Journal of Mathematical Imaging and Vision

, Volume 24, Issue 3, pp 341–348 | Cite as

Some Results on Minimal Euclidean Reconstruction from Four Points

  • Long Quan
  • Bill Triggs
  • Bernard Mourrain
Article

Abstract

Methods for reconstruction and camera estimation from miminal data are often used to boot-strap robust (RANSAC and LMS) and optimal (bundle adjustment) structure and motion estimates. Minimal methods are known for projective reconstruction from two or more uncalibrated images, and for “5 point” relative orientation and Euclidean reconstruction from two calibrated parameters, but we know of no efficient minimal method for three or more calibrated cameras except the uniqueness proof by Holt and Netravali. We reformulate the problem of Euclidean reconstruction from minimal data of four points in three or more calibrated images, and develop a random rational simulation method to show some new results on this problem. In addition to an alternative proof of the uniqueness of the solutions in general cases, we further show that unknown coplanar configurations are not singular, but the true solution is a double root. The solution from a known coplanar configuration is also generally unique. Some especially symmetric point-camera configurations lead to multiple solutions, but only symmetry of points or the cameras gives a unique solution.

Keywords

3D reconstruction relative orientation structure from motion polynomial methods algebraic geometry 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B.M. Bennet, D.D. Hoffman, J.E. Nicola, and C. Prakash, “Structure from two orthographic views of rigid motion,” Journal of the Optical Society of America, Vol. 6, No. 7, pp. 1052–1069, 1989.MathSciNetGoogle Scholar
  2. 2.
    S. Carlsson and D. Weinshall, “Dual computation of projective shape and camera positions from multiple images,” International Journal of Computer Vision, Vol. 27, No. 3, pp. 227–241, 1998.CrossRefGoogle Scholar
  3. 3.
    D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Springer, 1998.Google Scholar
  4. 4.
    D. Cox, J. Little, and D.O’Shea. Using Algebraic Geometry, Springer, 1998.Google Scholar
  5. 5.
    M. Demazure, “Sur deux problèmes de reconstruction,” Technical report, INRIA, 1988.Google Scholar
  6. 6.
    I.Z. Emeris, “A general solver based on sparse resultants: Numerical issues and kinematic applications,” Technical Report RR-3110, INRIA, 1997.Google Scholar
  7. 7.
    I.Z. Emeris, “A general solver based on sparse resultants: Numerical issues and kinematic applications,” Technical Report RR-3110 (http://www.inria.fr/RRRT/RR-3110.html), INRIA, Sophia Antipolis, France, Jan. 1997.
  8. 8.
    O. Faugeras, “What can be seen in three dimensions with an uncalibrated stereo rig?” in Proceedings of the 2nd European Conference on Computer Vision, Santa Margherita Ligure, Italy, G. Sandini (Ed.), Springer-Verlag, May 1992, pp. 563–578.Google Scholar
  9. 9.
    O. Faugeras and S. Maybank, “Motion from point matches: Multiplicity of solutions,” International Journal of Computer Vision, Vol. 3, No. 4, pp. 225–246, 1990.CrossRefGoogle Scholar
  10. 10.
    M.A. Fischler and R.C. Bolles, “Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography,” Graphics and Image Processing, Vol. 24, No. 6, pp. 381–395, 1981.MathSciNetGoogle Scholar
  11. 11.
    S.I. Granshaw, “Bundle adjustment methods in engineering photogrammetry,” Photogrammetric Record, Vol. 56, No. 10, pp. 181–207, 1980.Google Scholar
  12. 12.
    R.I. Hartley, R. Gupta, and T. Chang, “Stereo from uncalibrated cameras,” in Proceedings of the Conference on Computer Vision and Pattern Recognition, Urbana-Champaign, Illinois, USA, 1992, pp. 761–764.Google Scholar
  13. 13.
    A. Heyden and G. Sparr, “Reconstruction from calibrated cameras: a new proof of the kruppa-demazure theorem,” Journal of Mathematical Imaging and Vision, Vol. 10, pp. 1–20, 1999.CrossRefMathSciNetGoogle Scholar
  14. 14.
    R.J. Holt and A. N. Netravali, “Uniqueness of solutions to three perspective views of four points,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 3, pp. 303–307, 1995.CrossRefGoogle Scholar
  15. 15.
    B. Horn, “Relative orientation,” Int. J. Computer Vision, Vol. 4, pp. 59–78, 1990.CrossRefGoogle Scholar
  16. 16.
    T.S. Huang and C.H. Lee, “Motion and structure from orthographic projections,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 11, No. 5, pp. 536–540, 1989.CrossRefGoogle Scholar
  17. 17.
    J. Koenderink and A. van Doorn, “Affine structure from motion,” Journal of the Optical Society of America A, Vol. 8, No. 2, pp. 377–385, 1991.CrossRefGoogle Scholar
  18. 18.
    J. Krames, “Zur Ermittlung eines Objektes aus zwei Perspektiven (Ein Beitrag zur Theorie der “gefährlichen Örter”),” Monatshefte für Mathematik und Physik, Vol. 49, pp. 327–354, 1941.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    E. Kruppa, “Zur Ermittlung eines Objektes aus zwei Perspektiven mit innerer Orientierung,” Sitzungsberichte Österreichische Akademie der Wissenschaften, Mathematisch-naturwissenschaftliche Klasse, Abteilung II a, Vol. 122, pp. 1939–1948, 1913.MATHGoogle Scholar
  20. 20.
    C.H. Lee and T. Huang, “Finding point correspondences and determining motion of a rigid object from two weak perspective views,” Computer Vision, Graphics and Image Processing, Vol. 52, pp. 309–327, 1990.CrossRefGoogle Scholar
  21. 21.
    H.C. Longuet-Higgins, “A method of obtaining the relative positions of 4 points from 3 perspective projections,” in Proceedings of the British Machine Vision Conference, Glasgow, Scotland, pp. 86–94, 1991.Google Scholar
  22. 22.
    S.J. Maybank and O.D. Faugeras, “A theory of self calibration of a moving camera,” International Journal of Computer Vision, Vol. 8, No. 2, pp. 123–151, 1992.CrossRefGoogle Scholar
  23. 23.
    A.N. Netravali, T.S. Huang, A.S. Krishnakumar, and R.J. Holt, “Algebraic methods in 3D motion estimation from two-view point correspondences,” Int. J. Imaging Systems & Technology, Vol. 1, pp. 78–99, 1989.Google Scholar
  24. 24.
    D. Nister, “An efficient solution to the five-point relative pose problem,” in Proceedings of the Conference on Computer Vision and Pattern Recognition, Madison, USA, June 2003, pp. II–195–202.Google Scholar
  25. 25.
    J. Philip, “A non-iterative algorithm for determining all essential matrices corresponding to five point pairs,” Photogrammetric Record, Vol. 15, No. 88, pp. 589–599, 1996.CrossRefGoogle Scholar
  26. 26.
    L. Quan, “Invariants of six points and projective reconstruction from three uncalibrated images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 1, pp. 34–46, 1995.CrossRefGoogle Scholar
  27. 27.
    L.S. Shapiro, A. Zisserman, and M. Brady, “3D motion recovery via affine epipolar geometry,” International Journal of Computer Vision, Vol. 16, No. 2, pp. 147–182, 1995.CrossRefGoogle Scholar
  28. 28.
    A. Shashua, “Algebraic functions for recognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 8, pp. 779–789, 1995.CrossRefGoogle Scholar
  29. 29.
    C.C. Slama (ed), Manual of Photogrammetry, 4th edition. American Society of Photogrammetry and Remote Sensing, Falls Church, Virginia, USA, 1980.Google Scholar
  30. 30.
    M. Spetsakis and J. Aloimonos, “A unified theory of structure from motion,” in Proceedings of DARPA Image Understanding Workshop, 1990, pp. 271–283.Google Scholar
  31. 31.
    R. Sturm, “Das Problem der Projektivität und seine Anwendung auf die Flächen Zweiten Grades,” Math. Ann., Vol. 1, pp. 533–574, 1869.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    P.H.S. Torr and A. Zisserman, “Robust parameterization and computation of the trifocal tensor,” in R.B. Fisher and E. Trucco (Eds.), Proceedings of the seventh British Machine Vision Conference, Vol. 2, Edinburgh, Scotland, British Machine Vision Association, September 1996, pp. 655–664.Google Scholar
  33. 33.
    B. Triggs, “Routines for relative pose of two calibrated cameras from 5 points,” Technical report, INRIA, 2000.Google Scholar
  34. 34.
    S. Ullman, “The Interpretation of Visual Motion,” The MIT Press: Cambridge, MA, USA, 1979.Google Scholar
  35. 35.
    Z. Zhang, R. Deriche, O. Faugeras, and Q.T. Luong, “A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry,” Rapport de recherche 2273, INRIA, May 1994.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Long Quan
    • 1
  • Bill Triggs
    • 2
  • Bernard Mourrain
    • 3
  1. 1.Department of Computer ScienceHKUSTHong Kong
  2. 2.INRIA Rhône-AlpesFrance
  3. 3.INRIA Sophia AntipolisFrance

Personalised recommendations