How to draw the Euler angles in the xy plane for head pose

How to draw the Euler angles in the xy plane for head pose

  • 目次

    When I drew the head pose on the xy plane, I had no idea how to draw it. This article is just a summary.

    Head pose visualization

    Head pose is a representation of the human head orientation.
    To visualise head pose, this article draws the xyz unit vector on an image of a human head.
    X-axis is drawn in red
    Y-axis is drawn in green
    Z-axis (out of the screen) is drawn in blue

    1

    The Euler angles for head pose

    The definition of the Euler angles for head pose is shown in the following left figure.

    1


    The representation of roll, pitch and yaw in xyz space depends on a perspective (field).
    For example, a roll from the airplane's perspective rotates around the X-axis, but a roll from the head pose's perspective rotates around the Z-axis.

    1. Convert the Euler angles to the rotation matrix

    From this chapter, I explain how to draw the Euler angles in the xy plane.
    First of all, convert the Euler angles to rotation matirx.
    The following three basic rotation matrices rotate vectors by the Euler angles using the right-hand rule.
    The rotation matrix of the X-axis is represented by $\boldsymbol{R_x}$, the rotation matrix of the Y-axis by $\boldsymbol{R_y}$ and the rotation matrix of the Z-axis by $\boldsymbol{R_z}$.

    \begin{eqnarray} \boldsymbol{R_x} &=& \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos\theta_{x} & -\sin\theta_{x} \\ 0 & \sin\theta_{x} & \cos\theta_{x} \end{array} \right) \\ \end{eqnarray}
    \begin{eqnarray} \boldsymbol{R_y} &=& \left( \begin{array}{ccc} \cos\theta_{y} & 0 & \sin\theta_{y} \\ 0 & 1 & 0 \\ -\sin\theta_{y} & 0 & \cos\theta_{y} \end{array} \right) \\ \end{eqnarray}
    \begin{eqnarray} \boldsymbol{R_z} &=& \left( \begin{array}{ccc} \cos\theta_{z} & -\sin\theta_{z} & 0 \\ \sin\theta_{z} & \cos\theta_{z} & 0 \\ 0 & 0 & 1 \end{array} \right) \end{eqnarray}

    Note:
    Please replace $\theta_{x}$, $\theta_{y}$ and $\theta_{z}$ to the Euler angles.
    e.g. $\theta_{x}$ means $\theta_{pitch}$ for head pose.

    This article rotates in the order Z, Y, X (roll, yaw, pitch).
    Note that the order of rotation affects the final result.

    \begin{eqnarray} \boldsymbol{R_{zyx}} = \boldsymbol{R_x}\boldsymbol{R_y}\boldsymbol{R_z} = \left( \begin{array}{ccc} \cos\theta_{y}\cos\theta_{z} & -\cos\theta_{y}\sin\theta_{z} & \sin\theta_{y} \\ \sin\theta_{x}\sin\theta_{y}\cos\theta_{z} + \cos\theta_{x}\sin\theta_{z} & -\sin\theta_{x}\sin\theta_{y}\sin\theta_{z} + \cos\theta_{x}\cos\theta_{z} & -\sin\theta_{x}\cos\theta_{y} \\ -\cos\theta_{x}\sin\theta_{y}\cos\theta_{z} + \sin\theta_{x}\sin\theta_{z} & \cos\theta_{x}\sin\theta_{y}\sin\theta_{z} + \sin\theta_{x}\cos\theta_{z} & \cos\theta_{x}\cos\theta_{y} \end{array} \right) \end{eqnarray}

    The following equations use roll, pitch and yaw notation.

    \begin{eqnarray} \boldsymbol{R_{xyz}} = \left( \begin{array}{ccc} \cos\theta_{yaw}\cos\theta_{roll} & -\cos\theta_{yaw}\sin\theta_{roll} & \sin\theta_{yaw} \\ \sin\theta_{pitch}\sin\theta_{yaw}\cos\theta_{roll} + \cos\theta_{pitch}\sin\theta_{roll} & -\sin\theta_{pitch}\sin\theta_{yaw}\sin\theta_{roll} + \cos\theta_{pitch}\cos\theta_{roll} & -\sin\theta_{pitch}\cos\theta_{yaw} \\ -\cos\theta_{pitch}\sin\theta_{yaw}\cos\theta_{roll} + \sin\theta_{pitch}\sin\theta_{roll} & \cos\theta_{pitch}\sin\theta_{yaw}\sin\theta_{roll} + \sin\theta_{pitch}\cos\theta_{roll} & \cos\theta_{pitch}\cos\theta_{yaw} \end{array} \right) \end{eqnarray}

    2. Project the xyz unit vector onto the xy plane

    Next, to visualize the Euler angles, I draw the xyz unit vector in the xy plane (an image) as explained above.
    The xyz unit vectors are represented as follows.

    \begin{eqnarray} \boldsymbol{e_{x}} = \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right) \boldsymbol{e_{y}} = \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right) \boldsymbol{e_{z}} = \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) \end{eqnarray}

    To project the xyz unit vector onto the xy plane, multiply the rotation matrix as follows
    \begin{eqnarray} \widehat{\boldsymbol{e_{x}}} = \left( \begin{array}{rcc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \boldsymbol{R_{x}} \boldsymbol{R_{y}} \boldsymbol{R_{z}} \boldsymbol{e_{x}} = \left( \begin{array}{c} \cos\theta_{y}\cos\theta_{z} \\ \sin\theta_{x}\sin\theta_{y}\cos\theta_{z} + \cos\theta_{x}\sin\theta_{z} \\ 0 \end{array} \right) \\ \widehat{\boldsymbol{e_{y}}} = \left( \begin{array}{rcc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \boldsymbol{R_{x}} \boldsymbol{R_{y}} \boldsymbol{R_{z}} \boldsymbol{e_{y}} = \left( \begin{array}{c} -\cos\theta_{y}\sin\theta_{z} \\ -\sin\theta_{x}\sin\theta_{y}\sin\theta_{z} + \cos\theta_{x}\cos\theta_{z} \\ 0 \end{array} \right) \\ \widehat{\boldsymbol{e_{z}}} = \left( \begin{array}{rcc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right) \boldsymbol{R_{x}} \boldsymbol{R_{y}} \boldsymbol{R_{z}} \boldsymbol{e_{z}} = \left( \begin{array}{c} \sin\theta_{y} \\ -\sin\theta_{x}\cos\theta_{y} \\ 0 \end{array} \right) \\ \end{eqnarray}

    The Euler angles can be represented on the image by drawing the XY elements of the above equation (7) ~ (8) in the xy plane.

    Sample code (AFLW2000)

    Example of command line:

    python3 view.py --dataset_root <path/to/the/AFLW2000/root/directory> --filename image00078
    

    Example of execution:

    1

    Sample code (view.py):

     1import os
     2import argparse
     3import cv2
     4import numpy as np
     5import scipy.io as sio
     6from math import cos, sin
     7
     8def parse_args():
     9    parser = argparse.ArgumentParser()
    10    parser.add_argument('--datasets_root',
    11                        help='Path to the root directory of AFLW2000 datasets',
    12                        type=str)
    13    parser.add_argument('--filename',
    14                        help='Filename without extension to load the data',
    15                        type=str)
    16    args = parser.parse_args()
    17    return args
    18
    19
    20def get_R(x,y,z):
    21    ''' Get rotation matrix from three rotation angles (radians). right-handed.
    22    Args:
    23        angles: [3,]. x, y, z angles
    24    Returns:
    25        R: [3, 3]. rotation matrix.
    26    '''
    27    # x
    28    Rx = np.array([[1, 0, 0],
    29                   [0, np.cos(x), -np.sin(x)],
    30                   [0, np.sin(x), np.cos(x)]])
    31    # y
    32    Ry = np.array([[np.cos(y), 0, np.sin(y)],
    33                   [0, 1, 0],
    34                   [-np.sin(y), 0, np.cos(y)]])
    35    # z
    36    Rz = np.array([[np.cos(z), -np.sin(z), 0],
    37                   [np.sin(z), np.cos(z), 0],
    38                   [0, 0, 1]])
    39
    40    R = Rx.dot(Ry.dot(Rz))
    41    return R
    42
    43
    44if __name__ == "__main__":
    45    args = parse_args()
    46
    47    image_file = os.path.join(args.datasets_root, '{}.jpg'.format(args.filename))
    48    mat_file = os.path.join(args.datasets_root, '{}.mat'.format(args.filename))
    49
    50    # Load .jpg
    51    img = cv2.imread(image_file)
    52    # Load .mat
    53    mat = sio.loadmat(mat_file)
    54
    55    # [pitch yaw roll tdx tdy tdz scale_factor]
    56    pre_pose_params = mat['Pose_Para'][0]
    57    # Get [pitch, yaw, roll]
    58    pitch, yaw, roll = pre_pose_params[:3]
    59
    60    # Covert to R
    61    x = pitch
    62    y = -yaw
    63    z = roll
    64    R = get_R(x, y, z)
    65    print("Rotation matrix:\n", R)
    66
    67    # Draw Axis
    68    cx = img.shape[1]//2
    69    cy = img.shape[0]//2
    70    size = 100
    71    # X-axis pointing to right. drawn in red
    72    x1 = size * R[0, 0] + cx
    73    y1 = size * R[1, 0] + cy
    74    # Y-axis drawn in green
    75    x2 = size * R[0, 1] + cx
    76    y2 = size * R[1, 1] + cy
    77    # Z-axis (out of the screen) drawn in blue
    78    x3 = size * R[0, 2] + cx
    79    y3 = size * R[1, 2] + cy
    80
    81    cv2.line(img, (int(cx), int(cy)), (int(x1),int(y1)), (0,0,255), 2)
    82    cv2.line(img, (int(cx), int(cy)), (int(x2),int(y2)), (0,255,0), 2)
    83    cv2.line(img, (int(cx), int(cy)), (int(x3),int(y3)), (255,0,0), 2)
    84
    85    cv2.imshow("View", img)
    86    cv2.waitKey(0)
    87    cv2.destroyAllWindows()
    88    print("End")

    Summary

    The representation of roll, pitch and yaw in xyz space depends on a perspective.
    Therefore I need to be concerned about which field I am dealing with, e.g.head pose, airplane.

    Reference

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