Pose-Free Structure From Motion Using Depth From Motion Constraints

Ji Zhang, Mireille Boutin, Daniel G. Aliaga
2011 IEEE Transactions on Image Processing  
Structure from motion (SFM) is the problem of recovering the geometry of a scene from a stream of images taken from unknown viewpoints. One popular approach to estimate the geometry of a scene is to track scene features on several images and reconstruct their position in 3D. During this process, the unknown camera pose must also be recovered. Unfortunately recovering the pose can be an ill-conditioned problem which, in turn, can make the SFM problem difficult to solve accurately. We propose an
more » ... lternative formulation of the SFM problem with fixed internal camera parameters known a priori. In this formulation, obtained by algebraic variable elimination, the external camera pose parameters do not appear. As a result, the problem is better conditioned in addition to involving much fewer variables. Variable elimination is done in three steps. First, we take the standard SFM equations in projective coordinates and eliminate the camera orientations from the equations. We then further eliminate the camera center positions. Finally, we also eliminate all 3D point positions coordinates, except for their depths with respect to the camera center, thus obtaining a set of simple polynomial equations of degree two and three. We show that, when there are merely a few points and pictures, these "depthonly equations" can be solved in a global fashion using homotopy methods. We also show that, in general, these same equations can be used to formulate a pose-free cost function to refine SFM solutions in a way that is more accurate than by minimizing the total reprojection error, as done when using the bundle adjustment method. The generalization of our approach to the case of varying internal camera parameters is briefly discussed.
doi:10.1109/tip.2011.2147322 pmid:21521669 fatcat:m7tpjoxlhreqdobqrz4sylhi2y