Mirrors in motion: epipolar geometry and motion estimation

In this paper we consider the images taken from pairs of parabolic catadioptric cameras separated by discrete motions. Despite the nonlinearity of the projection model, the epipolar geometry arising from such a system, like the perspective case, can be encoded in a bilinear form, the catadioptric fundamental matrix. We show that all such matrices have equal Lorentzian singular values, and they define a nine-dimensional manifold in the space of 4 × 4 matrices. Furthermore, this manifold can be identified with a quotient of two Lie groups. We present a method to estimate a matrix in this space, so as to obtain an estimate of the motion. We show that the estimation procedures are robust to modest deviations from the ideal assumptions.