Abstract
Regression analysis (fitting a model to noisy data) is a basic technique in computer vision, Robust regression methods that remain reliable in the presence of various types of noise are therefore of considerable importance. We review several robust estimation techniques and describe in detail the least-median-of-squares (LMedS) method. The method yields the correct result even when half of the data is severely corrupted. Its efficiency in the presence of Gaussian noise can be improved by complementing it with a weighted least-squares-based procedure. The high time-complexity of the LMedS algorithm can be reduced by a Monte Carlo type speed-up technique. We discuss the relationship of LMedS with the RANSAC paradigm and its limitations in the presence of noise corrupting all the data, and we compare its performance with the class of robust M-estimators. References to published applications of robust techniques in computer vision are also given.
Similar content being viewed by others
References
Andrews, D.F. 1974. A robust method for multiple linear regression. Technometrics 16:523–531.
Beaton, A.E., and Tukey, J.W. 1974. The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics 16:147–185.
Besl, P.J., Birch, J.B., and Watson, L.T. 1988. Robust window operators. Proc. 2nd Intern. Conf. Comput. Vision, Tampa, FL., pp. 591–600. See also, Mach. Vision Appl. 2:179–214, 1989.
Bolles, R.C., and Fischler, M.A. 1981. A RANSAC-based approach to model fitting and its application to finding cylinders in range data. Proc. 7th Intern. Joint Conf. Artif. Intell., Vancouver, Canada, pp. 637–643.
Bovik, A.C., Huang, T.S., and MunsonJr., D.C. 1987. The effect of median filtering on edge estimation and detection. IEEE Trans. Patt. Anal. Mach. Intell. PAMI-9:181–194.
Brown, G.W., and Mood, A.M. 1951. On median tests for linear hypotheses. Proc. 2nd Berkeley Symp. Math. Stat. Prob. J., Neyman (ed.), University of California Press: Berkeley and Los Angeles, pp. 159–166.
Cheng, K.S., and Hettmansperger, T.P. 1983. Weighted least-squares rank estimates. Commun. Stat. A12:1069–1086.
Coyle, E.J., Lin, J.H., and Gabbouj, M. 1989. Optimal stack filtering and the estimation and structural approaches to image processing. IEEE Trans. Acoust. Speech. Sig. Process. 37:2037–2066.
Edelsbrunner, H., and Souvaine, D.L. 1988. Computing median-of-squares regression lines and guided topological sweep. Report UIUCDCS-R-88-1483. Department of Computer Science, University of Illinois at Urbana-Champaign.
Efron, B. 1988. Computer-intensive methods in statistical regression. SIAM Review 30:421–449.
Fischler, M.A., and Bolles, R.C. 1981. Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Commu. ACM 24:381–395.
Fitch, J.P., Coyle, E.J., and GallagherJr., N.C. 1985. Root properties and convergence rates of median filters,” IEEE Trans. Acoust., Speech, Sig. Process. 33:230–240.
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., and Stahel, W.A. 1986. Statistics: An Approach Based on Influence Functions. Wiley: New York.
Haralick, R.M., and Watson, L., 1981. A facet model for image data. Comput. Graph. Image Process. 15:113–129.
Haralick, R.M., and Joo, H. 1988. 2D–3D pose estimation. Proc. 9th Intern. Conf. Patt. Recog. Rome, pp. 385–391. See also, IEEE Trans. Systems, Man, Cybern. 19:1426–1446, 1989.
Heiler, S. 1981. Robust estimates in linear regression—A simulation approach. In Computational Statistics. H., Büning and P., Naeve (eds.), De Gruyter, Berlin, pp. 115–136.
Holland, P.W., and Welsch, R.E. 1977. Robust regression using iteratively reweighted least squares. Commun. Stat. A6:813–828.
Huber, P.J. 1981. Robust Statistics. Wiley: New York.
Jaeckel, L.A. 1972. Estimating regression coefficients by minimizing the dispersion of residuals. Ann. Math. Stat. 43:1449–1458.
Johnstone, I.M., and Velleman, P.F. 1985. The resistant line and related regression methods. J. Amer. Stat. Assoc. 80:1041–1059.
Jolion, J.M., Meer, P., and Rosenfeld, A. 1990a. Generalized minimum volume ellipsoid clustering with applications in computer vision. Proc. Intern. Workshop Robust Comput. Vision, Seattle, WA, pp. 339–351.
Jolion, J.M., Meer, P., and Bataouche, S., 1990b. Range image segmentation by robust clustering. CAR-TR-500. Computer Vision Laboratory, University of Maryland, College Park.
Kamgar-Paris, B., Kamgar-Paris, B., and Netanyahu, N.S. 1989. A nonparametric method for fitting a straight line to a noisy image. IEEE Trans. Patt. Anal. Mach. Intell. PAMI-11:998–1001.
Kashyap, R.L., and Eom, K.B. 1988. Robust image models and their applications. In Advances in Electronics and Electron Physics, vol. 70, Academic Press: San Diego, CA, pp. 79–157.
Kim, D.Y., Kim, J.J., Meer, P., Mintz, D., and Rosenfeld, A. 1989. Robust computer vision: A least median of squares based approach. Proc. DARPA Image Understanding Workshop, Palo Alto, CA, pp. 1117–1134.
Koivo, A.J., and Kim, C.W. 1989. Robust image modeling for classification of surface defects on wood boards. IEEE Trans. Syst. Man, Cybern. 19:1659–1666.
Kumar, R., and Hanson, A.R. 1989. Robust estimation of camera location and orientation from noisy data having outliers. Proc. Workshop on Interpretation of 3D Scenes, Austin, TX, pp. 52–60.
Lee, C.N., Haralick, R.M., and Zhuang, X. 1989. Recovering 3-D motion parameters from image sequences with gross errors. Proc. Workshop on Visual Motion, Irvine, CA, pp. 46–53.
Li, G. 1985. Robust regression: In D.C., Hoaglin, F., Mosteller, and J.W., Tukey (eds.), Exploring Data Tables, Trends and Shapes. Wiley: New York, pp. 281–343.
Meer, P., Mintz, D., and Rosenfeld, A. 1990. Least median of squares based robust analysis of image structure. Proc. DARPA Image Understanding Workshop, Pittsburgh, PA, pp. 231–254.
Mintz, D., and Amir, A. 1989. Generalized random sampling. CARTR-441. Computer Vision Laboratory, University of Maryland, College Park.
Mintz, D., Meer, P., and Rosenfeld, A. 1990. A fast, high breakdown point robust estimator for computer vision applications. Proc. DARPA Image Understanding Workshop, Pittsburgh, PA, pp. 255–258.
Mosteller, F., and Tukey, J.W. 1977. Data Analysis and Regression. Addison-Wesley: Reading, MA.
Owens, R., Venkatesh, S., and Ross, J. 1989. Edge detection is a projection. Patt. Recog. Lett. 9:233–244.
Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W.T. 1988. Numerical Recipes. Cambridge University Press, Cambridge, England.
Rousseeuw, P.J. 1984. Least median of squares regression. J. Amer. Stat. Assoc. 79:871–880.
Rousseeuw, P.J., and Leroy, A.M. 1987. Robust Regression & Outlier Detection. Wiley: New York.
Siegel, A.F. 1982. Robust regression using repeated medians. Biometrika 69:242–244.
Steele, J.M., and Steiger, W.L. 1986. Algorithms and complexity for least median of squares regression. Discrete Appl. Math. 14:93–100.
Theil, H. 1950. A rank-invariant method of linear and polynomial regression analysis (parts 1–3). Ned. okad. Wetensch. Proc. ser. A53:386–392, 521–525, 1397–1412.
Tirumalai, A., and Schunck, B.G. 1988. Robust surface approximation using least median squares regression. CSE-TR-13–89. Artificial Intelligence Laboratory, University of Michigan, Ann Arbor.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Meer, P., Mintz, D., Rosenfeld, A. et al. Robust regression methods for computer vision: A review. Int J Comput Vision 6, 59–70 (1991). https://doi.org/10.1007/BF00127126
Issue Date:
DOI: https://doi.org/10.1007/BF00127126