Degeneracy from quadric cones for projective reconstruction from two views

Fundamental matrix encapsulates all the geometric information in two views, and it plays an important role in many applications of three-dimensional computer vision. However, some configurations have an inherent ambiguity, namely, degeneracy, and no matter how many matched image points are used, the fundamental matrix could not be determined uniquely. It is well-known that degeneracies occur when all the space points and both the camera centers belong to a ruled quadric. But it is not so well-known how great the degenerate degrees in different configurations are. In this paper, we discuss the degeneracies caused by a quadric cone and give the corresponding degenerate degrees. We parameterize all the points on a quadric cone by a twisted cubic lying on the cone, and obtain a parametric coefficient matrix of the equations for estimating the fundamental matrix. By analyzing the coefficient matrix, we get the rank of it, i.e. the degenerate degree of the given configuration. It gives a more intuitive degeneracy and the degenerate degrees of the configurations, and reveals the full picture of the degeneracies on a cone.