A Convex Approach to Low Rank Matrix Approximation with Missing Data

  • Carl Olsson
  • Magnus Oskarsson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5575)

Abstract

Many computer vision problems can be formulated as low rank bilinear minimization problems. One reason for the success of these problems is that they can be efficiently solved using singular value decomposition. However this approach fails if the measurement matrix contains missing data.

In this paper we propose a new method for estimating missing data. Our approach is similar to that of L 1 approximation schemes that have been successfully used for recovering sparse solutions of under-determined linear systems. We use the nuclear norm to formulate a convex approximation of the missing data problem. The method has been tested on real and synthetic images with promising results.

Keywords

Singular Value Decomposition Measurement Matrix Nuclear Norm Photometric Stereo Miss Data Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Tardif, J., Bartoli, A., Trudeau, M., Guilbert, N., Roy, S.: Algorithms for batch matrix factorization with application to structure-from-motion. In: Int. Conf. on Computer Vision and Pattern Recognition, Minneapolis, USA (2007)Google Scholar
  2. 2.
    Sturm, P., Triggs, B.: A factorization bases algorithm for multi-image projective structure and motion. In: European Conference on Computer Vision, Cambridge, UK (1996)Google Scholar
  3. 3.
    Tomasi, C., Kanade, T.: Shape and motion from image sttreams under orthography: a factorization method. Int. Journal of Computer Vision 9 (1992)Google Scholar
  4. 4.
    Bregler, C., Hertzmann, A., Biermann, H.: Recovering non-rigid 3D shape from image steams. In: Int. Conf. on Computer Vision and Pattern Recognition, Hilton Head, SC, USA (2000)Google Scholar
  5. 5.
    Xiao, J., Kanade, T.: A closed form solution to non-rigid shape and motion recovery. International Journal of Computer Vision 67, 233–246 (2006)CrossRefGoogle Scholar
  6. 6.
    Yan, J., Pollefeys, M.: A factorization approach to articulated motion recovery. In: IEEE Conf. on Computer Vision and Pattern Recognition, San Diego, USA (2005)Google Scholar
  7. 7.
    Basri, R., Jacobs, D., Kemelmacher, I.: Photometric stereo with general, unknown lighting. Int. Journal of Computer Vision 72, 239–257 (2007)CrossRefGoogle Scholar
  8. 8.
    Hartley, R., Schaffalitzky, F.: Powerfactoriztion: An approach to affine reconstruction with missing and uncertain data. In: Australia-Japan Advanced Workshop on Computer Vision, Adelaide, Australia (2003)Google Scholar
  9. 9.
    Buchanan, A., Fitzgibbon, A.: Damped newton algorithms for matrix factorization with missing data. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR 2005, June 20-25, 2005, vol. 2, pp. 316–322 (20)Google Scholar
  10. 10.
    Recht, B., Fazel, M., Parrilo, P.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization (2007), http://arxiv.org/abs/0706.4138v1
  11. 11.
    Fazel, M., Hindi, H., Boyd, S.: A rank minimization heuristic with application to minimum order system identification. In: Proceedings of the American Control Conference (2003)Google Scholar
  12. 12.
    El Ghaoui, L., Gahinet, P.: Rank minimization under lmi constraints: A framework for output feedback problems. In: Proceedings of the European Control Conference (1993)Google Scholar
  13. 13.
    Tropp, J.: Just relax: convex programming methods for identifying sparse signals in noise. IEEE Transactions on Information Theory 52, 1030–1051 (2006)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Donoho, D., Elad, M., Temlyakov, V.: Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Transactions on Information Theory 52, 6–18 (2006)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Candes, E., Romberg, J., Tao, T.: Stable signal recovery from incomplete and inaccurate measurments. Communications of Pure and Applied Mathematics 59, 1207–1223 (2005)CrossRefMATHGoogle Scholar
  16. 16.
    Golub, G., van Loan, C.: Matrix Computations. The Johns Hopkins University Press (1996)Google Scholar
  17. 17.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  18. 18.
    Sturm, J.F.: Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones (1998)Google Scholar
  19. 19.
    Torresani, L., Hertzmann, A., Bregler, C.: Non-rigid structure-from-motion: Estimating shape and motion with hierarchical priors. IEEE Transactions on Pattern Analysis and Machine Intelligence 30 (2008)Google Scholar
  20. 20.
    Raiko, T., Ilin, A., Karhunen, J.: Principal component analysis for sparse high-dimensional data. In: 14th International Conference on Neural Information Processing, Kitakyushu, Japan, pp. 566–575 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Carl Olsson
    • 1
  • Magnus Oskarsson
    • 1
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden

Personalised recommendations