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Your approach with pair wise calibration looks ok, but you loose much of the possible accuracy if you just use two cameras to get the position of a point even if you see it in 3 or even four cameras. What you probably need is a better optimizer like g2o or ceres. With these solvers, you can optimize the position of all cameras globally. Otherwise, you will have contradictions in your system. With your pairwise calibrations, you should be able to compute the Trafo T14 (from camera 1 to camera 4) as concatenation of T12, T23 and T34. But this product will not be the same as if you just calibrate the offset between Camera 1 and Camera 4.

So you can approach this problem with these methods but this is not the way how motion capture systems work and you will need a better approach if you want to get accurate results.