How can I derive relative translation and rotation from two extrinsic camera matrices?

Question

I have two sets of data associated with a video sequence. One contains relative rotation and translation data generated using one algorithm. The other is comprised of ground-truth extrinsic matrices associated with each frame.

I would like to compare the data-sets to determine the disparity between them. My question is, how can I derive the relative translation and rotation from the two extrinsic camera matrices?

Answer 1

If you have camera1 pose P1 = [R1|T1] and camera2 pose P2 = [R2|T2] then P1to2 = P2 * P1^-1.

Intuitively, imagine a simple case in which both cameras have translation zero, camera1 has rotation on X axis +30 degrees and camera2 rotation on X axis of +60 degrees.

P1 = [R1|0] P2 = [R2|0]

So they both differ of a +30 degrees rotation on the X axis:

P1to2 = P2 * P1^-1 = [R2|0] * [R1|0]^-1 = [R2|0] * [R1^1|0]

R2 * R1^1 = 60 - 30 on X = rotation of +30 degrees on X

Answer 2

The extrinsic matrix is comprised of the rotation matrix 3x3 and the translation vector 3x1.

So, M 3x4, is simply a concatenation of the two [R t]. Unless I misunderstand your question, getting the rotation and translation is trivial.