find Camera Center with opencv

Question

I'm trying to get the camera center from a calibrated camera. I have 4 measured 3D objectPoints and its images and trying to get the center (translation) from the projective matrix with no acceptable results. Any advise regarding the accuracy I should expect with opencv? Should I increase the number of points?

These are the results I got:

TrueCenter in mm for XYZ
 [[4680.]
 [5180.]
 [1621.]] 
Center
 [[-2508.791]
 [ 6015.98 ]
 [-1096.674]]

import numpy as np
import cv2
from scipy.linalg import inv

TrueCameraCenter = np.array([4680., 5180, 1621]).reshape(-1,1)

objectPoints = np.array(
        [[   0., 5783., 1970.],
           [   0., 5750., 1261.],
           [   0., 6412., 1968.],
           [1017., 9809., 1547.]], dtype=np.float32)

imagePoints=np.array(
        [[ 833.75, 1097.25],
           [ 798.  , 1592.25],
           [1323.  , 1133.5 ],
           [3425.5 , 1495.5 ]], dtype=np.float32)

cameraMatrix= np.array(
        [[3115.104,   -7.3  , 2027.605],
           [   0.   , 3077.283, 1504.034],
           [   0.   ,    0.   ,    1.   ]])

retval, rvec, tvec = cv2.solvePnP(objectPoints, imagePoints,cameraMatrix,None, None, None, False, cv2.SOLVEPNP_ITERATIVE)
R,jac= cv2.Rodrigues(rvec)
imagePoints2,jac= cv2.projectPoints(objectPoints, rvec, tvec, cameraMatrix,None)
print('TrueCenter in mm for XYZ\n', TrueCameraCenter, '\nCenter\n', -inv(R).dot(tvec))

Answer 1

I've found this interesting presentation regarding the Location Determination Problem by Bill Wolfe. Perspective View Of 3 Points

So, using 4 non-coplanar points (non 3 colinear) the solution improved.

import numpy as np
import cv2
from scipy.linalg import inv,norm


TrueCameraCenter = np.array([4680., 5180, 1621])
objectPoints = np.array(
        [[   0., 5783., 1970.],
       [   0., 5750., 1261.],
       [   0., 6412., 1968.],
       [   0., 6449., 1288.]])

imagePoints=np.array(
        [[ 497.5 ,  674.75],
       [ 523.75, 1272.5 ],
       [1087.75,  696.75],
       [1120.  , 1212.5 ]])

cameraMatrix= np.array(
        [[3189.096,    0.   , 2064.431],
         [   0.   , 3177.615, 1482.859],
         [   0.   ,    0.   ,    1.   ]])
dist_coefs=np.array([[ 0.232, -1.215, -0.002,  0.011,  1.268]])

retval, rvec, tvec = cv2.solvePnP(objectPoints, imagePoints,cameraMatrix,dist_coefs, 
                                  None, None, False, cv2.SOLVEPNP_ITERATIVE)
R,_= cv2.Rodrigues(rvec)
C=-inv(R).dot(tvec).flatten()
print('TrueCenter in mm for XYZ\n', TrueCameraCenter, '\nCenter\n',C.astype(int) )
print('Distance:', int(norm(TrueCameraCenter-C)))