Extraction of focal lengths from the fundamental matrix

The 8-point algorithm is a well known method for solving the relative orientation and scene reconstruction problem for calibrated cameras. In that algorithm, a matrix, called the fundamental matrix of the two calibrated cameras is computed using 8 or more point correspondences between the pair of images. The relative orientation of the two cameras may be computed from the fundamental matrix. In the present paper it is shown that the assumption of completely known calibration is unnecessary. In fact, the fundamental matrix may be used to solve the relative orientation problem for a pair of pinhole cameras with known principal points, but unknown (possibly different)focal lengths. A simple non-iterative algorithm is given for finding the two focal lengths. Once this is done, known techniques may be used to find the relative orientation of the cameras, and to reconstruct the scene.