cv2.solvePnP的返回值实际上在哪里保持相机的位置？

Question

现实世界中有一个立方体和一个相机。每个点坐标如下图所示。很明显，相机的坐标位于[0,0,1]。

我们可以在屏幕上计算每个位置的显示坐标。

import numpy as np
import cv2
import math

world = np.array(\
[\
( 5.00,  0.00,  0.00), \
( 5.00,  0.00,  0.50), \
( 6.00, -1.00,  0.00), \
( 6.00, -1.00,  0.50), \
( 6.00,  1.00,  0.00), \
( 6.00,  1.00,  0.50), \
( 7.00,  0.00,  0.50), \
])

# Camera Extrinsic Parameter
xRadEuler_C2W = -120 / 180 * math.pi
yRadEuler_C2W = 0 / 180 * math.pi
zRadEuler_C2W = -90 / 180 * math.pi

Rx = np.matrix([[1, 0, 0], [0, math.cos(xRadEuler_C2W), -   math.sin(xRadEuler_C2W)], [0, math.sin(xRadEuler_C2W),  math.cos(xRadEuler_C2W)]])
Ry = np.matrix([[ math.cos(yRadEuler_C2W), 0, math.sin(yRadEuler_C2W)], [0, 1, 0], [-math.sin(yRadEuler_C2W), 0, math.cos(yRadEuler_C2W)]])
Rz = np.matrix([[ math.cos(zRadEuler_C2W), -math.sin(zRadEuler_C2W), 0], [ math.sin(zRadEuler_C2W),  math.cos(zRadEuler_C2W), 0], [0, 0, 1]])

# Notice : Rotation Matrix from Euler Angle.
R = Rx * Ry * Rz

# tvec is expressed wrt camra coord.
tvec = R * np.matrix((0, 0, -1)).T

# Camera Intrinsic Paramter
dist_coeffs = np.zeros((5, 1))
width = 640
height = 480
focal_length = 160
center = (width / 2, height / 2)

camera_matrix = np.array([[focal_length, 0, center[0]], 
                          [0, focal_length, center[1]],
                          [0, 0, 1]], dtype = "double")

if __name__ == "__main__":
  print("\nProject Point on Screen")
  result = cv2.projectPoints(world, rvec, tvec, camera_matrix, None)

  for n in range(len(world)):
    print(world[n], '==>', result[0][n])

结果，我们得到了

Project Point on Screen
[ 5.   0.  0.   ] ==> [[ 320.                  294.12609945]]
[ 5.   0.  0.5 ] ==> [[ 320.                  312.2071607]]
[ 6.  -1.  0.   ] ==> [[ 291.91086401  299.94150262]]
[ 6.  -1.  0.5 ] ==> [[ 290.62146125  315.41433581]]
[ 6.   1.  0.   ] ==> [[ 348.08913599  299.94150262]]
[ 6.   1.  0.5 ] ==> [[ 349.37853875  315.41433581]]
[ 7.   0.  0.5 ] ==> [[ 320.                  317.74146755]]

现在，我想计算相机的位置，我们将其定义为[0,0,1]。

import numpy as np
import cv2
import math

world = np.array(\
[\
( 5.00,  0.00,  0.00), \
( 5.00,  0.00,  0.50), \
( 6.00, -1.00,  0.00), \
( 6.00, -1.00,  0.50), \
( 6.00,  1.00,  0.00), \
( 6.00,  1.00,  0.50), \
( 7.00,  0.00,  0.50), \
])

img_pnts = np.array(\
[\
(320.                , 294.12609945), \
(320.                , 312.2071607), \
(291.91086401, 299.94150262), \
(290.62146125, 315.41433581), \
(348.08913599, 299.94150262), \
(349.37853875, 315.41433581), \
(320.                , 317.74146755), \
])

# Camera Intrinsic Paramter
dist_coeffs = np.zeros((5, 1))
width = 640
height = 480
focal_length = 160
center = (width / 2, height / 2)

camera_matrix = np.array(
                    [[focal_length, 0, center[0]], 
                    [0, focal_length, center[1]],
                    [0, 0, 1]], dtype = "double"
                    )

if __name__ == "__main__":
  (success, rot_vec, trans_vec) = cv2.solvePnP(world, img_pnts, camera_matrix, dist_coeffs, flags=cv2.SOLVEPNP_ITERATIVE)

  print("\nTranslation Vector")
  print(trans_vec)

  print("\nRotation Vector")
  print(rot_vec)

  print("\nRotation Matrix")
  R, jacob = cv2.Rodrigues(rot_vec)
  print(R)

结果看起来像这样。

Translation Vector
[[ -8.87481507e-11]
 [ -8.66025403e-01]
 [  4.99999999e-01]]

Rotation Vector
[[-1.58351453]
 [-1.58351453]
 [-0.91424254]]

Rotation Matrix
[[  1.53020929e-11   1.00000000e+00  -5.93469718e-13]
 [  5.00000000e-01  -7.13717974e-12   8.66025404e-01]
 [  8.66025404e-01  -1.35487732e-11  -5.00000000e-01]]

[0,0,1]去了哪里？

免责声明：我借用this article中的数字和代码。