Recovering unknown focal lengths in self-calibration: an essentially linear algorithm and degenerate configurations

Abstract

If sufficiently many pairs of corresponding points in a stereo image pair are available to construct the associated fundamental matrix, then it has been shown that 5 relative orientation parameters and 2 focal lengths can be recovered from this fundamental matrix. This paper presents a new and essentially linear algorithm for recovering focal lengths. Moreover the derivation of the algorithm also provides a complete characterisation of all degenerate configurations in which focal lengths cannot be uniquely recovered. There are two classes of degenerate configurations: either one of the optical axes of the cameras lies in the plane spanned by the baseline and the other optical axis; or one optical axis lies in the plane spanned by the baseline and the vector that is orthogonal to both the baseline and the other axis. The result that the first class of configurations (i.e. ones in which the optical axes are coplanar) is degenerate is of some practical importance since it shows that self-calibration of unknown focal lengths is not possible in certain stereo heads, a configuration widely used for binocular vision systems in robotics.