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

Amit Agrawal, Yuichi Taguchi and Srikumar Ramalingam

Mitsubishi Electric Research Labs (MERL)

ECCV 2010 (oral presentation)

Given the 3D point and perspective camera location, can we analytically compute the point on the mirror (refractive sphere/ball lens) where reflection (refraction) happens?


Exact modeling of non-central catadioptric cameras with rotationally symmetric mirror in axial configuration. Analytical equations for fast projection of 3D points observed via reflection through a quadric mirror or refraction through a sphere (ball lens).


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.

Paper pdf, Talk Slides

History of Forward Projection for Spherical Mirror
(Alhazen problem)

Forward projection for 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 ancient Greeks and is described by Ptolemy. Alhazen, who is widely regarded as the father of optics, talked extensively about this problem 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. The problem of computing analytical solutions of forward projection for general conic mirrors and refractive sphere can be considered as an extension of Alhazen problem. Our paper provides the analytical solutions and thus provide an exact non-central model when catadioptric systems with rotationally symmetric mirrors are used in axial configuration. We also provide the analytical solution for looking through a glass sphere or a ball lens (crystal ball photography).


The analytical equations allow us to use exact non-central model without using any approximations such as (a) central approximation and (b) General Linear Cameras (GLC) approximation when using these non-perspective cameras. It allows fast 3D reconstruction using bundle adjustment, similar to perspective cameras and offer a speed up of 100X over previous approaches.

Our CVPR 2011 paper describes the analytical projection model for general case (off-axis camera placement).


1. Alhazen problem, Wikipedia
2. Alhazen's billiard problem, Wolfram Math World
3. Marcus Baker, "Alhazen problem", American Journal of Mathematics, Vol 4, No. 1, 1881
4. Heinrich Dorrie, "100 Great problems of elementary Mathematics"

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