可变光照和遮挡条件下的人脸识别技术研究及其应用
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摘要
人脸识别研究是近年来模式识别领域的一大研究热点,具有广阔的应用前景。本文主要致力于基于静态图像的可变光照和遮挡条件下的人脸识别方法研究,重点研究了基于子空间、基于LBP纹理特征和基于稀疏表示的人脸识别方法。
本文研究内容主要包括以下几个方面：
(1)研究了基于子空间的人脸识别方法的原理和在人脸识别应用,该方法通过将图像从数据空间映射到特征空间,达到数据降维的目的同时提取出有利于识别的信息,具体分析了PCA、Fisherface、SLPP、2DPCA、2DLDA、KPCA和KFDA七种子空间方法的优缺点及内在联系。
(2)针对光照变化问题,本文提出了分块LBP纹理特征的人脸识别方法,首先提取每个窗口内的LBP纹理特征直方图,然后将每块的直方图叠加,最终获得人脸描述特征。实验结果证明了该方法对光照变化稳健并且计算量低。
(3)针对带遮挡或受噪声干扰的正面人脸图像,本文给出了一种自动人脸识别方法,即基于稀疏表示的人脸识别方法。该方法将识别问题当作多元线性回归模型中的一种分类问题来研究同时有关稀疏信号恢复的新理论对阐述这个问题起了关键作用。利用L1范数最小化获得的稀疏表示,本文得到了一种用于物体识别的统一算法。本文对公开的人脸数据库做了大量的实验,证明了该方法的有效性。
最后,在实验室内自然环境下,本文采用基于LBP纹理特征的方法搭建了一个实时的人脸识别系统并且取得了良好的识别效果。
As one of the most heated research spot in the field of pattern recognition, the human face recognition applies to tremendous aspects and has a promising future. This dissertation focuses on face recognition algorithms based on statci image under variation of illumination and occlusion, special attention has been paid to the subspace, LBP, as well as spare representation.
The main research results are as following:
(1) Considering that, by casting image from data space to feature space, subplace based face recognition methods can absorb information that is useful for recognition from reducted date dimension, this dissertation studies its application in face recognition. The merits and demerits of seven subspace based methods are presented and the relationship between the seven methods are disclosed.
(2) To alleviate the impact of variation of illumination, we bring about the method based on LBP texture feature of sub windows. Firstly we extract the LBP texture feature histogram of every window, and then we pile up every histogram, at last we acquire the characters of human face. The result confirms that this method has a steady effectiveness to the variation of illumination and requires low computational complexity.
(3) This dissertation finds the method of automatically recognizing human faces from frontal views with disguise or occlusion, this is face recognition based on spare representation. The method casts the recognition problem as one of classifying among multiple linear regression models and argue that new theory from sparse signal representation offers the key to addressing this problem. Based on a sparse representation computed by L1-minimization, we propose a general classification algorithm for (image-based) object recognition. We conduct extensive experiments on publicly available databases to verify the efficacy of the proposed algorithm.
At the same time, under natural environment, we build a real-time face recognition system using LBP texture feature which achieves good results in the laboratory.
本文研究内容主要包括以下几个方面：
(1)研究了基于子空间的人脸识别方法的原理和在人脸识别应用,该方法通过将图像从数据空间映射到特征空间,达到数据降维的目的同时提取出有利于识别的信息,具体分析了PCA、Fisherface、SLPP、2DPCA、2DLDA、KPCA和KFDA七种子空间方法的优缺点及内在联系。
(2)针对光照变化问题,本文提出了分块LBP纹理特征的人脸识别方法,首先提取每个窗口内的LBP纹理特征直方图,然后将每块的直方图叠加,最终获得人脸描述特征。实验结果证明了该方法对光照变化稳健并且计算量低。
(3)针对带遮挡或受噪声干扰的正面人脸图像,本文给出了一种自动人脸识别方法,即基于稀疏表示的人脸识别方法。该方法将识别问题当作多元线性回归模型中的一种分类问题来研究同时有关稀疏信号恢复的新理论对阐述这个问题起了关键作用。利用L1范数最小化获得的稀疏表示,本文得到了一种用于物体识别的统一算法。本文对公开的人脸数据库做了大量的实验,证明了该方法的有效性。
最后,在实验室内自然环境下,本文采用基于LBP纹理特征的方法搭建了一个实时的人脸识别系统并且取得了良好的识别效果。
As one of the most heated research spot in the field of pattern recognition, the human face recognition applies to tremendous aspects and has a promising future. This dissertation focuses on face recognition algorithms based on statci image under variation of illumination and occlusion, special attention has been paid to the subspace, LBP, as well as spare representation.
The main research results are as following:
(1) Considering that, by casting image from data space to feature space, subplace based face recognition methods can absorb information that is useful for recognition from reducted date dimension, this dissertation studies its application in face recognition. The merits and demerits of seven subspace based methods are presented and the relationship between the seven methods are disclosed.
(2) To alleviate the impact of variation of illumination, we bring about the method based on LBP texture feature of sub windows. Firstly we extract the LBP texture feature histogram of every window, and then we pile up every histogram, at last we acquire the characters of human face. The result confirms that this method has a steady effectiveness to the variation of illumination and requires low computational complexity.
(3) This dissertation finds the method of automatically recognizing human faces from frontal views with disguise or occlusion, this is face recognition based on spare representation. The method casts the recognition problem as one of classifying among multiple linear regression models and argue that new theory from sparse signal representation offers the key to addressing this problem. Based on a sparse representation computed by L1-minimization, we propose a general classification algorithm for (image-based) object recognition. We conduct extensive experiments on publicly available databases to verify the efficacy of the proposed algorithm.
At the same time, under natural environment, we build a real-time face recognition system using LBP texture feature which achieves good results in the laboratory.
引文
[1]谭铁牛.中国生物特征认证动态[M].西安：西北大学出版社,2003
[2]周杰,卢春雨,张长水等.人脸自动识别方法综述[J].电子学报,2000,28(4)：102-106
[3]高文.人脸识别研究概述[C].北京：生物特征识别技术高级研讨班,2003
[4]Phillips P. J., Wechsler H., Huang J.. The FERET database and evaluation procedure for face recognition algorithms. Image and Vision Computing,1998,16(5):295-306
[5]Allen A. L.. Personal Description[M]. London:Butterworth,1950
[6]Harmon L. D.. The recognition of faces[J]. Science of America,1973, (229):71-82
[7]Kanade T. Picture processing system by computer complex and recognition of human faces[D]. Kyoto:Kyoto University,1973
[8]Kaya Y.. A basic study on human recognition[M]. New York:Academic,1971,265-289
[9]Bledsoe W.. Man-Machine Facial Recognition. Technical Report,1996
[10]Kohenen T. Self-organization and associative memory.1998
[11]Von der Malsburg C. Face recognition by elastic bunch graph matching. IEEE Transactions on PAMI,1997,19(7):775-779
[12]Samal A., Iyengar P. A.. Automatic Recognition and Analysis of Human Faces and Facial Expressions:A survey. Pattern Recognition,1992,25(1):65-67
[13]Nefian A. V, Hayes M. H.. Hidden Markov Models for Face Recognition. Proc of the IEEE ICASSP,1998,2721-2724
[14]Dempster A. P., Laird N. M., Rubin D. B.. Maximum Likelihood from Incomplete Data via the EM algorithm. Journal of the Royal Statistical Society-Series,1977,39(1):1-38
[15]Lanitis A., Taylor C., Cootes T. Automatic interpretation and coding of face images using flexible models. IEEE Trans. on PAMI,1997,19(7):743-756
[16]Cootes T. F., Edwards G. J., Taylor C. J.. Active appearance model. Proc. European conference on computer vision,1998,484-498
[17]Kass M., Witdin A., Terzopoulos D.. Snakes:Active Contour Model. Proc. first international conference on computer vision,1987,259-268
[18]Jolliffe I. T.. Principal Component Analysis.1986
[19]Blanz V, Vetter T. Face recognition based on fitting a 3D morphable model. IEEE Trans. On PAMI,2003,25(9):1063-1075
[20]Blanz V, Romdhani S., Vetter T.. Face identification across different poses and illuminations with a 3D morphable model. Proc.of the 5th Int. Conference on AFGR,2002, 202-207
[21]Kirby M., Sirovich L.. Application of the Karhunen-Loeve procedure for the characterization of human faces. IEEE Tran. on PAMI,1990,12 (1):103-108
[22]Turk M., Pentland A.. Eigenfaces For Recognition. Journal of Cognitive Neuroscience, 1991,3(1):71-86
[23]Belhumeur P. N., Hespanha J. P., Kriegman D. J.. Eigenfaces vs Fisherfaces:Recognition using class specific linear projection. IEEE Tran. on PAMI,1997, 19(7):711-720
[24]Liu C., Wechsler H.. Probabilistic reasoning models for face recognition. Proceedings of the 1998, IEEE Computer Society Conference on CVPR,1998,827-832
[25]Bartlett M. S.. Independent components of face images:A representation for face recognition. Processings of the Fourth AJSNC, Pasadena, USA,1997
[26]Moghaddam B., Jebara T., Pentland A.. Bayesian face recognition. Pattern Recognition, 2000,33(11):1771-1782
[27]Hong Z. Q.. Algebraic feature extraction of image for recognition. Pattern Recognition, 1991,24(3):211-219
[28]Lades M., Vorbruggen J. C., Buhmann J.. Distortion invariant object recognition in the dynamic link architecture. IEEE Trans. on Computer Vision,1992,42(3):300-311
[29]Lawrence S., Giles C. L., Tsoi A. C. Face recognition:A convolutional neural-network approach. IEEE transactions on Neural Networks,1997,8(1):98-113
[30]Lin S. H., 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
[31]赖剑煌.频谱脸：一种基于小波变换和Fourier变换的人像识别新方法.中国图象图形学报,1999,4(10)：811-816
[32]Ziad M., Martin D.. Face recognition using the discrete cosine transform. International Journal of Computer Vision,2001,43(3):167-188
[33]Lee Tai Sing. Image representation using 2D Gabor wavelets. IEEE Trans. On PAMI, 1996,18(10):959-971
[34]Viola Paul, Jones M.. Rapid Object Detection Using a Boosted Cascade of simple Features. IEEE Conference on CVPR,2001
[35]Zhang Lei, Li Stan Z.. Boosting Local Feature Based Classifiers for Face Recognition. Proceedings of the 2004 IEEE Computer Society Conference on CVPR Workshops,2004
[36]Li Stan Z., Zhang ZhenQiu. Float Boost Learning and Statistical Face Detection. IEEE Transactions on PAMI,2004,26(9)
[37]罗家洪.矩阵分析引论.广州：华南理工大学出版社,1996
[38]Fisher R. A.. The use of multiple measurements in taxonomic problems. Annals of Eugenics,1936,7:179-188
[39]Swets D., Weng J.. Using discriminant eigenfeatures for image retrieval. IEEE Transaction on PAMI,1996,18(8):831-836
[40]Yang J., Zhang D., Yang J. Y.. Two-dimensional PCA:a new approach to appearance-based face representation and recognition. IEEE PAMI,2004,18:831-866
[41]Bellhumer P. N., Hespanha J.. Eigenfaces vs. fisherfaces:Recognition using class specific linear projection. IEEE Transactions on PAMI, Special Issue on Face Recognition, 1997,17(7):711-720
[42]Chen L., Liao H.. A new LDA-based face recognition system which can solve the small sample size problem[J]. Pattern Recognition,2000,33(10):1713-1726
[43]Hong Z. Q.. Optimal Discriminant Plane For a small number of samples and design method of classifier on the plane[J]. Pattern Recognition,1991,24(4):317-324
[44]Yang J., Yang J. Y. Two-Dimensional PCA:A New Approach to Appearance-Based Face Representation and Recognition[J]. IEEE Transaction on PAMI,2004,26(1):131-137
[45]Li Ming, Yuan Baozong.2D-LDA:A statistical linear discriminant analysis for image matrix[J]. Pattern Recognition Letters,2005,26(5):527-532
[46]He Xiaofei, Niyogi Partha. Locality preserving projection. Pattern Recognition,2004, 37(4):781-788
[47]Alexander Smola. Nonlinear component analysis as a kernel eigenvalue problem. Neural problem,1998,10:1299-1319
[48]白鹏,张喜斌等编.支持向量机理论及工程应用实例[M].西安：西安电子科技大学出版社,2008
[49]Mika S., Ratsch G.. Fisher discriminant analysis with kernels. IEEE Neural Network for Signal Processing Workshop,1999,41-48
[50]He X., Yan S.. Face recognition using laplacianfaces. IEEE Trans. PAMI,2005,27(3): 328-340
[51]Ojala T., Pietikaeinen M.. A Comparative Study of Texture Measures with classification Based on Feature Distributions. Pattern Recognition,1996,29:51-59
[52]Ojala T., Pietikaeinen M.. Multiresolution Gray-Scale and Rotation Invariant Texture Classification with Local Binary Patterns. IEEE Trans. PAMI,2002,24:971-987
[53]Topi M.. The local binary pattern approach to texture analysis extensions and applications. University of Oulu,2003
[54]Timo Ahonen, Abdenour Hadid. Face Recognition with Local Binary Patterns. ECCV, 2004,469-481
[55]Chen S., Donoho D., Saunders M.. Atomic Decomposition by Basis Pursuit[J]. SIAM Review,2001,43(1):129-159
[56]Cande E.. Compressive Sampling[C]. Int'l Congress of mathematicians,2006
[57]Marcellin M. W., Gormish M. J., Bilgin A.. An overview of JPEG-2000[C]. Data Compression Conf.,2000:523-541
[58]Donoho D.. Compressed sensing. IEEE Trans. Inform. Theory,2006,52(4):1289-1306
[59]Mallat S.. A Wavelet Tour of Signal Processing. New York:Academic,1999
[60]Belhumeur P., Hespanda J., Kriegman D.. Eigenfaces versus Fisherfaces:Recognition Using Class Specific Linear Projection. IEEE Trans. PAMI,1997,19(7):711-720
[61]Donoho D., Elad M.. Optimal Sparse Representation in General Dictionaries via L1 Minimization. Proc. Natl Academy of Sciences,2003,2197-2202
[62]Amaldi E., Kann V.. On the Approximability of Minimizing Nonzero Variables or Unsatisfied Relations in Linear Systems. Theoretical Computer Science,1998,209:237-260
[63]Donoho D.. For Most Large Underdetermined Systems of Linear Equations the Minimal L1-Norm Solution Is Also the Sparsest Solution. Comm. Pure and Applied Math., 2006,59(6):797-829
[64]Chen S., Donoho D.. Atomic Decomposition by Basis Pursuit. SIAMR ev.,2001, 43(1):129-159
[65]Donoho D.. Neighborly Polytopes and Sparse Solution of Underdetermined Linear Equations. Technical Report 2005-4, Dept. of Statistics, Stanford Univ.,2005
[66]Liu C. Capitalize on Dimensionality Increasing Techniques for Improving Face Recognition Grand Challenge Performance. IEEE Trans. PAMI,2006,28(5):725-737
[67]Candes E., Tao T.. Near-Optimal Signal Recovery from Random Projections:Universal Encoding Strategies. IEEE Trans. Information Theory,2006,52(12):5406-5425
[68]Donoho D., Tanner J.. Counting Faces of Randomly Projected Polytopes When the Projection Radically Lowers Dimension. preprint, http://www.math.utah.edu/-tanner/,2007
[69]Wright J., Yang Y.. Robust face recognition via sparse representation [J]. PAMI,2009, 31(2):210-227
[70]Valiant L. G.. A Theory of Learnable[J]. Communication of ACM,1984,27:1134
[71]Schapire R. E.. The Strength of Weak ability[J]. Machine Learning,1990,5:197
[72]Freund Y, Schapire R. E.. A Decision-theoretic Generalization of online Learning and an Application to Boosting[J]. Journal of Computer and System Sciences,1997,55(1):119
[73]Cootes T. F., Taylor C. J.. Active Shape Models-Their Training and Application. Computer Vision and Image Understanding,1995,61(1):38-59
[74]Cootes T. F., Taylor C. J.. Active Shape Models:Evaluation of a Multi-Resolution Method for Improving Image Search. In Proc. British Machine Vision Conference,1994, 327-336
[2]周杰,卢春雨,张长水等.人脸自动识别方法综述[J].电子学报,2000,28(4)：102-106
[3]高文.人脸识别研究概述[C].北京：生物特征识别技术高级研讨班,2003
[4]Phillips P. J., Wechsler H., Huang J.. The FERET database and evaluation procedure for face recognition algorithms. Image and Vision Computing,1998,16(5):295-306
[5]Allen A. L.. Personal Description[M]. London:Butterworth,1950
[6]Harmon L. D.. The recognition of faces[J]. Science of America,1973, (229):71-82
[7]Kanade T. Picture processing system by computer complex and recognition of human faces[D]. Kyoto:Kyoto University,1973
[8]Kaya Y.. A basic study on human recognition[M]. New York:Academic,1971,265-289
[9]Bledsoe W.. Man-Machine Facial Recognition. Technical Report,1996
[10]Kohenen T. Self-organization and associative memory.1998
[11]Von der Malsburg C. Face recognition by elastic bunch graph matching. IEEE Transactions on PAMI,1997,19(7):775-779
[12]Samal A., Iyengar P. A.. Automatic Recognition and Analysis of Human Faces and Facial Expressions:A survey. Pattern Recognition,1992,25(1):65-67
[13]Nefian A. V, Hayes M. H.. Hidden Markov Models for Face Recognition. Proc of the IEEE ICASSP,1998,2721-2724
[14]Dempster A. P., Laird N. M., Rubin D. B.. Maximum Likelihood from Incomplete Data via the EM algorithm. Journal of the Royal Statistical Society-Series,1977,39(1):1-38
[15]Lanitis A., Taylor C., Cootes T. Automatic interpretation and coding of face images using flexible models. IEEE Trans. on PAMI,1997,19(7):743-756
[16]Cootes T. F., Edwards G. J., Taylor C. J.. Active appearance model. Proc. European conference on computer vision,1998,484-498
[17]Kass M., Witdin A., Terzopoulos D.. Snakes:Active Contour Model. Proc. first international conference on computer vision,1987,259-268
[18]Jolliffe I. T.. Principal Component Analysis.1986
[19]Blanz V, Vetter T. Face recognition based on fitting a 3D morphable model. IEEE Trans. On PAMI,2003,25(9):1063-1075
[20]Blanz V, Romdhani S., Vetter T.. Face identification across different poses and illuminations with a 3D morphable model. Proc.of the 5th Int. Conference on AFGR,2002, 202-207
[21]Kirby M., Sirovich L.. Application of the Karhunen-Loeve procedure for the characterization of human faces. IEEE Tran. on PAMI,1990,12 (1):103-108
[22]Turk M., Pentland A.. Eigenfaces For Recognition. Journal of Cognitive Neuroscience, 1991,3(1):71-86
[23]Belhumeur P. N., Hespanha J. P., Kriegman D. J.. Eigenfaces vs Fisherfaces:Recognition using class specific linear projection. IEEE Tran. on PAMI,1997, 19(7):711-720
[24]Liu C., Wechsler H.. Probabilistic reasoning models for face recognition. Proceedings of the 1998, IEEE Computer Society Conference on CVPR,1998,827-832
[25]Bartlett M. S.. Independent components of face images:A representation for face recognition. Processings of the Fourth AJSNC, Pasadena, USA,1997
[26]Moghaddam B., Jebara T., Pentland A.. Bayesian face recognition. Pattern Recognition, 2000,33(11):1771-1782
[27]Hong Z. Q.. Algebraic feature extraction of image for recognition. Pattern Recognition, 1991,24(3):211-219
[28]Lades M., Vorbruggen J. C., Buhmann J.. Distortion invariant object recognition in the dynamic link architecture. IEEE Trans. on Computer Vision,1992,42(3):300-311
[29]Lawrence S., Giles C. L., Tsoi A. C. Face recognition:A convolutional neural-network approach. IEEE transactions on Neural Networks,1997,8(1):98-113
[30]Lin S. H., 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
[31]赖剑煌.频谱脸：一种基于小波变换和Fourier变换的人像识别新方法.中国图象图形学报,1999,4(10)：811-816
[32]Ziad M., Martin D.. Face recognition using the discrete cosine transform. International Journal of Computer Vision,2001,43(3):167-188
[33]Lee Tai Sing. Image representation using 2D Gabor wavelets. IEEE Trans. On PAMI, 1996,18(10):959-971
[34]Viola Paul, Jones M.. Rapid Object Detection Using a Boosted Cascade of simple Features. IEEE Conference on CVPR,2001
[35]Zhang Lei, Li Stan Z.. Boosting Local Feature Based Classifiers for Face Recognition. Proceedings of the 2004 IEEE Computer Society Conference on CVPR Workshops,2004
[36]Li Stan Z., Zhang ZhenQiu. Float Boost Learning and Statistical Face Detection. IEEE Transactions on PAMI,2004,26(9)
[37]罗家洪.矩阵分析引论.广州：华南理工大学出版社,1996
[38]Fisher R. A.. The use of multiple measurements in taxonomic problems. Annals of Eugenics,1936,7:179-188
[39]Swets D., Weng J.. Using discriminant eigenfeatures for image retrieval. IEEE Transaction on PAMI,1996,18(8):831-836
[40]Yang J., Zhang D., Yang J. Y.. Two-dimensional PCA:a new approach to appearance-based face representation and recognition. IEEE PAMI,2004,18:831-866
[41]Bellhumer P. N., Hespanha J.. Eigenfaces vs. fisherfaces:Recognition using class specific linear projection. IEEE Transactions on PAMI, Special Issue on Face Recognition, 1997,17(7):711-720
[42]Chen L., Liao H.. A new LDA-based face recognition system which can solve the small sample size problem[J]. Pattern Recognition,2000,33(10):1713-1726
[43]Hong Z. Q.. Optimal Discriminant Plane For a small number of samples and design method of classifier on the plane[J]. Pattern Recognition,1991,24(4):317-324
[44]Yang J., Yang J. Y. Two-Dimensional PCA:A New Approach to Appearance-Based Face Representation and Recognition[J]. IEEE Transaction on PAMI,2004,26(1):131-137
[45]Li Ming, Yuan Baozong.2D-LDA:A statistical linear discriminant analysis for image matrix[J]. Pattern Recognition Letters,2005,26(5):527-532
[46]He Xiaofei, Niyogi Partha. Locality preserving projection. Pattern Recognition,2004, 37(4):781-788
[47]Alexander Smola. Nonlinear component analysis as a kernel eigenvalue problem. Neural problem,1998,10:1299-1319
[48]白鹏,张喜斌等编.支持向量机理论及工程应用实例[M].西安：西安电子科技大学出版社,2008
[49]Mika S., Ratsch G.. Fisher discriminant analysis with kernels. IEEE Neural Network for Signal Processing Workshop,1999,41-48
[50]He X., Yan S.. Face recognition using laplacianfaces. IEEE Trans. PAMI,2005,27(3): 328-340
[51]Ojala T., Pietikaeinen M.. A Comparative Study of Texture Measures with classification Based on Feature Distributions. Pattern Recognition,1996,29:51-59
[52]Ojala T., Pietikaeinen M.. Multiresolution Gray-Scale and Rotation Invariant Texture Classification with Local Binary Patterns. IEEE Trans. PAMI,2002,24:971-987
[53]Topi M.. The local binary pattern approach to texture analysis extensions and applications. University of Oulu,2003
[54]Timo Ahonen, Abdenour Hadid. Face Recognition with Local Binary Patterns. ECCV, 2004,469-481
[55]Chen S., Donoho D., Saunders M.. Atomic Decomposition by Basis Pursuit[J]. SIAM Review,2001,43(1):129-159
[56]Cande E.. Compressive Sampling[C]. Int'l Congress of mathematicians,2006
[57]Marcellin M. W., Gormish M. J., Bilgin A.. An overview of JPEG-2000[C]. Data Compression Conf.,2000:523-541
[58]Donoho D.. Compressed sensing. IEEE Trans. Inform. Theory,2006,52(4):1289-1306
[59]Mallat S.. A Wavelet Tour of Signal Processing. New York:Academic,1999
[60]Belhumeur P., Hespanda J., Kriegman D.. Eigenfaces versus Fisherfaces:Recognition Using Class Specific Linear Projection. IEEE Trans. PAMI,1997,19(7):711-720
[61]Donoho D., Elad M.. Optimal Sparse Representation in General Dictionaries via L1 Minimization. Proc. Natl Academy of Sciences,2003,2197-2202
[62]Amaldi E., Kann V.. On the Approximability of Minimizing Nonzero Variables or Unsatisfied Relations in Linear Systems. Theoretical Computer Science,1998,209:237-260
[63]Donoho D.. For Most Large Underdetermined Systems of Linear Equations the Minimal L1-Norm Solution Is Also the Sparsest Solution. Comm. Pure and Applied Math., 2006,59(6):797-829
[64]Chen S., Donoho D.. Atomic Decomposition by Basis Pursuit. SIAMR ev.,2001, 43(1):129-159
[65]Donoho D.. Neighborly Polytopes and Sparse Solution of Underdetermined Linear Equations. Technical Report 2005-4, Dept. of Statistics, Stanford Univ.,2005
[66]Liu C. Capitalize on Dimensionality Increasing Techniques for Improving Face Recognition Grand Challenge Performance. IEEE Trans. PAMI,2006,28(5):725-737
[67]Candes E., Tao T.. Near-Optimal Signal Recovery from Random Projections:Universal Encoding Strategies. IEEE Trans. Information Theory,2006,52(12):5406-5425
[68]Donoho D., Tanner J.. Counting Faces of Randomly Projected Polytopes When the Projection Radically Lowers Dimension. preprint, http://www.math.utah.edu/-tanner/,2007
[69]Wright J., Yang Y.. Robust face recognition via sparse representation [J]. PAMI,2009, 31(2):210-227
[70]Valiant L. G.. A Theory of Learnable[J]. Communication of ACM,1984,27:1134
[71]Schapire R. E.. The Strength of Weak ability[J]. Machine Learning,1990,5:197
[72]Freund Y, Schapire R. E.. A Decision-theoretic Generalization of online Learning and an Application to Boosting[J]. Journal of Computer and System Sciences,1997,55(1):119
[73]Cootes T. F., Taylor C. J.. Active Shape Models-Their Training and Application. Computer Vision and Image Understanding,1995,61(1):38-59
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