Abstract

We overview techniques for optimal geometric estimation from noisy observations for computer vision applications. We first describe estimation techniques based on minimization of given cost functions: least squares (LS), maximum likelihood (ML), which includes reprojection error minimization (Gold Standard) as a special case, and Sampson error minimization. We then formulate estimation techniques not based on minimization of any cost function: iterative reweight, renormalization, and hyper-renormalization. Showing numerical examples, we conclude that hyper-renormalization is robust to noise and currently is the best method.

Keywords

Mahalanobis Distance Generalize Eigenvalue Problem Bundle Adjustment Total Little Square Geometric Estimation 
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.

References

  1. 1.
    Al-Sharadqah, A., Chernov, N.: A doubly optimal ellipse fit. Comp. Stat. Data Anal. 56(9), 2771–2781 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bartoli, A., Sturm, P.: Nonlinear estimation of fundamental matrix with minimal parameters. IEEE Trans. Patt. Anal. Mach. Intell. 26(3), 426–432 (2004)CrossRefGoogle Scholar
  3. 3.
    Chernov, N., Lesort, C.: Statistical efficiency of curve fitting algorithms. Comp. Stat. Data Anal. 47(4), 713–728 (2004)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chojnacki, W., Brooks, M.J., van den Hengel, A.: Rationalising the renormalization method of Kanatani. J. Math. Imaging Vis. 21(11), 21–38 (2001)CrossRefGoogle Scholar
  5. 5.
    Chojnacki, W., Brooks, M.J., van den Hengel, A., Gawley, D.: On the fitting of surfaces to data with covariances. IEEE Trans. Patt. Anal. Mach. Intell. 22(11), 1294–1303 (2000)CrossRefGoogle Scholar
  6. 6.
    Godambe, V.P. (ed.): Estimating Functions. Oxford University Press, New York (1991)MATHGoogle Scholar
  7. 7.
    Hartley, R., Kahl, F.: Optimal algorithms in multiview geometry. In: Proc. 8th Asian Conf. Comput. Vis., Tokyo, Japan, vol. 1, pp. 13–34 (November 2007)Google Scholar
  8. 8.
    Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004)MATHCrossRefGoogle Scholar
  9. 9.
    Kahl, F., Agarwal, S., Chandraker, M.K., Kriegman, D., Belongie, S.: Practical global optimization for multiview geometry. Int. J. Comput. Vis. 79(3), 271–284 (2008)CrossRefGoogle Scholar
  10. 10.
    Kanatani, K.: Renormalization for unbiased estimation. In: Proc. 4th Int. Conf. Comput. Vis., Berlin, Germany, pp. 599–606 (May 1993)Google Scholar
  11. 11.
    Kanatani, K.: Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier, Amsterdam (1996); reprinted, Dover, New York (2005) MATHGoogle Scholar
  12. 12.
    Kanatani, K.: Cramer-Rao lower bounds for curve fitting. Graphical Models Image Process. 60(2), 93–99 (1998)CrossRefGoogle Scholar
  13. 13.
    Kanatani, K.: Ellipse fitting with hyperaccuracy. IEICE Trans. Inf. & Syst. E89-D(10), 2653–2660 (2006)Google Scholar
  14. 14.
    Kanatani, K.: Statistical optimization for geometric fitting: Theoretical accuracy analysis and high order error analysis. Int. J. Comput. Vis. 80(2), 167–188 (2008) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kanatani, K., Al-Sharadqah, A., Chernov, N., Sugaya, Y.: Renormalization Returns: Hyper-renormalization and Its Applications. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part III. LNCS, vol. 7574, pp. 384–397. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  16. 16.
    Kanatani, K., Rangarajan, P.: Hyper least squares fitting of circles and ellipses. Comput. Stat. Data Anal. 55(6), 2197–2208 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kanatani, K., Rangarajan, P., Sugaya, Y., Niitsuma, H.: HyperLS and its applications. IPSJ Trans. Comput. Vis. Appl. 3, 80–94 (2011)CrossRefGoogle Scholar
  18. 18.
    Kanatani, K., Sugaya, Y.: Performance evaluation of iterative geometric fitting algorithms. Comp. Stat. Data Anal. 52(2), 1208–1222 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kanatani, K., Sugaya, Y.: Compact algorithm for strictly ML ellipse fitting. In: Proc. 19th Int. Conf. Patt. Recog., Tampa, FL, U.S.A (December 2008)Google Scholar
  20. 20.
    Kanatani, K., Sugaya, Y.: Compact fundamental matrix computation. IPSJ Tran. Comput. Vis. Appl. 2, 59–70 (2010)CrossRefGoogle Scholar
  21. 21.
    Kanatani, K., Sugaya, Y.: Unified computation of strict maximum likelihood for geometric fitting. J. Math. Imaging Vis. 38(1), 1–13 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Leedan, Y., Meer, P.: Heteroscedastic regression in computer vision: Problems with bilinear constraint. Int. J. Comput. Vis. 37(2), 127–150 (2000)MATHCrossRefGoogle Scholar
  23. 23.
    Lourakis, M.I.A., Argyros, A.A.: SBA: A software package for generic sparse bundle adjustment. ACM Trans. Math. Software 36(1), 2, 1–30 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Matei, J., Meer, P.: Estimation of nonlinear errors-in-variables models for computer vision applications. IEEE Trans. Patt. Anal. Mach. Intell. 28(10), 1537–1552 (2006)CrossRefGoogle Scholar
  25. 25.
    Okatani, T., Deguchi, K.: On bias correction for geometric parameter estimation in computer vision. In: Proc. IEEE Conf. Comput. Vis. Patt. Recog., Miami Beach, FL, U.S.A, pp. 959–966 (June 2009)Google Scholar
  26. 26.
    Okatani, T., Deguchi, K.: Improving accuracy of geometric parameter estimation using projected score method. In: Proc. Int. Conf. Comput. Vis., Kyoto, Japan, pp. 1733–1740 (September/October 2009)Google Scholar
  27. 27.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)Google Scholar
  28. 28.
    Rangarajan, P., Kanatani, K.: Improved algebraic methods for circle fitting. Electronic J. Stat. 3, 1075–1082 (2009)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sampson, P.D.: Fitting conic sections to “very scattered” data: An iterative refinement of the Bookstein algorithm. Comput. Graphics Image Process. 18(1), 97–108 (1982)CrossRefGoogle Scholar
  30. 30.
    Sturm, P., Gargallo, P.: Conic fitting using the geometric distance. In: Proc. 8th Asian Conf. Comput. Vis., Tokyo, Japan, vol. 2, pp. 784–795 (November 2007)Google Scholar
  31. 31.
    Taubin, G.: Estimation of planar curves, surfaces, and non-planar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Trans. Patt. Anal. Mach. Intell. 13(11), 1115–1138 (1991)CrossRefGoogle Scholar
  32. 32.
    Triggs, B., McLauchlan, P.F., Hartley, R.I., Fitzgibbon, A.W.: Bundle Adjustment – A Modern Synthesis. In: Triggs, B., Zisserman, A., Szeliski, R. (eds.) ICCV-WS 1999. LNCS, vol. 1883, pp. 298–375. Springer, Heidelberg (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Kenichi Kanatani
    • 1
  1. 1.Department of Computer ScienceOkayama UniversityOkayamaJapan

Personalised recommendations