[0]
    # focal length
    f = np.array([focal_length, focal_length])
    # optical center
    center = np.array([img_size[1] / 2., img_size[0] / 2.])

    # transformations
    Z = np.reshape(np.tile(S[:, 2], (2, 1)).T, -1)
    XY = np.reshape(S[:, 0:2], -1)
    O = np.tile(center, num_joints)
    F = np.tile(f, num_joints)
    weight2 = np.reshape(np.tile(np.sqrt(joints_conf), (2, 1)).T, -1)

    # least squares
    Q = np.array(
        [
            F * np.tile(np.array([1, 0]), num_joints), F * np.tile(np.array([0, 1]), num_joints),
            O - np.reshape(joints_2d, -1)
        ]
    ).T
    c = (np.reshape(joints_2d, -1) - O) * Z - F * XY

    # weighted least squares
    W = np.diagflat(weight2)
    Q = np.dot(W, Q)
    c = np.dot(W, c)

    # square matrix
    A = np.dot(Q.T, Q)
    b = np.dot(Q.T, c)

    # solution
    trans = np.linalg.solve(A, b)

    return trans


def estimate_translation(S, joints_2d, focal_length=5000., img_size=224., use_all_kps=False):
    """Find camera translation that brings 3D joints S closest to 2D the corresponding joints_2d.
    Input:
        S: (B, 49, 3) 3D joint locations
        joints: (B, 49, 3) 2D joint locations and confidence
    Returns:
        (B, 3) camera translation vectors
    """
    if isinstance(focal_length, numbers.Number):
        focal_length = [
            focal_length,
        ] * S.shape[0]
        # print(len(focal_length), focal_length)

    if isinstance(img_size, numbers.Number):
        img_size = [
            (img_size, img_size),
        ] * S.shape[0]
        # print(len(img_size), img_size)

    device = S.device
    if use_all_kps:
        S = S.cpu().numpy()
        joints_2d = joints_2d.cpu().numpy()
    else:
        # Use only joints 25:49 (GT joints)
        S = S[:, 25:, :].cpu().numpy()
        joints_2d = joints_2d[:, 25:, :].cpu().numpy()
    joints_conf = joints_2d[:, :, -1]
    joints_2d = joints_2d[:, :, :-1]
    trans = np.zeros((S.shape[0], 3), dtype=np.float32)
    # Find the translation for each example in the batch
    for i in range(S.shape[0]):
        S_i = S[i]
        joints_i = joints_2d[i]
        conf_i = joints_conf[i]
        trans[i] = estimate_translation_np(
            S_i, joints_i, conf_i, focal_length=focal_length[i], img_size=img_size[i]
        )
    return torch.from_numpy(trans).to(device)


def Rot_y(angle, category='torch', prepend_dim=True, device=None):
    '''Rotate around y-axis by angle
	Args:
		category: 'torch' or 'numpy'
		prepend_dim: prepend an extra dimension
	Return: Rotation matrix with shape [1, 3, 3] (prepend_dim=True)
	'''
    m = np.array(
        [[np.cos(angle), 0., np.sin(angle)], [0., 1., 0.], [-np.sin(angle), 0.,
                                                            np.cos(angle)]]
    )
    if category == 'torch':
        if prepend_dim:
            return torch.tensor(m, dtype=torch.float, device=device).unsqueeze(0)
        else:
            return torch.tensor(m, dtype=torch.float, device=device)
    elif category == 'numpy':
        if prepend_dim:
            return np.expand_dims(m, 0)
        else:
            return m
    else:
        raise ValueError("category must be 'torch' or 'numpy'")


def Rot_x(angle, category='torch', prepend_dim=True, device=None):
    '''Rotate around x-axis by angle
	Args:
		category: 'torch' or 'numpy'
		prepend_dim: prepend an extra dimension
	Return: Rotation matrix with shape [1, 3, 3] (prepend_dim=True)
	'''
    m = np.array(
        [[1., 0., 0.], [0., np.cos(angle), -np.sin(angle)], [0., np.sin(angle),
                                                             np.cos(angle)]]
    )
    if category == 'torch':
        if prepend_dim:
            return torch.tensor(m, dtype=torch.float, device=device).unsqueeze(0)
        else:
            return torch.tensor(m, dtype=torch.float, device=device)
    elif category == 'numpy':
        if prepend_dim:
            return np.expand_dims(m, 0)
        else:
            return m
    else:
        raise ValueError("category must be 'torch' or 'numpy'")


def Rot_z(angle, category='torch', prepend_dim=True, device=None):
    '''Rotate around z-axis by angle
	Args:
		category: 'torch' or 'numpy'
		prepend_dim: prepend an extra dimension
	Return: Rotation matrix with shape [1, 3, 3] (prepend_dim=True)
	'''
    m = np.array(
        [[np.cos(angle), -np.sin(angle), 0.], [np.sin(angle), np.cos(angle), 0.], [0., 0., 1.]]
    )
    if category == 'torch':
        if prepend_dim:
            return torch.tensor(m, dtype=torch.float, device=device).unsqueeze(0)
        else:
            return torch.tensor(m, dtype=torch.float, device=device)
    elif category == 'numpy':
        if prepend_dim:
            return np.expand_dims(m, 0)
        else:
            return m
    else:
        raise ValueError("category must be 'torch' or 'numpy'")


def compute_twist_rotation(rotation_matrix, twist_axis):
    '''
    Compute the twist component of given rotation and twist axis
    https://stackoverflow.com/questions/3684269/component-of-a-quaternion-rotation-around-an-axis
    Parameters
    ----------
    rotation_matrix : Tensor (B, 3, 3,)
        The rotation to convert
    twist_axis : Tensor (B, 3,)
        The twist axis
    Returns
    -------
    Tensor (B, 3, 3)
        The twist rotation
    '''
    quaternion = rotation_matrix_to_quaternion(rotation_matrix)

    twist_axis = twist_axis / (torch.norm(twist_axis, dim=1, keepdim=True) + 1e-9)

    projection = torch.einsum('bi,bi->b', twist_axis, quaternion[:, 1:]).unsqueeze(-1) * twist_axis

    twist_quaternion = torch.cat([quaternion[:, 0:1], projection], dim=1)
    twist_quaternion = twist_quaternion / (torch.norm(twist_quaternion, dim=1, keepdim=True) + 1e-9)

    twist_rotation = quaternion_to_rotation_matrix(twist_quaternion)
    twist_aa = quaternion_to_angle_axis(twist_quaternion)

    twist_angle = torch.sum(twist_aa, dim=1,
                            keepdim=True) / torch.sum(twist_axis, dim=1, keepdim=True)

    return twist_rotation, twist_angle