Camera Calibration Using Principal-Axes Aligned Conics

  • Xianghua Ying
  • Hongbin Zha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4843)


The projective geometric properties of two principal-axes aligned (PAA) conics in a model plane are investigated in this paper by utilized the generalized eigenvalue decomposition (GED). We demonstrate that one constraint on the image of the absolute conic (IAC) can be obtained from a single image of two PAA conics even if their parameters are unknown. And if the eccentricity of one of the two conics is given, two constraints on the IAC can be obtained. An important merit of the algorithm using PAA is that it can be employed to avoid the ambiguities when estimating extrinsic parameters in the calibration algorithms using concentric circles. We evaluate the characteristics and robustness of the proposed algorithm in experiments with synthetic and real data.


Camera calibration Generalized eigenvalue decomposition Principal-axes aligned conics Image of the absolute conic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fitzgibbon, A.W., Pilu, M., Fisher, R.B.: Direct least squares fitting of ellipses. IEEE Trans. Pattern Analysis and Machine Intelligence 21(5), 476–480 (1999)CrossRefGoogle Scholar
  2. 2.
    Forsyth, D., Mundy, J.L., Zisserman, A., Coelho, C., Heller, A., Rothwell, C.: Invariant descriptors for 3-D object recognition and pose. IEEE Trans. Pattern Analysis and Machine Intelligence 13(10), 971–991 (1991)CrossRefGoogle Scholar
  3. 3.
    Gurdjos, P., Kim, J.-S., Kweon, I.-S.: Euclidean Structure from Confocal Conics: Theory and Application to Camera Calibration. In: Proc. IEEE. Conf. Computer Vision and Pattern Recognition, vol. 1, pp. 1214–1222. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  4. 4.
    Hartley, R., Zisserman, A.: Multiple View Geometry in computer vision, 2nd edn. Cambridge University Press, Cambridge, UK (2003)Google Scholar
  5. 5.
    Heisterkamp, D., Bhattacharya, P.: Invariants of families of coplanar conics and their applications to object recognition. Journal of Mathematical Imaging and Vision 7(3), 253–267 (1997)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Jiang, G., Quan, L.: Detection of Concentric Circles for Camera Calibration. In: Proc. Int’l. Conf. Computer Vision, pp. 333–340 (2005)Google Scholar
  7. 7.
    Kahl, F., Heyden, A.: Using conic correspondence in two images to estimate the epipolar geometry. In: Proc. Int’l. Conf. Computer Vision, pp. 761–766 (1998)Google Scholar
  8. 8.
    Kanatani, K., Liu, W.: 3D Interpretation of Conics and Orthogonality. Computer Vision and Image Understanding 58(3), 286–301 (1993)CrossRefGoogle Scholar
  9. 9.
    Kim, J.-S., Gurdjos, P., Kweon, I.-S.: Geometric and Algebraic Constraints of Projected Concentric Circles and Their Applications to Camera Calibration. IEEE Trans. Pattern Analysis and Machine Intelligence 27(4), 637–642 (2005)CrossRefGoogle Scholar
  10. 10.
    Ma, S.: Conics-Based Stereo, Motion Estimation, and Pose Determination. Int’l J. Computer Vision 10(1), 7–25 (1993)CrossRefGoogle Scholar
  11. 11.
    Ma, S., Si, S., Chen, Z.: Quadric curve based stereo. In: Proc. of The 11th Int’l. Conf. Pattern Recognition, vol. 1, pp. 1–4 (1992)Google Scholar
  12. 12.
    Mundy, J.L., Zisserman, A. (eds.): Geometric Invariance in Computer Vision. MIT Press, Cambridge (1992)Google Scholar
  13. 13.
    Mudigonda, P., Jawahar, C.V., Narayanan, P.J.: Geometric structure computation from conics. In: Proc. Indian Conf. Computer Vison, Graphics and Image Processing (ICVGIP), pp. 9–14 (2004)Google Scholar
  14. 14.
    Quan, L.: Algebraic and geometric invariant of a pair of noncoplanar conics in space. Journal of Mathematical Imaging and Vision 5(3), 263–267 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Quan, L.: Conic reconstruction and correspondence from two views. IEEE Transactions on Pattern Analysis and Machine Intelligence 18(2), 151–160 (1996)CrossRefGoogle Scholar
  16. 16.
    Semple, J.G., Kneebone, G.T.: Algebraic Projective Geometry. Oxford University Press, Oxford (1952)zbMATHGoogle Scholar
  17. 17.
    Sugimoto, A.: A linear algorithm for computing the homography from conics in correspondence. Journal of Mathematical Imaging and Vision 13, 115–130 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Weiss, I.: 3-D curve reconstruction from uncalibrated cameras. In: Proc. of Int’l. Conf. Pattern Recognition, vol. 1, pp. 323–327 (1996)Google Scholar
  19. 19.
    Yang, C., Sun, F., Hu, Z.: Planar Conic Based Camera Calibration. In: Proc. of Int’l. Conf. Pattern Recognition, vol. 1, pp. 555–558 (2000)Google Scholar
  20. 20.
    Zhang, Z.: A flexible new technique for camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(11), 1330–1334 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Xianghua Ying
    • 1
  • Hongbin Zha
    • 1
  1. 1.National Laboratory on Machine Perception, Peking University, Beijing, 100871P.R. China

Personalised recommendations