Planar motion of a parabolic catadioptric camera

Camera motion is said to be planar if the direction of translation is perpendicular to the axis of rotation. A parabolic catadioptric camera is a camera realizing the orthogonal projection of rays reflected on a parabolic mirror. We consider the planar motion of a parabolic catadioptric camera, especially the motion restricted to a plane perpendicular to the optical axis, a common case in mobile robots working in urban environments. We begin by deriving the catadioptric fundamental matrix for such a motion and the intrinsic degrees of freedom in this matrix, which turn out to be 8. We show that the camera intrinsics and the 3D motion can be recovered from the fundamental matrix. We derive the necessary and sufficient condition for a fundamental matrix to be induced by a planar motion. Based on the additional constraint for a planar motion, we present an algorithm to compute epipolar geometry and recover the camera parameters and motion.