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 constrain 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 8th 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 sweep. 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 off-axis perspective camera are demonstrated.

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 can be considered an extension of the Alhazen problem.

In this paper, we solve this fundamental problem for quadric mirrors where the camera can be placed anywhere, including off-axis. Our ECCV 2010 paper provides the analytical model only for the axial case.

The analytical equations allow us to use an exact non-central model without approximations such as (a) a central approximation and (b) General Linear Cameras (GLC) approximations when using these non-perspective cameras.

Most catadioptric imaging techniques minimize the 3D error instead of the reprojection error for 3D reconstruction, which is sub-optimal. Using our model, one can minimize the reprojection error. Thus, we can use the familiar bundle adjustment pipeline for 3D reconstruction for non-central catadioptric cameras, simply by replacing the perspective projection equation with our analytical model.

1. Alhazen problem, Wikipedia.
2. Alhazen’s billiard problem, Wolfram MathWorld.
3. Marcus Baker, “Alhazen problem,” American Journal of Mathematics, Vol. 4, No. 1, 1881.
4. Heinrich Dörrie, 100 Great Problems of Elementary Mathematics.
5. A. Agrawal, Y. Taguchi, and S. Ramalingam, Analytical Forward Projection for Axial Non-Central Dioptric and Catadioptric Cameras, ECCV 2010.