Analytical Forward Projection for Axial Non-Central Dioptric and Catadioptric Cameras

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 projection, we present an analytical solution. This allows computation of the optical path from a given 3D point to the given viewpoint by solving 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.

Forward projection for a spherical mirror is a classical problem in geometry known as the Alhazen problem or the circular billiard problem. The problem can be traced back to the ancient Greeks and is described by Ptolemy. Alhazen, who is widely regarded as the father of optics, discussed this problem extensively in his Book of Optics around 1000 A.D.

Several mathematicians have formulated algebraic, trigonometric, and analytical solutions to this problem and have shown that there exist four solutions. However, Alhazen only considered spherical and cylindrical mirrors. Computing analytical solutions of forward projection for general conic mirrors and refractive spheres can be considered an extension of the Alhazen problem.

Our paper provides analytical solutions and an exact non-central model for catadioptric systems with rotationally symmetric mirrors in axial configurations. We also provide an analytical solution for looking through a glass sphere or ball lens, as in crystal ball photography.

The analytical equations allow an exact non-central model without central or General Linear Cameras (GLC) approximations. This enables fast 3D reconstruction using bundle adjustment, similar to perspective cameras, with a 100× speedup over iterative forward projection for spherical mirrors.

For the general case, including off-axis camera placement, see our CVPR 2011 paper, Beyond Alhazen’s Problem.

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