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:

2 Answers

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])

C = K[R|t] = K(R|Rt) = (KR|KRt) = (M | Mt)The fourth column of C is c4.

Mt = c4.
t = M^(-1)c4

ref

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