lib/utils/geometry.py

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


def perspective_projection(points, rotation, translation,
                           focal_length, camera_center, distortion=None):
    """
    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]
    
    # Extrinsic
    if rotation is not None:
        points = torch.einsum('bij,bkj->bki', rotation, points)

    if translation is not None:
        points = points + translation.unsqueeze(1)

    if distortion is not None:
        kc = distortion
        points = points[:,:,:2] / points[:,:,2:]
        
        r2 = points[:,:,0]**2 + points[:,:,1]**2
        dx = (2 * kc[:,[2]] * points[:,:,0] * points[:,:,1] 
                + kc[:,[3]] * (r2 + 2*points[:,:,0]**2))

        dy = (2 * kc[:,[3]] * points[:,:,0] * points[:,:,1] 
                + kc[:,[2]] * (r2 + 2*points[:,:,1]**2))
        
        x = (1 + kc[:,[0]]*r2 + kc[:,[1]]*r2.pow(2) + kc[:,[4]]*r2.pow(3)) * points[:,:,0] + dx
        y = (1 + kc[:,[0]]*r2 + kc[:,[1]]*r2.pow(2) + kc[:,[4]]*r2.pow(3)) * points[:,:,1] + dy
        
        points = torch.stack([x, y, torch.ones_like(x)], dim=-1)
        
    # Intrinsic
    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

    # Apply camera intrinsicsrf
    points = points / points[:,:,-1].unsqueeze(-1)
    projected_points = torch.einsum('bij,bkj->bki', K, points)
    projected_points = projected_points[:, :, :-1]

    return projected_points


def avg_rot(rot):
    # input [B,...,3,3] --> output [...,3,3]
    rot = rot.mean(dim=0)
    U, _, V = torch.svd(rot)
    rot = U @ V.transpose(-1, -2)
    return rot
    

def rot9d_to_rotmat(x):
    """Convert 9D rotation representation to 3x3 rotation matrix.
    Based on Levinson et al., "An Analysis of SVD for Deep Rotation Estimation"
    Input:
        (B,9) or (B,J*9) Batch of 9D rotation (interpreted as 3x3 est rotmat)
    Output:
        (B,3,3) or (B*J,3,3) Batch of corresponding rotation matrices
    """
    x = x.view(-1,3,3)
    u, _, vh = torch.linalg.svd(x)

    sig = torch.eye(3).expand(len(x), 3, 3).clone()
    sig = sig.to(x.device)
    sig[:, -1, -1] = (u @ vh).det()

    R = u @ sig @ vh

    return R


"""
Deprecated in favor of: rotation_conversions.py

Useful geometric operations, e.g. 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 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
    """
    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)
    return torch.stack((b1, b2, b3), dim=-1)

def rot6d_to_rotmat_hmr2(x: torch.Tensor) -> torch.Tensor:
    """
    Convert 6D rotation representation to 3x3 rotation matrix.
    Based on Zhou et al., "On the Continuity of Rotation Representations in Neural Networks", CVPR 2019
    Args:
        x (torch.Tensor): (B,6) Batch of 6-D rotation representations.
    Returns:
        torch.Tensor: Batch of corresponding rotation matrices with shape (B,3,3).
    """
    x = x.reshape(-1,2,3).permute(0, 2, 1).contiguous()
    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)
    return torch.stack((b1, b2, b3), dim=-1)

def rotmat_to_rot6d(rotmat):
    """ Inverse function of the above.
    Input:
        (B,3,3) Batch of corresponding rotation matrices 
    Output:
        (B,6) Batch of 6-D rotation representations
    """
    # rot6d = rotmat[:, :, :2]
    rot6d = rotmat[...,:2]
    rot6d = rot6d.reshape(rot6d.size(0), -1)
    return rot6d


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 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 estimate_translation_np(S, joints_2d, joints_conf, focal_length=5000., img_size=224.):
    """
    This function is borrowed from https://github.com/nkolot/SPIN/utils/geometry.py

    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/2., img_size/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.):
    """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
    """

    device = S.device
    # Use only joints 25:49 (GT joints)
    S = S[:, -24:, :3].cpu().numpy()
    joints_2d = joints_2d[:, -24:, :].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, img_size=img_size)
    return torch.from_numpy(trans).to(device)