International Journal of Computer Vision

, Volume 24, Issue 3, pp 271–300 | Cite as

The Development and Comparison of Robust Methods for Estimating the Fundamental Matrix

  • P.H.S. Torr
  • D.W. Murray


This paper has two goals. The first is to develop a variety of robust methods for the computation of the Fundamental Matrix, the calibration-free representation of camera motion. The methods are drawn from the principal categories of robust estimators, viz. case deletion diagnostics, M-estimators and random sampling, and the paper develops the theory required to apply them to non-linear orthogonal regression problems. Although a considerable amount of interest has focussed on the application of robust estimation in computer vision, the relative merits of the many individual methods are unknown, leaving the potential practitioner to guess at their value. The second goal is therefore to compare and judge the methods.

Comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery. In contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth. A variety of error measures are examined. The experiments allow a statistically satisfying and quasi-optimal method to be synthesized, which is shown to be stable with up to 50 percent outlier contamination, and may still be used if there are more than 50 percent outliers. Performance bounds are established for the method, and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined.

The results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers.

robust methods fundamental matrix matching 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ballard, D. H. and Brown, C. M. 1982. Computer Vision. Prentice-Hall: New Jersey.Google Scholar
  2. Bar-Shalom, Y. and Fortmann, T. E. 1988. Tracking and Data Association. Academic Press.Google Scholar
  3. Beardsley, P. A., Torr, P. H. S., and Zisserman, A. P. 1996. 3D model acquisition from extended image sequences. OUEL Report 2089/96, Department of Engineering Science, University of Oxford.Google Scholar
  4. Bookstein, F. 1979. Fitting conic sections to scattered data. Computer Vision Graphics and Image Processing, 9:56-71.Google Scholar
  5. Chaterjee, S. and Hadi, A. S. 1988. Sensitivity Analysis in Linear Regression. John Wiley: New York.Google Scholar
  6. Cook, R. D. and Weisberg, S. 1980. Characterisations of an empirical influence function for detecting influential cases in regression. Technometrics, 22:337-344.Google Scholar
  7. Critchley, F. 1985. Influence in principal component analysis. Biometrika, 72:627-636.Google Scholar
  8. Dempster, A. P., Laird, N. M., and Rubin, D. B. 1977. Maximum likelihood from incomplete data via the em algorithm. J. Roy. Statist. Soc., 39 B:1-38.Google Scholar
  9. Devlin S. J., Gnanadesikan, R., and Kettering, J. R. 1981. Robust estimation of dispersion matrices and principal components. J. Amer. Stat. Assoc., 76:354-362.Google Scholar
  10. Faugeras, O. D. 1992. What can be seen in three dimensions with an uncalibrated stereo rig? In Proc. 2nd European Conference on Computer Vision, G. Sandini (Ed.), Santa Margherita Ligure, Italy, Springer-Verlag, vol. LNCS 588, pp. 563-578.Google Scholar
  11. Fischler, M. A. and Bolles, R. C. 1981. Random sample consensus: A paradigm for model fitting with application to image analysis and automated cartography. Commun. Assoc. Comp. Mach., 24:381- 395.Google Scholar
  12. Gill, P. E. and Murray, W. 1978. Algorithms for the solution of the nonlinear least-squares problem. SIAM J. Num. Anal., 15(5):977- 992.Google Scholar
  13. Golub, G. H. 1973. Some modified eigenvalue problems. SIAM Review, 15(2):318-335.Google Scholar
  14. Golub, G. H. and van Loan, C. F. 1989. Matrix Computations. The John Hopkins University Press.Google Scholar
  15. Gu, M. and Eisenstat, S. C. 1995. Downdating the singular value decomposition. SIAM J. Matrix Analysis and Applications, 16:793- 810.Google Scholar
  16. Hampel J. P., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A. 1986. Robust Statistics: An Approach Based on Influence Functions. Wiley: New York.Google Scholar
  17. Hartley, R. I. 1992. Estimation of relative camera positions for uncalibrated cameras. In Proc. 2nd European Conference on Computer Vision, G. Sandini (Ed.), Santa Margherita Ligure, Italy, Springer-Verlag, vol. LNCS 588, pp. 579-587.Google Scholar
  18. Hartley, R. I. 1995. In defence of the 8-point algorithm. Proc. 5th Int. Conf. on Computer Vision, Boston, MA, pp. 1064-1070. IEEE Computer Society Press: Los Alamitos CA.Google Scholar
  19. Hartley, R. I. and Sturm, Y. 1994. Triangulation. In Proc. ARPA Image Understanding Workshop, pp. 957-966. and see Proc. Computer Analysis of Images and Patterns, Prague, vol. LNCS 970, Springer Verlag, 1995, pp. 190-197.Google Scholar
  20. Hoaglin, D. C., Mosteller, F., and Tukey, J. W. (Eds.), 1985. Robust Regression. John Wiley and Sons.Google Scholar
  21. Huber, P. J. 1981. Robust Statistics. John Wiley and Sons.Google Scholar
  22. Kanatani, K. 1992. Geometric Computation for Machine Vision. Oxford University Press.Google Scholar
  23. Kanatani, K. 1994. Statistical bias of conic fitting and renormalization. IEEE Trans. Pattern Analysis and Machine Intelligence, 16(3):320-326.Google Scholar
  24. Kanatani, K. 1996. Statistical Optimization for Geometric Computation: Theory and Practice. Elsevier Science: Amsterdam.Google Scholar
  25. Kendall, M. and Stuart, A. 1983. The Advanced Theory of Statistics. Charles Griffin and Company: London.Google Scholar
  26. Kumar, R. and Hanson, A. R. 1994. Robust methods for estimating pose and a sensitivity analysis. Computer Vision, Graphics and Image Processing, 60(3):313-342.Google Scholar
  27. Li, G. 1985. Exploring data tables, trends and shapes. In Robust Regression, D. C. Hoaglin, F. Mosteller, and J. W. Tukey (Eds.), John Wiley and Sons, pp. 281-343.Google Scholar
  28. Li, H., Lavin, M. A., and LeMaster, R. J. 1986. Fast Hough transforms: A hierarchical approach. Computer Vision, Graphics and Image Processing, 36:139-161.Google Scholar
  29. Longuet-Higgins, H. C. 1981. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133-135.Google Scholar
  30. Luong, Q. T. 1992. Matrice Fondamentale et Calibration Visuelle sur l'environnement: Vers use plus grande autonomie des systemes robotiques. Ph. D. Thesis, Paris University.Google Scholar
  31. Luong, Q. T., Deriche, R., Faugeras, O. D., and Papadopoulo, T. 1993. On determining the fundamental matrix: Analysis of different methods and experimental results. INRIA Technical Report 1894, INRIA-Sophia Antipolis.Google Scholar
  32. Maronna, R. A. 1976. Robust M-estimators of multivariate location and scatter. Ann. Stat., 4:51-67.Google Scholar
  33. McLauchlan, P. F. 1990. Describing Textured Surfaces Using Stereo Vision. Ph. D. Thesis, AI Vision Research Unit, University of Sheffield.Google Scholar
  34. Meer, M., Mintz, D., and Rosenfeld, A. (1991). Robust regression methods for computer vision: A review. International Journal of Computer Vision, 6:59-70.Google Scholar
  35. Mosteller, F. and Tukey, J. W. 1977. Data and Analysis and Regression. Addison-Wesley: Reading, MA. Numerical Algorithms Group, 1988. NAG Fortran Library, vol 7.Google Scholar
  36. Olsen, S. I. 1992. Epipolar line estimation. In Proc. 2nd European Conference on Computer Vision, G. Sandini (Ed.), Santa Margherita Ligure, Italy, Springer-Verlag, vol. LNCS 588, pp. 307-311.Google Scholar
  37. Pearson, K. 1901. On lines and planes of closest fit to systems of points in space. Philos. Mag. Ser. 6, 2:559.Google Scholar
  38. Pratt, V. 1987. Direct least squares fitting of algebraic surfaces. Computer Graphics, 21(4):145-152.Google Scholar
  39. Roth, G. and Levine, M. D. 1993. Extracting geometric primitives. Computer Vision, Graphics, and Image Processing, 58(1):1-22.Google Scholar
  40. Rousseeuw, P. J. 1987. Robust Regression and Outlier Detection. Wiley: New York.Google Scholar
  41. Sampson, P. D. 1982. Fitting conic sections to 'very scattered' data: An iterative refinement of the Bookstein algorithm. Computer Graphics and Image Processing, 18:97-108.Google Scholar
  42. Shapiro, L. S. and Brady, J. M. 1995. Rejecting outliers and estimating errors in an orthogonal regression framework. Phil. Trans. R. Soc. Lond. A, 350:407-439.Google Scholar
  43. Spetsakis, M. and Aloimonos, Y. 1991. Amulti-frame approach to visual motion perception. International Journal of Computer Vision, 6:245-255.Google Scholar
  44. Sprent, P. 1989. Applied Nonparametric Statistical Methods. Chapman and Hall: London.Google Scholar
  45. Stewart, C. V. 1995. MINPRAN, a new robust estimator for computer vision. IEEE Trans. on Pattern Analysis and Machine Intelligence, 17(10):925-938.Google Scholar
  46. Taubin, G. 1991. Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 13(11):1115-1138.Google Scholar
  47. Thisted, R. A. 1988. Elements of Statistical Computing. Chapman and Hall: New York.Google Scholar
  48. Torr, P. H. S. 1995. Outlier Detection and Motion Segmentation. D. Phil. Thesis, University of Oxford.Google Scholar
  49. Torr, P. H. S. and Murray, D. W. 1992. Statistical detection of non-rigid motion. In Proc. 3rd British Machine Vision Conference, Leeds, D. Hogg (Ed.), Springer-Verlag, pp. 79-88.Google Scholar
  50. Torr, P. H. S. and Murray, D. W. 1993a. Statistical detection of independent movement from a moving camera. Image and Vision Computing, 1(4):180-187.Google Scholar
  51. Torr, P. H. S. and Murray, D. W. 1993b. Outlier detection and motion segmentation. In Proc. Sensor Fusion VI, Boston, MA, P. S. Schenker (Ed.), vol. SPIE 2059, pp. 432-443.Google Scholar
  52. Torr, P. H. S. and Murray, D. W. 1994. Stochastic motion segmentation. In Proc. 3rd European Conference on Computer Vision, Stockholm, J.-O. Ecklundh (Ed.), Springer-Verlag, pp. 328- 338.Google Scholar
  53. Torr, P. H. S., Zisserman, A., and Maybank, S. 1995a. Robust detection of degeneracy. Proc. 5th Int. Conf. on Computer Vision, Boston, MA, IEEE Computer Society Press: Los Alamitos CA, pp. 1037-1044.Google Scholar
  54. Torr, P. H. S, Zisserman, A., and Murray, D. W. 1995b. Motion clustering using the trilinear constraint over three views. In Europe-China Workshop on Geometrical Modelling and Invariants for Computer Vision, R. Mohr and C. Wu (Eds.), Springer-Verlag, pp. 118- 125.Google Scholar
  55. Torr, P. H. S, Maybank, S., and Zisserman, A. 1996. Robust detection of degenerate configurations for the fundamental matrix. OUEL Report 2090/96, Department of Engineering Science, University of Oxford.Google Scholar
  56. Teukolsky, S. A., Press, W. H., Flannery, B. P., and Vetterling, W. T. 1988. Numerical Recipes in C, the Art of Scientific Computing. Cambridge University Press: Cambridge.Google Scholar
  57. Tsai, R. Y. and Huang, T. S. 1984. Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:13-27.Google Scholar
  58. Weng, J., Huang, T. S., and Ahuja, N. 1989. Motion and structure from two perspective views: Algorithms, error analysis, and error estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11:451-476.Google Scholar
  59. Weng, J., Ahuja, N., and Huang, T. S. 1993. Optimal motion and structure estimation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 15(9):864-884.Google Scholar
  60. Zhang, Z., Deriche, R., Faugeras, O. D., and Luong, Q. T. 1994. A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. AI Journal, 78:87- 119.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • P.H.S. Torr
    • 1
  • D.W. Murray
    • 1
  1. 1.Department of Engineering ScienceUniversity of OxfordOxfordUK

Personalised recommendations