Analytical Forward Projection for Axial

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 ap-proximation and/or iterative methods for forward projection, we present an analytical solution. This allows computation of the optical path from a given 3D point to the given viewpoint by solv-ing a 6th degree forward projection equation for general conic mirrors. For a spherical mirror, the forward projection reduces to a 4th degree equation, resulting in a closed form solution. We also derive the forward projection equation for imaging 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 analytical forward projection 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 projection for 3D reconstruction using spherical mirrors