Linear Solvability in the Viewing Graph

  • Alessandro Rudi
  • Matia Pizzoli
  • Fiora Pirri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6494)


The Viewing Graph [1] represents several views linked by the corresponding fundamental matrices, estimated pairwise. Given a Viewing Graph, the tuples of consistent camera matrices form a family that we call the Solution Set.

This paper provides a theoretical framework that formalizes different properties of the topology, linear solvability and number of solutions of multi-camera systems. We systematically characterize the topology of the Viewing Graph in terms of its solution set by means of the associated algebraic bilinear system. Based on this characterization, we provide conditions about the linearity and the number of solutions and define an inductively constructible set of topologies which admit a unique linear solution. Camera matrices can thus be retrieved efficiently and large viewing graphs can be handled in a recursive fashion. The results apply to problems such as the projective reconstruction from multiple views or the calibration of camera networks.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Levi, N., Werman, M.: The viewing graph. In: CVPR (2003)Google Scholar
  2. 2.
    Luong, Q.T., Faugeras, O.: The fundamental matrix: theory, algorithms, and stability analysis. International Journal of Computer Vision 17, 43–75 (1995)CrossRefGoogle Scholar
  3. 3.
    Torr, P., Zisserman, A.: Robust parameterization and computation of the trifocal tensor. Image and Vision Computing 15, 591–605 (1997)CrossRefGoogle Scholar
  4. 4.
    Hartley, R.I.: Cheirality. Int. J. Comput. Vision 26, 41–61 (1998)CrossRefGoogle Scholar
  5. 5.
    Goldberger, J.: Reconstructing camera projection matrices from multiple pairwise overlapping views. Computer Vision and Image Understanding 97, 283–296 (2005)CrossRefGoogle Scholar
  6. 6.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004) ISBN: 0521540518 CrossRefzbMATHGoogle Scholar
  7. 7.
    Triggs, B., McLauchlan, P.F., Hartley, R.I., Fitzgibbon, A.W.: Bundle adjustment - a modern synthesis. In: ICCV (2000)Google Scholar
  8. 8.
    Tomasi, C., Kanade, T.: Shape and motion from image streams under orthography: a factorization method. Int. J. Comput. Vision 9, 137–154 (1992)CrossRefGoogle Scholar
  9. 9.
    Sturm, P.F., Triggs, B.: A factorization based algorithm for multi-image projective structure and motion. In: Buxton, B.F., Cipolla, R. (eds.) ECCV 1996. LNCS, vol. 1065. Springer, Heidelberg (1996)Google Scholar
  10. 10.
    Fitzgibbon, A.W., Zisserman, A.: Automatic camera recovery for closed or open image sequences. In: Burkhardt, H.-J., Neumann, B. (eds.) ECCV 1998. LNCS, vol. 1406, p. 311. Springer, Heidelberg (1998)Google Scholar
  11. 11.
    Sinha, S.N., Pollefeys, M., McMillan, L.: Camera network calibration from dynamic silhouettes. In: CVPR (2004)Google Scholar
  12. 12.
    Triggs, B.: Linear projective reconstruction from matching tensors. Image and Vision Computing 15, 617–625 (1997)CrossRefGoogle Scholar
  13. 13.
    Cohen, S., Tomasi, C.: Systems of bilinear equations (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alessandro Rudi
    • 1
  • Matia Pizzoli
    • 1
  • Fiora Pirri
    • 1
  1. 1.Department of Computer and System SciencesSapienza University of RomeItaly

Personalised recommendations