Advertisement

An Exact Robust Method to Localize a Known Sphere by Means of One Image

  • Rudi Penne
  • Bart Ribbens
  • Pedro Roios
Article

Abstract

In this article we provide a very robust algorithm to compute the position of the center of a sphere with known radius from one image by a calibrated camera. To our knowledge it is the first time that an exact sphere localization formula is published that only uses the (pixel) area and the ellipse center of the sphere image. Other authors either derived an approximation formula or followed the less robust and more time consuming procedure of fitting an ellipse through the detected edge pixels. Our method is analytic and deterministic, making use of the unique positive real tool of a cubic equation. We observe that the proposed area method is significantly more accurate and precise than an ellipse fitting method. Furthermore, we investigate in what conditions for sphere images the proposed exact method is preferable to the robust approximation method. These observations are validated by virtual, synthetic and real experiments.

Keywords

Extrinsic calibration Robust localization Projective geometry 

Notes

References

  1. Agrawal, M., & Davis, L. S. (2003). Camera calibration using spheres: A semi-definite programming approach. In: IEEE international conference on computer vision (pp. 782–789).Google Scholar
  2. Arun, K. S., Huang, T. S., & Blostein, S. D. (1987). Least-squares fitting of two 3-D point sets. IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(5), 698–700.CrossRefGoogle Scholar
  3. Ballard, D. H. (1981). Generalizing the hough transform to detect arbitrary shapes. Pattern Recognition, 13(2), 111–122.CrossRefGoogle Scholar
  4. Barbeau, E. (1989). Polynomials. New York: Springer.CrossRefGoogle Scholar
  5. Beardsley, P., Murray, D., & Zisserman, A. (1992). Camera calibration using multiple images. In Proceedings of the second European conference on computer vision, ECCV (pp. 312–320).Google Scholar
  6. Canny, J. (1986). A computational approach to edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(6), 679–698.CrossRefGoogle Scholar
  7. Dandelin, G. P. (1822). Mémoire sur quelques propriétés remarquables de la focale parabolique. Nouveaux mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles T, II, 171–202.Google Scholar
  8. Daucher, D., Dhome, M., & Lapreste, J. (1994). Camera calibration from spheres images. In Proceedings European conference computer vision (pp. 449–454).Google Scholar
  9. Fitzgibbon, A., Pilu, M., & Fisher, R. (1996). Direct least-square fitting of ellipses. In Proceedings international conference on pattern recognition (pp. 253–257).Google Scholar
  10. Guan, J., Deboeverie, F., Slembrouck, M., van Haerenborgh, D., van Cauwelaert, D., Veelaert, P., et al. (2015). Extrinsic calibration of camera networks using a sphere. Sensors, 15(8), 18985–19005.  https://doi.org/10.3390/s150818985.CrossRefGoogle Scholar
  11. Ho, C., & Chen, L. (1995). A fast ellipse/circle detector using geometric symmetry. Pattern Recognition, 28(1), 117–124.CrossRefGoogle Scholar
  12. Lu, Y., & Payandeh, S. (2010). On the sensitivity analysis of camera calibration from images of spheres. Computer Vision and Image Understanding, 114(1), 8–20.CrossRefGoogle Scholar
  13. Ouellet, J. N., & Hébert, P. (2008). Precise ellipse estimation without contour point extraction. Machine Vision and Applications, 21(1), 59.  https://doi.org/10.1007/s00138-008-0141-3.CrossRefGoogle Scholar
  14. Penna, M. (1991). Camera calibration: A quick and easy way to determine the scale factor. IEEE Transactions on Pattern Analysis & Machine Intelligence, 13(12), 1240–1245.CrossRefGoogle Scholar
  15. Penne, R., Ribbens, B., Mertens, L., & Levrie, P. (2015). What does one image of one ball tell us about the focal length? Advanced Concepts for Intelligent Vision Systems, Lecture Notes in Computer Science, 9386, 501–509.MathSciNetCrossRefGoogle Scholar
  16. Sun, J., Chen, X., Gong, Z., Liu, Z., & Zhao, Y. (2015). Accurate camera calibration with distortion models using sphere images. Optics Laser Technology, 65, 83–87.CrossRefGoogle Scholar
  17. Teramoto, H., & Xu, G. (2002). Camera calibration by a single image of balls: From conics to the absolute conic. In Asian conference on computer vision (pp. 499–506).Google Scholar
  18. Xie, S., & Tu, Z. (2015). Holistically-nested edge detection. CoRR. abs/1504.06375. http://arxiv.org/abs/1504.06375.
  19. Xie, Y., & Ji, Q. (2002). A new efficient ellipse detection method. In International conference on pattern recognition (pp. 957–960).Google Scholar
  20. Yin, P., & Chen, L. H. (1994). A new method for ellipse detection using symmetry. Journal of Electronic Imaging, 3, 20–29.CrossRefGoogle Scholar
  21. Zhang, H., Wong, K., & Zhang, G. (2007). Camera calibration from images of spheres. IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(3), 499–503.CrossRefGoogle Scholar
  22. Zhang, Z. (2000). A flexible new technique for camera calibration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(11), 1330–1334.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Applied EngineeringUniversity of AntwerpAntwerpBelgium
  2. 2.Faculty of Applied EngineeringUniversity of AntwerpAntwerpBelgium

Personalised recommendations