Journal of Mathematical Imaging and Vision

, Volume 33, Issue 3, pp 281–295 | Cite as

Error Analysis in Homography Estimation by First Order Approximation Tools: A General Technique

  • Pei Chen
  • David Suter


This paper shows how to analytically calculate the statistical properties of the errors in estimated parameters. The basic tools to achieve this aim include first order approximation/perturbation techniques, such as matrix perturbation theory and Taylor Series. This analysis applies for a general class of parameter estimation problems that can be abstracted as a linear (or linearized) homogeneous equation.

Of course there may be many reasons why one might which to have such estimates. Here, we concentrate on the situation where one might use the estimated parameters to carry out some further statistical fitting or (optimal) refinement. In order to make the problem concrete, we take homography estimation as a specific problem. In particular, we show how the derived statistical errors in the homography coefficients, allow improved approaches to refining these coefficients through subspace constrained homography estimation (Chen and Suter in Int. J. Comput. Vis. 2008).

Indeed, having derived the statistical properties of the errors in the homography coefficients, before subspace constrained refinement, we do two things: we verify the correctness through statistical simulations but we also show how to use the knowledge of the errors to improve the subspace based refinement stage. Comparison with the straightforward subspace refinement approach (without taking into account the statistical properties of the homography coefficients) shows that our statistical characterization of these errors is both correct and useful.


Error analysis Matrix perturbation theory Singular value decomposition Low rank matrix approximation Homography First order approximation Mahalanobis distance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chen, P.: An investigation of statistical aspects of linear subspace analysis for computer vision applications. Ph.D. Thesis, Monash University (2004) Google Scholar
  2. 2.
    Chen, P., Suter, D.: An analysis of linear subspace approaches for computer vision and pattern recognition. Int. J. Comput. Vis. 68(1), 83–106 (2006) CrossRefGoogle Scholar
  3. 3.
    Chen, P., Suter, D.: A bilinear approach to the parameter estimation of a general heteroscedastic linear system, with application to conic fitting. J. Math. Imaging Vis. 28(3), 191–208 (2007) CrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen, P., Suter, D.: Rank constraints for homographies over two views: Revisiting the rank four constraint. Int J. Comput. Vis. (2008, to appear) Google Scholar
  5. 5.
    Chernov, N.: On the convergence of fitting algorithms in computer vision. J. Math. Imaging Vis. 27(3), 231–239 (2007) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chernov, N., Lesort, C.: Statistical efficiency of curve fitting algorithms. Comput. Stat. Data Anal. 47(4), 713–G728 (2004) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chojnacki, W., Brooks, M.J., van den Hengel, A., Gawley, D.: On the fitting of surfaces to data with covariances. IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1294–1303 (2000) CrossRefGoogle Scholar
  8. 8.
    Chojnacki, W., Brooks, M.J., van den Hengel, A., Gawley, D.: A new approach to constrained parameter estimation applicable to some computer vision problems. Image Vis. Comput. 22(2), 85–91 (2004) CrossRefGoogle Scholar
  9. 9.
    Chojnacki, W., Brooks, M.J., van den Hengel, A., Gawley, D.: From fns to heiv: a link between two vision parameter estimation methods. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 264–268 (2004) CrossRefGoogle Scholar
  10. 10.
    Chum, O., Pajdla, T., Sturm, P.: The geometric error for homographies. Comput. Vis. Image Underst. 97(1), 86–102 (2005) CrossRefGoogle Scholar
  11. 11.
    Chum, O., Werner, T., Matas, J.: Two-view geometry estimation unaffected by a dominant plane. In: Proc. Conf. Computer Vision and Pattern Recognition (1), pp. 772–779 (2005) Google Scholar
  12. 12.
    Golub, G.H., Loan, C.F.V.: Matrix Computations, 3nd edn. Johns Hopkins Press, Baltimore (1996) MATHGoogle Scholar
  13. 13.
    Haralick, R.M.: Propagating covariance in computer vision. In: Proc. of 12th ICPR, pp. 493–498 (1994) Google Scholar
  14. 14.
    Haralick, R.M.: Propagating covariance in computer vision. Int. J. Pattern Recogn. Artif. Intell. 10(5), 561–572 (1996) CrossRefGoogle Scholar
  15. 15.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge Univ Press, Cambridge (2003) Google Scholar
  16. 16.
    Jain, A.K., Mao, J., Duin, R.: Statistical pattern recognition: A review. IEEE Trans. Pattern Anal. Mach. Intell. 22(1), 4–37 (2000) CrossRefGoogle Scholar
  17. 17.
    Kanatani, K.: Unbiased estimation and statistical analysis of 3-d rigid motion from two views. IEEE Trans. Pattern Anal. Mach. Intell. 15(1), 37–50 (1993) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kanatani, K.: Statistical bias of conic fitting and renormalization. IEEE Trans. Pattern Anal. Mach. Intell. 16(3), 320–326 (1994) MATHCrossRefGoogle Scholar
  19. 19.
    Kanatani, K.: Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier, Amsterdam (1996) MATHGoogle Scholar
  20. 20.
    Kanatani, K.: Uncertainty modeling and model selection for geometric inference. IEEE Trans. Pattern Anal. Mach. Intell. 26(10), 1307–1319 (2004) CrossRefGoogle Scholar
  21. 21.
    Kanatani, K.: Statistical optimization for geometric fitting: Theoretical accuracy bound and high order error analysis. Int. J. Comput. Vis. (2008, in print) Google 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.
    Manton, J.H., Mahony, R., Hua, Y.: The geometry of weighted low-rank approximations. IEEE Trans. Signal Process. 51(2), 500–514 (2003) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Mühlich, M., Mester, R.: A considerable improvement in non-iterative homography estimation using tls and equilibration. Pattern Recogn. Lett. 22(11), 1181–1189 (2001) CrossRefGoogle Scholar
  25. 25.
    Mulich, M., Mester, R.: The role of total least squares in motion analysis. In: ECCV, pp. 305–321 (1998) Google Scholar
  26. 26.
    Mulich, M., Mester, R.: Subspace methods and equilibration in computer vision. In: Scandinavian Conference on Image Analysis (2001) Google Scholar
  27. 27.
    Mulich, M., Mester, R.: Unbiased errors-in-variables estimation using generalized eigensystem analysis. In: ECCV Workshop SMVP, pp. 38–49 (2004) Google Scholar
  28. 28.
    Nadabar, S.G., Jain, A.K.: Parameter estimation in Markov random field contextual models using geometric models of objects. IEEE Trans. Pattern Anal. Mach. Intell. 18(3), 326–329 (1996) CrossRefGoogle Scholar
  29. 29.
    Nayak, A., Trucco, E., Thacker, N.A.: When are simple ls estimators enough? An empirical study of ls, tls, and gtls. Int. J. Comput. Vis. 68(2), 203–216 (2006) CrossRefGoogle Scholar
  30. 30.
    Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Academic Press, San Diego (1990) MATHGoogle Scholar
  31. 31.
    Taubin, G.: Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 13(11), 1115–1138 (1991) CrossRefGoogle Scholar
  32. 32.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon, Oxford (1965) MATHGoogle Scholar
  33. 33.
    Zelnik-Manor, L., Irani, M.: Multi-view subspace constraints on homographies. In: Proc. Int’l Conf. Computer Vision, pp. 710–715 (1999) Google Scholar
  34. 34.
    Zelnik-Manor, L., Irani, M.: Multi-view subspace constraints on homographies. IEEE Trans. Pattern Anal. Mach. Intell. 24(2), 214–223 (2002) CrossRefGoogle Scholar
  35. 35.
    Zhang, Z.: Parameter estimation techniques: A tutorial with application to conic fitting. Image Vis. Comput. 15, 59–76 (1997) CrossRefGoogle Scholar
  36. 36.
    Zhang, Z.: Determining the epipolar geometry and its uncertainty: A review. Int. J. Comput. Vis. 27(2), 161–195 (1998) CrossRefGoogle Scholar
  37. 37.
    Zhang, Z.: On the optimization criteria used in two-view motion analysis. IEEE Trans. Pattern Anal. Mach. Intell. 20(7), 717–729 (1998) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.School of Information Science and TechnologySun Yat-sen UniversityGuangzhouChina
  2. 2.Shenzhen Institute of Advanced Integration TechnologyCAS/CUHKShenzhenChina
  3. 3.ARC Centre for Perceptive and Intelligent Machines in Complex Environments, Department of Electrical and Computer Systems EngineeringMonash UniversityMelbourneAustralia

Personalised recommendations