Abstract

We present a technique for modeling non-central catadioptric cameras consisting of a perspective camera and a rotationally symmetric conic reflector. While previous approaches use a central approximation and/or iterative methods for forward pro jection, we present an analyt- ical solution. This allows computation of the optical path from a given 3D point to the given viewpoint by solving a 6th degree forward pro- jection equation for general conic mirrors. For a spherical mirror, the forward pro jection reduces to a 4th degree equation, resulting in a closed form solution. We also derive the forward pro jection equation for imag- ing through a refractive sphere (non-central dioptric camera) and show that it is a 10th degree equation. While central catadioptric cameras lead to conic epipolar curves, we show the existence of a quartic epipolar curve for catadioptric systems using a spherical mirror. The analyti- cal forward pro jection leads to accurate and fast 3D reconstruction via bundle adjustment. Simulations and real results on single image sparse 3D reconstruction are presented. We demonstrate ? 100 times speed up using the analytical solution over iterative forward pro jection for 3D reconstruction using spherical mirrors.