International Journal of Computer Vision

, Volume 45, Issue 3, pp 223–243

Catadioptric Projective Geometry

  • Christopher Geyer
  • Kostas Daniilidis

DOI: 10.1023/A:1013610201135

Cite this article as:
Geyer, C. & Daniilidis, K. International Journal of Computer Vision (2001) 45: 223. doi:10.1023/A:1013610201135


Catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera. Central catadioptric sensors are the class of these devices having a single effective viewpoint. In this paper, we propose a unifying model for the projective geometry induced by these devices and we study its properties as well as its practical implications. We show that a central catadioptric projection is equivalent to a two-step mapping via the sphere. The second step is equivalent to a stereographic projection in the case of parabolic mirrors. Conventional lens-based perspective cameras are also central catadioptric devices with a virtual planar mirror and are, thus, covered by the unifying model. We prove that for each catadioptric projection there exists a dual catadioptric projection based on the duality between points and line images (conics). It turns out that planar and parabolic mirrors build a dual catadioptric projection pair. As a practical example we describe a procedure to estimate focal length and image center from a single view of lines in arbitrary position for a parabolic catadioptric system.

omnidirectional visioncatadioptric systemsconic sectionsdualitystereographic projectioncalibration

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Christopher Geyer
    • 1
  • Kostas Daniilidis
    • 1
  1. 1.GRASP LaboratoryUniversity of PennsylvaniaPhiladelphiaUSA