Robust Tensor Classifiers for Color Object Recognition

  • Christian Bauckhage
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4633)


This paper presents an extension of linear discriminant analysis to higher order tensors that enables robust color object recognition. Given a labeled sample of training images, the basic idea is to consider a parallel factor model of a corresponding projection tensor. In contrast to other recent approaches, we do not compute a higher order singular value decomposition of the optimal projection. Instead, we directly derive a suitable approximation from the training data. Applying an alternating least squares procedure to repeated tensor contractions allows us to compute templates or binary classifiers alike. Moreover, we show how to incorporate a regularization method and the kernel trick in order to better cope with variations in the data. Experiments on face recognition from color images demonstrate that our approach performs very reliably, even if just a few examples are available for training.


Face Recognition Linear Discriminant Analysis Training Image Alternate Little Square Projection Tensor 
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.


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  1. 1.
    Shashua, A., Levin, A.: Linear Image Coding for Regression and Classification using the Tensor-rank Principle. In: Proc. CVPR, vol. I, pp. 40–42 (2001)Google Scholar
  2. 2.
    Vasilescu, M., Terzopoulos, D.: Multilinear Analsysis of Image Ensembles: Tensorfaces. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 447–460. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    De Lathauwer, L., De Moor, B., Vanderwalle, J.: A Multilinear Singular Value Decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1235–1278 (2000)Google Scholar
  4. 4.
    Shashua, A., Hazan, T.: Non-Negative Tensor Factorization with Applications to Statistics and Computer Vision. In: Proc. ICML, pp. 792–799 (2005)Google Scholar
  5. 5.
    Wang, H., Ahuja, N.: Compact representation of multidimensional data using tensor rank-one decomposition. In: Proc. ICPR, vol. I, pp. 44–47 (2004)Google Scholar
  6. 6.
    Vlasic, D., Brand, M., Pfister, H., Popović, J.: Face transfer with multilinear models. ACM Trans. on Graphics (Proc. SIGGRAPH’05) 24(3), 426–433 (2005)CrossRefGoogle Scholar
  7. 7.
    Wang, H., Wu, Q., Shi, L., Yu, Y., Ahuja, N.: Out-of-core tensor approximation of multi-dimensional matrices of visual data. ACM Trans. on Graphics (Proc. SIGGRAPH 2005) 24(3), 527–535 (2005)zbMATHGoogle Scholar
  8. 8.
    Tenenbaum, J., Freeman, W.: Separating Style and Content with Bilinear Models. Neural Computing 12, 1247–1283 (2000)CrossRefGoogle Scholar
  9. 9.
    Kienzle, W., Bakir, G., Franz, M., Schölkopf, B.: Face Detection – Efficient and Rank Deficient. In: Proc. NIPS, pp. 673–680 (2005)Google Scholar
  10. 10.
    Bauckhage, C., Tsotsos, J.: Separable Linear Discriminant Classification. In: Kropatsch, W.G., Sablatnig, R., Hanbury, A. (eds.) Pattern Recognition. LNCS, vol. 3663, pp. 318–325. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Bauckhage, C., Käster, T., Tsotsos, J.: Applying Ensembles of Multilinear Classifiers in the Frequency Domain. In: Proc. CVPR, vol. I, pp. 95–102 (2006)Google Scholar
  12. 12.
    Ye, J., Janardan, R., Li, Q.: Two-Dimensional Linear Discriminant Analysis. In: Proc. NIPS, pp. 1569–1576 (2005)Google Scholar
  13. 13.
    Yan, S., Xu, D., Zhang, L., Tang, X., Zhang, H.J.: Discriminant Analysis with Tensor Representation. In: Proc. CVPR, vol. 1, pp. 526–532 (2005)Google Scholar
  14. 14.
    Kolda, T.: Orthogonal Tensor Decompositions. SIAM J. Matrix Anal. Appl. 23(1), 243–255 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Zhang, T., Golub, G.: Rank-One Approximation to High Order Tensors. SIAM J. Matrix Anal. Appl. 23(2), 534–550 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Tomasi, G., Bro, R.: A Comparision of Algorithms for Fitting the PARAFAC Model. Comp. Statistics & Data Analysis 50, 1700–1734 (2006)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Fisher, R.: The Use of Multiple Measurements in Taxonomic Problems. Ann. Eugenics 7, 179–188 (1936)Google Scholar
  18. 18.
    Bishop, C.: Pattern Recognition and Machine Learning. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  19. 19.
    Martínez, A.M., Kak, A.: PCA versus LDA. IEEE Trans. Pattern Anal. and Machine Intelli. 23(2), 228–233 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Christian Bauckhage
    • 1
  1. 1.Deutsche Telekom Laboratories 10587 BerlinGermany

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