Abstract
The bootstrap is a numerical technique, with solid theoretical foundations, to obtain statistical measures about the quality of an estimate by using only the available data. Performance assessment through bootstrap provides the same or better accuracy than the traditional error propagation approach, most often without requiring complex analytical derivations. In many computer vision tasks a regression problem in which the measurement errors are point dependent has to be solved. Such regression problems are called heteroscedastic and appear in the linearization of quadratic forms in ellipse fitting and epipolar geometry, in camera calibration, or in 3D rigid motion estimation. The performance of these complex vision tasks is difficult to evaluate analytically, therefore we propose in this paper the use of bootstrap. The technique is illustrated for 3D rigid motion and fundamental matrix estimation. Experiments with real and synthetic data show the validity of bootstrap as an evaluation tool and the importance of taking the heteroscedasticity into account.
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References
K.S. Arun, T.S. Huang and S.D. Blostein, “Least-squares fitting of two 3D point sets”, IEEE Transactions on Pattern Analysis and Machine Intelligence. vol. 9, pp. 698–700, 1987.
S.D. Blostein and T.S. Huang, “Error analysis in stereo determination of 3D point positions”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 9, pp. 752–765, 1987.
G. Csurka, C. Zeller, Z. Zhang and O. Faugeras, “Characterizing the Uncertainty of the Fundamental Matrix”, Computer Vision and Image Understanding, Vol. 68, pp. 18–36, 1997.
A.C. Davison and D.V. Hinkley, Bootstrap Methods and their Application, Cambridge University Press, 1998.
B. Efron and R.J. Tibshirani, An Introduction to the Bootstrap, Chapman&Hall, 1993.
D.W. Eggert, A. Lorusso and R.B. Fisher, “Estimating 3-D rigid body transformations: A comparison of four major algorithms”, Machine Vision and Applications, Vol. 9, pp. 272–290, 1997.
O. Faugeras, Three-dimensional Computer Vision. A Geometric Viewpoint, MIT Press, 1993.
S. Yi, R.H. Haralick and L. Shapiro, “Error propagation in machine vision”, Machine Vision and Applications, vol. 7, pp. 93–114, 1994.
R.I. Hartley, “In Defense of the 8-Point Algorithm”, Proceedings of the 5th International Conference on Computer Vision, Cambridge (MA), pp. 1064–1070, 1995.
R. I. Hartley, “Triangulation”, Computer Vision and Image Understanding, vol. 68, pp. 146–157, 1997.
B.K.P. Horn, H.M. Hilden and S. Negahdaripour, “Closed-form solution of absolute orientation using orthonormal matrices”, J. Opt. Soc. Am. vol. 5, pp. 1127–1135, 1988.
K. Kanatani, Geometric Computation for Machine Vision, Oxford Science Publications, 1993.
K. Kanatani, “Analysis of 3-D rotation fitting”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, No. 5, pp. 543–549, 1994.
K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier, 1996.
Y. Leedan and P. Meer, “Estimation with bilinear constraints in computer vision”, Proceedings of the 5th International Conference on Computer Vision, Bombay, India, pp. 733–738, 1998.
B. Matei and P. Meer, “Optimal rigid motion estimation and performance evaluation with bootstrap”, Proceedings of the Computer Vision and Pattern Recognition 99, Fort Collins Co., vol 1, pp. 339–345, 1999.
N. Ohta and K. Kanatani, “Optimal estimation of three-dimensional rotation and reliability evaluation”, Computer Vision-ECCV 98’, H. Burkhardt, B. Neumann Eds., Lecture Notes in Computer Science, Springer, pp. 175–187, 1998.
T. Kato, A Short Introduction to Perturbation Theory for Linear Operators, Springer-Verlag, 1982.
H.C. Longuet-Higgins, “A Computer Algorithm for Reconstructing a Scene from Two Projections”, Nature, Vol. 293, pp. 133–135, 1981.
X. Pennec and J.P. Thirion, “A framework for uncertainty and validation of 3-D registration methods based on points and frames”, International Journal on Computer Vision, vol. 25, pp. 203–229, 1997.
H. Sahibi and A. Basu, “Analysis of error in depth perception with vergence and spatially varying sensing”, Computer Vision and Image Understanding vol. 63, pp. 447–461, 1996.
L.L. Scharf, Statistical Signal Processing, Addison-Wesley, 1990.
E. Trucco and A. Verri, Introductory Techniques for 3-D Computer Vision, Prentice Hall, 1998.
S. Umeyama, “Least-squares estimation of transformation parameters between two point patterns”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, pp. 376–380, 1991.
Z. Zhang, R. Deriche, O. Faugeras, Q.-T. Luong, “A robust technique for matching two uncalibrated images through the recovery of unknown epipolar geometry”, Artificial Intelligence, vol. 78, 87–119, 1995.
Z. Zhang, “On the Optimization Criteria Used in Two View Motion Analysis”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 20, pp. 717–729, 1998.
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Matei, B., Meer, P. (2000). Bootstrapping Errors-in-Variables Models. In: Triggs, B., Zisserman, A., Szeliski, R. (eds) Vision Algorithms: Theory and Practice. IWVA 1999. Lecture Notes in Computer Science, vol 1883. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44480-7_15
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DOI: https://doi.org/10.1007/3-540-44480-7_15
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