On Screw-Transform Manifolds

Abstract

This paper describes the mathematical theory of screw-transform manifolds and their use in camera self calibration. When a camera with fixed internal parameters views a scene from two different locations, the physical transformation that moves the camera from the first location to the second location is equivalent to a screw transformation. The fundamental matrix between the two views has a representation in terms of this screw transformation. The same fundamental matrix can be generated by different cameras undergoing different screw transformations. The set of all cameras that could generate a particular fundamental matrix in this way is the screw-transform manifold for the fundamental matrix. The screw-transform manifold can be generated directly from the fundamental matrix by varying the parameters of the underlying screw transformation. When several fundamental matrices are generated using the same camera, each screw-transform manifold arising from these fundamental matrices must contain the camera. Hence by finding the mutual intersection point of all the manifolds, the original camera can be recovered; this forms a technique for self calibration. We describe two types of screw-transform manifolds: Kruppu-constraint munifolds and modulus-constraint manifolds. The first type can be generated directly from fundamental matrices, but are three-dimensional manifolds embedded in a five-dimensional space making them more difficult to use. The latter type are simpler two-dimensional manifolds embedded in a three-dimensional space, but require an initial projective reconstruction of the cameras, which is not always possible or desirable to attain, to be used in self calibration. We also describe three algorithms for finding the mutual intersection point of a set of manifolds and provide extensive experimental results for the performance of these algorithms.