Statistics and Computing

, Volume 19, Issue 3, pp 229–238 | Cite as

Factored principal components analysis, with applications to face recognition

  • Ian L. DrydenEmail author
  • Li Bai
  • Christopher J. Brignell
  • Linlin Shen


A dimension reduction technique is proposed for matrix data, with applications to face recognition from images. In particular, we propose a factored covariance model for the data under study, estimate the parameters using maximum likelihood, and then carry out eigendecompositions of the estimated covariance matrix. We call the resulting method factored principal components analysis. We also develop a method for classification using a likelihood ratio criterion, which has previously been used for evaluating the strength of forensic evidence. The methodology is illustrated with applications in face recognition.


Face recognition Forensic identification Gabor wavelets Kernel density estimator Likelihood ratio Multivariate normal Principal components analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aitken, C.G.G., Lucy, D.: Evaluation of trace evidence in the form of multivariate data. J. R. Stat. Soc. Ser. C 53(1), 109–122 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  2. Aitken, C.G.G., Lucy, D., Zadora, G., Curran, J.M.: Evaluation of transfer evidence for three-level multivariate data with the use of graphical models. Comput. Stat. Data Anal. 50(10), 2571–2588 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  3. Aitken, C.G.G., Zadora, G., Lucy, D.: A two-level model for evidence evaluation. J. Forensic Sci. 52, 412–419 (2007) CrossRefGoogle Scholar
  4. Bai, L., Shen, L., Wang, Y.: A novel eye location algorithm based on radial symmetry transform. In: 18th International Conference on Pattern Recognition (ICPR 2006), 20–24 August 2006, Hong Kong, China, vol. 3, pp. 511–514. IEEE Computer Society (2006). ISBN 0-7695-2521-0 Google Scholar
  5. Bailly-Bailliére, E., Bengio, S., Bimbot, F., Hamouz, M., Kittler, J., Mariéthoz, J., Matas, J., Messer, K., Popovici, V., Porée, F., Ruíz, B., Thiran, J.-P.: The Banca database and evaluation protocol. In: Kittler, J., Nixon, M.S. (eds.) Audio- and Video-Based Biometric Person Authentication, 4th International Conference (AVBPA 2003) Proceedings, Guildford, UK, 9–11 June 2003. Lecture Notes in Computer Science, vol. 2688, pp. 625–638. Springer, Berlin (2003). ISBN 3-540-40302-7 CrossRefGoogle Scholar
  6. Brignell, C.J.: Shape analysis and statistical modelling in brain imaging. Ph.D. thesis, University of Nottingham (2007) Google Scholar
  7. Bruce, V., Young, A.: Understanding face recognition. Br. J. Psychol. 77, 305–327 (1986) Google Scholar
  8. Chang, C.-C., Lin, C.-J.: LIBSVM: a library for support vector machines, Software available at (2001)
  9. Dutilleul, P.: The MLE algorithm for the matrix normal distribution, J. Stat. Comput. Simul., 105–123 (1999) Google Scholar
  10. Galecki, A.: General class of covariance structures for two or more repeated factors in longitudinal data analysis. Commun. Stat. Theory Methods 23, 3105–3119 (1994) zbMATHCrossRefGoogle Scholar
  11. Gao, Q.-X.: Is two-dimensional PCA equivalent to a special case of modular PCA? Pattern Recognit. Lett. 28(10), 1250–1251 (2007) CrossRefGoogle Scholar
  12. Gottumukkal, R., Asari, V.K.: An improved face recognition technique based on modular PCA approach. Pattern Recognit. Lett. 25(4), 429–436 (2004) CrossRefGoogle Scholar
  13. Kong, H., Wang, L., Teoh, E.K., Li, X., Wang, J.-G., Venkateswarlu, R.: Generalized 2D principal component analysis for face image representation and recognition. Neural Netw. 18(56), 585–594 (2005) CrossRefGoogle Scholar
  14. Lanitis, A., Taylor, C.J., Cootes, T.F.: Automatic interpretation and coding of face images using flexible models. IEEE Trans. Pattern Anal. Mach. Intell. 19(7), 743–756 (1997) CrossRefGoogle Scholar
  15. Mardia, K.V., Goodall, C.R.: Spatial-temporal analysis of multivariate environmental monitoring data. In: Patil, G.P., Rao, C.R. (eds.) Multivariate Environmental Statistics. North-Holland, Amsterdam (1993) Google Scholar
  16. Mardia, K.V., Kent, J.T., Bibby, J.M.: Multivariate Analysis. Academic Press, London (1979) zbMATHGoogle Scholar
  17. Martin, R.J.: A subclass of lattice processes applied to a problem in planar sampling. Biometrika 66(2), 209–217 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  18. Phillips, P.J., Moon, H., Rauss, P.J., Rizvi, S.: The FERET evaluation methodology for face recognition algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 22, 1090–1104 (2000) CrossRefGoogle Scholar
  19. Rangarajan, A.: Learning matrix space image representations. In: Figueiredo, M.A.T., Zerubia, J., Jain, A.K. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition, Third International Workshop (EMMCVPR 2001) Proccedings, Sophia Antipolis, France, 3–5 September 2001. Lecture Notes in Computer Science, vol. 2134, pp. 153–168. Springer, Berlin (2001). ISBN 3-540-42523-3 CrossRefGoogle Scholar
  20. Shen, L., Bai, L.: Mutualboost learning for selecting Gabor features for face recognition. Pattern Recognit. Lett. 27, 1758–1767 (2006) CrossRefGoogle Scholar
  21. Sinha, P., Balas, B.J., Ostrovsky, Y., Russel, R.: Face recognition by humans. In: Zhao, W., Chellappa, R. (eds.) Face Processing: Advanced Modeling and Methods. Academic Press, San Diego (2006) Google Scholar
  22. Vasilescu, M.A.O., Terzopoulos, D.: Multilinear subspace analysis of image ensembles. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2003), Madison, WI, USA, 16–22 June 2003, vol. 2, pp. 93–99. IEEE Computer Society (2003). ISBN 0-7695-1900-8 Google Scholar
  23. Wiskott, L., Fellous, J.-M., Kruger, N., von der Malsburg, C.: Face recognition by elastic bunch graph matching. IEEE Trans. Pattern Anal. Mach. Intell. 19(7), 775–779 (1997) CrossRefGoogle Scholar
  24. Yang, J., Zhang, D., Frangi, A.F., Yang, J.-Y.: Two-dimensional PCA: A new approach to appearance-based face representation and recognition. IEEE Trans. Pattern Anal. Mach. Intell. 26(1), 131–137 (2004) CrossRefGoogle Scholar
  25. Ye, J.: Generalized low rank approximations of matrices. Mach. Learn. 61, 167–191 (2005) zbMATHCrossRefGoogle Scholar
  26. Zhao, W., Chellappa, R., Phillips, J., Rosenfeld, A.: Face recognition: A literature survey. ACM Comput. Surv. 35, 399–458 (2003) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Ian L. Dryden
    • 1
    Email author
  • Li Bai
    • 2
  • Christopher J. Brignell
    • 1
  • Linlin Shen
    • 3
  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.School of Computer ScienceUniversity of NottinghamNottinghamUK
  3. 3.School of Information and EngineeringShen Zhen UniversityShen ZhenChina

Personalised recommendations