On the Solvability of Viewing Graphs

  • Matthew TragerEmail author
  • Brian Osserman
  • Jean Ponce
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11220)


A set of fundamental matrices relating pairs of cameras in some configuration can be represented as edges of a “viewing graph”. Whether or not these fundamental matrices are generically sufficient to recover the global camera configuration depends on the structure of this graph. We study characterizations of “solvable” viewing graphs, and present several new results that can be applied to determine which pairs of views may be used to recover all camera parameters. We also discuss strategies for verifying the solvability of a graph computationally.


Viewing graph Fundamental matrix 3D reconstruction 

Supplementary material

474218_1_En_20_MOESM1_ESM.pdf (239 kb)
Supplementary material 1 (pdf 239 KB)


  1. 1.
    Thompson, M., Eller, R., Radlinski, W., Speert, J. (eds.): Manual of Photogrammetry, 3rd edn. American Society of Photogrammetry, California (1966)Google Scholar
  2. 2.
    Longuet-Higgins, H.C.: A computer algorithm for reconstructing a scene from two projections. Nature 293(5828), 133 (1981)CrossRefGoogle Scholar
  3. 3.
    Luong, Q.T., Faugeras, O.: The fundamental matrix: theory, algorithms, and stability analysis 17(1), 43–76 (1996)Google Scholar
  4. 4.
    Shashua, A.: Algebraic functions for recognition 17(8), 779–789 (1995)Google Scholar
  5. 5.
    Hartley, R.I.: Lines and points in three views and the trifocal tensor. Int. J. Comput. Vis. 22(2), 125–140 (1997)CrossRefGoogle Scholar
  6. 6.
    Hartley, R.: Computation of the quadrifocal tensor, pp. 20–35 (1998)CrossRefGoogle Scholar
  7. 7.
    Faugeras, O., Mourrain, B.: On the geometry and algebra of the point and line correspondences between n images. In: Proceedings of the Fifth International Conference on Computer Vision, pp. 951–956. IEEE (1995)Google Scholar
  8. 8.
    Triggs, B.: Matching constraints and the joint image. In: Proceedings of the Fifth International Conference on Computer Vision, pp. 338–343. IEEE (1995)Google Scholar
  9. 9.
    Heyden, A., Åström, K.: Algebraic properties of multilinear constraints. Math. Methods Appl. Sci. 20(13), 1135–1162 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Levi, N., Werman, M.: The viewing graph. In: Proceedings of the 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. I-I. IEEE (2003)Google Scholar
  11. 11.
    Rudi, A., Pizzoli, M., Pirri, F.: Linear solvability in the viewing graph. In: Kimmel, R., Klette, R., Sugimoto, A. (eds.) ACCV 2010. LNCS, vol. 6494, pp. 369–381. Springer, Heidelberg (2011). Scholar
  12. 12.
    Snavely, N., Seitz, S., Szeliski, R.: Photo tourism: exploring image collections in 3D. In: SIGGRAPH (2006)Google Scholar
  13. 13.
    Trager, M., Hebert, M., Ponce, J.: The joint image handbook. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 909–917 (2015)Google Scholar
  14. 14.
    Ozyesil, O., Singer, A.: Robust camera location estimation by convex programming. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2674–2683 (2015)Google Scholar
  15. 15.
    Sweeney, C., Sattler, T., Hollerer, T., Turk, M., Pollefeys, M.: Optimizing the viewing graph for structure-from-motion. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 801–809 (2015)Google Scholar
  16. 16.
    Sinha, S.N., Pollefeys, M.: Camera network calibration and synchronization from Silhouettes in archived video. Int. J. Comput. Vis. 87(3), 266–283 (2010)CrossRefGoogle Scholar
  17. 17.
    Hartley, R., Zisserman, A.: Multiple view geometry in computer vision. Cambridge University Press, Cambridge (2003)Google Scholar
  18. 18.
    Heyden, A.: Tensorial properties of multiple view constraints. Math. Methods Appl. Sci. 23(2), 169–202 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Aholt, C., Sturmfels, B., Thomas, R.: A Hilbert scheme in computer vision. Can. J. Math. 65(5), 961–988 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mumford, D.: Abelian Varieties. Studies in Mathematics. Hindustan Book Agency, Gurgaon (2008)zbMATHGoogle Scholar
  21. 21.
    Developers, T.S.: SageMath, the sage mathematics software system (Version 8.0.0) (2017).

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.InriaParisFrance
  2. 2.École Normale SupérieureCNRS, PSL Research UniversityParisFrance
  3. 3.UC DavisDavisUSA

Personalised recommendations