Line-Images in Cone Mirror Catadioptric Systems

The projection surface of a 3D line in a non-central camera is a ruled surface, containing the complete information of the 3D line. The resulting line-image is a curve which contains the 4 degrees of freedom of the 3D line. In this paper we investigate the properties of the line-image in conical catadioptric systems. This curve is a particular quartic that can be described by only six homogeneous parameters. We present the relation between the line-image description and the geometry of the mirror. This result reveals the coupling between the depth of the line and the distance from the camera to the mirror. If this distance is unknown the 3D information of a projected line can be recovered up to scale. Knowing this distance allows obtaining the 3D metric reconstruction. The proposed parametrization also allows to simultaneously reconstruct the 3D line and computing the aperture angle of the mirror from five projected points on the line-image. We analytically solve the metric distance from a point to a line-image and we evaluate the proposal with real images.