邻域结构保持投影及应用
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摘要
人脸识别作为一种典型的生物特征识别技术,以其自身的优势已经成为一个热点研究问题。在人脸识别中,特征提取是非常关键的一步,影响着后续分析和分类系统的性能。因此,如何有效地提取出反映数据本质结构的特征,方便后续分类,已成为目前需要解决的核心问题之一。在众多方法中,子空间分析是目前非常活跃的研究方向之一。本论文从子空间分析中的流形学习技术入手,深入研究了基于图论的图像空间几何结构描述,主要内容和贡献有:
1.针对NPE不能较好地保持模式之间的差异信息,尤其是非线性数据的差异信息,提出了邻域结构保持嵌入(Neighborhood Structure Preserving Embedding, NSPE)。NSPE分别利用差异邻接图和相似邻接图描述数据空间的差异几何关系和相似几何关系,并给出了度量差异信息和相似信息的离散度矩阵;在此基础上,给出了一个意义明确、简洁的特征提取准则。
2.针对NSPE需要将图像数据转换成向量,导致计算复杂和小样本等问题,提出了二维邻域结构保持嵌入算法(Two-dimensional Neighborhood Structure Preserving Embedding ,2DNSPE)。2DNSPE直接用图像矩阵来计算度量图象差异信息和和相似信息的离散度矩阵;在此基础上,给出了简洁、明确的特征提取准则。
3.针对2DNPE和2DNSPE不能很好地保持图像像素的空间几何关系,导致冗余比较多和识别率不是足够的好等问题,分别提出了二维方向邻域保持嵌入(Directional Two-dimensional Neighborhood Preserving Embedding,Dir-2DNPE)和二维方向邻域结构保持嵌入(Directional Two-dimensional Neighborhood Structure Preserving Embedding, Dir-2DNSPE)算法。两种算法借助方向图对图象像素进行重组,有效地利用了像素的空间几何关系,然后分别利用Dir-2DNPE和Dir-2DNSPE提取图像特征,实验结果证实了所提方法的有效性。
Face recognition is one of the representative biometric feature recognition techniques and has becoming an active research field in pattern recognition and computer vision. In face recognition, feature extraction is one of the very important key steps and involves the performance of the subsequent anslysis and classification system. So, how to effectively extract the intrinsic geometric structure embedded in data, which facilitates the subsequent analysis such as classification, has become one of the key problems to be solved. Many approaches have been proposed to slove it, and one of the most active approaches is subspace analysis methods (SAM). The dissertation mainly studies how to characterize the spatial geometric structures including similarity and diversity of data by combining graph theory and manifold learning belongs to SAM. The main contributions and work are as follows:
1. Neighborhood Structure Preserving Embedding (NSPE) is proposed. NSPE defines two adjacency graphs, namely similarity graph and diversity graph, over the training data to model the spatial similarity and diversity structures of data, respectively. Similarity scatter and diversity scatter are calculated from the two graphs. Based on the two scatters, a concise feature extraction criterion is then raised by maximizing the ratio of the diversity scatter to similarity scatter.
2. Two-dimensional Neighborhood Structure Preserving Embedding(2DNSPE) is proposed. 2DNSPE directly calculates the similarity scatter and diversity scatter from the image matrices. Based on the two scatters, a concise feature extraction criterion is raised by maximizing the ratio of the diversity scatter to similarity scatter. Different from NSPE, 2DNSPE avoids transforming the image matric into a vector, thus reducing the computational complexity and alleviating the small sample problem.
3. Two novel methods, namely Directional Two-dimensional Neighborhood Preserving Embedding (Dir-2DNPE) and Directional Two-dimensional Neighborhood Structure Preserving Embedding (Dir-2DNSPE), are proposed to reduce dimension of images. Two methods rearrange pixels in images by using directional image and then performe Dir-2DNPE and Dir-2DNSPE to reduce dimension of images respectively. Experiment results show the efficiency of the two methods.
1.针对NPE不能较好地保持模式之间的差异信息,尤其是非线性数据的差异信息,提出了邻域结构保持嵌入(Neighborhood Structure Preserving Embedding, NSPE)。NSPE分别利用差异邻接图和相似邻接图描述数据空间的差异几何关系和相似几何关系,并给出了度量差异信息和相似信息的离散度矩阵;在此基础上,给出了一个意义明确、简洁的特征提取准则。
2.针对NSPE需要将图像数据转换成向量,导致计算复杂和小样本等问题,提出了二维邻域结构保持嵌入算法(Two-dimensional Neighborhood Structure Preserving Embedding ,2DNSPE)。2DNSPE直接用图像矩阵来计算度量图象差异信息和和相似信息的离散度矩阵;在此基础上,给出了简洁、明确的特征提取准则。
3.针对2DNPE和2DNSPE不能很好地保持图像像素的空间几何关系,导致冗余比较多和识别率不是足够的好等问题,分别提出了二维方向邻域保持嵌入(Directional Two-dimensional Neighborhood Preserving Embedding,Dir-2DNPE)和二维方向邻域结构保持嵌入(Directional Two-dimensional Neighborhood Structure Preserving Embedding, Dir-2DNSPE)算法。两种算法借助方向图对图象像素进行重组,有效地利用了像素的空间几何关系,然后分别利用Dir-2DNPE和Dir-2DNSPE提取图像特征,实验结果证实了所提方法的有效性。
Face recognition is one of the representative biometric feature recognition techniques and has becoming an active research field in pattern recognition and computer vision. In face recognition, feature extraction is one of the very important key steps and involves the performance of the subsequent anslysis and classification system. So, how to effectively extract the intrinsic geometric structure embedded in data, which facilitates the subsequent analysis such as classification, has become one of the key problems to be solved. Many approaches have been proposed to slove it, and one of the most active approaches is subspace analysis methods (SAM). The dissertation mainly studies how to characterize the spatial geometric structures including similarity and diversity of data by combining graph theory and manifold learning belongs to SAM. The main contributions and work are as follows:
1. Neighborhood Structure Preserving Embedding (NSPE) is proposed. NSPE defines two adjacency graphs, namely similarity graph and diversity graph, over the training data to model the spatial similarity and diversity structures of data, respectively. Similarity scatter and diversity scatter are calculated from the two graphs. Based on the two scatters, a concise feature extraction criterion is then raised by maximizing the ratio of the diversity scatter to similarity scatter.
2. Two-dimensional Neighborhood Structure Preserving Embedding(2DNSPE) is proposed. 2DNSPE directly calculates the similarity scatter and diversity scatter from the image matrices. Based on the two scatters, a concise feature extraction criterion is raised by maximizing the ratio of the diversity scatter to similarity scatter. Different from NSPE, 2DNSPE avoids transforming the image matric into a vector, thus reducing the computational complexity and alleviating the small sample problem.
3. Two novel methods, namely Directional Two-dimensional Neighborhood Preserving Embedding (Dir-2DNPE) and Directional Two-dimensional Neighborhood Structure Preserving Embedding (Dir-2DNSPE), are proposed to reduce dimension of images. Two methods rearrange pixels in images by using directional image and then performe Dir-2DNPE and Dir-2DNSPE to reduce dimension of images respectively. Experiment results show the efficiency of the two methods.
引文
[1]谭铁牛.生物识别研究新进展(一).北京:清华大学出版社,2002.
[2] Jain A, Ross A, Prabhakar S. An introduction to biometric recognition. IEEE Transactionon Circuit and System for Video Technology. 2004, 14(1):4-20.
[3] Pankanti S, Bolle R M, Jain A. Biometrics: the future of identification. Computer, 2000, 33(2):46-49.
[4] Woodward J D, Orlans J N M, Higgins P T著,陈菊明.邓启威等译.生物认证.北京:清华大学出版社, 2004:21-46.
[5] Bahl L R, Jelinek F. A Maximum likelihood approach to continuous speech recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1983, 5:179-190.
[6] Hanson B A, Applehaum T H. Robust Speaker-independent Word Recognition Using Static. Dynamic and Acceleration Features: Experiments with Lombard and Noisy Speech. International conference on Acoustics, Speech and Signal Processing 1990, 857-860.
[7] Bledssoe W W. Man-machine facial recognition: Report on a Large-Scale Experiment.Technical Report PRI22, Panoramic Research Inc. Palo Alto, Californi, 1996.
[8]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.
[9]张翠萍,苏光大.人脸识别技术综述.中国图像图形学报. 2000, 5(11):855-894.
[10] Samal A, Iyengar P A. Automatic recognition and analysis of human faces and facial expression:A survey.Pattern Recognition.1992,25(1):65-67.
[11] Chellappa R, Wilson C L, Sirohey S. Human and machine recognition of faces:A survey. IEEE Proc.1995, 83(5):705-740.
[12]侯晓宇.基于流形学习的特征提取方法研究.大连理工大学硕士论文. 2009, 6.
[13]黄凤岗,宋克欧.模式识别.哈尔滨:哈尔滨工程大学出版社. 1998.
[14]边肇祺,张学工.模式识别.北京:清华大学出版社. 2000. 1-271.
[15]高全学.基于子空间分析的人脸识别研究.西北工业大学博士学位论文. 2005.
[16] Cavalcanti G D C, Filho E C B C. Eigenbands Fusion for Frontal Face Recognition. Proc. IEEE Intel Conf. on Image Processing. 2003: 665-668.
[17] Wiskott L, Fellous J M, Kruger N, etc. Face recognition by elastic bunch graph matching. IEEE Trans, Pattern Analysis and Machine Intelligence. 1997,19(7):775-779.
[18] Samaria F S. Face recognition using hidden Markov model [Ph.Ddissertation]. University of Cambridge, 1994.
[19] Kohonen T. Self-Organizing maps. Springer Series in Information Sciences. Berlin.1995, 30.
[20] Lin S, Kung S Y, Lin L J. Face recognition/detection by probabilistic decision-based neural network. IEEE Transactions on Neural Networks. 1997, 8(1).114-132.
[21] Lee S Y, Ham Y K, Park R H. Recognition of human front faces using knowledge-based feature extraction and neuro-fuzzy algorithm. Pattern Recognition. 1996, 29(11).1863-1876.
[22] Yuille A L, Hallinan P W, Cohn D S. Feature extraction from faces using deformable templates. International Journal of Computer Vision. 1992, 8(2). 99-111.
[23] Turk M, Pentland A. Eigenfaces for recognition. Journal of Cognitive Neuroscience, 1991, 3(1): 72-86.
[24] Bartlett M S, Lades H M, Sejnowski T J. Independent component representations for face recognition. Proceeding of SPIE. 1998, 2399(3): 528-539.
[25] Belhumeur P N, Hespanha J P, Kriegman D J. Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection. IEEE Trans, Pattern Analysis and Machine Intelligence. 1997.
[26] Fisher R A.The use of multiple measurements in taxonomic problems.Annals of Eugenics. 1936, 7:179-188.
[27] Yang J, Zhang D, Frangi A F, etc. Two-Dimensional PCA: A New Approach to Appearance-Based Face Representation and Recognition. IEEE Trans. 2004.
[28] Vasilescu M A O, Terzopoulos D. Multilinear Analysis of Image Ensembles: TensorFaces. Proc. 7th European Conference on Computer Vision. 2002.
[29] Ye J P, Janardan R, Li Q. Two-Dimensional Linear Discriminant Analysis. Advances in Neural Information Processing Systems. 2004.
[30] Yan S C, Xu D, Yang Q, etc. Multilinear Discriminant Analysis for Face Recognition. IEEE Trans, Image Processing. 2007, Vol.16, No.1.
[31] Tenenbaum J B, Silva V D, Langford J C. A global geometric framework for nonlinear dimensionality reduction. Science. 2000, 290(5500). 2319-2323.
[32] Roweis S T, Saul L K. Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science. 2000, Vol.290 22.
[33] Belkin M, Niyogi P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation. 2003, 15(6). 1373-1396.
[34] Seung H S, Lee D D. The manifold ways of perception. Science. 2000, 290: 2268-2269.
[35] He X F, Yan S C, Hu Y X, etc. Face Recognition Using Laplacianfaces. IEEE Trans, Pattern Analysis and Machine Intelligence. 2005.
[36] He X F, Cai D, Yan S C, etc. Neighborhood Preserving Embedding. Proceedings of the Tenth IEEE International Conference on Computer Vision, 2005.
[37] Cai D, He X F, Han J. Isometric Projection. Association for the Advancement of Artificial Intelligence. 2007.
[38] Hu D W, Feng G Y, Zhou Z T. Two-dimensional locality preserving projections(2DLPP) with its application to palmprint recognition. Pattern Recognition. 2007, 40: 339-342.
[39] Zhao H T, Sun S Y, Jing Z L, etc. Local structure based supervised feature extraction. Pattern Recognition. 2006, (39) :1546-1550.
[40] Zhi R C, Ruan Q Q. Facial expression recognition based on two dimensional discriminant locality preserving projections. Neurocomputing. 2008, (71): 1730-1734.
[41] Yan S C, Xu D, Zhang B Y, etc. Graph Embedding and Extensions: A General Framework for Dimensionality Reduction. IEEE Trans, Pattern Analysis and Machine Intelligence. 2007.
[42] Gao Q X, Xu H, Li Y Y, etc.Two-Dimensional Surprised Locally Similarity and Discriminant Information Preserving Projection. 2010, 43(10)3359-3363.
[43]高全学.谢德艳.徐辉等.融合局部结构和差异信息的监督特征提取算法.自动化学报.2010.
[44] Scholkopf B, Smola A, Muller K R. Nonlinear component analysis as a kernel eigenface problem. Neural Computation. 1998, 10(5): 1299-1319.
[45] Liu Q, Huang R, Lu H etc. Face Recognition Using Kernel Based Fisher Discriminant Analysis. Automatic Face and Gesture Recognition. 2002.
[46] Dai G, Yeung D Y. Tensor embedding methods. Proceedings of the National Conference on Artificial Intelligence. 2006, 330-335.
[47] He X, Niyogi P. Locality Preserving Projections. Advance in Neural Information Processing Systems, 2003.
[48] Silva D, Tenenbaum B. Global Versus Local Methods in Nonlinear Dimensionality Reduction. Neural Information Processing Systems. 2003,705-712.
[49] Torre F D L, Black M J. A Framework for Robust Subspace Learning, International Journal of Computer Vision. 2003, 54(3):117-142.
[50]高全学,梁彦,潘泉等.基于描述特征的人脸识别研究.自动化学报. 2006, 32(3):386-392.
[51] Kwak N. Principal component analysis based on L1-norm maximization. IEEE Trans. Pattern Analysis and Machine Intelligence. 2008, 30(9): 1672-1688.
[52] Zhang S, Sim T. Discriminant Subspace Analysis: a Fukunaga-koontz Approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2007, 29(10): 1732-1745.
[53]程云鹏.矩阵论.西安:西北工业大学. 1989.
[54] Zhang D Q, Zhou Z H, etc. Diagonal principal component analysis for face recgnition. Pattern Recognition. 2006, 39(1):140-142.
[55] Xu D, Yan S C, Lin S, etc. Enhancing Bilinear Subspace Learning by Element Rearrangement. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2009,vol 31,1913-1920.
[56] Yan S C, Xu D, Lin S, etc.Element Rearrangement for Tensor-Based Subspace Learning.Computer Vision and Pattern Recognition.2007, pp.1-8.
[57] Gao Q X, Zhang L, Zhang D, etc. Directional independent component analysis with tensor representation. In proc. of IEEE conference on Computer Vision and Pattern Recognition. 2008, pp.1-7.
[58] Gao Q X, Zhang L, Zhang D, etc. Independent components extraction from image matrix. Pattern Recognition Letters. 2010,31(3), pp.171-178.
[59] Li Y Y, Tian Q C, Gao Q X, etc. Directional two-dimensional neighborhood preserving projection for face recognition.International Conference on Computational Aspects of Social Networks. 2010, pp.357-360.
[60] Gao Q X, Li Y Y, Liu Y M, etc. Directional principal component analysis for image matrix. International Conference on Computational Aspects of Social Networks. 2010, pp.374-377.
[61] Gao Q X, Hou J, Hao X J, etc. Directional multimode subspace analysis with tensor representation- Discriminant feature extraction. 2010, pp. 361-364.
[62] Yale Univ. Face Database, [2009.7], http://cvc.yale.edu/projects/yalefaces/
[63] UMIST Face Database, [2009.7] http://images.ee.umist.ac.uk/danny/ database.html, 2005.
[64] The AR Face Database, [2009.7] http://rvll.ecn.purdue.edu/ ~aleix/ aleix_face_DB.
[65] The ORL database.[2009.7]http://www.cam-orl.co.uk.
[2] Jain A, Ross A, Prabhakar S. An introduction to biometric recognition. IEEE Transactionon Circuit and System for Video Technology. 2004, 14(1):4-20.
[3] Pankanti S, Bolle R M, Jain A. Biometrics: the future of identification. Computer, 2000, 33(2):46-49.
[4] Woodward J D, Orlans J N M, Higgins P T著,陈菊明.邓启威等译.生物认证.北京:清华大学出版社, 2004:21-46.
[5] Bahl L R, Jelinek F. A Maximum likelihood approach to continuous speech recognition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 1983, 5:179-190.
[6] Hanson B A, Applehaum T H. Robust Speaker-independent Word Recognition Using Static. Dynamic and Acceleration Features: Experiments with Lombard and Noisy Speech. International conference on Acoustics, Speech and Signal Processing 1990, 857-860.
[7] Bledssoe W W. Man-machine facial recognition: Report on a Large-Scale Experiment.Technical Report PRI22, Panoramic Research Inc. Palo Alto, Californi, 1996.
[8]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.
[9]张翠萍,苏光大.人脸识别技术综述.中国图像图形学报. 2000, 5(11):855-894.
[10] Samal A, Iyengar P A. Automatic recognition and analysis of human faces and facial expression:A survey.Pattern Recognition.1992,25(1):65-67.
[11] Chellappa R, Wilson C L, Sirohey S. Human and machine recognition of faces:A survey. IEEE Proc.1995, 83(5):705-740.
[12]侯晓宇.基于流形学习的特征提取方法研究.大连理工大学硕士论文. 2009, 6.
[13]黄凤岗,宋克欧.模式识别.哈尔滨:哈尔滨工程大学出版社. 1998.
[14]边肇祺,张学工.模式识别.北京:清华大学出版社. 2000. 1-271.
[15]高全学.基于子空间分析的人脸识别研究.西北工业大学博士学位论文. 2005.
[16] Cavalcanti G D C, Filho E C B C. Eigenbands Fusion for Frontal Face Recognition. Proc. IEEE Intel Conf. on Image Processing. 2003: 665-668.
[17] Wiskott L, Fellous J M, Kruger N, etc. Face recognition by elastic bunch graph matching. IEEE Trans, Pattern Analysis and Machine Intelligence. 1997,19(7):775-779.
[18] Samaria F S. Face recognition using hidden Markov model [Ph.Ddissertation]. University of Cambridge, 1994.
[19] Kohonen T. Self-Organizing maps. Springer Series in Information Sciences. Berlin.1995, 30.
[20] Lin S, Kung S Y, Lin L J. Face recognition/detection by probabilistic decision-based neural network. IEEE Transactions on Neural Networks. 1997, 8(1).114-132.
[21] Lee S Y, Ham Y K, Park R H. Recognition of human front faces using knowledge-based feature extraction and neuro-fuzzy algorithm. Pattern Recognition. 1996, 29(11).1863-1876.
[22] Yuille A L, Hallinan P W, Cohn D S. Feature extraction from faces using deformable templates. International Journal of Computer Vision. 1992, 8(2). 99-111.
[23] Turk M, Pentland A. Eigenfaces for recognition. Journal of Cognitive Neuroscience, 1991, 3(1): 72-86.
[24] Bartlett M S, Lades H M, Sejnowski T J. Independent component representations for face recognition. Proceeding of SPIE. 1998, 2399(3): 528-539.
[25] Belhumeur P N, Hespanha J P, Kriegman D J. Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear Projection. IEEE Trans, Pattern Analysis and Machine Intelligence. 1997.
[26] Fisher R A.The use of multiple measurements in taxonomic problems.Annals of Eugenics. 1936, 7:179-188.
[27] Yang J, Zhang D, Frangi A F, etc. Two-Dimensional PCA: A New Approach to Appearance-Based Face Representation and Recognition. IEEE Trans. 2004.
[28] Vasilescu M A O, Terzopoulos D. Multilinear Analysis of Image Ensembles: TensorFaces. Proc. 7th European Conference on Computer Vision. 2002.
[29] Ye J P, Janardan R, Li Q. Two-Dimensional Linear Discriminant Analysis. Advances in Neural Information Processing Systems. 2004.
[30] Yan S C, Xu D, Yang Q, etc. Multilinear Discriminant Analysis for Face Recognition. IEEE Trans, Image Processing. 2007, Vol.16, No.1.
[31] Tenenbaum J B, Silva V D, Langford J C. A global geometric framework for nonlinear dimensionality reduction. Science. 2000, 290(5500). 2319-2323.
[32] Roweis S T, Saul L K. Nonlinear Dimensionality Reduction by Locally Linear Embedding. Science. 2000, Vol.290 22.
[33] Belkin M, Niyogi P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation. 2003, 15(6). 1373-1396.
[34] Seung H S, Lee D D. The manifold ways of perception. Science. 2000, 290: 2268-2269.
[35] He X F, Yan S C, Hu Y X, etc. Face Recognition Using Laplacianfaces. IEEE Trans, Pattern Analysis and Machine Intelligence. 2005.
[36] He X F, Cai D, Yan S C, etc. Neighborhood Preserving Embedding. Proceedings of the Tenth IEEE International Conference on Computer Vision, 2005.
[37] Cai D, He X F, Han J. Isometric Projection. Association for the Advancement of Artificial Intelligence. 2007.
[38] Hu D W, Feng G Y, Zhou Z T. Two-dimensional locality preserving projections(2DLPP) with its application to palmprint recognition. Pattern Recognition. 2007, 40: 339-342.
[39] Zhao H T, Sun S Y, Jing Z L, etc. Local structure based supervised feature extraction. Pattern Recognition. 2006, (39) :1546-1550.
[40] Zhi R C, Ruan Q Q. Facial expression recognition based on two dimensional discriminant locality preserving projections. Neurocomputing. 2008, (71): 1730-1734.
[41] Yan S C, Xu D, Zhang B Y, etc. Graph Embedding and Extensions: A General Framework for Dimensionality Reduction. IEEE Trans, Pattern Analysis and Machine Intelligence. 2007.
[42] Gao Q X, Xu H, Li Y Y, etc.Two-Dimensional Surprised Locally Similarity and Discriminant Information Preserving Projection. 2010, 43(10)3359-3363.
[43]高全学.谢德艳.徐辉等.融合局部结构和差异信息的监督特征提取算法.自动化学报.2010.
[44] Scholkopf B, Smola A, Muller K R. Nonlinear component analysis as a kernel eigenface problem. Neural Computation. 1998, 10(5): 1299-1319.
[45] Liu Q, Huang R, Lu H etc. Face Recognition Using Kernel Based Fisher Discriminant Analysis. Automatic Face and Gesture Recognition. 2002.
[46] Dai G, Yeung D Y. Tensor embedding methods. Proceedings of the National Conference on Artificial Intelligence. 2006, 330-335.
[47] He X, Niyogi P. Locality Preserving Projections. Advance in Neural Information Processing Systems, 2003.
[48] Silva D, Tenenbaum B. Global Versus Local Methods in Nonlinear Dimensionality Reduction. Neural Information Processing Systems. 2003,705-712.
[49] Torre F D L, Black M J. A Framework for Robust Subspace Learning, International Journal of Computer Vision. 2003, 54(3):117-142.
[50]高全学,梁彦,潘泉等.基于描述特征的人脸识别研究.自动化学报. 2006, 32(3):386-392.
[51] Kwak N. Principal component analysis based on L1-norm maximization. IEEE Trans. Pattern Analysis and Machine Intelligence. 2008, 30(9): 1672-1688.
[52] Zhang S, Sim T. Discriminant Subspace Analysis: a Fukunaga-koontz Approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2007, 29(10): 1732-1745.
[53]程云鹏.矩阵论.西安:西北工业大学. 1989.
[54] Zhang D Q, Zhou Z H, etc. Diagonal principal component analysis for face recgnition. Pattern Recognition. 2006, 39(1):140-142.
[55] Xu D, Yan S C, Lin S, etc. Enhancing Bilinear Subspace Learning by Element Rearrangement. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2009,vol 31,1913-1920.
[56] Yan S C, Xu D, Lin S, etc.Element Rearrangement for Tensor-Based Subspace Learning.Computer Vision and Pattern Recognition.2007, pp.1-8.
[57] Gao Q X, Zhang L, Zhang D, etc. Directional independent component analysis with tensor representation. In proc. of IEEE conference on Computer Vision and Pattern Recognition. 2008, pp.1-7.
[58] Gao Q X, Zhang L, Zhang D, etc. Independent components extraction from image matrix. Pattern Recognition Letters. 2010,31(3), pp.171-178.
[59] Li Y Y, Tian Q C, Gao Q X, etc. Directional two-dimensional neighborhood preserving projection for face recognition.International Conference on Computational Aspects of Social Networks. 2010, pp.357-360.
[60] Gao Q X, Li Y Y, Liu Y M, etc. Directional principal component analysis for image matrix. International Conference on Computational Aspects of Social Networks. 2010, pp.374-377.
[61] Gao Q X, Hou J, Hao X J, etc. Directional multimode subspace analysis with tensor representation- Discriminant feature extraction. 2010, pp. 361-364.
[62] Yale Univ. Face Database, [2009.7], http://cvc.yale.edu/projects/yalefaces/
[63] UMIST Face Database, [2009.7] http://images.ee.umist.ac.uk/danny/ database.html, 2005.
[64] The AR Face Database, [2009.7] http://rvll.ecn.purdue.edu/ ~aleix/ aleix_face_DB.
[65] The ORL database.[2009.7]http://www.cam-orl.co.uk.