How to plot the camera and image positions from camera calibration data?

Question

I have the intrisic and extrinsic parameters of the camera. The extrinsic is a 4 x 4 matrix with rotation and translation.

I have sample data as under, I have this one per camera image taken.

2.11e-001 -3.06e-001 -9.28e-001 7.89e-001 
6.62e-001 7.42e-001 -9.47e-002 1.47e-001 
7.18e-001 -5.95e-001 3.60e-001 3.26e+000 
0.00e+000 0.00e+000 0.00e+000 1.00e+000

I would like to plot the image as given on the Matlab calibration toolkit page or However I'm unable to figure out the Math of how to plot these 2 images.

The only lead I have is from this page http://en.wikipedia.org/wiki/Camera_resectioning. Which tells me that the camera position can be found by C = − R` . T

Any idea how to achieve this task?

Answer 1

This is how you can plot the cameras. Red arrows are camera directions (-z or z), blue arrows are cameras orientations. All of example cameras point at [0,0,0].

Code:

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

def plot_cameras(extrinsic_matrices):
    """
    Visualizes the extrinsics of multiple cameras in 3D space.

    Parameters:
    - extrinsic_matrices (numpy.ndarray): Array of camera extrinsics matrices (Nx4x4).
    """
    ax = plt.figure().add_subplot(projection='3d')

    for camera_extrinsics in extrinsic_matrices:
        # Extract translation and rotation from camera extrinsics matrix
        translation = camera_extrinsics[:3, 3]
        rotation_matrix = camera_extrinsics[:3, :3]

        # Plot camera position
        ax.scatter(*translation, marker='o')

        # Plot camera orientation axes
        origin = translation
        for i in range(3):
            axis_direction = rotation_matrix[:,i] 
            if i == 0:
                ax.quiver(*origin, *axis_direction, length=0.5, normalize=True)
            else:
                ax.quiver(*origin, *axis_direction, length=1, normalize=True)
        # Plot camera direction
        z = -1 * rotation_matrix[:,2]
        ax.quiver(*origin, *z, length=1, normalize=True, color='r', alpha=0.5)

    ax.set_xlabel('X')
    ax.set_ylabel('Y')
    ax.set_zlabel('Z')
    ax.set_title('Multiple Cameras Extrinsics Visualization')

    ax.set_zlim(-2,2)

    plt.show()


# List of extrinsic matrices or poses
# Example poses all pointing at [0,0,0]
poses = np.array([[
        [-9.9990219e-01,  4.1922452e-03, -1.3345719e-02, -5.3798322e-02],
        [-1.3988681e-02, -2.9965907e-01,  9.5394367e-01,  3.8454704e+00],
        [-4.6566129e-10,  9.5403719e-01,  2.9968831e-01,  1.2080823e+00],
        [ 0.0000000e+00,  0.0000000e+00,  0.0000000e+00,  1.0000000e+00]],

       [[-9.3054223e-01,  1.1707554e-01, -3.4696460e-01, -1.3986591e+00],
        [-3.6618456e-01, -2.9751042e-01,  8.8170075e-01,  3.5542498e+00],
        [ 7.4505806e-09,  9.4751304e-01,  3.1971723e-01,  1.2888215e+00],
        [ 0.0000000e+00,  0.0000000e+00,  0.0000000e+00,  1.0000000e+00]],

       [[ 4.4296363e-01,  3.1377721e-01, -8.3983749e-01, -3.3854935e+00],
        [-8.9653969e-01,  1.5503149e-01, -4.1494811e-01, -1.6727095e+00],
        [ 0.0000000e+00,  9.3675458e-01,  3.4998694e-01,  1.4108427e+00],
        [ 0.0000000e+00,  0.0000000e+00,  0.0000000e+00,  1.0000000e+00]],

       [[ 7.9563183e-01,  5.2092606e-01, -3.0920234e-01, -1.2464346e+00],
        [-6.0578054e-01,  6.8418401e-01, -4.0610620e-01, -1.6370665e+00],
        [-1.4901161e-08,  5.1041979e-01,  8.5992539e-01,  3.4664700e+00],
        [ 0.0000000e+00,  0.0000000e+00,  0.0000000e+00,  1.0000000e+00]],

       [[-6.5665424e-01, -2.3678228e-01,  7.1605819e-01,  2.8865230e+00],
        [ 7.5419182e-01, -2.0615987e-01,  6.2345237e-01,  2.5132170e+00],
        [-1.4901163e-08,  9.4943780e-01,  3.1395501e-01,  1.2655932e+00],
        [ 0.0000000e+00,  0.0000000e+00,  0.0000000e+00,  1.0000000e+00]]])

plot_cameras(poses)

Explanation why Z is where camera is pointing.

Answer 2

There is now an example in opencv for visualizing the extrinsics generated from their camera calibration example

It outputs something similar to the original questions ask:

Answer 3

Assume the corners of the plane that you want to draw are 3x1 column vectors, a = [0 0 0]', b = [w 0 0]', c = [w h 0]' and d = [0 h 0]'.

Assume that the calibration matrix that you provide is A and consists of a rotation matrix R = A(1:3, 1:3) and a translation T = A(1:3, 4).

To draw the first view For every pose A_i with rotation R_i and translation T_i, transform each corner x_w (that is a, b, c or d) of the plane to its coordinates x_c in the camera by

x_c = R_i*x_w + T_i

Then draw the plane with transformed corners.

To draw the camera, its centre of projection in camera coordinates is [0 0 0]' and the camera x axis is [1 0 0]', y axis is [0 1 0]' and z axis is [0 0 1]'.

Note that in the drawing, the camera y-axis is pointing down, so you might want to apply an additional rotation on all the computed coordinates by multiplication with B = [1 0 0; 0 0 1; 0 -1 0].

Draw the second view Drawing the plane is trivial since we are in world coordinates. Just draw the plane using a, b, c and d.

To draw the cameras, each camera centre is c = -R'*T. The camera axes are the rows of the rotation matrix R, so for instance, in the matrix you provided, the x-axis is [2.11e-001 -3.06e-001 -9.28e-001]'. You can also draw a camera by transforming each point x_c given in camera coordinates to world coordinates x_w by x_w = R'*(x_c - T) and draw it.