International Journal of Computer Vision

, Volume 101, Issue 3, pp 498–518 | Cite as

Euler Principal Component Analysis

  • Stephan Liwicki
  • Georgios Tzimiropoulos
  • Stefanos Zafeiriou
  • Maja Pantic


Principal Component Analysis (PCA) is perhaps the most prominent learning tool for dimensionality reduction in pattern recognition and computer vision. However, the 2-norm employed by standard PCA is not robust to outliers. In this paper, we propose a kernel PCA method for fast and robust PCA, which we call Euler-PCA (e-PCA). In particular, our algorithm utilizes a robust dissimilarity measure based on the Euler representation of complex numbers. We show that Euler-PCA retains PCA’s desirable properties while suppressing outliers. Moreover, we formulate Euler-PCA in an incremental learning framework which allows for efficient computation. In our experiments we apply Euler-PCA to three different computer vision applications for which our method performs comparably with other state-of-the-art approaches.


Euler PCA Robust subspace Online learning Tracking Background modeling 



The research presented in this paper is supported in part by the European Research Council (ERC) under the ERC Starting Grant Agreement ERC-2007- StG-203143 (MAHNOB). The work of S. Liwicki is supported by the Engineering and Physical Science Research Council DTA Studentship. The work of G. Tzimiropoulos is currently supported in part by the European Community’s 7th Framework Programme FP7/2007-2013 under Grant Agreement 288235 (FROG).

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  1. Avidan, S. (2004). Support vector tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1064–1072. doi: 10.1109/TPAMI.2004.53.
  2. Babenko, B., Yang, M., & Belongie, S. (2009). Visual tracking with online multiple instance learning. In CVPR’09 (pp. 983–990). Google Scholar
  3. Babenko, B., Yang, M., & Belongie, S. (2011). Robust object tracking with online multiple instance learning. IEEE Transactions on Pattern Analysis and Machine Intelligence. doi: 10.1109/TPAMI.2010.226.
  4. Candés, E., Li, X., Ma, Y., & Wright, J. (2009). Robust principal component analysis? Available at:
  5. Chin, T. J., & Suter, D. (2007). Incremental kernel principal component analysis. IEEE Transactions on Image Processing, 1662–1674. doi: 10.1109/TIP.2007.896668.
  6. Chin, T., Schindler, K., & Suter, D. (2006). Incremental kernel SVD for face recognition with image sets. In FG’06 (pp. 461–466). Google Scholar
  7. Cohn, J., Zlochower, A., Lien, J., & Kanade, T. (1999). Automated face analysis by feature point tracking has high concurrent validity with manual FACS coding. Psychophysiology, 35–43. doi: 10.1017/S0048577299971184.
  8. Collins, R., Lipton, J., Fujiyoshi, H., & Kanade, T. (2001). Algorithms for cooperative multisensor surveillance. In The IEEE (p. 89). doi: 10.1109/5.959341. Google Scholar
  9. Comaniciu, D., Ramesh, V., & Meer, P. (2003). Kernel-based object tracking. IEEE Transactions on Pattern Analysis and Machine Intelligence, 564–577. doi: 10.1109/TPAMI.2003.1195991.
  10. de la Torre, F., & Black, M. (2003). A framework for robust subspace learning. International Journal of Computer Vision, 117–142. doi: 10.1023/A:1023709501986.
  11. Ding, D., Zhou, D., He, X., & Zha, H. (2006). R1-PCA: rotational invariant L1-norm principal component analysis for robust subspace factorization. In ACM (pp. 281–288). doi: 10.1145/1143844.1143880. Google Scholar
  12. Fitch, A., Kadyrov, A., Christmas, W., & Kittler, J. (2005). Fast robust correlation. IEEE Transactions on Image Processing, 1063–1073. doi: 10.1109/TIP.2005.849767.
  13. Fraundorfer, F., Engels, C., & Nistér, D. (2007). Topological mapping, localization and navigation using image collections. In Intell. robots and systems (pp. 3872–3877). Google Scholar
  14. Gunawan, H., Neswan, O., & Budhi, W. (2005). A formula for angles between subspaces of inner product spaces. Contributions to Algebra and Geometry, 46(2), 311–320. MathSciNetMATHGoogle Scholar
  15. Gunes, H., & Pantic, M. (2010). Automatic, dimensional and continuous emotion recognition. International Journal of Synthetic Emotion, 68–99. doi: 10.4018/jse.2010101605.
  16. Handmann, U., Kalinke, T., & Tzomakas, C. (1998). Computer vision for driver assistance systems. Proc. SPIE, 136–147. doi: 10.1117/12.317463.
  17. Haritaoglu, I., Harwood, D., & Davis, L. (2000). W4: real-time surveillance of people and their activities. IEEE Transactions on Pattern Analysis and Machine Intelligence, 809–830. doi: 10.1109/34.868683.
  18. Hashima, M., Hasegawa, F., Kanda, S., Maruyama, T., & Uchiyama, T. (1997). Localization and obstacle detection for a robot for carrying food trays. In Intell. robots and systems (pp. 345–351). Google Scholar
  19. He, R., Hu, B., Zheng, W., & Kong, X. (2011). Robust principal component analysis based on maximum correntropy criterion. IEEE Transactions on Image Processing, 1485–1494. doi: 10.1109/TIP.2010.2103949.
  20. Honeine, P., & Richard, C. (2011). Preimage problem in kernel-based machine learning. IEEE Signal Processing Magazine, 28(2), 77–88. CrossRefGoogle Scholar
  21. Hsieh, J., Yu, S., Chen, Y., & Hu, W. (2006). Automatic traffic surveillance system for vehicle tracking and classification. IEEE Transactions on Intelligent Transportation Systems, 175–187. doi: 10.1109/TITS.2006.874722.
  22. Jolliffe, T. (2002). Principal component analysis (2nd edn.). Berlin: Springer. MATHGoogle Scholar
  23. Kamijo, S., Matsushita, Y., Ikeuchi, K., & Sakauchi, M. (2000). Traffic monitoring and accident detection at intersections. IEEE Transactions on Intelligent Transportation Systems, 108–118. doi: 10.1109/6979.880968.
  24. Ke, Q., & Kanade, T. (2003). Robust subspace computation using L1 norm (Tech. Rep. CMU-CS-03-172). Computer Science Department, Carnegie Mellon University. Google Scholar
  25. Ke, Q., & Kanade, T. (2005). Robust L1 norm factorization in the presence of outliers and missing data by alternative convex programming. In CVPR’05 (pp. 739–746). Google Scholar
  26. Krzanowski, W. (1979). Between-groups comparison of principal components. Journal of the American Statistical Association, 703–707. doi: 10.1080/01621459.1979.10481674.
  27. Kwak, N. (2008). Principal component analysis based on L1-norm maximization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1672–1680. doi: 10.1109/TPAMI.2008.114.
  28. Kwok, J., & Tsang, I. (2004). The pre-image problem in kernel methods. IEEE Transactions on Neural Networks, 15(6), 1517–1525. CrossRefGoogle Scholar
  29. Levy, A., & Lindenbaum, M. (2000). Sequential Karhunen-Loeve basis extraction and its application to images. IEEE Transactions on Image Processing, 1371–1374. doi: 10.1109/83.855432.
  30. Li, Y. (2004). On incremental and robust subspace learning. Pattern Recognition, 1509–1518. doi: 10.1016/j.patcog.2003.11.010.
  31. Li, L., Huang, W., Gu, I., & Tian, Q. (2004). Statistical modeling of complex backgrounds for foreground object detection. IEEE Transactions on Image Processing, 13(11), 1459–1472. CrossRefGoogle Scholar
  32. Liu, W., Pokharel, P., & Principe, J. (2007). Correntropy: properties and applications in non-Gaussian signal processing. IEEE Transactions on Signal Processing, 5286–5298. doi: 10.1109/TSP.2007.896065.
  33. Liwicki, S., & Everingham, M. (2009). Automatic recognition of fingerspelled words in British sign language. In CVPR4HB’09, in conj. with CVPR’09 (pp. 50–57). Google Scholar
  34. Liwicki, S., et al. (2012). doi: 10.1109/TNNLS.2012.2208654.
  35. Luo, Y., Wu, T., & Hwang, J. (2003). Object-based analysis and interpretation of human motion in sports video sequences by dynamic Bayesian networks. Computer Vision and Image Understanding, 196–216. doi: 10.1016/j.cviu.2003.08.001.
  36. Maddalena, L., & Petrosino, A. (2008). A self-organizing approach to background subtraction for visual surveillance applications. IEEE Transactions on Image Processing, 1168–1177. doi: 10.1109/TIP.2008.924285.
  37. Martinez, A., & Benavente, R. (1998). The AR face database (Tech. Rep. #24). The Ohio State University. Google Scholar
  38. Mei, X., & Ling, H. (2009). Robust visual tracking using L1 minimization. In ICCV’09. Google Scholar
  39. Mika, S., Schölkopf, B., Smola, A., Müller, K., Scholz, M., & Rätsch, G. (1999). Kernel pca and de-noising in feature spaces. Advances in Neural Information Processing Systems, 11(1), 536–542. Google Scholar
  40. Nguyen, M., & de la Torre, F. (2009). Robust kernel principal component analysis. In Advances in NIPS (pp. 1185–1192). Google Scholar
  41. Oliver, N., Rosario, B., & Pentland, A. (2000). A Bayesian computer vision system for modeling human interactions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 831–843. doi: 10.1109/34.868684.
  42. Paulsen, V. (2009). An introduction to the theory of reproducing kernel Hilbert spaces. Available at:
  43. Ross, D., Lim, J., Lin, R., & Yang, M. (2008). Incremental learning for robust visual tracking. International Journal of Computer Vision, 125–141. doi: 10.1007/s11263-007-0075-7.
  44. Tzimiropoulos, G. (2010). IEEE Transactions on Pattern Analysis and Machine Intelligence. doi: 10.1109/TPAMI.2010.107.
  45. Tzimiropoulos, G. (2012). IEEE Transactions on Pattern Analysis and Machine Intelligence. doi: 10.1109/TPAMI.2012.40.
  46. Turk, M., & Pentland, A. (1991). Eigenfaces for recognition. Journal of Cognitive Neuroscience, 71–86. doi: 10.1162/jocn.1991.3.1.71.
  47. Wren, C., Azarbayejani, A., Darrel, T., & Pentland, A. (1997). Pfinder: real-time tracking of the human body. IEEE Transactions on Pattern Analysis and Machine Intelligence, 780–785. doi: 10.1109/34.598236.

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Stephan Liwicki
    • 1
  • Georgios Tzimiropoulos
    • 1
    • 2
  • Stefanos Zafeiriou
    • 1
  • Maja Pantic
    • 1
    • 3
  1. 1.Department of ComputingImperial College LondonLondonUK
  2. 2.School of Computer ScienceUniversity of LincolnLincolnUK
  3. 3.Faculty of Electrical Engineering, Mathematics and Computer ScienceUniversity of TwenteEnschedeThe Netherlands

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