A Unifying Theory for Central Panoramic Systems and Practical Applications

Omnidirectional vision systems can provide panoramic alertness in surveillance, improve navigational capabilities, and produce panoramic images for multimedia. Catadioptric realizations of omnidirectional vision combine reflective surfaces and lenses. A particular class of them, the central panoramic systems, preserve the uniqueness of the projection viewpoint. In fact, every central projection system including the well known perspective projection on a plane falls into this category. In this paper, we provide a unifying theory for all central catadioptric systems. We show that all of them are isomorphic to projective mappings from the sphere to a plane with a projection center on the perpendicular to the plane. Subcases are the stereographic projection equivalent to parabolic projection and the central planar projection equivalent to every conventional camera. We define a duality among projections of points and lines as well as among different mappings. This unification is novel and has a a significant impact on the 3D interpretation of images. We present new invariances inherent in parabolic projections and a unifying calibration scheme from one view. We describe the implied advantages of catadioptric systems and explain why images arising in central catadioptric systems contain more information than images from conventional cameras. One example is that intrinsic calibration from a single view is possible for parabolic catadioptric systems given only three lines. Another example is metric rectification using only affine information about the scene.