Reconstruction from calibrated cameras - A new proof of the Kruppa-Demazure theorem

This paper deals with the problem of reconstructing the locations of five p oints in space from two different images taken by calibrated cameras. Equiv alently, the problem can be formulated as finding the possible relative loc ations and orientations, in three-dimensional Euclidean space, of two label ed stars, of five lines each, such that corresponding lines intersect. The problem was first treated by Kruppa more than 50 years ago. He found th at there were at most eleven solutions. Later Demazure and also Maybank sho wed that there were actually ten solutions. In this article will be given a nother proof of this theorem based on a different parameterisation of the p roblem neither using the epipoles nor the essential matrix. This is within the same point of view as direct structure recovery in the uncalibrated cas e. Instead of the essential matrix we use the kinetic depth vectors, which has shown to be were useful in the uncalibrated case. We will also present an algorithm that in most cases calculates the ten different solutions, alt hough some may be complex and some may not be physically realisable. The al gorithm is based on a homotopy method and tracks solutions on the so called Chasles' manifold. One of the major contributions of this paper is to brid ge the gap between reconstruction methods for calibrated and uncalibrated c ameras. Furthermore, we show that the twisted pair solutions are natural in this context because the kinetic depths are the same for both components.