人脸图像的子空间表示研究
|
摘要
上世纪90年代以来,以主成分分析(Principal Component Analysis, PCA)和线性判别分析(Linear Discriminant Analysis, LDA)为代表的子空间方法有力地促进了人脸识别技术的发展,并成为事实上的基准方法。在子空间人脸识别中,两个主要问题是小样本和面部大变化。本文旨在深入分析和比较一些子空间方法以利于后续研究,并提出一些克服小样本和大变化问题的有效算法。本文的主要贡献在于:
(1)对基于主成分分析的方法展开了四个方面研究。第一,深入分析了二维主成分分析(two-Dimensional PCA, 2DPCA),指出2DPCA是具有Kronecker积约束的PCA;第二,重新审视了推广低秩逼近(Generalized Low Rank Approximations of Matrices, GLRAM),主要研究结果包括:(a)给出了GLRAM的一些基本性质,(b)导出了GLRAM准则函数的一个下界,并回答了Ye提出的两个公开问题,(c)探讨了GLRAM在什么时候和为什么能取得较好的压缩(重建)性能。第三,提出了非迭代的GLRAM算法,主要工作在于,构造并优化近似的GLRAM准则以导出非迭代算法,并给出一个确定投影方向数的准则。第四,提出了一个渐进主成分分析算法,主要特点在于,利用图像子模式之间的上下文信息,渐进地读取图像样本以计算PCA的投影向量。
(2)对基于线性判别分析的方法展开了四个方面研究。第一,揭示了判别公共向量(Discriminant Common Vectors, DCV)、近邻成分分析(Neighborhood Component Analysis, NCA)、拉普拉斯脸(Laplacianfaces, LAP)和加权最大间隔准则(weighted Maximal Margin Criterion, wMMC)的性质。第二,为正则化判别分析(Regularized Discriminant Analysis, RDA)、wMMC、伪逆法线性判别分析(Pseudo-inverse Linear Discriminant Analysis, PLDA)和核PLDA提出了快速算法。第三,揭示了DCV与NCA、LAP、wMMC和RDA的关系,主要结论是:(a) DCV能取得NCA和LAP准则函数的最优解;(b) DCV是wMMC和RDA的特殊情形;(c)当平均标准差异准则(Mean Standard Variance, MSV)较小时,DCV能取得好的识别性能。第四,运用重采样技术提高了Fisherfaces和基于QR分解的线性判别分析(LDA via QR decomposition, LDA/QR)的识别性能。
(3)提出了人脸图像的分数阶奇异值分解表示(Fractional order Singular Value Decomposition Representation, FSVDR)。FSVDR能缓和人脸面部大变化,并可作为人脸图像和子空间方法之间的中间级表示。实验表明,FSVDR能显著提高PCA、DCV和LDA/QR等子空间方法在人脸面部大变化下的识别性能。
(4)提出了单图像子空间(Single Image Subspace, SIS)表示方法。在SIS中,把每幅(训练和测试)人脸图像表示成由其虚拟样本所张成的子空间,并通过所定义的能处理不同子空间维数的子空间距离来度量人脸图像的不相似度。此外,为了更好地对付人脸面部大变化,SIS采用了将人脸图像划分为若干子模式的方法。最后,基于单图像子空间表示,定义了几种人脸图像之间的核。
Since the 1990s, the face recognition technology has been greatly promoted by subspace methods such as PCA (Principal Component Analysis) and LDA (Linear Discriminant Analysis), which have become the de facto baseline methods. Two major problems with regard to face recognition by subspace methods are Small Sample Size (SSS) and Great Facial Variations (GFV). This paper is focused on two aspects, namely, on one hand, conducting comparative studies among some existing subspace methods to facilitate future studies, and on the other hand, proposing some effective methods to deal with the aforementioned problems. The major contributions in this paper are that:
(1) On PCA, the following four researches are conducted. First, an in-depth analysis is given to 2DPCA (two-dimensional PCA), with the main result being that 2DPCA is indeed PCA subjected to the Kronecker product constraint. Second, the GLRAM (Generalized Low Rank Approximations of Matrices) method is revisited, with the main results being that: (a) some basic properties of GLRAM are revealed, (b) a lower bound of the objective function of GLRAM is derived to answer the two open problems raised by Ye, and (c) when and why GLRAM can obtain good compression (reconstruction) performance is explored. Third, the GLRAM method is extended to its non-iterative version, NIGLRAM (Non-Iterative GLRAM). NIGLRAM constructs and optimizes an approximate objective function of GLRAM, and enjoys an automatic criterion in determining the number of projection vectors. Fourth, a PrPCA (Progressive PCA) method is proposed. PrPCA makes use of the contextual information among the image blocks, and reads face images progressively to compute the principal components
(2) On LDA, the following four researches are carried out. First, some properties of DCV (Discriminant Common Vectors), NCA (Neighborhood Component Analysis), LAP (Laplacianfaces) and wMMC (weighted Maximal Margin Criterion) are revealed. Second, efficient algorithms are proposed for RDA (Regularized Discriminant Analysis), wMMC, PLDA (Pseudo-inverse LDA) and KPLDA (Kernelized PLDA). Third, DCV’s relationships with NCA, LAP, wMMC and RDA are revealed, with the main results being that: (a) DCV can obtain the optimal result of the objective functions of both NCA and LAP, (b) DCV is a special case of both wMMC and RDA, and (c) when the MSV (Mean Standard Variation) criterion is relatively small, DCV can obtain good classification performance. Fourth, the resampling technique is employed to improve the classification performance of Fisherfaces (or PCA+LDA) and LDA/QR ((LDA via QR decomposition).
(3) The FSVDR (Fractional order Singular Value Decomposition Representation) is proposed to alleviate the great facial variations, and to offer an intermediate representation between the original face images and subspace methods. Empirical results show that, FSVD can obviously improve the classification performance of subspace methods such as PCA, DCV and LDA/QR under great facial variations.
(4) The SIS (Single Image Subspace) representation is proposed. The core of SIS is to represent each (training, testing) image as a subspace spanned by its synthesized virtual images. To measure the dissimilarity of face images under SIS representation, a new subspace distance that can deal with unequal number of subspace dimensions is developed. To deal with the great facial variations, face images are divided into a set of sub-images, on which SIS is employed. Finally, some new kernels based on SIS are defined.
(1)对基于主成分分析的方法展开了四个方面研究。第一,深入分析了二维主成分分析(two-Dimensional PCA, 2DPCA),指出2DPCA是具有Kronecker积约束的PCA;第二,重新审视了推广低秩逼近(Generalized Low Rank Approximations of Matrices, GLRAM),主要研究结果包括:(a)给出了GLRAM的一些基本性质,(b)导出了GLRAM准则函数的一个下界,并回答了Ye提出的两个公开问题,(c)探讨了GLRAM在什么时候和为什么能取得较好的压缩(重建)性能。第三,提出了非迭代的GLRAM算法,主要工作在于,构造并优化近似的GLRAM准则以导出非迭代算法,并给出一个确定投影方向数的准则。第四,提出了一个渐进主成分分析算法,主要特点在于,利用图像子模式之间的上下文信息,渐进地读取图像样本以计算PCA的投影向量。
(2)对基于线性判别分析的方法展开了四个方面研究。第一,揭示了判别公共向量(Discriminant Common Vectors, DCV)、近邻成分分析(Neighborhood Component Analysis, NCA)、拉普拉斯脸(Laplacianfaces, LAP)和加权最大间隔准则(weighted Maximal Margin Criterion, wMMC)的性质。第二,为正则化判别分析(Regularized Discriminant Analysis, RDA)、wMMC、伪逆法线性判别分析(Pseudo-inverse Linear Discriminant Analysis, PLDA)和核PLDA提出了快速算法。第三,揭示了DCV与NCA、LAP、wMMC和RDA的关系,主要结论是:(a) DCV能取得NCA和LAP准则函数的最优解;(b) DCV是wMMC和RDA的特殊情形;(c)当平均标准差异准则(Mean Standard Variance, MSV)较小时,DCV能取得好的识别性能。第四,运用重采样技术提高了Fisherfaces和基于QR分解的线性判别分析(LDA via QR decomposition, LDA/QR)的识别性能。
(3)提出了人脸图像的分数阶奇异值分解表示(Fractional order Singular Value Decomposition Representation, FSVDR)。FSVDR能缓和人脸面部大变化,并可作为人脸图像和子空间方法之间的中间级表示。实验表明,FSVDR能显著提高PCA、DCV和LDA/QR等子空间方法在人脸面部大变化下的识别性能。
(4)提出了单图像子空间(Single Image Subspace, SIS)表示方法。在SIS中,把每幅(训练和测试)人脸图像表示成由其虚拟样本所张成的子空间,并通过所定义的能处理不同子空间维数的子空间距离来度量人脸图像的不相似度。此外,为了更好地对付人脸面部大变化,SIS采用了将人脸图像划分为若干子模式的方法。最后,基于单图像子空间表示,定义了几种人脸图像之间的核。
Since the 1990s, the face recognition technology has been greatly promoted by subspace methods such as PCA (Principal Component Analysis) and LDA (Linear Discriminant Analysis), which have become the de facto baseline methods. Two major problems with regard to face recognition by subspace methods are Small Sample Size (SSS) and Great Facial Variations (GFV). This paper is focused on two aspects, namely, on one hand, conducting comparative studies among some existing subspace methods to facilitate future studies, and on the other hand, proposing some effective methods to deal with the aforementioned problems. The major contributions in this paper are that:
(1) On PCA, the following four researches are conducted. First, an in-depth analysis is given to 2DPCA (two-dimensional PCA), with the main result being that 2DPCA is indeed PCA subjected to the Kronecker product constraint. Second, the GLRAM (Generalized Low Rank Approximations of Matrices) method is revisited, with the main results being that: (a) some basic properties of GLRAM are revealed, (b) a lower bound of the objective function of GLRAM is derived to answer the two open problems raised by Ye, and (c) when and why GLRAM can obtain good compression (reconstruction) performance is explored. Third, the GLRAM method is extended to its non-iterative version, NIGLRAM (Non-Iterative GLRAM). NIGLRAM constructs and optimizes an approximate objective function of GLRAM, and enjoys an automatic criterion in determining the number of projection vectors. Fourth, a PrPCA (Progressive PCA) method is proposed. PrPCA makes use of the contextual information among the image blocks, and reads face images progressively to compute the principal components
(2) On LDA, the following four researches are carried out. First, some properties of DCV (Discriminant Common Vectors), NCA (Neighborhood Component Analysis), LAP (Laplacianfaces) and wMMC (weighted Maximal Margin Criterion) are revealed. Second, efficient algorithms are proposed for RDA (Regularized Discriminant Analysis), wMMC, PLDA (Pseudo-inverse LDA) and KPLDA (Kernelized PLDA). Third, DCV’s relationships with NCA, LAP, wMMC and RDA are revealed, with the main results being that: (a) DCV can obtain the optimal result of the objective functions of both NCA and LAP, (b) DCV is a special case of both wMMC and RDA, and (c) when the MSV (Mean Standard Variation) criterion is relatively small, DCV can obtain good classification performance. Fourth, the resampling technique is employed to improve the classification performance of Fisherfaces (or PCA+LDA) and LDA/QR ((LDA via QR decomposition).
(3) The FSVDR (Fractional order Singular Value Decomposition Representation) is proposed to alleviate the great facial variations, and to offer an intermediate representation between the original face images and subspace methods. Empirical results show that, FSVD can obviously improve the classification performance of subspace methods such as PCA, DCV and LDA/QR under great facial variations.
(4) The SIS (Single Image Subspace) representation is proposed. The core of SIS is to represent each (training, testing) image as a subspace spanned by its synthesized virtual images. To measure the dissimilarity of face images under SIS representation, a new subspace distance that can deal with unequal number of subspace dimensions is developed. To deal with the great facial variations, face images are divided into a set of sub-images, on which SIS is employed. Finally, some new kernels based on SIS are defined.
引文
[1] V. Bruce, P.J.B. Hancock, and A.M. Burton, Human Face Perception and Identification, in Face Recognition: From Theory to Applications, H. Wechsler, et al., Editors. 1998, Springer-Verlag. p. 51-72.
[2] R. Brunell and T. Poggio, "Face recognition: Features vs templates", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1993, 15(10), 1042-1053.
[3] R. Chellappa, C.L. Wilson, and S. Sirohey, "Human and Machine Recognition of Faces: A survey", Proceedings of the IEEE, 1995, 83(5), 705-740.
[4] L. Harmon, M. Kaun, R. Lasch, and P. Ramig, "Machine identification of human faces", Pattern Recognition, 1981, 13(2), 97-110.
[5] W. Zhao, R. Chellappa, P.J. hillips, and A. Rosenfeld, "Face Recognition: A Literature Survey", ACM Computing Survey, 2003, 35(4), 399-458.
[6] S.G. Kong, J. Heo, B.R. Abidi, J. Paik, and M.A. Abidi, "Recent advances in visual and infrared face recognition-a review", Computer Vision and Image Understanding, 2005, 97(1), 103-135.
[7] X. Tan, S.C. Chen, Z.H. Zhou, and h. F., "Face recognition from a single image per person: A survey", Pattern Recognition, 2006, 39(9), 1725-1745.
[8] T. Kanade. Picture Processing by Computer Complex and Recognition of Human Faces. PhD thesis, Kyoto University, 1973
[9] G. Kaufman and K. Breeding, "The automatic recognition of human faces from profile silhouettes", IEEE Transactions on Systems, Man and Cybernetics, 1976, 6(2), 113-121.
[10]边肇祺,张学工, "模式识别",清华大学出版社, 2000,
[11] A.K. Hrechak and J.A. McHugh, "Automated fingerprint recognition using structural matching", Pattern Recognition 1990, 23(8), 893-904.
[12] R.P. Wildes, "Iris recognition: an emerging biometric technology", Proceedings of the IEEE, 1997, 85(9), 1348-1363.
[13] G. Lu, D. Zhang, and K. Wang, "Palmprint recognition using eigenpalms features ", Pattern Recognition Letters, 2003, 24(9-10), 1463 - 1467.
[14] E. Hjelmas and B.K. Low, "Face detection: A Survey", Computer Vision and Image Understanding, 2001, 83(3), 236-274.
[15] M.H. Yang, D. Kriegman, and N. Ahuja, "Detecting faces in images: A survey", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24(1), 34-58.
[16]刘青山,卢汉清,马颂德, "综述人脸识别中的子空间方法",自动化学报, 2003, 29(6), 900-911.
[17] R.O. Duda and P.E.S.D.G. Hart, Pattern Classification, Wiley-Interscience Press, 2nd Edition, 2000.
[18] K. Fukunaga, Introduction to Statistical Pattern Recognition, Academic Press, 2nd Edition, 1990.
[19] C. Cortes and V.N. Vapnik, "Support-vector networks", Machine Learning, 1995, 20(3), 273-297.
[20] J.J. Atick, P.A. Griffin, and A.N. Redlich, "Statistical Approach to Shape from Shading: Reconstruction of 3D Face Surfaces from Single 2D Images", Neural Computation, 1996, 8(6), 1321-1340.
[21] C. Beumier and M. Acheroy, "Automatic 3D face authentication", Image and Vision Computing, 2000, 18(4), 315-321.
[22] A.M. Bronstein, M.M. Bronstein, and R. Kimmel, "Expression-invariant 3D face recognition", International Conference on Audio and Video-based Biometric Person Authentication, 2003, 62-70.
[23] D.A. Socolinsky and A. Selinger, "Thermal Face Recognition in an Operational Scenario", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004, 1012-.
[24] G. Bebis, A. Gyaourova, S. Singh, and I. Pavlidis, "Face recognition by fusing thermal infrared and visible imagery", Image and Vision Computing, 2006, 24(7), 727-742.
[25] D.O. Gorodnichy, "Seeing faces in video by computers. Editorial for Special Issue on Face Processing in Video Sequences ", Image and Vision Computing 2006, 24(6), 551-556.
[26] A. Yip and P. Sinn, "Role of Color in Face Recognition ", AI Memo, 2001, 2001-035.
[27] A. Goldstein, L. Harmon, and A. Lesk, "Identification of human faces", Proceedings of the IEEE, 1971, 59(5), 748-760.
[28] A. Goldstein and P. Iyengar, "Automatic recognition and analysis of human faces and facial expressions: A survey", Pattern Recognition, 1992, 25(1), 65-77.
[29] L. Harmon, S. Kuo, P. Ramig, and U. Raudkivi, "Identification of human face profiles by computer", Pattern Recognition, 1978, 10(5-6), 301-312.
[30] M. Turk and A. Pentland, "Eigenfaces for recognition", Journal of Cognitive Neuroscience, 1991, 3(1), 71-96.
[31] D. Beymer and T. Poggio, "Face recognition from one example view", International Conference on Computer Vision, 1995, 500-507.
[32] A.M. Martinez, "Recognizing imprecisely localized, partially occluded, and expression variant faces from a single sample per class", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 25(6), 748-763.
[33] P. Niyogi, F. Girosi, and T. Poggio, "Incorporating prior information in machine learning by creating virtual examples", Proceedings of the IEEE, 1998, 86(11), 2196-2209.
[34] J. Wu and Z.H. Zhou, "Face recognition with one training image per person", Pattern Recognition Letters, 2002, 23(14), 1711-1719.
[35] S. Shan, W. Gao, and D. Zhao, "Face Identification Based on Face-Specific Subspace", International Journal of Imaging Systems and Technology, 2003, 13(1), 23-32.
[36] F. Torre, R. Gross, S. Baker, and V. Kumar, "Representational oriented component analysis (ROCA) for face recognition with one sample image per training class", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 266-273.
[37] H.C. Jung, B.W. Hwang, and S.W. Lee, "Authenticating corrupted face image based on noise model", IEEE International Workshop on Analysis and Modeling of Faces and Gestures, 2004, 120-126.
[38] X. Liu and T. Chen, "Pose-Robust Face Recognition Using Geometry Assisted Probabilistic Modeling", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 502-509.
[39] J. Liu, S.C. Chen, Z.H. Zhou, and X. Tan, "Single Image Subspace For Face Recognition", IEEE International Workshop on Analysis and Modeling of Faces and Gestures, 2007, 205-219.
[40] 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, 1997, 19(7), 711-720.
[41] R.P.W. Duin, M. Loog, and T.K. Ho, "Editorial: Recent submissions in linear dimensionality reduction and face recognition", Pattern Recognition Letters, 2006, 27(7), 707-708.
[42] M. Kirby and L. Sirovich, "Application of the Karhunen-Loeve procedure for the characterization ofhuman faces", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, 12(1), 103-108.
[43] A. Leonardis and H. Bischof, "Kernel and subspace for computer vision", Pattern Recognition, 2003, 36(9), 1925-1927.
[44] T. Ahonen, A. Hadid, and M. Pietikainen, "Face Description with Local Binary Patterns: Application to Face Recognition", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2006, 28(12), 2037-2041.
[45] S.C. Chen, J. Liu, and Z.H. Zhou, "Making FLDA applicable to face recognition with one sample per person", Pattern Recognition, 2004, 37(7), 1553-1555.
[46] X. Tan, J. Liu, and S.C. Chen, "Sub-intrapersonal space analysis for face recognition", Neurocomputing, 2006, 69(13-15), 1796-1801.
[47] X. Tan, S.C. Chen, Z.H. Zhou, and J. Liu, "Learning Non-Metric Partial Similarity Based on Maximal Margin Criterion", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2006, 138-145.
[48] A. Pentland, B. Moghaddam, and T. Starner, "View-based and modular eigenspaces for face recognition", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1994, 84-91.
[49] T.K. Ho, "The Random Subspace Method for Constructing Decision Forests", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1998, 20(8), 832-844.
[50] C. Ding and J. Ye, "2-dimensional singular value decomosition for 2d maps and images", SIAM International Conference on Data Mining, 2005, 22-34.
[51] J. Liu and S.C. Chen, "Non-iterative generalized low rank approximation of matrices", Pattern Recognition Letters, 2006, 27(9), 1002-1008.
[52] H. Wang and N. Ahuja, "Rank-R Approximation of Tensors Using Image-as-Matrix Representation", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 346-353.
[53] D. Xu, S. Yan, L. Zhang, H. Zhang, Z. Liu, and H. Shum, "Concurrent subspaces analysis", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 203-208.
[54] J. Yang, D. Zhang, A.F. Frangi, and J.Y. Yang, "Two-dimensional PCA: A new approach to appearance-based face representation and recognition", IEEE Transactions on Pattern Analysisand Machine Intelligence, 2004, 26(1), 131-137.
[55] J. Ye, "Generalized low rank approximation of matrices", International Conference on Machine Learning, 2004, 887-894.
[56] J. Ye, "Generalized low rank approximations of matrices", Machine Learning, 2005, 61(1-3), 167-191.
[57] J. Ye, R. Janardan, and Q. Li, Two-Dimensional Linear Discriminant Analysis, in Advances in Neural Information Processing Systems. 2004. p. 354-363.
[58] D. Zhang, S.C. Chen, and J. Liu, "Representing image matrices: Eigenimages versus eigenvectors", International Symposium on Neural Networks, 2005, 659-664.
[59] Z. Wang, S.C. Chen, J. Liu, and D. Zhang, "Pattern Representation in Feature Extraction and Classification- Matrix Versus Vector", IEEE Transactions on Neural Networks, 2007, accepted.
[60] H. Sellahewa and S. Jassim, "Face Recognition in the Presence of Expression and/or Illumination Variation", IEEE Workshop on Automatic Identification Advanced Technologies, 2005, 144-148.
[61] Y. Adini, Y. Moses, and S. Ullman, "Face Recognition: The Problem of Compensating for Changes in Illumination Direction", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19(7), 721-732.
[62] V. Blanz, S. Romdhami, and T. Vetter, "Face identification across different poses and illuminations with a 3D morphable model", International Conference on Automatic Face and Gesture Recognition, 2002, 202-207.
[63] K. Lee, J. Ho, and D. Kriegman, "Acquiring Linear Subspaces for Face Recognition under Variable Lighting Illumination-Insensitive Face Recognition Using Symmetric Shape-from-Shading", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(5), 1-15.
[64] W. Zhao and R. Chellappa, "Illumination-Insensitive Face Recognition Using Symmetric Shape-from-Shading", International Conference on Computer Vision and Pattern Recognition, 2000, 286-549.
[65] A.M. Martinez and R. Benavente, "The AR Face Database", Technical Report, CVC, 1998,
[66] W. Zhang, S. Shan, W. Gao, X. Chen, and H. Zhang, "Local Gabor Binary Pattern Histogram Sequence (LGBPHS): A Novel Non-Statistical Model for Face Recognition", International Conference on Computer Vision, 2005, 786-791.
[67] J. Liu, S.C. Chen, and X. Tan, "Fractional order singular value decomposition representation for facerecognition", Pattern Recognition, 2008, 41(1), 378-395.
[68] G. Donato, M.S. Bartlett, J.C. Hager, P. Ekman, and T.J. Sejnowski, "Classifying facial actions", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1999, 21(10), 974-989.
[69] C. Liu and H. Wechsler, "Independent Component Analysis of Gabor Features for Face Recognition", IEEE Transactions on Neural Networks, 2003, 14(4), 919-928.
[70] R.C. Gonzalez and R.E. Woods, Digital Image Processing, Prentice Hall Press, 2nd Edition, 2004.
[71] M.J. Er, W. Chen, and S. Wu, "High-Speed Face Recognition Based on Discrete Cosine Transform and RBF Neural Networks", IEEE Transactions on Neural Networks, 2005, 16(3), 679-691.
[72] Z.M. Hafed and M.D. Levine, "Face recognition using the discrete cosine transform", International Journal of Computer Vision, 2001, 43(3), 167-188.
[73] P. Comon, "Independent component analysis, A new concept?" Signal Processing 1994, 36(3), 287-314.
[74] R. Gottumukkal and V.K. Asari, "An improved face recognition technique based on modular PCA approach", Pattern Recognition Letters, 2004, 25(4), 429-436.
[75] S.C. Chen and Y. Zhu, "Subpattern-based principle component analysis", Pattern Recognition, 2004, 37(5), 1081-1083.
[76] K.R. Tan and S.C. Chen, "Adaptively weighted sub-pattern PCA for face recognition", Neurocomputing, 2005, 64, 505-511.
[77] D.S. Bolme, J.R. Beveridge, M. Teixeira, and B.A. Draper, "The CSU face identification evaluation system: Its purpose, features and structure", International Conference on Computer Vision Systems, 2003, 303-311.
[78] K. Inoue and K. Urahama, "Dsvd: A tensor-based image compression and recognition method", IEEE International Symposium on Circuits and Systems, 2005, 6308-6311.
[79] J. Weng, Y. Zhang, and W.S. Hwang, "Candid covariance-free incremental principal component analysis ", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2003, 25(8), 1034-1040.
[80] H. Zhao, P. Yuen, and K. J., "Incremental Principal Component Analysis and its Application for Face Recognition", IEEE Transactions on Systems, Man and Cybernetics (Part B), 2006, 36(4), 873-886.
[81] K.I. Kim, K. Jung, and H.J. Kim, "Face recognition using kernel principal component analysis", IEEE Transactions on Signal Processing Letters, 2002, 9(2), 40-42.
[82] M.H. Yang, "Kernel Eigenfaces vs. Kernel Fisherfaces: Face Recognition Using Kernel Methods", IEEE International Conference on Automatic Face and Gesture Recognition 2002, 215-220.
[83] R.A. Fisher, "The use of multiple measures in taxonomic problems", Annual Eugenics, 1936, 7, 179-188.
[84] J.H. Friedman, "Regularized discriminant analysis", Journal of The American Statistical Association, 1989, 84(405), 165-175.
[85] Q. Tian, M. Barbero, Z.H. Gu, and S.H. Lee, "Image classification by the Foley-Sammon transform", Optical Engineering, 1986, 25(7), 834-840.
[86] G.H. Golub and C.F. Van Loan, Matrix Computations. Baltimore, MD, The Johns Hopkins University Pres, 3rd Edition, 1996.
[87] W.J. Krzanowski, P. Jonathan, W.V. McCarthy, and M.R. Thomas, "Discriminant analysis with singular covariance matrices: methods and applications to spectroscopic data", Applied Statistics, 1995, 44(1), 101-115.
[88] M. Skurichina and R.P.W. Duin, "Regularization of linear classifiers by adding redundant features", Pattern Analysis and Applications, 1999, 2(1), 44-52.
[89] M. Skurichina and R.P.W. Duin, "Stabilizing classifiers for very small sample size", International Conference on Pattern Recognition, 1996, 891-896.
[90] J. Liu, S.C. Chen, X. Tan, and D. Zhang, "Efficient Pseudo-inverse Linear Discriminant Analysis and its Nonlinear Form for Face Recognition", International Conference on Neural Information Processing 2006,
[91] J. Ye and Q. Li, "A two-stage Linear Discriminant Analysis via QR-Decomposition", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(6), 929-941.
[92] L.F. Chen, H.Y.M. iao, M.T. Ko, J.C. Lin, and G.J. Yu, "A new LDA-based face recognition system which can solve the small sample size problem", Pattern Recognition, 2000, 33(10), 1713-1726.
[93] H. Cevikalp, M. Neamtu, M. Wilke, and A. Barkana, "Discriminative common vectors for face recognition", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(1), 4-13.
[94] R. Huang, Q. Liu, H. Lu, and S. Ma, "Solving the small sample size problem of LDA", InternationalConference on Pattern Recognition, 2002, 29-32.
[95] J. Liu and S.C. Chen, "Discriminant Common Vecotors Versus Neighbourhood Components Analysis and Laplacianfaces: A comparative study in small sample size problem", Image and Vision Computing, 2006, 24(3), 249-262.
[96] X. Wang and X. Tang, "Random Sampling for Subspace Face Recognition", International Journal of Computer Vision, 2006, 70(1), 94-104.
[97] S.C. Chen, Y.L. Zhu, D. Zhang, and J. Yang, "Feature extraction approaches based on matrix pattern: MatPCA and MatFLDA", Pattern Recognition Letters, 2005, 26(8), 1157-1167.
[98] H. Li, T. Jiang, and K. Zhang, "Efficient and Robust Feature Extraction by Maximum Margin Criterion", IEEE Transactions on Neural Networks, 2006, 17(1), 157-165.
[99] H. Li, T. Jiang, and K. Zhang, "Efficient and Robust Feature Extraction by Maximum Margin Criterion", Advances in Neural Information Processing Systems, 2003, 97-104.
[100] M. Loog, R.P.W. Duin, and R. Haeb-Umbach, "Multiclass linear dimension reduction by weighted pairwise Fisher criteria", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2001, 23(7), 762-766.
[101] X. He, S. Yan, Y. Hu, and P. Niyogi, "Face recognition using Laplacianfaces", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(3), 328-340.
[102] J. Liu and S.C. Chen, "Resampling LDA/QR and PCA+LDA for face recognition", Australian Joint Conference on Artificial Intelligence, 2005, 1221-1224.
[103] X. Lu and A.K. Jain, "Resampling for face recognition", International Conference on Audio- and Video-based Biometric Person Authentication, 2003, 869-877.
[104] G. Baudat and F. Anouar, "Generalized discriminant analysis using a kernel approach", Neural Computation, 2000, 12(10), 2385-2404.
[105] S. Mika, G. Ratsch, J. Weston, B. Scholkpf, and K.R. Muller, "Fisher discriminant analysis with kernels", Neural networks for signal processing IX, 1999, 41-48.
[106] W. Zheng, C. Zou, and L. Zhao, "Weighted maximum margin discriminant analysis with kernels", Neurocomputing, 2005, 67, 357-362.
[107] A.M. Martinez, "PCA versus LDA", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2001, 23(2), 228-233.
[108] A.M. Martinez and M. Zhu, "Where Are Linear Feature Extraction Methods Applicable", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(12), 1934-1944.
[109] J. Liu, S.C. Chen, and X. Tan, "A Study on three Linear Discriminant Analysis Based Methods in Small Sample Size Problem", Pattern Recognition, 2008, 41(1), 102-116.
[110] J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov, "Neighbourhood Component Analysis", Advances in Neural Information Processing Systems, 2004, 512-520.
[111] L. Wang, X. Wang, X. Zhang, and J. Feng, "The equivalence of two-dimensional PCA to line-based PCA", Pattern Recognition Letters, 2005, 26(1), 57-60.
[112] Q. Gao, "Is two-dimensional PCA equivalent to a special case of modular PCA?" Pattern Recognition Letters 2007, 28(10), 1250-1251.
[113] Z. Liang and P. Shi, "An analytical algorithm for generalized low-rank approximations of matrices", Pattern Recognition, 2005, 38(11), 2217-2219.
[114] K. Inoue and K. Urahama, "Equivalence of Non-Iterative Algorithms for Simultaneous Low Rank Approximations of Matrices", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2006, 154-159.
[115] J. Liu, S.C. Chen, and Z.H. Zhou, "Progressive Principal Component Analysis", International Symposium on Neural Networks, 2004, 768-773.
[116] M. Berry, S. Dumais, and G. O'Brien, "Using linear algebra for intelligent information retrieval", SIAM Review, 1995, 37, 573-595.
[117] S. Deerwester, S.T. Dumais, T.K. Landauer, G.W. Furnas, and L. Beck, "Improving information retrieval using Latent Semantic Indexing", Annual meeting of the American Society for Information Science, 1988, 36-40.
[118] M. Brand, "Incremental Singular Value Decomposition of Uncertain Data with Missing Values", European Conference on Computer Vision, 2002, 707-720.
[119] M. Gu and S.C. Eisenstat, A fast and stable algorithm for updating the singular value decomposition. 1993, Department of Computer Science, Yale University, Technical Report, YALEU/DCS/RR-966.
[120] K.V.R. Kanth, D. Agrawal, and A. Singh, "Dimensionality Reduction for Similarity Searching in Dynamic Databases", Computer Vision and Image Understanding, 1999, 75(1-2), 59-72.
[121] D. Achlioptas and F. McSherry, "Fast Computation of Low Rank Matrix Approximations", Annual ACM symposium on the theory of computing, 2001, 611-618.
[122] P. Drineas, A. Frieze, R. Kannan, S. Vempala, and V. Vinay, "Clustering in large graphs and matrices", Annual ACM-SIAM Symposium on Discrete Algorithms, 1999, 291-299.
[123] A. Frieze, R. Kannan, and S. Vempala, "Fast monte-carlo algorithms for finding low-rank approximations", Journal of the ACM, 2004, 51(6), 1025-1041.
[124] S.A. Nene, S.K. Nayar, and H. Murase, Columbia Object Image Library (COIL-20). 1996, Department of Computer Science, Columbia University, Technical Report, CUCS-006- 96.
[125] M.A.O. Vasilescu and D. Terzopoulos, "Multilinear analysis of image ensembles: Tensorfaces", European Conference on Computer Vision, 2002, 447-460.
[126] P.J. Phillips, H. Wechsler, J. Huang, and P. Rauss, "The FERET database and evaluation procedure for face recognition algorithms", Image and Vision Computing, 1998, 16(5), 295-306.
[127] S.C. Chen, D. Zhang, and Z.H. Zhou, "Enhanced (PC)(2)A for face recognition with one training image per person", Pattern Recognition Letters, 2004, 25(10), 1173-1181.
[128] S. Dasgupta, "Experiments with random projection", Uncertainty in Artificial Intelligence, 2000,
[129] J. Liu, S.C. Chen, X. Tan, and D. Zhang, "Comments on "Efficient and Robust Feature Extraction by Maximum Margin Criterion"", IEEE Transactions on Neural Networks, 2007, 18(6), 1862-1864.
[130] S. Raudys and R.P.W. Duin, "On expected classification error of the fish linear classifier with pseudo-inverse covariance matrix", Pattern Recognition Letters, 1998, 19(5-6), 385-392.
[131] D.L. Swets and J. Weng, "Using Discriminant Eigenfeatures for Image Retrieval", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1996, 18(8), 831-836.
[132] D.L. Swets and J. Weng, "Discriminant Analysis and Eigenspace Partition Tree for Face and Object Recognition from Views", International Conference on Automatic Face and Gesture Recognition, 1996, 192-197.
[133] M.B. Gulmezoglu, V. Dzhafarov, and A. Barkana, "The common vector approach and its relation to principal component analysis", IEEE Transactions on Speech and Audio Processing, 2001, 9(6), 655-662.
[134] M.B. Gulmezoglu, V. Dzhafarov, M. Keskin, and A. Barkana, "A novel approach to isolated word recognition", IEEE Transactions on Speech and Audio Processing, 1999, 7(6), 620-628.
[135] L. Wolf and A. Shashua, "Learning over sets using kernel principal angles", Journal of Machine Learning Research, 2004, 4(6), 913-931.
[136] B. Scholkpf, R. Herbrich, and A. Smola, "A generalized representer theorem", Annual Conference on Computational Learning Theory, 2001, 416-426.
[137] B. Scholkpf and A. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond, MIT Press, 2002.
[138] T. Hastie and R. Tibshirani, "Efficient quadratic regularization for expression arrays", Biostatistics, 2004, 5(3), 329-340.
[139] J. Yang and J.Y. Yang, "Why can LDA be performed in PCA transformed space", Pattern Recognition, 2003, 36(2), 563-566.
[140] J. Ye, "Least Squares Linear Discriminant Analysis", International Conference on Machine Learning, 2007, 1087-1093.
[141] L. Wiskott, J. Fellous, N. Kruger, and C. Malsburg, "Face Recognition by Elastic Bunch Graph Matching", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19(7), 775-779.
[142] Y.Q. Cheng, K. Liu, J. ang, Y. Zhuang, and N. Gu, "Human face recognition method based on the statistical model of small sample size", Intelligent Robots and Computer Vision X: Algorithms and Techniques, 1991, 85-95.
[143] Y. Tian, T. Tan, Y. Wang, and Y. Fang, "Do singular values contain adequate information for face recognition", Pattern Recognition, 2003, 36(3), 649-655.
[144] X. Tan, S.C. Chen, Z.H. Zhou, and F. Zhang, "Recognizing partially occluded, expression variant faces from single training image per person with SOM and soft kNN ensemble", IEEE Transactions on Neural Networks, 2005, 16(4), 875-886.
[145] J. Blitzer, S. Ben-David, K. Crammer, and F. Pereira, "Analysis of Representations for Domain Adaptation", Advances in Neural Information Processing Systems, 2006, 137--144.
[146] B. Moghaddam, C. Nastar, and A. Pentland, "A bayesian similarity measure for direct image matching", International Conference on Pattern Recognition, 1996, 350-358.
[147] A.S. Georghiades, P.N. Belhumeur, and D.J. Kriegman, "From few to many: Illumination cone models for face recognition under variable lighting and pose", IEEE Transactions on PatternAnalysis and Machine Intelligence, 2001, 23(6), 643-660.
[148] X. Tan, J. Liu, and S.C. Chen, "Recognition from a Single Sample per Person with Multiple SOM Fusion", International Symposium on Neural Networks, 2006, 128-133.
[149] X. Tan, J. Liu, and S.C. Chen, "Weighted SOM-Face: Selecting Local Features for Recognition from Individual Face Image", International Conference on Intelligence Data Engineering And Automated Learning, 2005, 351-358.
[150]谭晓阳,刘俊,张福炎, "基于SOM脸的选择性单训练样本人脸识别",南京航空航天大学学报, 2005, 37(1), 44-47.
[151] L. Wolf and A. Shashua, "Kernel principal angles for classification machines with applications to image sequence interpretation", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003, 635-640.
[152] H. Wang, Y. Zhou, X. Ge, and J. Yang, "Subspace evolution analysis for face representation and recognition", Pattern Recognition, 2007, 40(1), 335-338.
[153] O. Chapelle, P. Haffner, and V.N. Vapnik, "Support Vector Machines for Histogram-based Image Classification", IEEE Transactions on Neural Networks, 1999, 10(5), 1055-1064.
[154] J. Shawe-Taylor and N. Cristianini, "Kernel Methods for Pattern Analysis", Cambridge University Press, 2004,
[155] K. Mikolajczyk and C. Schmid, "A Performance Evaluation of Local Descriptors", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(10), 1615-1630.
[2] R. Brunell and T. Poggio, "Face recognition: Features vs templates", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1993, 15(10), 1042-1053.
[3] R. Chellappa, C.L. Wilson, and S. Sirohey, "Human and Machine Recognition of Faces: A survey", Proceedings of the IEEE, 1995, 83(5), 705-740.
[4] L. Harmon, M. Kaun, R. Lasch, and P. Ramig, "Machine identification of human faces", Pattern Recognition, 1981, 13(2), 97-110.
[5] W. Zhao, R. Chellappa, P.J. hillips, and A. Rosenfeld, "Face Recognition: A Literature Survey", ACM Computing Survey, 2003, 35(4), 399-458.
[6] S.G. Kong, J. Heo, B.R. Abidi, J. Paik, and M.A. Abidi, "Recent advances in visual and infrared face recognition-a review", Computer Vision and Image Understanding, 2005, 97(1), 103-135.
[7] X. Tan, S.C. Chen, Z.H. Zhou, and h. F., "Face recognition from a single image per person: A survey", Pattern Recognition, 2006, 39(9), 1725-1745.
[8] T. Kanade. Picture Processing by Computer Complex and Recognition of Human Faces. PhD thesis, Kyoto University, 1973
[9] G. Kaufman and K. Breeding, "The automatic recognition of human faces from profile silhouettes", IEEE Transactions on Systems, Man and Cybernetics, 1976, 6(2), 113-121.
[10]边肇祺,张学工, "模式识别",清华大学出版社, 2000,
[11] A.K. Hrechak and J.A. McHugh, "Automated fingerprint recognition using structural matching", Pattern Recognition 1990, 23(8), 893-904.
[12] R.P. Wildes, "Iris recognition: an emerging biometric technology", Proceedings of the IEEE, 1997, 85(9), 1348-1363.
[13] G. Lu, D. Zhang, and K. Wang, "Palmprint recognition using eigenpalms features ", Pattern Recognition Letters, 2003, 24(9-10), 1463 - 1467.
[14] E. Hjelmas and B.K. Low, "Face detection: A Survey", Computer Vision and Image Understanding, 2001, 83(3), 236-274.
[15] M.H. Yang, D. Kriegman, and N. Ahuja, "Detecting faces in images: A survey", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 24(1), 34-58.
[16]刘青山,卢汉清,马颂德, "综述人脸识别中的子空间方法",自动化学报, 2003, 29(6), 900-911.
[17] R.O. Duda and P.E.S.D.G. Hart, Pattern Classification, Wiley-Interscience Press, 2nd Edition, 2000.
[18] K. Fukunaga, Introduction to Statistical Pattern Recognition, Academic Press, 2nd Edition, 1990.
[19] C. Cortes and V.N. Vapnik, "Support-vector networks", Machine Learning, 1995, 20(3), 273-297.
[20] J.J. Atick, P.A. Griffin, and A.N. Redlich, "Statistical Approach to Shape from Shading: Reconstruction of 3D Face Surfaces from Single 2D Images", Neural Computation, 1996, 8(6), 1321-1340.
[21] C. Beumier and M. Acheroy, "Automatic 3D face authentication", Image and Vision Computing, 2000, 18(4), 315-321.
[22] A.M. Bronstein, M.M. Bronstein, and R. Kimmel, "Expression-invariant 3D face recognition", International Conference on Audio and Video-based Biometric Person Authentication, 2003, 62-70.
[23] D.A. Socolinsky and A. Selinger, "Thermal Face Recognition in an Operational Scenario", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004, 1012-.
[24] G. Bebis, A. Gyaourova, S. Singh, and I. Pavlidis, "Face recognition by fusing thermal infrared and visible imagery", Image and Vision Computing, 2006, 24(7), 727-742.
[25] D.O. Gorodnichy, "Seeing faces in video by computers. Editorial for Special Issue on Face Processing in Video Sequences ", Image and Vision Computing 2006, 24(6), 551-556.
[26] A. Yip and P. Sinn, "Role of Color in Face Recognition ", AI Memo, 2001, 2001-035.
[27] A. Goldstein, L. Harmon, and A. Lesk, "Identification of human faces", Proceedings of the IEEE, 1971, 59(5), 748-760.
[28] A. Goldstein and P. Iyengar, "Automatic recognition and analysis of human faces and facial expressions: A survey", Pattern Recognition, 1992, 25(1), 65-77.
[29] L. Harmon, S. Kuo, P. Ramig, and U. Raudkivi, "Identification of human face profiles by computer", Pattern Recognition, 1978, 10(5-6), 301-312.
[30] M. Turk and A. Pentland, "Eigenfaces for recognition", Journal of Cognitive Neuroscience, 1991, 3(1), 71-96.
[31] D. Beymer and T. Poggio, "Face recognition from one example view", International Conference on Computer Vision, 1995, 500-507.
[32] A.M. Martinez, "Recognizing imprecisely localized, partially occluded, and expression variant faces from a single sample per class", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2002, 25(6), 748-763.
[33] P. Niyogi, F. Girosi, and T. Poggio, "Incorporating prior information in machine learning by creating virtual examples", Proceedings of the IEEE, 1998, 86(11), 2196-2209.
[34] J. Wu and Z.H. Zhou, "Face recognition with one training image per person", Pattern Recognition Letters, 2002, 23(14), 1711-1719.
[35] S. Shan, W. Gao, and D. Zhao, "Face Identification Based on Face-Specific Subspace", International Journal of Imaging Systems and Technology, 2003, 13(1), 23-32.
[36] F. Torre, R. Gross, S. Baker, and V. Kumar, "Representational oriented component analysis (ROCA) for face recognition with one sample image per training class", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 266-273.
[37] H.C. Jung, B.W. Hwang, and S.W. Lee, "Authenticating corrupted face image based on noise model", IEEE International Workshop on Analysis and Modeling of Faces and Gestures, 2004, 120-126.
[38] X. Liu and T. Chen, "Pose-Robust Face Recognition Using Geometry Assisted Probabilistic Modeling", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 502-509.
[39] J. Liu, S.C. Chen, Z.H. Zhou, and X. Tan, "Single Image Subspace For Face Recognition", IEEE International Workshop on Analysis and Modeling of Faces and Gestures, 2007, 205-219.
[40] 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, 1997, 19(7), 711-720.
[41] R.P.W. Duin, M. Loog, and T.K. Ho, "Editorial: Recent submissions in linear dimensionality reduction and face recognition", Pattern Recognition Letters, 2006, 27(7), 707-708.
[42] M. Kirby and L. Sirovich, "Application of the Karhunen-Loeve procedure for the characterization ofhuman faces", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1990, 12(1), 103-108.
[43] A. Leonardis and H. Bischof, "Kernel and subspace for computer vision", Pattern Recognition, 2003, 36(9), 1925-1927.
[44] T. Ahonen, A. Hadid, and M. Pietikainen, "Face Description with Local Binary Patterns: Application to Face Recognition", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2006, 28(12), 2037-2041.
[45] S.C. Chen, J. Liu, and Z.H. Zhou, "Making FLDA applicable to face recognition with one sample per person", Pattern Recognition, 2004, 37(7), 1553-1555.
[46] X. Tan, J. Liu, and S.C. Chen, "Sub-intrapersonal space analysis for face recognition", Neurocomputing, 2006, 69(13-15), 1796-1801.
[47] X. Tan, S.C. Chen, Z.H. Zhou, and J. Liu, "Learning Non-Metric Partial Similarity Based on Maximal Margin Criterion", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2006, 138-145.
[48] A. Pentland, B. Moghaddam, and T. Starner, "View-based and modular eigenspaces for face recognition", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1994, 84-91.
[49] T.K. Ho, "The Random Subspace Method for Constructing Decision Forests", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1998, 20(8), 832-844.
[50] C. Ding and J. Ye, "2-dimensional singular value decomosition for 2d maps and images", SIAM International Conference on Data Mining, 2005, 22-34.
[51] J. Liu and S.C. Chen, "Non-iterative generalized low rank approximation of matrices", Pattern Recognition Letters, 2006, 27(9), 1002-1008.
[52] H. Wang and N. Ahuja, "Rank-R Approximation of Tensors Using Image-as-Matrix Representation", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 346-353.
[53] D. Xu, S. Yan, L. Zhang, H. Zhang, Z. Liu, and H. Shum, "Concurrent subspaces analysis", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005, 203-208.
[54] J. Yang, D. Zhang, A.F. Frangi, and J.Y. Yang, "Two-dimensional PCA: A new approach to appearance-based face representation and recognition", IEEE Transactions on Pattern Analysisand Machine Intelligence, 2004, 26(1), 131-137.
[55] J. Ye, "Generalized low rank approximation of matrices", International Conference on Machine Learning, 2004, 887-894.
[56] J. Ye, "Generalized low rank approximations of matrices", Machine Learning, 2005, 61(1-3), 167-191.
[57] J. Ye, R. Janardan, and Q. Li, Two-Dimensional Linear Discriminant Analysis, in Advances in Neural Information Processing Systems. 2004. p. 354-363.
[58] D. Zhang, S.C. Chen, and J. Liu, "Representing image matrices: Eigenimages versus eigenvectors", International Symposium on Neural Networks, 2005, 659-664.
[59] Z. Wang, S.C. Chen, J. Liu, and D. Zhang, "Pattern Representation in Feature Extraction and Classification- Matrix Versus Vector", IEEE Transactions on Neural Networks, 2007, accepted.
[60] H. Sellahewa and S. Jassim, "Face Recognition in the Presence of Expression and/or Illumination Variation", IEEE Workshop on Automatic Identification Advanced Technologies, 2005, 144-148.
[61] Y. Adini, Y. Moses, and S. Ullman, "Face Recognition: The Problem of Compensating for Changes in Illumination Direction", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19(7), 721-732.
[62] V. Blanz, S. Romdhami, and T. Vetter, "Face identification across different poses and illuminations with a 3D morphable model", International Conference on Automatic Face and Gesture Recognition, 2002, 202-207.
[63] K. Lee, J. Ho, and D. Kriegman, "Acquiring Linear Subspaces for Face Recognition under Variable Lighting Illumination-Insensitive Face Recognition Using Symmetric Shape-from-Shading", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(5), 1-15.
[64] W. Zhao and R. Chellappa, "Illumination-Insensitive Face Recognition Using Symmetric Shape-from-Shading", International Conference on Computer Vision and Pattern Recognition, 2000, 286-549.
[65] A.M. Martinez and R. Benavente, "The AR Face Database", Technical Report, CVC, 1998,
[66] W. Zhang, S. Shan, W. Gao, X. Chen, and H. Zhang, "Local Gabor Binary Pattern Histogram Sequence (LGBPHS): A Novel Non-Statistical Model for Face Recognition", International Conference on Computer Vision, 2005, 786-791.
[67] J. Liu, S.C. Chen, and X. Tan, "Fractional order singular value decomposition representation for facerecognition", Pattern Recognition, 2008, 41(1), 378-395.
[68] G. Donato, M.S. Bartlett, J.C. Hager, P. Ekman, and T.J. Sejnowski, "Classifying facial actions", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1999, 21(10), 974-989.
[69] C. Liu and H. Wechsler, "Independent Component Analysis of Gabor Features for Face Recognition", IEEE Transactions on Neural Networks, 2003, 14(4), 919-928.
[70] R.C. Gonzalez and R.E. Woods, Digital Image Processing, Prentice Hall Press, 2nd Edition, 2004.
[71] M.J. Er, W. Chen, and S. Wu, "High-Speed Face Recognition Based on Discrete Cosine Transform and RBF Neural Networks", IEEE Transactions on Neural Networks, 2005, 16(3), 679-691.
[72] Z.M. Hafed and M.D. Levine, "Face recognition using the discrete cosine transform", International Journal of Computer Vision, 2001, 43(3), 167-188.
[73] P. Comon, "Independent component analysis, A new concept?" Signal Processing 1994, 36(3), 287-314.
[74] R. Gottumukkal and V.K. Asari, "An improved face recognition technique based on modular PCA approach", Pattern Recognition Letters, 2004, 25(4), 429-436.
[75] S.C. Chen and Y. Zhu, "Subpattern-based principle component analysis", Pattern Recognition, 2004, 37(5), 1081-1083.
[76] K.R. Tan and S.C. Chen, "Adaptively weighted sub-pattern PCA for face recognition", Neurocomputing, 2005, 64, 505-511.
[77] D.S. Bolme, J.R. Beveridge, M. Teixeira, and B.A. Draper, "The CSU face identification evaluation system: Its purpose, features and structure", International Conference on Computer Vision Systems, 2003, 303-311.
[78] K. Inoue and K. Urahama, "Dsvd: A tensor-based image compression and recognition method", IEEE International Symposium on Circuits and Systems, 2005, 6308-6311.
[79] J. Weng, Y. Zhang, and W.S. Hwang, "Candid covariance-free incremental principal component analysis ", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2003, 25(8), 1034-1040.
[80] H. Zhao, P. Yuen, and K. J., "Incremental Principal Component Analysis and its Application for Face Recognition", IEEE Transactions on Systems, Man and Cybernetics (Part B), 2006, 36(4), 873-886.
[81] K.I. Kim, K. Jung, and H.J. Kim, "Face recognition using kernel principal component analysis", IEEE Transactions on Signal Processing Letters, 2002, 9(2), 40-42.
[82] M.H. Yang, "Kernel Eigenfaces vs. Kernel Fisherfaces: Face Recognition Using Kernel Methods", IEEE International Conference on Automatic Face and Gesture Recognition 2002, 215-220.
[83] R.A. Fisher, "The use of multiple measures in taxonomic problems", Annual Eugenics, 1936, 7, 179-188.
[84] J.H. Friedman, "Regularized discriminant analysis", Journal of The American Statistical Association, 1989, 84(405), 165-175.
[85] Q. Tian, M. Barbero, Z.H. Gu, and S.H. Lee, "Image classification by the Foley-Sammon transform", Optical Engineering, 1986, 25(7), 834-840.
[86] G.H. Golub and C.F. Van Loan, Matrix Computations. Baltimore, MD, The Johns Hopkins University Pres, 3rd Edition, 1996.
[87] W.J. Krzanowski, P. Jonathan, W.V. McCarthy, and M.R. Thomas, "Discriminant analysis with singular covariance matrices: methods and applications to spectroscopic data", Applied Statistics, 1995, 44(1), 101-115.
[88] M. Skurichina and R.P.W. Duin, "Regularization of linear classifiers by adding redundant features", Pattern Analysis and Applications, 1999, 2(1), 44-52.
[89] M. Skurichina and R.P.W. Duin, "Stabilizing classifiers for very small sample size", International Conference on Pattern Recognition, 1996, 891-896.
[90] J. Liu, S.C. Chen, X. Tan, and D. Zhang, "Efficient Pseudo-inverse Linear Discriminant Analysis and its Nonlinear Form for Face Recognition", International Conference on Neural Information Processing 2006,
[91] J. Ye and Q. Li, "A two-stage Linear Discriminant Analysis via QR-Decomposition", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(6), 929-941.
[92] L.F. Chen, H.Y.M. iao, M.T. Ko, J.C. Lin, and G.J. Yu, "A new LDA-based face recognition system which can solve the small sample size problem", Pattern Recognition, 2000, 33(10), 1713-1726.
[93] H. Cevikalp, M. Neamtu, M. Wilke, and A. Barkana, "Discriminative common vectors for face recognition", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(1), 4-13.
[94] R. Huang, Q. Liu, H. Lu, and S. Ma, "Solving the small sample size problem of LDA", InternationalConference on Pattern Recognition, 2002, 29-32.
[95] J. Liu and S.C. Chen, "Discriminant Common Vecotors Versus Neighbourhood Components Analysis and Laplacianfaces: A comparative study in small sample size problem", Image and Vision Computing, 2006, 24(3), 249-262.
[96] X. Wang and X. Tang, "Random Sampling for Subspace Face Recognition", International Journal of Computer Vision, 2006, 70(1), 94-104.
[97] S.C. Chen, Y.L. Zhu, D. Zhang, and J. Yang, "Feature extraction approaches based on matrix pattern: MatPCA and MatFLDA", Pattern Recognition Letters, 2005, 26(8), 1157-1167.
[98] H. Li, T. Jiang, and K. Zhang, "Efficient and Robust Feature Extraction by Maximum Margin Criterion", IEEE Transactions on Neural Networks, 2006, 17(1), 157-165.
[99] H. Li, T. Jiang, and K. Zhang, "Efficient and Robust Feature Extraction by Maximum Margin Criterion", Advances in Neural Information Processing Systems, 2003, 97-104.
[100] M. Loog, R.P.W. Duin, and R. Haeb-Umbach, "Multiclass linear dimension reduction by weighted pairwise Fisher criteria", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2001, 23(7), 762-766.
[101] X. He, S. Yan, Y. Hu, and P. Niyogi, "Face recognition using Laplacianfaces", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(3), 328-340.
[102] J. Liu and S.C. Chen, "Resampling LDA/QR and PCA+LDA for face recognition", Australian Joint Conference on Artificial Intelligence, 2005, 1221-1224.
[103] X. Lu and A.K. Jain, "Resampling for face recognition", International Conference on Audio- and Video-based Biometric Person Authentication, 2003, 869-877.
[104] G. Baudat and F. Anouar, "Generalized discriminant analysis using a kernel approach", Neural Computation, 2000, 12(10), 2385-2404.
[105] S. Mika, G. Ratsch, J. Weston, B. Scholkpf, and K.R. Muller, "Fisher discriminant analysis with kernels", Neural networks for signal processing IX, 1999, 41-48.
[106] W. Zheng, C. Zou, and L. Zhao, "Weighted maximum margin discriminant analysis with kernels", Neurocomputing, 2005, 67, 357-362.
[107] A.M. Martinez, "PCA versus LDA", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2001, 23(2), 228-233.
[108] A.M. Martinez and M. Zhu, "Where Are Linear Feature Extraction Methods Applicable", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(12), 1934-1944.
[109] J. Liu, S.C. Chen, and X. Tan, "A Study on three Linear Discriminant Analysis Based Methods in Small Sample Size Problem", Pattern Recognition, 2008, 41(1), 102-116.
[110] J. Goldberger, S. Roweis, G. Hinton, and R. Salakhutdinov, "Neighbourhood Component Analysis", Advances in Neural Information Processing Systems, 2004, 512-520.
[111] L. Wang, X. Wang, X. Zhang, and J. Feng, "The equivalence of two-dimensional PCA to line-based PCA", Pattern Recognition Letters, 2005, 26(1), 57-60.
[112] Q. Gao, "Is two-dimensional PCA equivalent to a special case of modular PCA?" Pattern Recognition Letters 2007, 28(10), 1250-1251.
[113] Z. Liang and P. Shi, "An analytical algorithm for generalized low-rank approximations of matrices", Pattern Recognition, 2005, 38(11), 2217-2219.
[114] K. Inoue and K. Urahama, "Equivalence of Non-Iterative Algorithms for Simultaneous Low Rank Approximations of Matrices", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2006, 154-159.
[115] J. Liu, S.C. Chen, and Z.H. Zhou, "Progressive Principal Component Analysis", International Symposium on Neural Networks, 2004, 768-773.
[116] M. Berry, S. Dumais, and G. O'Brien, "Using linear algebra for intelligent information retrieval", SIAM Review, 1995, 37, 573-595.
[117] S. Deerwester, S.T. Dumais, T.K. Landauer, G.W. Furnas, and L. Beck, "Improving information retrieval using Latent Semantic Indexing", Annual meeting of the American Society for Information Science, 1988, 36-40.
[118] M. Brand, "Incremental Singular Value Decomposition of Uncertain Data with Missing Values", European Conference on Computer Vision, 2002, 707-720.
[119] M. Gu and S.C. Eisenstat, A fast and stable algorithm for updating the singular value decomposition. 1993, Department of Computer Science, Yale University, Technical Report, YALEU/DCS/RR-966.
[120] K.V.R. Kanth, D. Agrawal, and A. Singh, "Dimensionality Reduction for Similarity Searching in Dynamic Databases", Computer Vision and Image Understanding, 1999, 75(1-2), 59-72.
[121] D. Achlioptas and F. McSherry, "Fast Computation of Low Rank Matrix Approximations", Annual ACM symposium on the theory of computing, 2001, 611-618.
[122] P. Drineas, A. Frieze, R. Kannan, S. Vempala, and V. Vinay, "Clustering in large graphs and matrices", Annual ACM-SIAM Symposium on Discrete Algorithms, 1999, 291-299.
[123] A. Frieze, R. Kannan, and S. Vempala, "Fast monte-carlo algorithms for finding low-rank approximations", Journal of the ACM, 2004, 51(6), 1025-1041.
[124] S.A. Nene, S.K. Nayar, and H. Murase, Columbia Object Image Library (COIL-20). 1996, Department of Computer Science, Columbia University, Technical Report, CUCS-006- 96.
[125] M.A.O. Vasilescu and D. Terzopoulos, "Multilinear analysis of image ensembles: Tensorfaces", European Conference on Computer Vision, 2002, 447-460.
[126] P.J. Phillips, H. Wechsler, J. Huang, and P. Rauss, "The FERET database and evaluation procedure for face recognition algorithms", Image and Vision Computing, 1998, 16(5), 295-306.
[127] S.C. Chen, D. Zhang, and Z.H. Zhou, "Enhanced (PC)(2)A for face recognition with one training image per person", Pattern Recognition Letters, 2004, 25(10), 1173-1181.
[128] S. Dasgupta, "Experiments with random projection", Uncertainty in Artificial Intelligence, 2000,
[129] J. Liu, S.C. Chen, X. Tan, and D. Zhang, "Comments on "Efficient and Robust Feature Extraction by Maximum Margin Criterion"", IEEE Transactions on Neural Networks, 2007, 18(6), 1862-1864.
[130] S. Raudys and R.P.W. Duin, "On expected classification error of the fish linear classifier with pseudo-inverse covariance matrix", Pattern Recognition Letters, 1998, 19(5-6), 385-392.
[131] D.L. Swets and J. Weng, "Using Discriminant Eigenfeatures for Image Retrieval", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1996, 18(8), 831-836.
[132] D.L. Swets and J. Weng, "Discriminant Analysis and Eigenspace Partition Tree for Face and Object Recognition from Views", International Conference on Automatic Face and Gesture Recognition, 1996, 192-197.
[133] M.B. Gulmezoglu, V. Dzhafarov, and A. Barkana, "The common vector approach and its relation to principal component analysis", IEEE Transactions on Speech and Audio Processing, 2001, 9(6), 655-662.
[134] M.B. Gulmezoglu, V. Dzhafarov, M. Keskin, and A. Barkana, "A novel approach to isolated word recognition", IEEE Transactions on Speech and Audio Processing, 1999, 7(6), 620-628.
[135] L. Wolf and A. Shashua, "Learning over sets using kernel principal angles", Journal of Machine Learning Research, 2004, 4(6), 913-931.
[136] B. Scholkpf, R. Herbrich, and A. Smola, "A generalized representer theorem", Annual Conference on Computational Learning Theory, 2001, 416-426.
[137] B. Scholkpf and A. Smola, Learning with Kernels: Support Vector Machines, Regularization, Optimization and Beyond, MIT Press, 2002.
[138] T. Hastie and R. Tibshirani, "Efficient quadratic regularization for expression arrays", Biostatistics, 2004, 5(3), 329-340.
[139] J. Yang and J.Y. Yang, "Why can LDA be performed in PCA transformed space", Pattern Recognition, 2003, 36(2), 563-566.
[140] J. Ye, "Least Squares Linear Discriminant Analysis", International Conference on Machine Learning, 2007, 1087-1093.
[141] L. Wiskott, J. Fellous, N. Kruger, and C. Malsburg, "Face Recognition by Elastic Bunch Graph Matching", IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19(7), 775-779.
[142] Y.Q. Cheng, K. Liu, J. ang, Y. Zhuang, and N. Gu, "Human face recognition method based on the statistical model of small sample size", Intelligent Robots and Computer Vision X: Algorithms and Techniques, 1991, 85-95.
[143] Y. Tian, T. Tan, Y. Wang, and Y. Fang, "Do singular values contain adequate information for face recognition", Pattern Recognition, 2003, 36(3), 649-655.
[144] X. Tan, S.C. Chen, Z.H. Zhou, and F. Zhang, "Recognizing partially occluded, expression variant faces from single training image per person with SOM and soft kNN ensemble", IEEE Transactions on Neural Networks, 2005, 16(4), 875-886.
[145] J. Blitzer, S. Ben-David, K. Crammer, and F. Pereira, "Analysis of Representations for Domain Adaptation", Advances in Neural Information Processing Systems, 2006, 137--144.
[146] B. Moghaddam, C. Nastar, and A. Pentland, "A bayesian similarity measure for direct image matching", International Conference on Pattern Recognition, 1996, 350-358.
[147] A.S. Georghiades, P.N. Belhumeur, and D.J. Kriegman, "From few to many: Illumination cone models for face recognition under variable lighting and pose", IEEE Transactions on PatternAnalysis and Machine Intelligence, 2001, 23(6), 643-660.
[148] X. Tan, J. Liu, and S.C. Chen, "Recognition from a Single Sample per Person with Multiple SOM Fusion", International Symposium on Neural Networks, 2006, 128-133.
[149] X. Tan, J. Liu, and S.C. Chen, "Weighted SOM-Face: Selecting Local Features for Recognition from Individual Face Image", International Conference on Intelligence Data Engineering And Automated Learning, 2005, 351-358.
[150]谭晓阳,刘俊,张福炎, "基于SOM脸的选择性单训练样本人脸识别",南京航空航天大学学报, 2005, 37(1), 44-47.
[151] L. Wolf and A. Shashua, "Kernel principal angles for classification machines with applications to image sequence interpretation", IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003, 635-640.
[152] H. Wang, Y. Zhou, X. Ge, and J. Yang, "Subspace evolution analysis for face representation and recognition", Pattern Recognition, 2007, 40(1), 335-338.
[153] O. Chapelle, P. Haffner, and V.N. Vapnik, "Support Vector Machines for Histogram-based Image Classification", IEEE Transactions on Neural Networks, 1999, 10(5), 1055-1064.
[154] J. Shawe-Taylor and N. Cristianini, "Kernel Methods for Pattern Analysis", Cambridge University Press, 2004,
[155] K. Mikolajczyk and C. Schmid, "A Performance Evaluation of Local Descriptors", IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(10), 1615-1630.