Among linear DR methods, principal component analysis (PCA) perhaps is the most important one. In linear DR, the dissimilarity of two points in a data set is defined by the Euclidean distance between them, and correspondingly, the similarity is described by their inner product. Linear DR methods adopt the global neighborhood system: the neighbors of a point in the data set consist of all of other points. Let the original data set be H= {x 1 ··· x n ⊂ ℝ D and the DR data set of H be a d-dimensional set Y. Under the Euclidean measure, PCA finds a linear projection T: ℝ D → ℝ d so that the DR data Y = T(H) maximize the data energy. PCA is widely used in many applications. The present chapter is organized as follows. In Section 5.1, we discuss the description of PCA. In Section 5.2, we present the PCA algorithms. Some real-world applications of PCA are introduced in Section 5.3.


Principal Component Analysis Face Recognition Singular Value Decomposition Face Image Principal Direction 
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  1. [1]
    Pearson, K.: On lines and planes of closest fit to systems of points in space. Philosophical Magazine 2(6), 559–572 (1901).Google Scholar
  2. [2]
    Jolliffe, I.T.: Principal Component Analysis. Springer Series in Statistics. Springer-Verlag, Berlin (1986).Google Scholar
  3. [3]
    Rao, C., Rao, M.: Matrix Algebra and Its Applications to Statistics and Econometric. World Scientific, Singapore (1998).Google Scholar
  4. [4]
    Lehoucq, R., Sorensen, D.: Deflation techniques for an implicitly re-started arnoldi iteration. SIAM J. Matrix Analysis and Applications 17, 789–821 (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Barrett, R., Berry, M.W., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Publisher (1993).Google Scholar
  6. [6]
    Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quarterly of AppliedMathematics 9, 17–29 (1951).MathSciNetzbMATHGoogle Scholar
  7. [7]
    Lehoucq, R., Sorensen, D., Yang, C.: ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. SIAM Publications, Philadelphia (1998).Google Scholar
  8. [8]
    Sorensen, D.: Implicit application of polynomial filters in a k-step arnoldi method. SIAM J. Matrix Analysis and Applications 13, 357–385 (1992).MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Roweis, S.: EM algorithms for PCA and SPCA. In: anips, vol. 10, pp. 626–632 (1998).Google Scholar
  10. [10]
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, B 39(1), 1–38 (1977).MathSciNetzbMATHGoogle Scholar
  11. [11]
    Belhumeur, P.N., Hespanha, J.P., Kriegman, D.J.: Eigenfaces vs. fisherfaces: Recognition using class specific linear projection. TPAMI 7(19), 711–720 (1997).CrossRefGoogle Scholar
  12. [12]
    Georghiades, A., Belhumeur, P., Kriegman, D.: From few to many: Illumination cone models for face recognition under variable lighting and pose. TPAMI 6(23), 643–660 (2001).CrossRefGoogle Scholar
  13. [13]
    Wang, W., Song, J., Yang, Z., Chi, Z.: Wavelet-based illumination compensation for face recognition using eigenface method. In: Proceedings of Intelligent Control and Automation. Dalian, China (2006).Google Scholar
  14. [14]
    Chen, T., Zhou, X.S., Comaniciu, D., Huang, T.: Total variation models for variable lighting face recognition. TPAMI 9(28), 1519–1524 (2006).CrossRefGoogle Scholar
  15. [15]
    Wang, H.T., Li, S.Z., Wang, Y.S.: Face recognition under varying lighting conditions using self quotient image. In: Automatic Face and Gesture Recognition, International Conference on FGR, pp. 819–824 (2004).Google Scholar
  16. [16]
    Basri, R., Jacobs, D.: Photometric stereo with general, unknown lighting. In: IEEE Conference on CVPR, 374–381 (2001).Google Scholar
  17. [17]
    Horn, B.: The Psychology of Computer Vision, chap. 4. Obtaining Shape from Shading Information and chap 6. Shape from Shading, pp. 115–155. McGraw-Hill, New York (1975).Google Scholar
  18. [18]
    Xie, X.D., Lam, K.: Face recognition under varying illumination based on a 2D face shape model. Journal of Pattern Recognition 2(38), 221–230 (2005).Google Scholar
  19. [19]
    Hu, Y.K., Wang, Z.: A low-dimensional illumination space representation of human faces for arbitrary lighting conditions. In: Proceedings of ICPR, pp. 1147–1150. Hong Kong (2006).Google Scholar
  20. [20]
    Ramamoorthi, R.: Analytic PCA construction for theoretical analysis of lighting variability in images of a lambertian object. TPAMI 10(24), 1322–1333 (2002).CrossRefGoogle Scholar
  21. [21]
    Wang, H.T., Li, S.Z., Wang, Y.: Generalized quotient image. In: Proceeding of CVPR (2004).Google Scholar
  22. [22]
    Xie, X., Zheng, W.S., Lai, J., Yuen, P.C.: Face illumination normalization on large and small scale features. In: IEEE Coference on CVPR (2008).Google Scholar
  23. [23]
    Xie, X., Lai, J., Zheng, W.S.: Extraction of illumination invariant facial features from a single image using nonsubsampled contourlet transform. Pattern Recognition 43(12), 4177–4189 (2010).zbMATHCrossRefGoogle Scholar
  24. [24]
    Xie, X., Zheng, W.S., Lai, J., Yuen, P.C., Suen, C.Y.: Normalization of face illumination based on large-and small-scale features. IEEE Trans. on Image Processing (2010). Accepted.Google Scholar
  25. [25]
    Zheng, W.S., Lai, J., Yuen, P.C.: Penalized pre-image learning in kernel principal component analysis. IEEE Trans. on Neural Networks 21(4), 551–570 (2010).CrossRefGoogle Scholar
  26. [26]
    Turk, M., Pentland, A.: Eigenfaces for recognition. The Journal of Cognitive Neuroscience 3(1), 71–86 (1991).CrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jianzhong Wang
    • 1
  1. 1.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA

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