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A maximum uncertainty LDA-based approach for limited sample size problems — with application to face recognition
Journal of the Brazilian Computer Society volume 12, pages 7–18 (2006)
Abstract
A critical issue of applying Linear Discriminant Analysis (LDA) is both the singularity and instability of the within-class scatter matrix. In practice, particularly in image recognition applications such as face recognition, there are often a large number of pixels or pre-processed features available, but the total number of training patterns is limited and commonly less than the dimension of the feature space. In this study, a new LDA-based method is proposed. It is based on a straightforward stabilisation approach for the within-class scatter matrix. In order to evaluate its effectiveness, experiments on face recognition using the well-known ORL and FERET face databases were carried out and compared with other LDA-based methods. The classification results indicate that our method improves the LDA classification performance when the within-class scatter matrix is not only singular but also poorly estimated, with or without a Principal Component Analysis intermediate step and using less linear discriminant features. Since statistical discrimination methods are suitable not only for classification but also for characterisation of differences between groups of patterns, further experiments were carried out in order to extend the new LDA-based method to visually analyse the most discriminating hyper-plane separating two populations. The additional results based on frontal face images indicate that the new LDA-based mapping provides an intuitive interpretation of the two-group classification tasks performed, highlighting the group differences captured by the multivariate statistical approach proposed.
References
P. N. Belhumeur, J. P. Hespanha, and D. J. Kriegman, “Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection”,IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 7, pp. 711–720, 1997.
N.A. Campbell, “Shrunken estimator in discriminant and canonical variate analysis”,Applied Statistics, vol. 29, pp. 5–14, 1980.
L. Chen, H. Liao, M. Ko, J. Lin, and G. Yu, “A new LDA-based face recognition system which can solve the small sample size problem”,Pattern Recognition, 33 (10), pp. 1713–1726, 2000.
P.J. Di Pillo, “Biased Discriminant Analysis: Evaluation of the optimum probability of misclassification”,Communications in Statistics-Theory and Methods, vol. A8, no. 14, pp. 1447–1457, 1979.
P.A. Devijver and J. Kittler,Pattern Classification: A Statistical Approach. Prentice-Hall, Englewood Cliffs, N. J., 1982.
J.H. Friedman, “Reguralized Discriminant Analysis”,Journal of the American Statistical Association, vol. 84, no. 405, pp. 165–175, March 1989.
K. Fukunaga,Introduction to Statistical Pattern Recognition, second edition. Boston: Academic Press, 1990.
T. Greene and W.S. Rayens, “Covariance pooling and stabilization for classification”,Computational Statistics & Data Analysis, vol. 11, pp. 17–42, 1991.
A. K. Jain and B. Chandrasekaran, “Dimensionality and SampleSize Considerations in Pattern Recognition Practice”,Handbook of Statistics, P.R. Krishnaiah and L.N. Kanal Eds, vol. 2, pp. 835–855, North Holland, 1982.
A. K. Jain, R. P. W. Duin and J. Mao, “Statistical Pattern Recognition: A Review”,IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, no. 1, pp. 4–37, January 2000.
R.A. Johnson and D.W. Wichern,Applied Multivariate Statistical Analysis, fourth edition. New Jersey: Prentice Hall, 1998.
K. Liu, Y. Cheng, and J. Yang, “Algebraic feature extraction for image recognition based on an optimal discriminant criterion”,Pattern Recognition, 26 (6), pp. 903–911, 1993.
Y. Li, J. Kittler, and J. Matas, “Effective Implementation of Linear Discriminant Analysis for Face Recognition and Verification”,Computer Analysis of Images and Patterns: 8th International Conference CAIPV9, Springer-Verlag LNCS 1689, pp. 232–242, Ljubljana, Slovenia, September 1999.
S.L. Marple,Digital Spectral Analysis with Applications. Englewood Cliffs, N.J: Prentice-Hall, 1987.
J.R. Magnus and H. Neudecker,Matrix Differential Calculus with Applications in Statistics and Econometrics, revised edition. Chichester: John Wiley & Sons Ltd., 1999.
S. Mika, G. Ratsch, J. Weston, B. Scholkopf, and K.-R. Muller, “Fisher discriminant analysis with kernels”,IEEE Neural Networks for Signal Processing IX, pp. 41–48, 1999.
R. Peck and J. Van Ness, “The use of shrinkage estimators in linear discriminant analysis”,IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 4, no. 5, pp. 531–537, September 1982.
P. J. Phillips, H. Wechsler, J. Huang and P. Rauss, “The FERET database and evaluation procedure for face recognition algorithms”,Image and Vision Computing Journal, vol. 16, no. 5, pp. 295–306, 1998.
W.S. Rayens, “A Role for Covariance Stabilization in the Construction of the Classical Mixture Surface”,Journal of Chemometrics, vol. 4, pp. 159–169, 1990.
A. Samal and P. Iyengar, “Automatic Recognition and Analysis of Human Faces and Facial Expressions: A Survey”,Pattern Recognition, 25 (1), pp. 65–77, 1992.
D. L. Swets and J. J. Weng, “Using Discriminant Eigenfeatures for Image Retrieval”,IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 8, pp. 831–836, 1996.
S. Tadjudin, “Classification of High Dimensional Data With Limited Training Samples”, PhD thesis, Purdue University, West Lafayette, Indiana, 1998.
C. E. Thomaz, D. F. Gillies and R. Q. Feitosa. “A New Covariance Estimate for Bayesian Classifiers in Biometrie Recognition”,IEEE Transactions on Circuits and Systems for Video Technology, Special Issue on Image- and Video-Based Biometrics, vol. 14, no. 2, pp. 214–223, February 2004.
M. Turk and A. Pentland, “Eigenfaces for Recognition”,Journal of Cognitive Neuroscience, vol. 3, pp. 71–86, 1991.
J. Yang and J. Yang, “Optimal FLD algorithm for facial feature extraction”,SPIE Proceedings of the Intelligent Robots and Computer Vision XX: Algorithms, Techniques, and Active Vision, vol. 4572, pp. 438–444, 2001.
J. Yang and J. Yang, “Why can LDA be performed in PCA transfoimed space?”,Pattern Recognition, vol. 36, pp. 563–566, 2003.
H. Yu and J. Yang, “A direct LDA algorithm for high dimensional data — with application to face recognition”,Pattern Recognition, vol. 34, pp. 2067–2070, 2001.
W. Zhao, R. Chellappa and A. Krishnaswamy, “Discriminant Analysis of Principal Components for Face Recognition”, inProc. 2 nd International Conference on Automatic Face and Gesture Recognition, 336–341,1998.
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Thomaz, C.E., Kitani, E.C. & Gillies, D.F. A maximum uncertainty LDA-based approach for limited sample size problems — with application to face recognition. J Braz Comp Soc 12, 7–18 (2006). https://doi.org/10.1007/BF03192391
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DOI: https://doi.org/10.1007/BF03192391