Catadioptric Projective Geometry: Theory and Applications

CATADIOPTRIC PROJECTIVE GEOMETRY: THEORY AND APPLICATIONS Christopher Michael Geyer Supervisor: Kostas Daniilidis Catadioptric cameras are devices which use both mirrors (catadioptric elements) and lenses (dioptric elements) to form images. In computer vision and robotics, catadioptric cameras have been used for their wide field of view. Such cameras often have a field of view greater than 180Æ, which is unobtainable from perspective cameras. The cost of this extra-wide field of view is a distortion in the image when compared with perspective images. Unless this distortion is known, it is not possible to apply now classical computer vision algorithms to solve the structure-from-motion (SFM) problem, to use just one example. In this dissertation we study the geometry induced by such devices so that we may solve the SFM problem for catadioptric devices. In the first part of the thesis we prove that all central catadioptric devices, those whose mirrors have conic sections as cross-sections, can be decomposed into a central projection to a sphere centered about the viewpoint, followed by central projection from a point on the sphere’s axis. This result introduces a unifying model for describing all central catadioptric projections. The second principle result singles out those catadioptric devices with parabolic mirrors, which the previous result shows to be equivalent to projection to the sphere followed by stereographic projection. We demonstrate the existence of a bilinear constraint on the inverse stereographic projection of image points, even in the uncalibrated case. The coefficients of the bilinear constraint are shown to encode the intrinsic parameters of the projection, thus allowing for self-calibration from two views if the intrinsic parameters are equal. Using these results we implement a procedure to estimate the structure and motion from two parabolic views. These results are novel and have important applications in computer vision. In particular, a framework for studying vision in central catadioptric sensors is provided. Using this framework we introduce reconstruction and self-calibration algorithms solving the SFM problem for a class of catadioptric sensors, and furthermore we obtain results which are theoretically impossible to achieve in perspective cameras. The algorithms are shown to be robust to noise and deviations from ideal models.