Computer Vision Talks at Columbia University

THE GEOMETRY OF CENTRAL CATADIOTRIC VIEWS

KOSTAS DANIILIDIS

University of Pennsylvania

Wednesday, November/28, 11 AM

CS Conference Room, 4th Floor MUDD 

Host: Prof. Shree Nayar 

Abstract 

To enable surround perception, new omnidirectional systems were designed which gave a new impetus for rethinking the way images are acquired and analyzed. Based on insights gained from such designs, we formulate a novel unifying theory of imaging. We prove that all single viewpoint mirror-lens devices are equivalent to projective mappings from the sphere to the plane. These mappings are paired with a duality principle which relates points to line projections. The commonly used parabolic mirror projection is shown to be equivalent to the stereographic projection, providing therefore the invariants of a conformal mapping. It turns out that conventional cameras, which are only a special case in our theory, provide the barest minimum of information about the environment.

We present a new algorithm for structure from motion from point correspondences in catadioptric images. We study images obtained from catadioptri c systems with a single effective viewpoint via a parabolic mirror. We assume that the unknown intrinsic parameters are three: the combined focal length of the mirror and lens and the intersection of the optical axis with the image. We introduce a new representation for circles which are catadioptric images of lines. Such circles lie in a plane in a three dimensional circle space where all real circles lie outside of a paraboloid. We show that the pole of this plane with respect to this paraboloid is the point representation of the image of the absolute conic.