OpenCV Decompose projection matrix

Question

I got confused with the outputs of opencv decomposeProjectionMatrix function.

I have the projection matrix and the camera matrix "K" and I want to get the translation vector "t"and the Rotation matrix "R" from the projection matrix

As I know the projection matrix of dimension 3*4 = K[R|t] in which "t" is a 3*1 vector

cv2.decomposeProjectionMatrix returns R with dimension 3*3 which is correct but the transVect returned is of dimension 4*1 not 3*1

My question is how to get back the projection matrix from the function outputs?

documentation: https://docs.opencv.org/2.4/modules/calib3d/doc/camera_calibration_and_3d_reconstruction.html

Answer 1

You have to normalise it by the last number: t = t[:3] / t[3]

But for some reason, you need to multiply it by -1. Here is my example:

k = np.array([[631,   0, 384],
              [  0, 631, 288],
              [  0,   0,   1]])
r = np.array([[-0.30164902,  0.68282439, -0.66540117],
              [-0.63417301,  0.37743435,  0.67480953],
              [ 0.71192167,  0.6255351 ,  0.3191761 ]])
t = np.array([ 3.75082481, -1.18089565,  1.06138781])

C = np.eye(4)
C[:3, :3] = k @ r
C[:3, 3] = k @ r @ t

out = cv2.decomposeProjectionMatrix(C[:3, :])

and out gives the following:

out[0]:
array([[631.,  -0., 384.],
       [  0., 631., 288.],
       [  0.,   0.,   1.]])
out[1]:
array([[-0.30164902,  0.68282439, -0.66540117],
       [-0.63417301,  0.37743435,  0.67480953],
       [ 0.71192167,  0.6255351 ,  0.3191761 ]])
out[2]:
array([[-0.89432836],
       [ 0.28156699],
       [-0.25307213],
       [ 0.23843512]])

If you apply the normalisation, you get:

array([[-3.75082481],
       [ 1.18089565],
       [-1.06138781]])

what is actually a -t. However np.linalg.inv(C[:3, :3]) @ C[:3, 3] gives the correct result

array([ 3.75082481, -1.18089565,  1.06138781])