Advertisement

International Journal of Computer Vision

, Volume 127, Issue 2, pp 163–180 | Cite as

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

  • Qi Cai
  • Yuanxin WuEmail author
  • Lilian Zhang
  • Peike Zhang
Article
First Online:

Abstract

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.

Keywords

Relative pose Coplanar relationship Same-side constraint Intersection constraint Pure rotation 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

Thanks to anonymous reviewers for their constructive comments and Dr. Danping Zou for group talks. The work is funded by National Natural Science Foundation of China (61422311, 61673263, 61503403) and Hunan Provincial Natural Science Foundation of China (2015JJ1021).

References

  1. Bazin, J. C., Demonceaux, C., Vasseur, P., & Kweon, I. S. (2010). Motion estimation by decoupling rotation and translation in catadioptric vision. Computer Vision and Image Understanding, 114, 254–273.CrossRefGoogle Scholar
  2. Beardsley, P. A., & Zisserman, A. (1995). Affine calibration of mobile vehicles. In Europe-China Workshop on Geometrical Modelling and Invariants for Computer Vision, pp. 214–221.Google Scholar
  3. Faugeras, O. D., & Maybank, S. (1990). Motion from point matches: Multiplicity of solutions. International Journal of Computer Vision, 4, 225–246.CrossRefzbMATHGoogle Scholar
  4. Ferraz, L., Binefa, X., & Moreno-Noguer, F. (2014). Very fast solution to the PnP problem with algebraic outlier rejection. In Computer Vision and Pattern Recognition, pp. 501–508.Google Scholar
  5. Garro, V., Crosilla, F., & Fusiello, A. (2012). Solving the PnP problem with anisotropic orthogonal procrustes analysis. In Second International Conference on 3d Imaging, Modeling, Processing, Visualization and Transmission, pp. 262–269.Google Scholar
  6. Golub, G. H., & Van Loan, C. F. (1996) Matrix computations (3rd ed.). Johns Hopkins University Press.Google Scholar
  7. Hartley, R. I. (1992). Estimation of relative camera positions for uncalibrated cameras. Presented at the ECCV ‘92 Proceedings of the Second European Conference on Computer Vision.Google Scholar
  8. Hartley, R. I. (1995). An investigation of the essential matrix. Report Ge.Google Scholar
  9. Hartley, R., Gupta, R., & Chang, T. (1992). Stereo from uncalibrated cameras. In Proceedings CVPR ‘92., 1992 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 761–764.Google Scholar
  10. Hartley, R., & Zisserman, A. (2003). Multiple view geometry in computer vision. Cambridge: Cambridge University Press.zbMATHGoogle Scholar
  11. Horn, B. K. P. (1990). Relative orientation. International Journal of Computer Vision, 4, 59–78.CrossRefGoogle Scholar
  12. Huang, T. S., & Faugeras, O. D. (1989). Some properties of the E matrix in two-view motion estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 1310–1312.CrossRefGoogle Scholar
  13. Huang, T. S. & Shim, Y. S. (1988). Linear algorithm for motion estimation: How to handle degenerate cases. In International Conference on Pattern Recognition, pp. 439–447.Google Scholar
  14. Kneip, L., Siegwart, R., & Pollefeys, M. (2012). Finding the exact rotation between two images independently of the translation. In European Conference on Computer Vision, pp. 696–709.Google Scholar
  15. Ling, L., Cheng, E., & Burnett, I. S. (2011). Eight solutions of the essential matrix for continuous camera motion tracking in video augmented reality. In Proceedings of the 2011 IEEE International Conference on Multimedia and Expo (ICME 2011), Barcelona, Spain, pp. 1–6.Google Scholar
  16. Longuet-Higgins, H. C. (1981). A computer algorithm for reconstructing a scene from two projections. Nature, 293, 133–135.CrossRefGoogle Scholar
  17. Ma, Y., Soatto, S., Kosecka, J., & Sastry, S. S. (2004). An invitation to 3-D vision. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  18. Maybank, S. (1993). Theory of reconstruction from image motion (Vol. 28). Berlin: Springer.zbMATHGoogle Scholar
  19. Nister, D. (2004). An efficient solution to the five-point relative pose problem. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26, 756–777.CrossRefGoogle Scholar
  20. Stewénius, H., Engels, C., & Nistér, D. (2006). Recent developments on direct relative orientation. ISPRS Journal of Photogrammetry & Remote Sensing, 60, 284–294.CrossRefGoogle Scholar
  21. Vieville, T., & Lingrand, D. (1996). Using singular displacements for uncalibrated monocular visual systems. In European Conference on Computer Vision, pp. 207–216.Google Scholar
  22. Wang, W., & Tsui, H. T. (2000). An SVD decomposition of essential matrix with eight solutions for the relative positions of two perspective cameras. In International Conference on Pattern Recognition, Vol. 1, pp. 362–365.Google Scholar
  23. Zhang, Z. (1998). Determining the epipolar geometry and its uncertainty: A review. International Journal of Computer Vision, 27, 161–195.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Navigation and Location-Based Services, School of Electronic Information and Electrical EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.School of Aeronautics and AstronauticsCentral South UniversityChangshaChina
  3. 3.College of Intelligent Science and TechnologyNational University of Defense TechnologyChangshaChina

Personalised recommendations