Abstract
In this paper, we address the problem of robustly recovering several instances of a curve model from a single noisy data set with outliers. Using M-estimators revisited in a Lagrangian formalism, we derive an algorithm that we call Simultaneous Multiple Robust Fitting (SMRF), which extends the classical Iterative Reweighted Least Squares algorithm (IRLS). Compared to the IRLS, it features an extra probability ratio, which is classical in clustering algorithms, in the expression of the weights. Potential numerical issues are tackled by banning zero probabilities in the computation of the weights and by introducing a Gaussian prior on curves coefficients. Applications to camera calibration and lane-markings tracking show the effectiveness of the SMRF algorithm, which outperforms classical Gaussian mixture model algorithms in the presence of outliers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Tarel, J.-P., Ieng, S.-S., Charbonnier, P.: Using robust estimation algorithms for tracking explicit curves. In: European Conference on Computer Vision (ECCV 2002), Copenhagen, Danmark, vol. 1, pp. 492–507 (2002)
Mizera, I., Müller, C.: Breakdown points and variation exponents of robust M-estimators in linear models. The Annals of Statistics 27(4), 1164–1177 (1999)
Huber, P.J.: Robust Statistics. John Wiley and Sons, New York (1981)
Geman, D., Reynolds, G.: Constrained restoration and the recovery of discontinuities. IEEE Transactions on Pattern Analysis and Machine Intelligence 14(3), 367–383 (1992)
Charbonnier, P., Blanc-Féraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Transactions on Image Processing 6(2), 298–311 (1997)
Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B (Methodological) 39(1), 1–38 (1977)
Cambell, N.A.: Mixture models and atypical values. Mathematical Geology 16(5), 465–477 (1984)
Blake, A., Zisserman, A.: Visual Reconstruction. MIT Press, Cambridge (1987)
Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004); ISBN: 0521540518
Ieng, S.-S., Tarel, J.-P., Charbonnier, P.: Evaluation of robust fitting based detection. In: Pajdla, T., Matas, J(G.) (eds.) ECCV 2004. LNCS, vol. 3024, pp. 341–352. Springer, Heidelberg (2004)
Bigorgne, E., Tarel, J.-P.: Backward segmentation and region fitting for geometrical visibility range estimation. In: Yagi, Y., Kang, S.B., Kweon, I.S., Zha, H. (eds.) ACCV 2007, Part II. LNCS, vol. 4844, pp. 817–826. Springer, Heidelberg (2007)
Tarel, J.-P., Ieng, S.-S., Charbonnier, P.: Accurate and robust image alignment for road profile reconstruction. In: Proceedings of IEEE International Conference on Image Processing (ICIP 2007), San Antonio, Texas, USA, vol. V, pp. 365–368 (2007)
Tarel, J.-P., Ieng, S.-S., Charbonnier, P.: Robust Lane Marking Detection by the Half Quadratic Approach. Collection Etudes et Recherches des Laboratoires des Ponts et Chaussées, CR 49, LCPC (November 2007)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Minoux, M.: Mathematical Programming: Theory and Algorithms. John Wiley and Sons, Chichester (1986)
Luenberger, D.G.: Introduction to linear and nonlinear programming. Addison Wesley, Reading (1973)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tarel, JP., Charbonnier, P., Ieng, SS. (2008). A Revisited Half-Quadratic Approach for Simultaneous Robust Fitting of Multiple Curves. In: Braz, J., Ranchordas, A., Araújo, H.J., Pereira, J.M. (eds) Computer Vision and Computer Graphics. Theory and Applications. VISIGRAPP 2007. Communications in Computer and Information Science, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89682-1_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-89682-1_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89681-4
Online ISBN: 978-3-540-89682-1
eBook Packages: Computer ScienceComputer Science (R0)