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Degeneracy from Twisted Cubic Under Two Views

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Abstract

Fundamental matrix, drawing geometric relationship between two images, plays an important role in 3-dimensional computer vision. Degenerate configurations of the space points and the two camera optical centers affect stability of computation for fundamental matrix. In order to robustly estimate fundamental matrix, it is necessary to study these degenerate configurations. We analyze all the possible degenerate configurations caused by twisted cubic and give the corresponding degenerate rank for each case. Relationships with general degeneracies, the previous ruled quadric degeneracy and the homography degeneracy, are also reported.

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References

  1. Hartley R, Zisserman A. Multiple View Geometry in Computer Vision. Cambridge University Press, 2000.

  2. Hartley R, Kahl F. Critical configurations for projective reconstruction from multiple views. International Journal of Computer Vision, 2006, 71(1): 5–47.

    Article  Google Scholar 

  3. Horn B.K.P. Relative orientation. International Journal of Computer Vision, 1990, 4(1): 59–78.

    Article  Google Scholar 

  4. Kahl F, Triggs B, Astrom K. Critical motions for auto-calibration when some intrinsic parameters can vary. Journal of Mathematical Imaging and Vision, 2000, 13(2): 131–146.

    Article  MATH  MathSciNet  Google Scholar 

  5. Luong Q T, Faugeras O. Self-calibration of a moving camera from point correspondences. International Journal of Computer Vision, 1997, 22(3): 261–289.

    Article  Google Scholar 

  6. Maybank S, Faugeras O. A theory of self-calibration of a moving camera. International Journal of Computer Vision, 1992, 8(2): 123–151.

    Article  Google Scholar 

  7. Weng J, Huang T S, Ahuja N. Motion and structure from two perspective views: Algorithms, error analysis, and error estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1989, 11(5): 451–476.

    Article  Google Scholar 

  8. Xu G, Zhang Z. Epipolar Geometry in Stereo, Motion and Object Recognition: A Unified Approach. Kluwer Academic Publishers, Norwell, MA, USA, 1996.

    MATH  Google Scholar 

  9. Zhang Z, Deriche R, Faugeras O, Luong Q T. A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artificial Intelligence Journal, 1995, 78(1/2): 87–119.

    Article  Google Scholar 

  10. Bartoli A, Sturm P. Non-linear estimation of the fundamental matrix with minimal parameters. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004, 26(4): 426–432.

    Article  Google Scholar 

  11. Bober M, Georgis N, Kittler J. On accurate and robust estimation of fundamental matrix. Computer Vision and Image Understanding, 1998, 72(1): 39–53.

    Article  Google Scholar 

  12. Frahm J, Pollefeys M. RANSAC for (quasi-)degenerate data (QDEGSAC). IEEE Computer Society Conference on Computer Vision and Pattern Recognition, New York, USA, June 17-22, 2006, pp.453–460.

  13. Hartley R. In defense of the eight-point algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19(6): 580–593.

    Article  Google Scholar 

  14. Luong Q T, Faugeras O. The fundamental matrix: Theory, algorithms, and stability analysis. International Journal of Computer Vision, 1996, 17(1): 43–75.

    Article  Google Scholar 

  15. Torr P H S, Murray D W. The development and comparison of robust methods for estimating the fundamental matrix. International Journal of Computer Vision, 1997, 24(3): 271–300.

    Article  Google Scholar 

  16. Torr P H S, Zisserman A, Maybank S J. Robust detection of degenerate configurations while estimating the fundamental matrix. Computer Vision and Image Understanding, 1998, 71(3): 312–333.

    Article  Google Scholar 

  17. Zhang Z. Determining the epipolar geometry and its uncertainty: A review. International Journal of Computer Vision, 1998, 27(2): 161–195.

    Article  Google Scholar 

  18. Gurdjos P, Bartoli A, Sturm P. Is dual linear self-calibration artificially ambiguous? International Conference on Computer Vision, Kyoto, Japan, Sept. 29-Oct. 2, 2009, pp.88–95.

  19. Sturm P. A case against Kruppa's equations for camera self-calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000, 22(10): 1199–1204.

    Article  Google Scholar 

  20. Sturm P. Critical motion sequences for the self-calibration of cameras and stereo systems with variable focal length. Image and Vision Computing, 2002, 20(5/6): 415–426.

    Article  Google Scholar 

  21. Chum O, Werner T, Matas J. Two-view geometry estimation unaffected by a dominant plane. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, San Diego, CA, USA, June 20-26, 2005, pp.772–779.

  22. Buchanan T. The twisted cubic and camera calibration. Computer Vision, Graphics and Image Processing, 1988, 42(1): 130–132.

    Article  MATH  MathSciNet  Google Scholar 

  23. Wu Y, Li Y, Hu Z. Detecting and handling unreliable points for camera parameter estimation. International Journal of Computer Vision, 2008, 79(2): 209–223.

    Article  Google Scholar 

  24. Maybank S. Theory of Reconstruction from Image Motion. Springer-Verlag, 1992.

  25. Luong Q T, Faugeras O. A stability analysis of the fundamental matrix. In Proc. European Conference on Computer Vision, Prague, Czech, May 11-14, 1994, pp.577–588.

  26. Hartley R. Ambiguous configurations for 3-view projective reconstruction. In Proc. European Conference on Computer Vision, Dublin, Ireland, Jun. 26-Jul. 1, 2000, pp.922–935.

  27. Maybank S, Shashua A. Ambiguity in reconstruction from images of six points. In Proc. International Conference on Computer Vision, Bombay, India, Jan. 4-7, 1998, pp.703–708.

  28. Fischler M A, Bolles R C. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, 1981, 24(6): 381–395.

    Article  MathSciNet  Google Scholar 

  29. Semple J G, Kneebone G T. Algebraic Projective Geometry. Oxford University Press, 1952.

  30. Hoffman K, Kunze R. Linear Algebra, 2nd Edition. Englewood Cliffs, NJ: Prentice Hall, 1971.

    MATH  Google Scholar 

  31. Wu Y, Hu Z. Invariant representations of a quadric cone and a twisted cubic. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2003, 25(10): 1329–1332.

    Article  Google Scholar 

  32. Kahl F, Henrion D. Globally optimal estimates for geometric reconstruction problems. In Proc. ICCV 2005, Beijing, China, Oct. 17-20, 2005, pp.978-985. Also in International Journal of Computer Vision, 2007, 74(1): 3–15.

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Authors and Affiliations

  1. National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, China

    Yi-Hong Wu, Tian Lan & Zhan-Yi Hu

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  1. Yi-Hong Wu
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  2. Tian Lan
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  3. Zhan-Yi Hu
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Correspondence to Yi-Hong Wu.

Additional information

Supported by the National Natural Science Foundation of China under Grant Nos. 60835003 and 60773039.

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Wu, YH., Lan, T. & Hu, ZY. Degeneracy from Twisted Cubic Under Two Views. J. Comput. Sci. Technol. 25, 916–924 (2010). https://doi.org/10.1007/s11390-010-9376-3

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  • DOI: https://doi.org/10.1007/s11390-010-9376-3

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