Eigenfaces for Recognition 特征脸识别

人脸识别英文专业词汇gallery set参考图像集Probe set=test set测试图像集face renderingFacial Landmark Detection人脸特征点检测3D Morphable Model 3D形变模型AAM (Active Appearance Model)主动外观模型Aging modeling老化建模Aging simulation老化模拟Analysis by synthesis 综合分析Aperture stop孔径光标栏Appearance Feature表观特征Baseline基准系统Benchmarking 确定基准Bidirectional relighting 双向重光照Camera calibration摄像机标定（校正）Cascade of classifiers 级联分类器face detection 人脸检测Facial expression面部表情Depth of field 景深Edgelet 小边特征Eigen light-fields本征光场Eigenface特征脸Exposure time曝光时间Expression editing表情编辑Expression mapping表情映射Partial Expression Ratio Image局部表情比率图(,PERI) extrapersonal variations类间变化Eye localization,眼睛定位face image acquisition 人脸图像获取Face aging人脸老化Face alignment人脸对齐Face categorization人脸分类Frontal faces 正面人脸Face Identification人脸识别Face recognition vendor test人脸识别供应商测试Face tracking人脸跟踪Facial action coding system面部动作编码系统Facial aging面部老化Facial animation parameters脸部动画参数Facial expression analysis人脸表情分析Facial landmark面部特征点Facial Definition Parameters人脸定义参数Field of view视场Focal length焦距Geometric warping几何扭曲Street view街景Head pose estimation头部姿态估计Harmonic reflectances谐波反射Horizontal scaling水平伸缩Identification rate识别率Illumination cone光照锥Inverse rendering逆向绘制技术Iterative closest point迭代最近点Lambertian model朗伯模型Light-field光场Local binary patterns局部二值模式Mechanical vibration机械振动Multi-view videos多视点视频Band selection波段选择Capture systems获取系统Frontal lighting正面光照Open-set identification开集识别Operating point操作点Person detection行人检测Person tracking行人跟踪Photometric stereo光度立体技术Pixellation像素化Pose correction姿态校正Privacy concern隐私关注Privacy policies隐私策略Profile extraction轮廓提取Rigid transformation刚体变换Sequential importance sampling序贯重要性抽样Skin reflectance model,皮肤反射模型Specular reflectance镜面反射Stereo baseline 立体基线Super-resolution超分辨率Facial side-view面部侧视图Texture mapping纹理映射Texture pattern纹理模式Rama Chellappa读博计划：1.完成先前关于指纹细节点统计建模的相关工作。

DISCRIMINATIVE COMMON VECTORS FOR FACE RECOGNITION Hakan Cevikalp1, Marian Neamtu2, Mitch Wilkes1, and Atalay Barkana31Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, Tennessee, USA.2Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA.3Department of Electrical and Electronics Engineering, Osmangazi University, Eskisehir, Turkey.CORRESPONDENCE ADDRESS:Prof. Mitch WilkesDepartment of Electrical Engineering and Computer Science,Vanderbilt University, Nashville, Tennessee, USATel: (615) 343-6016Fax: (615) 322-7062e-mail: mitch.wilkes@AbstractIn face recognition tasks, the dimension of the sample space is typically larger than the number of the samples in the training set. As a consequence, the within-class scatter matrix is singular and the Linear Discriminant Analysis (LDA) method cannot be applied directly. This problem is known as the “small sample size” problem. In this paper, we propose a new face recognition method called the Discriminative Common Vector method based on a variation of Fisher’s Linear Discriminant Analysis for the small sample size case. Two different algorithms are given to extract the discriminative common vectors representing each person in the training set of the face database. One algorithm uses the within-class scatter matrix of the samples in the training set while the other uses the subspace methods and the Gram-Schmidt orthogonalization procedure to obtain the discriminative common vectors. Then the discriminative common vectors are used for classification of new faces. The proposed method yields an optimal solution for maximizing the modified Fisher’s Linear Discriminant criterion given in the paper. Our test results show that the Discriminative Common Vector method is superior to other methods in terms of recognition accuracy, efficiency, and numerical stability.Index Terms: Common Vectors, Discriminative Common Vectors, Face Recognition, Fisher’s Linear Discriminant Analysis, Principal Component Analysis, Small Sample Size, Subspace Methods.I. INTRODUCTIONRecently, due to military, commercial, and law enforcement applications, there has been much interest in automatically recognizing faces in still and video images. This research spans several disciplines such as image processing, pattern recognition, computer vision and neural networks. The data come from a wide variety of sources. One group of sources is the relatively controlled format images such as passports, credit cards, photo ID’s, driver’s licenses, and mug shots. A more challenging class of application imagery includes real-time detection and recognition of faces in surveillance video images, which present additional constraints in terms of speed and processing requirements [1].Face recognition can be defined as the identification of individuals from images of their faces by using a stored database of faces labeled with people’s identities. This task is complex and can be decomposed into the smaller steps of detection of faces in a cluttered background, localization of these faces followed by extraction of features from the face regions, and finally recognition and verification [2]. It is a difficult problem as there are numerous factors such as 3-D pose, facial expression, hair style, make up, and so on, which affect the appearance of an individual’s facial features.In addition to these varying factors, lighting, background, and scale changes make this task even more challenging. Additional problematic conditions include noise, occlusion, and many other possible factors.Many methods have been proposed for face recognition within the last two decades [1], [3]. Among these methods, appearance-based approaches operate directly on images or appearances of face objects, and process the images as two-dimensional (2-D) holistic patterns. In these approaches, a two-dimensional image of size w by h pixels is represented by a vector in a wh-dimensional space. Therefore, each facial image corresponds to a point in this space. This spaceis called the sample space or the image space, and its dimension typically is very high [4]. However, since face images have similar structure, the image vectors are correlated, and any image in the sample space can be represented in a lower-dimensional subspace without losing a significant amount of information. The Eigenface method has been proposed for finding such a lower-dimensional subspace [5]. The key idea behind the Eigenface method, which uses Principal Component Analysis (PCA), is to find the best set of projection directions in the sample space that will maximize the total scatter across all images such that ||max arg )(W S W W J T T Wopt PCA = is maximized. Here T S is the total scatter matrix of the trainingset samples, and W is the matrix whose columns are the orthonormal projection vectors. The projection directions are also called the eigenfaces. Any face image in the sample space can be approximated by a linear combination of the significant eigenfaces. The sum of the eigenvalues that correspond to the eigenfaces not used in reconstruction gives the mean square error of reconstruction. This method is an unsupervised technique, since it does not consider the classes within the training set data. In choosing a criterion that maximizes the total scatter, this approach tends to model unwanted within-class variations such as those resulting from differences in lighting, facial expression, and other factors [6], [7]. Additionally, since the criterion does not attempt to minimize within-class variation, the resulting classes may tend to have more overlap than other approaches. Thus, the projection vectors chosen for optimal reconstruction may obscure the existence of the separate classes.The Linear Discriminant Analysis (LDA) method is proposed in [6] and [7]. This method overcomes the limitations of the Eigenface method by applying the Fisher’s Linear Discriminant criterion. This criterion tries to maximize the ratio ||||maxarg )(W S W W S W W J W T B T W opt FLD =, where B S isthe between-class scatter matrix, and W S is the within-class scatter matrix. Thus, by applying this method, we find the projection directions that on one hand maximize the Euclidean distance between the face images of different classes and on the other minimize the distance between the face images of the same class. This ratio is maximized when the column vectors of the projection matrix W are the eigenvectors of B W S S 1−. In face recognition tasks, this method cannot be applied directly since the dimension of the sample space is typically larger than the number of samples in the training set. As a consequence, W S is singular in this case. This problem is also known as the “small sample size problem” [8].In the last decade numerous methods have been proposed to solve this problem. Tian et al . [9] used the Pseudo-Inverse method by replacing 1−W S with its pseudo-inverse. The Perturbation method is used in [2] and [10], where a small perturbation matrix ∆ is added to W S in order to make it nonsingular. Cheng et al . [11] proposed the Rank Decomposition method based on successive eigen-decompositions of the total scatter matrix T S and the between-class scatter matrix B S . However, the above methods are typically computationally expensive since the scatter matrices are very large (e.g., images of size 256 by 256 yield scatter matrices of size 65,536 by 65,536). Swets and Weng [7] proposed a two stage PCA+LDA method, also known as the Fisherface method, in which PCA is first used for dimension reduction so as to make W S nonsingular before the application of LDA. In this method the final optimal projection vector matrix becomes FLD PCA opt W W W =, where ||max arg W S W W T T W PCA =, and||||max arg W W S W W W W S W W W PCA W T PCA T PCA B T PCA T WFLD =. However, in order to make W S nonsingular, some directions corresponding to the small eigenvalues of T S are thrown away in the PCA step. Thus,applying PCA for dimensionality reduction has the potential to remove dimensions that contain discriminative information [12]-[16]. Chen et al . [17] proposed the Null Space method based on the modified Fisher’s Linear Discriminant criterion, W S W W S W W J T T B T W opt MFLD maxarg )(=. Thismethod was proposed to be used when the dimension of the sample space is larger than the rank of the within-class scatter matrix, W S . It has been shown that the original Fisher’s Linear Discriminant criterion can be replaced by the modified Fisher’s Linear Discriminant criterion in the course of solving the discriminant vectors of the optimal set in [18]. In this method, all image samples are first projected onto the null space of W S , resulting in a new within-class scatter that is a zero matrix. Then, PCA is applied to the projected samples to obtain the optimal projection vectors. Chen et al . also proved that by applying this method, the modified Fisher’s Linear Discriminant criterion attains its maximum. However, they did not propose an efficient algorithm for applying this method in the original sample space. Instead, a pixel grouping method is applied to extract geometric features and reduce the dimension of the sample space. Then they applied the Null Space method in this new reduced space. In our experiments, we observed that the performance of the Null Space method depends on the dimension of the null space of W S in the sense that larger dimension provides better performance. Thus, any kind of pre-processing that reduces the original sample space should be avoided.Another novel method, the PCA+Null Space method was proposed by Huang et al . in [15] for dealing with the small sample size problem. In this method, at first, PCA is applied to remove the null space of T S , which contains the intersection of the null spaces of B S and W S . Then, the optimal projection vectors are found in the remaining lower-dimensional space by using the Null Space method. The difference between the Fisherface method and the PCA+Null Space methodis that for the latter, the within-class scatter matrix in the reduced space is typically singular. This occurs because all eigenvectors corresponding to the nonzero eigenvalues of T S are used for dimension reduction. Yang et al . applied a variation of this method in [16]. After dimensionreduction, they split the new within-class scatter matrix, PCA W T PCAW P S P S =~ (where PCA P is the matrix whose columns are the orthonormal eigenvectors corresponding to the nonzeroeigenvalues of T S ), into its null space },...,{)~(1t r W span S N ξξ+= and orthogonal complement(i.e., range space) },...,{)~(1r W span S R ξξ= (where r is the rank of W S , and )(T S rank t = is thedimension of the reduced space). Then, all the projection vectors that maximize the between-class scatter in the null space are chosen. If, according to some criterion, more projection vectors are needed, the remaining projection vectors are obtained from the range space. Although the PCA+Null Space method and the variation proposed by Yang et al ., use the original sample space, applying PCA and using all eigenvectors corresponding to the nonzero eigenvalues make these methods impractical for face recognition applications when the training set size is large. This is due to the fact that the computational expense of training becomes very large.Lastly, the Direct-LDA method is proposed in [12]. This method uses the simultaneous diagonalization method [8]. First, the null space of B S is removed, and then the projection vectors that minimize the within-class scatter in the transformed space are selected from the range space of B S . However, removing the null space of B S by dimensionality reduction will also remove part of the null space of W S and may result in the loss of important discriminative information [13], [15], [16]. Furthermore, B S is whitened as a part of this method. This whitening process can be shown to be redundant and therefore should be skipped.In this paper, a new method is proposed which addresses the limitations of other methods that use the null space ofS to find the optimal projection vectors. Thus, the proposed method canWbe only used when the dimension of the sample space is larger than the rank ofS. TheW remainder of the paper is organized as follows. In Section II, the Discriminative Common Vector approach is introduced. In Section III, we describe the data sets and experimental results. Finally, we formulate our conclusions in Section IV.II. DISCRIMINATIVE COMMON VECTOR APPROACHThe idea of common vectors was originally introduced for isolated word recognition problems in the case where the number of samples in each class was less than or equal to the dimensionality of the sample space [19], [20]. These approaches extract the common properties of classes in the training set by eliminating the differences of the samples in each class. A common vector for each individual class is obtained by removing all the features that are in the direction of the eigenvectors corresponding to the nonzero eigenvalues of the scatter matrix of its own class. The common vectors are then used for recognition. In our case instead of using a given class’s own scatter matrix, we use the within-class scatter matrix of all classes to obtain the common vectors. We also give an alternative algorithm based on the subspace methods and the Gram-Schmidt orthogonalization procedure to obtain the common vectors. Then, a new set of vectors, called the discriminative common vectors, which will be used for classification are obtained from the common vectors. We introduce algorithms for obtaining the common vectors and the discriminative common vectors below.A. Obtaining the Discriminative Common Vectors by Using the Null Space of W S Let the training set be composed of C classes, where each class contains N samples, and let i mx be a d -dimensional column vector which denotes the m-th sample from the i-th class. There will be a total of M=NC samples in the training set. Suppose that d>M-C. In this case, W S , B S , and T S are defined as,T i i m i C i N m i m W x x S ))((11µµ−−=∑∑==, (1)T i C i i B N S )()(1µµµµ−−=∑=, (2)andB W T i mC i Nm i m T S S x x S +=−−∑∑===))((11µµ, (3) where µ is the mean of all samples, and i µ is the mean of samples in the i -th class.In the special case where 0=w S w W T and 0≠w S w B T , for all }0{\d R w ∈, the modified Fisher’s Linear Discriminant criterion attains a maximum. However, a projection vector w , satisfying the above conditions, does not necessarily maximize the between-class scatter. In this case, a better criterion is given in [6] and [13], namely||max arg ||max arg )(0||0||W S W W S W W J T T W S W B T W S W opt W T W T ====. (4)To find the optimal projection vectors w in the null space of W S , we project the face samples onto the null space of W S and then obtain the projection vectors by performing PCA. To do so, vectors that span the null space of W S must first be computed. However, this task is computationally intractable since the dimension of this null space can be very large. A moreefficient way to accomplish this task is by using the orthogonal complement of the null space of W S , which typically is a significantly lower-dimensional space.Let d R be the original sample space, V be the range space of W S , and ⊥V be the null space of W S . Equivalently,},...,1,0|{r k S span V k W k =≠=αα (5)and},...,1,0|{d r k S span V k W k +===⊥αα, 6)where d r < is the rank of W S , },....,{1d αα is an orthonormal set, and },....,{1r αα is the set of orthonormal eigen vectors corresponding to the nonzero eigenvalues of W S .Consider the matrices ]....[1r Q αα= and ]....[1d r Q αα+=. Since⊥⊕=V V R d , every face image d i m R x ∈ has a unique decomposition of the formi m i m i m z y x +=, (7)where V x QQ Px y i m T i m i m ∈==, ⊥∈==V x Q Q x P z i m T i m i m , and P and P are the orthogonal projection operators onto V and ⊥V , respectively. Our goal is to computei m i m i m i m i m Px x y x z −=−=. (8)To do this, we need to find a basis for V , which can be accomplished by an eigen-analysis of W S . In particular, the normalized eigenvectors k α corresponding to the nonzero eigenvalues of W S will be an orthonormal basis for V. The eigenvectors can be obtained by calculating the eigenvectors of the smaller M by M matrix, A A T , defined such that T W AA S =, where A is a d by M matrix of the form]........[22111111C C N N x x x x A µµµµ−−−−=. (9)Let k λand k v be the k -th nonzero eigenvalue and the corresponding eigenvector of A A T , where C M k −≤. Then k k Av =α will be the eigenvector that corresponds to the k -th nonzero eigenvalue of W S . The sought-for projection onto ⊥V is achieved by using (8). In this way, it turns out, we obtain the same unique vector for all samples of the same class,i m T i m T i m i com x Q Q x QQ x x =−=, m=1,…,N , i=1,...,C , (10)i.e., the vector on the right-hand side of (10) is independent of the sample index m . We refer tothe vectors i com x as the common vectors. The above fact is proved in the following theorem.Theorem 1: Suppose Q is a matrix whose column vectors are the orthonormal vectors thatspan the null space ⊥V of W S . Then, the projections of the samples i mx of the class i onto ⊥V produce a unique common vector i com x such thati m T i com x Q Q x =, m =1,…,N , i=1,…,C . (11)Proof : By definition, a vector d R ∈α is in ⊥V if 0=αW S . Let i µ be the mean vector of the i -th class, G be the N by N matrix whose entries are all 1−N , and i X be the d by N matrix whosem -th column is the sample i m x . Thus, multiplying both sides of identity 0=αW S by T α andwriting∑==Ci i W S S 1, (12)withT i i i i T i i m i Nm i m i G X X G X X x x S ))(())((1−−=−−=∑=µµ, (13)immediately leads to 211||))((||)())((0ααα∑∑==−=−−=Ci T i T i T i C i T X G I X G I G I X , (14)where ||.|| denotes the Euclidean norm. Thus, (14) holds if 0))((=−k T i X G I α, or k T i k T i X G X αα)()(=. From this relation we can see that,d r k C i N m x k T i k T i m ,...,1,,...,1,, (1))()(+====αµα. (15) Thus, the projection of i m x onto ⊥V ,k k d r k i k k d r k i m i com x x ααµαα〉〈=〉〈=∑∑+=+=,,11, (16)is independent of m , which proves the theorem.The theorem states that it is enough to project a single sample from each class. This will greatly reduce the computational burden of the calculations. This computational savings has not been previously reported in the literature.After obtaining the common vectors i com x , optimal projection vectors will be those thatmaximize the total scatter of the common vectors,||max arg ||max arg ||max arg )(0||0||W S W W S W W S W W J com T WT T W S W B T W S W opt W T W T =====, (17)where W is a matrix whose columns are the orthonormal optimal projection vectors k w , and com S is the scatter matrix of the common vectors,T com i com com Ci i com com x x S ))((1µµ−−=∑=, i =1,…,C , (18) where com µ is the mean of all common vectors, ∑==C i i com com x C 11µ.In this case optimal projection vectors k w can be found by an eigen-analysis of com S . In particular, all eigenvectors corresponding to the nonzero eigenvalues of com S will be the optimal projection vectors. com S is typically a large d by d matrix and thus we can use the smaller matrix, com T com A A , of size C by C, to find nonzero eigenvalues and the corresponding eigenvectors ofT com com com A A S =, where com A is the d by C matrix of the form]....[1com C com com com com x x A µµ−−=. (19)There will be C -1 optimal projection vectors since the rank of com S is C -1 if all common vectors are linearly independent. If two common vectors are identical, then the two classes which are represented by this vector cannot be distinguished. Since the optimal projection vectors k wbelong to the null space of W S , it follows that when the image samples i m x of the i -th class areprojected onto the linear span of the projection vectors k w , the feature vectorT C i m i m i w x w x ],....,[11><><=Ω− of the projection coefficients ><k i m w x , will also be independent of the sample index m . Thus, we havei m T i x W =Ω, m=1,…,N, i=1,…,C. (20)We call the feature vectors i Ω discriminative common vectors , and they will be used for classification of face images. The fact that i Ω does not depend on the index m in (20) guarantees 100% accuracy in the recognition of the samples in the training set. This guarantee has not been reported in connection with other methods [15], [17].To recognize a test image test x , the feature vector of this test image is found bytest T test x W =Ω, (21)which is then compared with the discriminative common vector i Ω of each class using the Euclidean distance. The discriminative common vector found to be the closest to test Ω is used to identify the test image.Since test Ω is only compared to a single vector for each class, the recognition is very efficientfor real-time face recognition tasks. In the Eigenface, the Fisherface, and the Direct-LDA methods, the test sample feature vector test Ω is typically compared to all feature vectors ofsamples in the training set, making these methods impractical for real-time applications for large training sets.The above method can be summarized as follows:Step 1: Compute the nonzero eigenvalues and corresponding eigenvectors of W S by using the matrix A A T , where T W AA S = and A is given by (9). Set ]....[1r Q αα=, where r is therank of W S . Step 2: Choose any sample from each class and project it onto the null space of W S to obtain the common vectorsi m T i m i com x QQ x x −=, N m ,...,1=, C i ,...,1=. (22)Step 3: Compute the eigenvectors k w of com S , corresponding to the nonzero eigenvalues, byusing the matrix com T comA A , where T com com com A A S = and com A is given in (19). There are at most C -1 eigenvectors that correspond to the nonzero eigenvalues. Use these eigenvectors to form the projection matrix ]....[11−=C w w W , which will be used to obtain feature vectors in (20) and (21).B. Obtaining the Discriminative Common Vectors by Using Difference Subspaces and the Gram-Schmidt Orthogonalization ProcedureTo find an orthonormal basis for the range of W S , the algorithm described above uses the eigenvectors corresponding to the nonzero eigenvalues of the M by M matrix A A T , whereTW AA S =. Assuming that C M S rank W −=)(, then dC C M dM M M M l +−+−+)(2)234(233floating point operations (flops) are required to obtain an orthonormal basis set spanning the range of W S by using this approach. Here l represents the number of iterations required for convergence of the eigen-decomposition algorithm. However, the computations may become expensive and numerically unstable for large values of M . Since we do not need to find the eigenvalues (i.e., an explicit symmetric Schur decomposition) of W S , the following algorithm can be used for finding the common vectors efficiently. It requires only ))()(2(2C M d C M d −+− flops to find an orthonormal basis for the range of W S and is based on the subspace methods and the Gram-Schmidt orthogonalization procedure.Suppose that d >M-C . In this case, the subspace methods can be applied to obtain the commonvectors i com x for each class i . To do this, we choose any one of the image vectors from the i -th class as the subtrahend vector and then obtain the difference vectors i k b of the so-called difference subspace of the i -th class [20]. Thus, assuming that the first sample of each class istaken as the subtrahend vector, we have i i k i k x x b 11−=+, 1,...,1−=N k .The difference subspace i B of the i -th class is defined as },....,{11i N i i b b span B −=. Thesesubspaces can be summed up to form the complete difference subspace},....,,,....,{....12111111C N N C b b b b span B B B −−=++=. (23)The number of independent difference vectors i k b will be equal to the rank of W S . Forsimplicity, suppose there are M -C independent difference vectors. Since by Theorem 3, B and the range space V of W S , are the same spaces, the projection matrix onto B is the same as the matrix P (projection matrix onto the range space of W S ) defined previously in Section II-A. This matrix can be computed asT T D D D D P 1)(−=, (24)where ]........[1211111C N N b b b b D −−= is a d by M-C matrix [21]. This involvesfinding the inverse of an M -C by M -C nonsingular, positive definite symmetric matrix D D T . A computationally efficient method of applying the projection uses an orthonormal basis for B . Inparticular, the difference vectors i k b can be orthonormalized by using the Gram-Schmidtorthogonalization procedure to obtain orthonormal basis vectors C M −ββ,....,1. The complement of B is the indifference subspace ⊥B such that]....[1C M U −=ββ, T UU P =, (25) ]....[1d C M U ββ+−=, T U U P =, (26)where P and P are the orthogonal projection operators onto B and ⊥B , respectively. Thus matrices P and P are symmetric and idempotent, and satisfy I P P =+. Any sample from each class can now be projected onto the indifference subspace ⊥B to obtain the corresponding common vectors of the classes,.,...,1,,...,1,C i N m x UU x x U U Px x x P x i m T i m im T i mi m i m i com ==−==−== (27)The common vectors do not depend on the choice of the subtrahend vectors and they are identical to the common vectors obtained by using the null space of W S . This follows from Theorem 3 below, which uses the results of Lemma 1 and Theorem 2.Theorem 2: Let ⊥i V be the null space of the scatter matrix i S , and ⊥i B be the orthogonal complement of the difference subspace i B . Then ⊥⊥=i i B V and i i B V =.Proof : See [20].Lemma 1: Suppose that C S S ,....,1 are positive semi-definite scatter matrices. ThenI Ci i C S N S S N 11)()....(==++, (28) where N ( ) denotes the null space.Proof : The null space on the left-hand side of the above identity contains elements α such that0)....(1=++αC S S (29)or0....)....(11=++=++ααααααC T T C T S S S S , (30)by the positive semi-definiteness of C S S ++....1. Thus, again by the positive semi-definiteness, )....(1C S S N ++∈α if and only if0=ααi T S , i=1,...,C , (31)or, equivalently, I Ci i S N 1)(=∈α. Theorem 3: Let C S S ,....,1 be positive semi-definite scatter matrices. ThenC C C W B B S R S R S S R S R B ++=++=++==....)(....)()....()(111, (32)where R denotes the range.Proof : Since it is well known that the null space and the range of a matrix are complementary spaces, using the previous Lemma 1, we have,....)(....)())((....))(())(())....(()....(111111C C C C i i C C B B S R S R S N S N S N S S N S S R ++=++=++==++=++⊥⊥⊥=⊥I (33)where the last equality is a consequence of Theorem 2.After calculating the common vectors, the optimal projection vectors can be found by performing PCA as described previously in Section II-A. The eigenvectors corresponding to the nonzero eigenvalues of com S will be the optimal projection vectors. However, optimal projection vectors can also be obtained more efficiently by computing the basis of the difference subspace com B of the common vectors, since we are only interested in finding an orthonormal basis for the range of com S .The algorithm based on the Gram-Schmidt orthogonalization can be summarized as follows.Step 1: Find the linearly independent vectors i k b that span the difference subspace B and set},....,,,....,{1211111C N N b b b b span B −−=. There are totally r linearly independent vectors, where r is at most M -C .Step 2: Apply the Gram-Schmidt orthogonalization procedure to obtain an orthonormal basis r ββ,....,1 for B and set ]....[1r U ββ=.Step 3: Choose any sample from each class and project it onto B to obtain common vectors by using (27).Step 4: Find the difference vectors that span com B as11com k com k com x x b −=+, 1,...,1−=C k , (34)。

512长度的Face Recognition特征通常用于人脸识别系统的特征提取和比对。

它通过人脸特征提取算法从输入的人脸图像中提取出一组特征向量，这组特征向量可以用于表示该人脸的特征。

在人脸识别系统中，特征提取是至关重要的一步，它直接影响到人脸识别的准确率和可靠性。

512长度的特征向量通常包含了人脸的各种特征信息，如面部的几何特征、纹理特征、肤色特征等。

这些特征信息可以通过不同的算法进行提取，如主成分分析（PCA）、线性判别分析（LDA）等。

提取出的人脸特征向量可以用于比对和匹配，以实现人脸的识别和验证。

在比对过程中，系统会将输入的人脸特征向量与已存储的人脸特征向量进行比较，以找出最相似的人脸，从而实现人脸的识别或验证。

总之，512长度的Face Recognition特征是一种用于人脸识别系统的特征提取方法，它可以有效地提取出人脸的各种特征信息，并用于比对和匹配，从而实现人脸的识别和验证。

特征识别算法特征识别算法（Feature Recognition Algorithm）是一种通过对输入数据进行分析和处理，从中提取出具有特定意义的特征并进行识别的一种技术。

它可以应用于多个领域，如图像处理、语音识别、生物特征识别等。

在图像处理领域，特征识别算法被广泛应用于目标检测、物体识别等任务中。

通过提取图像的局部特征，比如边缘、纹理、颜色等，算法可以识别出图像中的目标物体，并进行分类或定位。

其中最常用的特征识别算法之一是SIFT（Scale-Invariant Feature Transform），它通过寻找图像中的关键点，并对这些关键点进行描述，从而实现图像特征的匹配和识别。

在语音识别领域，特征识别算法可以将语音信号转化为一系列特征向量，用于表示语音的特征。

常用的特征包括梅尔频率倒谱系数（MFCC）、线性预测编码（LPC）等。

这些特征可以用于语音识别任务中，比如说话人识别、语音指令识别等。

在生物特征识别领域，特征识别算法可以根据人体的生物特征进行身份识别。

常见的生物特征包括指纹、虹膜、面部等。

通过提取这些生物特征的特征向量，并与已知的特征进行比对，算法可以判断出一个人的身份。

特征识别算法的核心思想是将输入数据转化为一种可以被计算机处理的形式，并提取出具有特定意义的特征。

这些特征可以用于判断、分类或识别。

为了提取出有意义的特征，算法需要具备以下几个步骤：1. 数据预处理：对输入数据进行预处理，如去噪、归一化等。

这一步旨在减少数据中的噪声和冗余信息，提高特征的可靠性。

2. 特征提取：通过某种方法提取出数据中的特征。

常用的方法有统计分析、频域分析、小波变换等。

特征的选择应该具有一定的区分度和稳定性，能够准确地表达数据的特性。

3. 特征选择：根据特定的任务需求，选择出最相关的特征。

这一步旨在减少特征的维度，提高计算效率和准确性。

4. 特征匹配或分类：将提取到的特征与已知的特征进行比对，从而实现特征的匹配或分类。

信息科学①基金项目:本文获江苏省淮安市国际合作科技项目（项目编号:HAC201618）资助。

DOI：10.16660/ki.1674-098X.2018.09.150基于Emgu CV的GPU加速人脸识别①李刚 孙平(淮阴师范学院 江苏淮安 223300)摘 要：Emgu CV是.NET平台下对OpenCV图像处理库的封装，可以实现人脸识别的判断。

本文着重讨论了在.NET 下基于Emgu CV利用GPU加速技术实现了静态图像的人脸检测、人脸识别、人脸比对，以及视频流中的人脸识别。

该软件获得我校技术进步二等奖，实验结果证明本程序运行稳定，结果可靠，识别速度快。

关键词：Emgu CV GPU加速 人脸检测 人脸识别中图分类号：TP39 文献标识码：A 文章编号：1674-098X(2018)03(c)-0150-04 Abstract: Emgu CV is OpenCV image processing library encapsulation on .net platform, that can be used to realize face recognition judgment. This article emphatically introduced Emgu CV graphics library for face detection, face match, and face recognition in image and in video as well, with the assist of GPU speeding technology. This implemented procedure was granted 2nd prize in the Advanced Technologies in Huaiyin Normal University. Long-time running demos show the procedure worked stably, reliably and quickly.Key Words: Emgu CV; GPU; Face Detection; Face Recognition人脸识别作为一种成熟而新颖的生物特征识别技术，具有识别精度高，识别速度快，使用方便，不易被仿冒等优点，与虹膜识别、指纹扫描等这些生物特征识别技术相比，人脸识别技术在各个应用方面都表现出得天独厚的技术优势，而且成本低廉，易于推广使用，企业用户的接受程度非常高。

教师活动第2.1课《初识人脸识别——人脸识别在实际中的应用》板书课题：当你拿起智能手机对准自己的脸时，手机通过刷脸就能自动解锁；购物时，通过刷脸就能轻松支付；乘坐铁路交通时，用人脸对准闸机并刷身份证，就能完成身份识别……这些都离不开人脸识别技术的发展。

本节课，我们就一起来了解一下人脸识别吧！思考：同学们，想一想：生活中你们在哪些地方见过人脸识别技术的应用呢?你觉得人脸识别技术到底是什么呢?人脸识别技术的特点人脸识别在发展的过程中不断提升并优化自己的技术优点和特长，因其准确直观、非接触性、操作简单、可并发性等特点，在如今社会得到了广泛应用，如图 2.1.2 所示。

课堂活动说一说：机器识别人脸和人识别人脸有什么关系呢?机器识别人脸和人识别人脸之间的关系主要体现在技术的实现和应用上。

机器识别人脸是通过计算机视觉和深度学习等技术，自动识别和验证人脸特征，而人识别人脸则是人类通过视觉和经验来识别他人。

这两者的目标相同，但实现方式不同。

查找资料，说一说人脸识别技术还有什么特点？1. 高准确性：现代人脸识别技术通过深度学习算法，能够在复杂环境下实现高准确率的识别。

示。

只有识别出房屋主人的正确人脸，才能打开门，这样既方便使用，又能保证居住安全。

学校和企业等,还使用人脸识别门禁考勤系统，方便老师或管理员了解学生和员工的考勤情况，便于更好地进行管理。

人脸识别技术在出行中的应用外出通行是人们日常生活中不可缺少的一部分，各种交通工具为出行提供了便利。

如今，各大城市的火车站、飞机场等很多已经安装了人脸识别通行设备，通过将身份证和人脸进行比对，就能自动完成检索功能，如图2.1.4所示。

人脸识别技术在自助服务中的应用自助服务系统是信息时代技术发展涌现出来的新产物。

顾名思义，自助服务是指人们通过一些平台或设备能自己处理需要解决的问题。

自助服务通常会用到人脸识别技术。

利用银行的自助取款机取款时，自助取款机能通过刷脸轻松识别取款人的身份;在无人售货超市中，通过对顾客脸部进行识别，超市大门就能打开，一旦发生盗窃事件，超市中的摄像头也能轻松捕捉人脸，触动报警系统，如图2.1.5 所示。

收稿日期：2013-03-11；修回日期：2013-05-09基金项目：中央高校基本科研业务费专项资金资助项目（JUSＲP211A70）；国家自然科学基金资助项目（61373055）作者简介：陈达遥（1988-），男，湖南益阳人，硕士研究生，主要研究方向为人工智能和模式识别（chenda80@163．com ）；陈秀宏（1964-），男，江苏泰兴人，教授，硕导，博士，主要研究方向为数字图像处理、人脸识别；董昌剑（1991-），男，安徽淮北人，硕士研究生，主要研究方向为人工智能和模式识别．基于正交SＲDA 和SＲKDA 的人脸识别*陈达遥，陈秀宏，董昌剑（江南大学数字媒体学院，江苏无锡214122）摘要：利用正交投影技术进行降维可以更好地保留与度量结构有关的信息，提高人脸识别性能。

在谱回归判别分析（SＲDA ）和谱回归核判别分析（SＲKDA ）的基础上，提出正交SＲDA （OSＲDA ）和正交SＲKDA （OSＲKDA ）降维算法。

首先，给出基于Cholesky 分解求解正交鉴别矢量集的方法，然后，通过该方法对SＲDA 和SＲKDA 投影向量作正交化处理。

其简单、容易实现而且克服了迭代计算正交鉴别矢量集的方法不适应于谱回归（SＲ）降维的缺点。

OＲL 、Yale 和PIE 库上的实验结果表明了算法的有效性和效率，在有效降维的同时能进一步提高鉴别能力。

关键词：降维；人脸识别；谱回归；正交鉴别矢量；Cholesky 分解中图分类号：TP391.41文献标志码：A文章编号：1001-3695（2014）01-0299-05doi ：10．3969/j．issn．1001-3695．2014．01．071Face recognition based on orthogonal SＲDA and SＲKDACHEN Da-yao ，CHEN Xiu-hong ，DONG Chang-jian（School of Digital Media ，Jiangnan University ，Wuxi Jiangsu 214122，China ）Abstract ：The dimensionality reduction by orthogonal projection techniques helped preserve the information related to the metric structure and improved the recognition performance in face recognition．Based on spectral regression discriminant analy-sis （SＲDA ）and spectral regression kernel discriminant analysis （SＲKDA ），this paper proposed two dimensionality reduction algorithms named orthogonal SＲDA （OSＲDA ）and orthogonal OSＲKDA （OSＲKDA ）．Firstly ，it gave a set of orthogonal dis-criminant vectors obtained based on Cholesky decomposition．Then ，this paper orthogonalized the projection vectors of SＲDA and SＲKDA by this method．It was very simple and easy to implement．What ’s more ，it overcame the shortcoming that the iter-ative algorithm of orthogonal discriminant vectors was not suitable for spectral regression dimensionality reduction algorithms．Experiments on OＲL ，Yale and PIE demonstrate the effectiveness and efficiency of the algorithms ，and show that these algo-rithms can reduce the dimensions of the data and improve the discriminant ability．Key words ：dimensionality reduction ；face recognition ；spectral regression ；orthogonal discriminant vector ；Cholesky de-composition0引言在过去几十年里，人脸识别技术得到了很大的发展，特征提取是人脸识别的关键步骤，其主要目的是降维。