Exploiting Loops in the Graph of Trifocal Tensors for Calibrating a Network of Cameras

  • Jérôme Courchay
  • Arnak Dalalyan
  • Renaud Keriven
  • Peter Sturm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6312)


A technique for calibrating a network of perspective cameras based on their graph of trifocal tensors is presented. After estimating a set of reliable epipolar geometries, a parameterization of the graph of trifocal tensors is proposed in which each trifocal tensor is encoded by a 4-vector. The strength of this parameterization is that the homographies relating two adjacent trifocal tensors, as well as the projection matrices depend linearly on the parameters. A method for estimating these parameters in a global way benefiting from loops in the graph is developed. Experiments carried out on several real datasets demonstrate the efficiency of the proposed approach in distributing errors over the whole set of cameras.


Fundamental Matrix Less Square Estimator Projection Matrice Bundle Adjustment Fundamental Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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  1. 1.
    Hartley, R., Zisserman, A.: Multiple view geometry in computer vision, 2nd edn. Cambridge University Press, Cambridge (2003)Google Scholar
  2. 2.
    Faugeras, O., Luong, Q.T., Papadopoulou, T.: The Geometry of Multiple Images: The Laws That Govern The Formation of Images of A Scene and Some of Their Applications. MIT Press, Cambridge (2001)zbMATHGoogle Scholar
  3. 3.
    Snavely, N., Seitz, S.M., Szeliski, R.: Photo tourism: Exploring photo collections in 3D. ACM Press, New York (2006)Google Scholar
  4. 4.
    Martinec, D., Pajdla, T.: Robust rotation and translation estimation in multiview reconstruction. In: CVPR (2007)Google Scholar
  5. 5.
    Snavely, N., Seitz, S.M., Szeliski, R.: Modeling the world from Internet photo collections. Int. J. Comput. Vision 80, 189–210 (2008)CrossRefGoogle Scholar
  6. 6.
    Furukawa, Y., Ponce, J.: Accurate camera calibration from multi-view stereo and bundle adjustment. International Journal of Computer Vision 84, 257–268 (2009)CrossRefGoogle Scholar
  7. 7.
    Bujnak, M., Kukelova, Z., Pajdla, T.: 3D reconstruction from image collections with a single known focal length. In: ICCV, pp. 351–358 (2009)Google Scholar
  8. 8.
    Agarwal, S., Snavely, N., Simon, I., Seitz, S.M., Szeliski, R.: Building rome in a day. In: ICCV (2009)Google Scholar
  9. 9.
    Havlena, M., Torii, A., Knopp, J., Pajdla, T.: Randomized structure from motion based on atomic 3d models from camera triplets. In: CVPR (2009)Google Scholar
  10. 10.
    Mouragnon, E., Lhuillier, M., Dhome, M., Dekeyser, F., Sayd, P.: Generic and real-time sfm using local bundle adjustment. Image Vision Comput 27, 1178–1193 (2009)CrossRefGoogle Scholar
  11. 11.
    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, pp. 311–326. Springer, Heidelberg (1998)Google Scholar
  12. 12.
    Avidan, S., Shashua, A.: Threading fundamental matrices. IEEE Trans. Pattern Anal. Mach. Intell. 23, 73–77 (2001)CrossRefGoogle Scholar
  13. 13.
    Pollefeys, M., Van Gool, L., Vergauwen, M., Verbiest, F., Cornelis, K., Tops, J., Koch, R.: Visual modeling with a hand-held camera. Int. J. Comput. Vision 59, 207–232 (2004)CrossRefGoogle Scholar
  14. 14.
    Sinha, S.N., Pollefeys, M., McMillan, L.: Camera network calibration from dynamic silhouettes. In: CVPR (2004)Google Scholar
  15. 15.
    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
  16. 16.
    Sturm, P., 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, pp. 709–720. Springer, Heidelberg (1996)Google Scholar
  17. 17.
    Jacobs, D.: Linear fitting with missing data: Applications to structure-from-motion and to characterizing intensity images. In: CVPR, p. 206 (1997)Google Scholar
  18. 18.
    Martinec, D., Pajdla, T.: 3D reconstruction by fitting low-rank matrices with missing data. In: CVPR, pp. I: 198–205 (2005)Google Scholar
  19. 19.
    Triggs, B., McLauchlan, P., Hartley, R., Fitzgibbon, A.: Bundle adjustment - a modern synthesis. In: Triggs, B., Zisserman, A., Szeliski, R. (eds.) ICCV-WS 1999. LNCS, vol. 1883, pp. 298–372. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Klopschitz, M., Zach, C., Irschara, A., Schmalstieg, D.: Generalized detection and merging of loop closures for video sequences. In: 3DPVT (2008)Google Scholar
  21. 21.
    Scaramuzza, D., Fraundorfer, F., Pollefeys, M.: Closing the loop in appearance-guided omnidirectional visual odometry by using vocabulary trees. Robot. Auton. Syst. (to appear, 2010)Google Scholar
  22. 22.
    Cornelis, N., Cornelis, K., Van Gool, L.: Fast compact city modeling for navigation pre-visualization. In: CVPR (2006)Google Scholar
  23. 23.
    Tardif, J., Pavlidis, Y., Daniilidis, K.: Monocular visual odometry in urban environments using an omnidirectional camera. In: IROS, pp. 2531–2538 (2008)Google Scholar
  24. 24.
    Torii, A., Havlena, M., Pajdla, T.: From google street view to 3d city models. In: OMNIVIS (2009)Google Scholar
  25. 25.
    Vu, H., Keriven, R., Labatut, P., Pons, J.P.: Towards high-resolution large-scale multi-view stereo. In: CVPR (2009)Google Scholar
  26. 26.
    Chum, O., Matas, J.: Matching with PROSAC: Progressive sample consensus. In: CVPR, vol. I, pp. 220–226 (2005)Google Scholar
  27. 27.
    Ponce, J., McHenry, K., Papadopoulo, T., Teillaud, M., Triggs, B.: On the absolute quadratic complex and its application to autocalibration. In: CVPR, pp. 780–787 (2005)Google Scholar
  28. 28.
    Quan, L.: Invariants of six points and projective reconstruction from three uncalibrated images. IEEE Trans. Pattern Anal. Mach. Intell. 17, 34–46 (1995)CrossRefGoogle Scholar
  29. 29.
    Schaffalitzky, F., Zisserman, A., Hartley, R.I., Torr, P.H.S.: A six point solution for structure and motion. In: Vernon, D. (ed.) ECCV 2000. LNCS, vol. 1842, pp. 632–648. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  30. 30.
    Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  31. 31.
    Lowe, D.G.: Distinctive image features from scale-invariant keypoints. IJCV 60, 91–110 (2004)CrossRefGoogle Scholar
  32. 32.
    Chum, O., Werner, T., Matas, J.: Two-view geometry estimation unaffected by a dominant plane. In: CVPR, vol. I, pp. 772–779 (2005)Google Scholar
  33. 33.
    Sattler, T., Leibe, B., Kobbelt, L.: Scramsac: Improving ransac‘s efficiency with a spatial consistency filter. In: ICCV (2009)Google Scholar
  34. 34.
    Strecha, C., von Hansen, W., Van Gool, L., Fua, P., Thoennessen, U.: On Benchmarking Camera Calibration and Multi-View Stereo for High Resolution Imagery. In: CVPR (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jérôme Courchay
    • 1
  • Arnak Dalalyan
    • 1
  • Renaud Keriven
    • 1
  • Peter Sturm
    • 2
  1. 1.IMAGINE, LIGMUniversité Paris-Est 
  2. 2.Laboratoire Jean KuntzmannINRIA Grenoble - Rhône-Alpes 

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