Skip to main content
SpringerLink
Log in

A Framework for Robust Subspace Learning

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

Many computer vision, signal processing and statistical problems can be posed as problems of learning low dimensional linear or multi-linear models. These models have been widely used for the representation of shape, appearance, motion, etc., in computer vision applications. Methods for learning linear models can be seen as a special case of subspace fitting. One draw-back of previous learning methods is that they are based on least squares estimation techniques and hence fail to account for “outliers” which are common in realistic training sets. We review previous approaches for making linear learning methods robust to outliers and present a new method that uses an intra-sample outlier process to account for pixel outliers. We develop the theory of Robust Subspace Learning (RSL) for linear models within a continuous optimization framework based on robust M-estimation. The framework applies to a variety of linear learning problems in computer vision including eigen-analysis and structure from motion. Several synthetic and natural examples are used to develop and illustrate the theory and applications of robust subspace learning in computer vision.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aguiar, P. and Moura, J. 1999. Factorization as a rank 1 problem. In Conference on ComputerVision andPattern Recognition, pp. 178- 184.

  • Baldi, P. and Hornik, K. 1989. Neural networks and principal component analysis: Learning from examples without local minima. Neural Networks, 2:53-58.

    Google Scholar 

  • Beaton, A.E. and Tukey, J.W. 1974. The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics, 16(2):147-185.

    Google Scholar 

  • Black, M.J. and Anandan, P. 1996. The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields. Computer Vision and Image Understanding, 63(1):75-104.

    Google Scholar 

  • Black, M.J. and Jepson, A.D. 1998. Eigentracking: Robust matching and tracking of objects using view-based representation. International Journal of Computer Vision, 26(1):63-84.

    Google Scholar 

  • Black, M.J. and Rangarajan, A. 1996. On the unification of line processes, outlier rejection, and robust statistics with applications in early vision. International Journal of Computer Vision, 25(19):57- 92.

    Google Scholar 

  • Black, M.J., Sapiro, G., Marimont, D., and Heeger, D. 1998. Robust anisotropic diffusion. IEEE Transactions on Image Processing, 7:421-432.

    Google Scholar 

  • Black, M.J., Yaccob, Y., Jepson, A., and Fleet, D.J. 1997. Learning parameterized models of image motion. In Conference on Computer Vision and Pattern Recognition, pp. 561-567.

  • Blake, A. and Isard, M. 1998. Active Contours. Springer Verlag.

  • Blake, A. and Zisserman, A. 1987. Visual Reconstruction. MIT Press series, Massachusetts.

    Google Scholar 

  • Campbell, N.A. 1980. Robust procedures in multivariate analysis I: Robust covariance estimation. Applied Statistics, 29(3):231-2137.

    Google Scholar 

  • Cardoso, J.F. 1996. Independent component analysis, a survey of some algebraic methods. In International Symposium Circuits and Systems, vol. 2, pp. 93-96.

    Google Scholar 

  • Carroll, J. and Chang, J. 1970. Analysis of individual differences in multidimensional scaling via an n-way generalization eckartyoung decomposition. Psychometrika, 35:283-319.

    Google Scholar 

  • Cichocki, A., Unbehauen, R., and Rummert, E. 1993. Robust learning algorithm for blind separation of signals. Electronics Letters, 30(17):1386-1387.

    Google Scholar 

  • Cootes, T.F., Edwards, G.J., and Taylor, C.J. 1998. Active appearance models. In European Conference Computer Vision, pp. 484-498.

  • Croux, C. and Filzmoser, P. 1998. Robust factorization of a data matrix. In COMPSTAT, Proceedings in Computational Statistics, pp. 245-249.

  • De la Torre, F. and Black, M.J. 2001. Dynamic coupled component analysis. In Computer Vision and Pattern Recognition, pp. 643-650.

  • De la Torre, F. and Black, M.J. 2001. Robust principal component analysis for computer vision. In International Conference on Computer Vision, pp. 362-369.

  • De la Torre, F. and Black, M.J. 2002. Robust parameterized component analysis: Theory and applications to 2d facial modeling. In European Conference on Computer Vision, pp. 653-669.

  • Diamantaras, K.I. 1996. Principal Component Neural Networks (Therory and Applications). John Wiley & Sons.

  • Eckardt, C. and Young, G. 1936. The approximation of one matrix by another of lower rank. Psychometrika, 1:211-218.

    Google Scholar 

  • Everitt, B.S. 1984. An Introduction to Latent Variable Models. Chapman and Hall: London.

    Google Scholar 

  • Fukunaga, K. 1990. Introduction to Statistical Pattern Recognition, 2nd edn. Academic Press: Boston, MA.

    Google Scholar 

  • Gabriel, K.R. and Odoroff, C.L. 1984. Resistant lower rank approximation of matrices. In Data Analysis and Informatics, III, pp. 23- 30.

  • Gabriel, K.R. and Zamir, S. 1979. Lower rank approximation of matrices by least squares with any choice of weights. Technometrics, 21:489-498.

    Google Scholar 

  • Geiger, D. and Pereira, R. 1991. The outlier process. In IEEE Workshop on Neural Networks for Signal Proc., pp. 61-69.

  • Geman, S. and McClure, D. 1987. Statistical methods for tomographic image reconstruction. Bulletin of the International Statistical Institute, LII:4-5.

    Google Scholar 

  • Golub, G. and Van Loan, C.F. 1989. Matrix Computations. 2nd ed. The Johns Hopkins University Press.

  • Golub, G.H. and van der Vorst, H.A. 2000. Eigenvalue computation in the 20th century. Journal of Computational and Applied Mathematics, 123:35-65.

    Google Scholar 

  • Greenacre, M.J. 1984. Theory and Applications of Correspondence Analysis. Academic Press: London.

    Google Scholar 

  • Greenacre, M.J. 1988. Correspondence analysis of multivariate categorical data by weighted least squares. Biometrika, 75:457-467.

    Google Scholar 

  • Hampel, F., Ronchetti, E., Rousseeuw, P., and Stahel, W. 1986. Robust Statistics: The Approach Based on Influence Functions. Wiley: New York.

    Google Scholar 

  • Hartley, R.I. and Zisserman, A. 2000. Multiple View Geometry in Computer Vision. Cambridge University Press.

  • Holland, P.W. and Welsch, R.E. 1977. Robust regression using iteratively reweighted least-squares. Communications in Statistics, (A6):813-827.

    Google Scholar 

  • Huber, P.J. 1981. Robust Statistics. Wiley: New York.

    Google Scholar 

  • Irani, M. and Anandan, P. 2000. Factorization with uncertainty. In European Conference on Computer Vision, pp. 539-553.

  • Neudecker, H. and Magnus, J.R. 1999. Matrix Differential Calculus with Applications in Statistics and Econometrics. John Wiley.

  • Jolliffe, I.T. 1986. Principal Component Analysis. Springer-Verlag: New York.

    Google Scholar 

  • Karhunen, J. and Joutsensalo, J. 1995. Generalizations of principal component analysis, optimization problems, and neural networks. Neural Networks, 4(8):549-562.

    Google Scholar 

  • Kroonenberg, P. and de Leeuw, J. 1980. Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45:69-97.

    Google Scholar 

  • Lai, S.H. 2000. Robust image alignment under partial occlusion and spatially varying illumination change. Computer Vision and Image Understanding, 78:84-98.

    Google Scholar 

  • Li, G. 1985. Robust regression. In Exploring Data, Tables, Trends and Shapes. D.C. Hoaglin, F. Mosteller, and J.W. Tukey (Eds.). John Wiley & Sons.

  • MacLean, J., Jepson, A., and Frecker, R. 1994. Recovery of egomotion and segmentation of indepedent object motion using the EM-algorithm. In British Machine Vision Conference, pp. 175-184, Leeds, UK.

    Google Scholar 

  • Mardia, K., Kent, J., and Bibby, J. 1979. Multivariate Analysis. Academic Press: London.

    Google Scholar 

  • Meer, P., Mintz, D., Kim, D., and Rosenfeld, A. 1991. Robust regression methods in computer vision: A review. International Journal of Computer Vision, 6:59-70.

    Google Scholar 

  • Meer, P., Stewart, C., and Tyler, D. (Eds.). 2000. Special issue on robust statistics. Computer Vision and Image Understanding, 78(1).

  • Mirsky, L. 1960. Symmetric gauge functions and unitarily invariant norms. Quart. J. Marth. Oxford, 11:50-59.

    Google Scholar 

  • Moghaddam, B. and Pentland, A. 1997. Probabilistic visual learning for object representation. Pattern Analysis and Machine Intelligence, 19(7):137-143.

    Google Scholar 

  • Morris, D. and Kanade, T. 1998. A unified factorization algorithm for points, line segments and planes with uncertainty models. In International Conference on Computer Vision, pp. 696-702.

  • Murase, H. and Nayar, S.K. 1995. Visual learning and recognition of 3D objects from appearance. International Journal of Computer vision, 1(14):5-24.

    Google Scholar 

  • Oja, E. 1982. A simplified neuron model as principal component analyzer. Journal of Mathematical Biology, 15:267-273.

    Google Scholar 

  • Oliver, N., Rosario, B., and Pentland, A. 1999. A Bayesian computer vision system for modeling human interactions. In Int. Conf. Computer on Vision Systems, ICVS, H.I. Christensen (Ed.), vol. 1542 of LNCS-Series, Gran Canaria, Spain, Springer-Verlag, pp. 255-272.

    Google Scholar 

  • Parlett, B.N. 1980. The Symmetric Eigenvalue Problem. Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  • Poelman, C. and Kanade, T. 1994. A paraperspective factorization method for shape and motion recovery. In International Conference on Computer Vision, pp. 97-108.

  • Rao, R.P.N. 1999. An optimal estimation approach to visual perception and learning. Vision Research, 39(11):1963-1989.

    Google Scholar 

  • Rousseeuw, P.J. and Leroy, A.M. 1987. Robust Regression and Outlier Detection. John Wiley and Sons.

  • Roweis, S. 1997. EM algorithms for PCA and SPCA. In Neural Information Processing Systems, pp. 626-632.

  • Ruymagaart, F.H. 1981. A robust principal component analysis. Journal of Multivariate Analysis, 11:485-497.

    Google Scholar 

  • Sanger, T.D. 1989. Optimal unsupervised learning in a single-layer linear feedforward neural network. Neural Networks, 2:459-473.

    Google Scholar 

  • Shi, J. and Malik, J. 2000. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8).

  • Shum, H., Ikeuchi, K., and Reddy, R. 1995. Principal component analysis with missing data and its application to polyhedral object modeling. Pattern Analysis and Machine Intelligence, 17(9):855- 867.

    Google Scholar 

  • Sidenbladh, H., de la Torre, F., and Black, M.J. 2000. A framework for modeling the appearance of 3D articulated figures. In Face and Gesture Recognition, pp. 368-375.

  • Skoaj, D., Bischof, H., and Leonardis, A. 2002. A robust PCA algorithm for building representations from panoramic images. In European Conference Computer Vision, pp. 761-775.

  • Tenenbaum, J.B. and Freeman, W.T. 2000. Separating style and context with bilinear models. Neural Computation, 12(6):1247-1283.

    Google Scholar 

  • Tipping, M. and Bishop, C.M. 1999. Probabilistic principal component analysis. Journal of the Royal Statistical Society B, 61:611-622.

    Google Scholar 

  • Tomasi, C. and Kanade, T. 1992. Shape and motion from image streams under orthography: A factorization method. Int. Jorunal of Computer Vision., 9(2):137-154.

    Google Scholar 

  • Turk, M. and Pentland, A. 1991. Eigenfaces for recognition. Journal Cognitive Neuroscience, 3(1):71-86.

    Google Scholar 

  • Van Huffel, S. and Vandewalle, J. 1991. The Total Least Squares Problem: Computational Aspects and Analysis. Society for Industrial and Applied Mathematics, Philadelphia.

    Google Scholar 

  • Xu, L. 1993. Least mean square error recosntruction for self-organizing nerual nets. Neural Networks, 6:627-648.

    Google Scholar 

  • Xu, L. and Yuille, A. 1995. Robust principal component analysis by self-organizing rules based on statistical physics approach. IEEE Transactions on Neural Networks, 6(1):131-143.

    Google Scholar 

  • Yang, T.N. and Wang, S.D. 1999. Robust algorithms for principal component analysis. Pattern Recognition Letters, 20(9):927-933.

    Google Scholar 

  • Yuille, A.L., Snow, D., Epstein, R., and Belhumeur, P. 1999. Determining generative models for objects under varying illumination: Shape and albedo from multiple images using svd and integrability. International Journal of Computer Vision, 35(3):203-222.

    Google Scholar 

  • Zhang, Z. 1996. Parameter estimation techniques: A tutorial with application to conic fitting. Image and vision Computing, 15(1):59- 76.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Communications and Signal Theory, La Salle School of Engineering, Universitat Ramon LLull, Barcelona, 08022, Spain

    Fernando De la Torre

  2. Department of Computer Science, Brown University, Box 1910, Providence, RI, 02912, USA

    Michael J. Black

Authors
  1. Fernando De la Torre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael J. Black
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

De la Torre, F., Black, M.J. A Framework for Robust Subspace Learning. International Journal of Computer Vision 54, 117–142 (2003). https://doi.org/10.1023/A:1023709501986

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023709501986

Search

Navigation