Variable elimination for 3D from 2D Accurately reconstructing the 3D geometry of a scene or object observed on 2D images is a difficult problem: there are many unknowns involved (camera pose, scene structure, depth factors) and solving for all these unknowns simultaneously is computationally intensive and suffers from numerical instability. In this paper, we algebraically decouple some of the unknowns so that they can be solved for independently. Decoupling the pose from the other variables has been previously discussed in the literature. Unfortunately, pose estimation is an illconditioned problem. In this paper, we algebraically eliminate all the camera pose parameters (i.e., position and orientation) from the structurefrommotion equations for an internally calibrated camera. We then also fully eliminate the structure coordinates from the equations. This yields a very simple set of homogeneous polynomial equations of low degree involving only the depths of the observed points. When considering a small number of tracked points and pictures (e.g., five points on two pictures), these equations can be solved using the sparse resultant method.

Linebased Absolute and Relative Camera Pose Estimation in Structured Environments 3D lines in structured environments encode particular regularity like parallelism and orthogonality. We leverage this structural regularity to estimate the absolute and relative camera poses. We decouple the rotation and translation, and propose a novel rotation estimation method. We decompose the absolute and relative rotations and reformulate the problem as computing the rotation from the Manhattan frame to the camera frame. To compute this rotation, we propose an accurate and efficient twostep method. We first estimate its two degrees of freedom (DOF) by two image lines, and then estimate its third DOF by another image line. For these lines, we assume their associated 3D lines are mutually orthogonal, or two 3D lines are parallel to each other and orthogonal to the third. Thanks to our twostep DOF estimation, our absolute and relative pose estimation methods are accurate and efficient. Moreover, our relative pose estimation method relies on weaker assumptions or less correspondences than existing approaches. We also propose a novel strategy to reject outliers and identify dominant directions of the scene. We integrate it into our pose estimation methods, and show that it is more robust than RANSAC. Experiments on synthetic and realworld datasets demonstrated that our methods outperform stateoftheart approaches.