A unifying geometric representation for central projection systems

Abstract

In this paper, we study projection systems with a single effective viewpoint, including combinations of mirrors and lenses (catadioptric) as well as just lenses with or without radial distortion (dioptric systems). First, we extend a well-known unifying model for central catadioptric systems to incorporate a class of dioptric systems with radial distortion. Second, we provide a new representation for the image plane of central systems. This representation is the lifting through a Veronese map of the original image plane to the 5D projective space. We study how a collineation in the original image plane can be transferred to a collineation in the lifted space, and we prove that in the case of central parabolic systems and cameras with lens distortion the locus of the lifted points representing projections of world lines is a plane. The similarities between paracatadioptric systems and lens with radial distortion are emphasized by extending to the latter algorithms initially established for the former.