Beyond Alhazen´s problem: Analytical projection model for non-central catadioptric cameras with quadric mirrors

Catadioptric cameras are widely used to increase the field of view using mirrors. Central catadioptric systems having an effective single viewpoint are easy to model and use, but severely constraint the camera positioning with respect to the mirror. On the other hand, non-central catadioptric systems allow greater flexibility in camera placement, but are often approximated using central or linear models due to the lack of an exact model. We bridge this gap and describe an exact projection model for non-central catadioptric systems. We derive an analytical `forward projection´ equation for the projection of a 3D point reflected by a quadric mirror on the imaging plane of a perspective camera, with no restrictions on the camera placement, and show that it is an 8^th degree equation in a single unknown. While previous non-central catadioptric cameras primarily use an axial configuration where the camera is placed on the axis of a rotationally symmetric mirror, we allow off-axis (any) camera placement. Using this analytical model, a non-central catadioptric camera can be used for sparse as well as dense 3D reconstruction similar to perspective cameras, using well-known algorithms such as bundle adjustment and plane sweeping. Our paper is the first to show such results for off-axis placement of camera with multiple quadric mirrors. Simulation and real results using parabolic mirrors and an off-axis perspective camera are demonstrated.