# Equivalent Constraints for Two-View Geometry: Pose Solution/Pure Rotation Identification and 3D Reconstruction

arXiv: https://arxiv.org/abs/1810.05863 · HTML: https://mlsanity.com/paper/1810.05863/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2018.

## Built on by (most cited first, 9 of 9)

- [Learning to Segment Rigid Motions from Two Frames](https://mlsanity.com/paper/2101.03694/) (2021)
- [A Pose-only Solution to Visual Reconstruction and Navigation](https://mlsanity.com/paper/2103.01530/) (2021)
- [Six-Point Method for Multi-Camera Systems with Reduced Solution Space](https://mlsanity.com/paper/2402.18066/) (2024)
- [Linear Relative Pose Estimation Founded on Pose-only Imaging Geometry](https://mlsanity.com/paper/2401.13357/) (2024)
- [Consistent and Optimal Solution to Camera Motion Estimation](https://mlsanity.com/paper/2403.01174/) (2024)
- [PO-MSCKF: An Efficient Visual-Inertial Odometry by Reconstructing the Multi-State Constrained Kalman Filter with the Pose-only Theory](https://mlsanity.com/paper/2407.01888/) (2024)
- [A Pose-only Geometric Constraint for Multi-Camera Pose Adjustment](https://mlsanity.com/paper/2604.23704/) (2026)
- [A Birotation Solution for Relative Pose Problems](https://mlsanity.com/paper/2505.02025/) (2025)
- [Towards Rotation-only Imaging Geometry: Rotation Estimation](https://mlsanity.com/paper/2511.12415/) (2025)

## Abstract (arXiv)

Two-view relative pose estimation and structure reconstruction is a classical problem in computer vision. The typical methods usually employ the singular value decomposition of the essential matrix to get multiple solutions of the relative pose, from which the right solution is picked out by reconstructing the three-dimension (3D) feature points and imposing the constraint of positive depth. This paper revisits the two-view geometry problem and discovers that the two-view imaging geometry is equivalently governed by a Pair of new Pose-Only (PPO) constraints: the same-side constraint and the intersection constraint. From the perspective of solving equation, the complete pose solutions of the essential matrix are explicitly derived and we rigorously prove that the orientation part of the pose can still be recovered in the case of pure rotation. The PPO constraints are simplified and formulated in the form of inequalities to directly identify the right pose solution with no need of 3D reconstruction and the 3D reconstruction can be analytically achieved from the identified right pose. Furthermore, the intersection inequality also enables a robust criterion for pure rotation identification. Experiment results validate the correctness of analyses and the robustness of the derived pose solution/pure rotation identification and analytical 3D reconstruction.