A Projective Framework for Structure and Motion Recovery from Two Views of a Piecewise Planar Scene
Résumé
In this paper, we consider the problem of finding an optimal reconstruction from two views of a piecewise planar scene. We consider the general case of uncalibrated cameras, hence place us in a projective framework. In this case, there is no meaningful metric information about the object space that could be used to define optimization criteria. Taking into account that the images are then the only spaces where an optimization process makes sense, there is a need at each step of the reconstruction process, from the detection of planar structures to motion estimation and actual 3D reconstruction, of a consistent image level representation of geometric 3D structures. In our case, we need to represent camera motion and 3D points that are subject to coplanarity constraints. It is well known that camera motion between two views can be represented on the image level via the epipolar geometry (fundamental matrix). Coplanarity constraints can be expressed via a collection of 2D homographies. Unfortunately, these algebraic entities are over-parameterized in the sense that the 2D homographie- s must in addition obey constraints imposed by the epipolar geometry. We are thus looking for a minimal and consistent representation of motion (epipolar geometry) and structure (points+homographies) that in addition should be easy to use for minimizing reprojection error in a bundle adjustment manner. In this paper, we propose such a representation and use it to devise fast and accurate estimation methods for each step of the reconstructio- n process, including image point matching, plane detection and optimal triangulation of planes and points on planes. We make extensive use of the quasi-linear optimization principle. A great number of experimental results show that the new methods give superior results compared to approaches that do not estimate motion and multi-planar structure simultaneously and consistently, even in cases when the observed scene is not perfectly coplanar.