nips nips2005 nips2005-189 knowledge-graph by maker-knowledge-mining

189 nips-2005-Tensor Subspace Analysis

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Author: Xiaofei He, Deng Cai, Partha Niyogi

Abstract: Previous work has demonstrated that the image variations of many objects (human faces in particular) under variable lighting can be effectively modeled by low dimensional linear spaces. The typical linear subspace learning algorithms include Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), and Locality Preserving Projection (LPP). All of these methods consider an n1 × n2 image as a high dimensional vector in Rn1 ×n2 , while an image represented in the plane is intrinsically a matrix. In this paper, we propose a new algorithm called Tensor Subspace Analysis (TSA). TSA considers an image as the second order tensor in Rn1 ⊗ Rn2 , where Rn1 and Rn2 are two vector spaces. The relationship between the column vectors of the image matrix and that between the row vectors can be naturally characterized by TSA. TSA detects the intrinsic local geometrical structure of the tensor space by learning a lower dimensional tensor subspace. We compare our proposed approach with PCA, LDA and LPP methods on two standard databases. Experimental results demonstrate that TSA achieves better recognition rate, while being much more efﬁcient. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu 1 Abstract Previous work has demonstrated that the image variations of many objects (human faces in particular) under variable lighting can be effectively modeled by low dimensional linear spaces. [sent-4, score-0.117]

2 The typical linear subspace learning algorithms include Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), and Locality Preserving Projection (LPP). [sent-5, score-0.146]

3 All of these methods consider an n1 × n2 image as a high dimensional vector in Rn1 ×n2 , while an image represented in the plane is intrinsically a matrix. [sent-6, score-0.162]

4 TSA considers an image as the second order tensor in Rn1 ⊗ Rn2 , where Rn1 and Rn2 are two vector spaces. [sent-8, score-0.423]

5 The relationship between the column vectors of the image matrix and that between the row vectors can be naturally characterized by TSA. [sent-9, score-0.077]

6 TSA detects the intrinsic local geometrical structure of the tensor space by learning a lower dimensional tensor subspace. [sent-10, score-0.843]

7 Experimental results demonstrate that TSA achieves better recognition rate, while being much more efﬁcient. [sent-12, score-0.048]

8 1 Introduction There is currently a great deal of interest in appearance-based approaches to face recognition [1], [5], [8]. [sent-13, score-0.251]

9 When using appearance-based approaches, we usually represent an image of size n1 × n2 pixels by a vector in Rn1 ×n2 . [sent-14, score-0.06]

10 Throughout this paper, we denote by face space the set of all the face images. [sent-15, score-0.406]

11 The face space is generally a low dimensional manifold embedded in the ambient space [6], [7], [10]. [sent-16, score-0.317]

12 The typical linear algorithms for learning such a face manifold for recognition include Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA) and Locality Preserving Projection (LPP) [4]. [sent-17, score-0.338]

13 Most of previous works on statistical image analysis represent an image by a vector in high-dimensional space. [sent-18, score-0.137]

14 However, an image is intrinsically a matrix, or the second order tensor. [sent-19, score-0.083]

15 Recently, multilinear algebra, the algebra of higher-order tensors, was applied for analyzing the multifactor structure of image ensembles [9], [11], [12]. [sent-21, score-0.167]

16 Vasilescu and Terzopoulos have proposed a novel face representation algorithm called Tensorface [9]. [sent-22, score-0.219]

17 Tensorface represents the set of face images by a higher-order tensor and extends Singular Value Decomposition (SVD) to higher-order tensor data. [sent-23, score-0.985]

18 In this way, the multiple factors related to expression, illumination and pose can be separated from different dimensions of the tensor. [sent-24, score-0.048]

19 In this paper, we propose a new algorithm for image (human faces in particular) representation based on the considerations of multilinear algebra and differential geometry. [sent-25, score-0.162]

20 For an image of size n1 × n2 , it is represented as the second order tensor (or, matrix) in the tensor space Rn1 ⊗ Rn2 . [sent-27, score-0.786]

21 On the other hand, the face space is generally a submanifold embedded in Rn1 ⊗ Rn2 . [sent-28, score-0.304]

22 Given some images sampled from the face manifold, we can build an adjacency graph to model the local geometrical structure of the manifold. [sent-29, score-0.452]

23 TSA ﬁnds a projection that respects this graph structure. [sent-30, score-0.103]

24 The obtained tensor subspace provides an optimal linear approximation to the face manifold in the sense of local isometry. [sent-31, score-0.801]

25 Vasilescu shows how to extend SVD(PCA) to higher order tensor data. [sent-32, score-0.363]

26 While traditional linear dimensionality reduction algorithms like PCA, LDA and LPP ﬁnd a map from Rn to Rl (l < n), TSA ﬁnds a map from Rn1 ⊗ Rn2 to Rl1 ⊗ Rl2 (l1 < n1 , l2 < n2 ). [sent-35, score-0.11]

27 When label information is available, it can be easily incorporated into the graph structure. [sent-39, score-0.062]

28 The matrices in the eigen-problems are of size n1 ×n1 or n2 ×n2 , which are much smaller than the matrices of size n × n (n = n1 × n2 ) in PCA, LDA and LPP. [sent-44, score-0.044]

29 TSA explicitly takes into account the manifold structure of the image space. [sent-48, score-0.152]

30 The local geometrical structure is modeled by an adjacency graph. [sent-49, score-0.131]

31 2 Tensor Subspace Analysis In this section, we introduce a new algorithm called Tensor Subspace Analysis for learning a tensor subspace which respects the geometrical and discriminative structures of the original data space. [sent-53, score-0.584]

32 1 Laplacian based Dimensionality Reduction Problems of dimensionality reduction has been considered. [sent-55, score-0.092]

33 One general approach is based on graph Laplacian [2]. [sent-56, score-0.062]

34 The objective function of Laplacian eigenmap is as follows: 2 min f (f (xi ) − f (xj )) Sij ij where S is a similarity matrix. [sent-57, score-0.066]

35 These optimal functions are nonlinear but may be expensive to compute. [sent-58, score-0.044]

36 In this paper we will consider a more structured subset of linear functions that arise out of tensor analysis. [sent-61, score-0.381]

37 2 The Linear Dimensionality Reduction Problem in Tensor Space The generic problem of linear dimensionality reduction in the second order tensor space is the following. [sent-64, score-0.473]

38 Our method is of particular applicability in the special case where X1 , · · · , Xm ∈ M and M is a nonlinear submanifold embedded in Rn1 ⊗ Rn2 . [sent-66, score-0.125]

39 3 Optimal Linear Embeddings As we described previously, the face space is probably a nonlinear submanifold embedded in the tensor space. [sent-68, score-0.691]

40 One hopes then to estimate geometrical and topological properties of the submanifold from random points (“scattered data”) lying on this unknown submanifold. [sent-69, score-0.15]

41 In this section, we consider the particular question of ﬁnding a linear subspace approximation to the submanifold in the sense of local isometry. [sent-70, score-0.221]

42 Given m data points X = {X1 , · · · , Xm } sampled from the face submanifold M ∈ Rn1 ⊗ Rn1 , one can build a nearest neighbor graph G to model the local geometrical structure of M. [sent-72, score-0.484]

43 A possible deﬁnition of S is as follows:  2  e− Xi −Xj , if X is among the k nearest t  i  neighbors of Xj , or Xj is among (1) Sij =  the k nearest neighbors of Xi ;   0, otherwise. [sent-74, score-0.056]

44 The function exp(− Xi − Xj 2 /t) is the so called heat kernel which is intimately related to the manifold structure. [sent-76, score-0.069]

45 When the label information is available, it can be easily incorporated into the graph as follows: Sij = e− 0, Xi −Xj 2 t , if Xi and Xj share the same label; otherwise. [sent-80, score-0.062]

46 A reasonable transformation respecting the graph structure can be obtained by solving the following objective functions: U T Xi V − U T Xj V min U,V 2 Sij (3) ij The objective function incurs a heavy penalty if neighboring points Xi and Xj are mapped far apart. [sent-82, score-0.192]

47 = tr U T (DV − SV ) U T T where DV = i Dii Xi V V Xi and SV = T tr(A A), so we also have ij 1 2 2 U T Xi V − U T Xj V = Sij tr (Yi − Yj )T (Yi − Yj ) Sij ij 1 2 = 2 ij 1 2 = T Sij Xi V V T Xj . [sent-87, score-0.524]

48 Similarly, A tr YiT Yi + YjT Yj − YiT Yj − YjT Yi Sij ij Dii YiT Yi − = tr i = tr V Sij YiT Yj ij T T Dii Xi U U T Xi T Xi U U T Xj V − i ij . [sent-88, score-0.687]

49 = tr V T (DU − SU ) V T T where DU = i Dii Xi U U T Xi and SU = ij Sij Xi U U T Xj . [sent-89, score-0.229]

50 Therefore, we should T simultaneously minimize tr U (DV − SV ) U and tr V T (DU − SU ) V . [sent-90, score-0.326]

51 In addition to preserving the graph structure, we also aim at maximizing the global variance on the manifold. [sent-91, score-0.099]

52 By spectral graph theory [3], dP can be discretely estimated by T the diagonal matrix D(Dii = j Sij ) on the sample points. [sent-93, score-0.079]

53 Let Y = U XV denote the random variable in the tensor subspace and suppose the data points have a zero mean. [sent-94, score-0.491]

54 It is easy to see that the optimal U should be the generalized eigenvectors of (DV − SV , DV ) and the optimal V should be the generalized eigenvectors of (DU − SU , DU ). [sent-99, score-0.124]

55 3 Experimental Results In this section, several experiments are carried out to show the efﬁciency and effectiveness of our proposed algorithm for face recognition. [sent-105, score-0.203]

56 We compare our algorithm with the Eigenface (PCA) [8], Fisherface (LDA) [1], and Laplacianface (LPP) [5] methods, three of the most popular linear methods for face recognition. [sent-106, score-0.221]

57 The ﬁrst one is the PIE (Pose, Illumination, and Experience) database from CMU, and the second one is the ORL database. [sent-108, score-0.057]

58 Original images were normalized (in scale and orientation) such that the two eyes were aligned at the same position. [sent-110, score-0.073]

59 Then, the facial areas were cropped into the ﬁnal images for matching. [sent-111, score-0.095]

60 The size of each cropped image in all the experiments is 32×32 pixels, with 256 gray levels per pixel. [sent-112, score-0.08]

61 For the Eigenface, Fisherface, and Laplacianface methods, the image is represented as a 1024-dimensional vector, while in our algorithm the image is represented as a (32 × 32)-dimensional matrix, or the second order tensor. [sent-114, score-0.12]

62 The nearest neighbor classiﬁer is used for classiﬁcation for its simplicity. [sent-115, score-0.046]

63 First, we calculate the face subspace from the training set of face images; then the new face image to be identiﬁed is projected into d-dimensional subspace (PCA, LDA, and LPP) or (d × d)-dimensional tensor subspace (TSA); ﬁnally, the new face image is identiﬁed by nearest neighbor classiﬁer. [sent-117, score-1.725]

64 1 Experiments on PIE Database The CMU PIE face database contains 68 subjects with 41,368 face images as a whole. [sent-120, score-0.519]

65 The face images were captured by 13 synchronized cameras and 21 ﬂashes, under varying pose, illumination and expression. [sent-121, score-0.288]

66 dimensionality reduction on PIE database Table 1: Performance comparison on PIE database Method Baseline Eigenfaces Fisherfaces Laplacianfaces TSA error 69. [sent-123, score-0.234]

67 64% 5 Train dim 1024 338 67 67 112 20 Train dim 1024 889 67 146 132 time(s) 0. [sent-133, score-0.182]

68 88% 10 Train dim 1024 654 67 134 132 30 Train dim 1024 990 67 131 122 time(s) 5. [sent-151, score-0.182]

69 688 C29) and use all the images under different illuminations and expressions, thus we get 170 images for each individual. [sent-159, score-0.112]

70 For each individual, l(= 5, 10, 20, 30) images are randomly selected for training and the rest are used for testing. [sent-160, score-0.056]

71 The training set is utilized to learn the subspace representation of the face manifold by using Eigenface, Fisherface, Laplacianface and our algorithm. [sent-161, score-0.416]

72 The testing images are projected into the face subspace in which recognition is then performed. [sent-162, score-0.435]

73 It would be important to note that the Laplacianface algorithm and our algorithm share the same graph structure as deﬁned in Eqn. [sent-164, score-0.085]

74 Figure 1 shows the plots of error rate versus dimensionality reduction for the Eigenface, Fisherface, Laplacianface, TSA and baseline methods. [sent-166, score-0.233]

75 For the baseline method, the recognition is simply performed in the original 1024-dimensional image space without any dimensionality reduction. [sent-167, score-0.247]

76 Note that, the upper bound of the dimensionality of Fisherface is c − 1 where c is the number of individuals. [sent-168, score-0.051]

77 For our TSA algorithm, we only show its performance in the (d × d)-dimensional tensor subspace, say, 1, 4, 9, etc. [sent-169, score-0.363]

78 We show the best results obtained by them in Table 1 and the corresponding face subspaces are called optimal face subspace for each method. [sent-171, score-0.57]

79 It does not obtain any improvement over the baseline method. [sent-174, score-0.088]

80 dimensionality reduction on ORL database Table 2: Performance comparison on ORL database Method Baseline Eigenfaces Fisherfaces Laplacianfaces TSA error 30. [sent-180, score-0.234]

81 12% 2 Train dim 1024 79 23 39 102 4 Train dim 1024 122 39 39 102 time 38. [sent-190, score-0.182]

82 75% 3 Train dim 1024 113 39 39 112 5 Train dim 1024 182 39 40 102 time 85. [sent-208, score-0.182]

83 2 Experiments on ORL Database The ORL (Olivetti Research Laboratory) face database is used in this test. [sent-221, score-0.26]

84 It consists of a total of 400 face images, of a total of 40 people (10 samples per person). [sent-222, score-0.203]

85 The images were captured at different times and have different variations including expressions (open or closed eyes, smiling or non-smiling) and facial details (glasses or no glasses). [sent-223, score-0.075]

86 The images were taken with a tolerance for some tilting and rotation of the face up to 20 degrees. [sent-224, score-0.259]

87 For each individual, l(= 2, 3, 4, 5) images are randomly selected for training and the rest are used for testing. [sent-225, score-0.056]

88 2 shows the plots of error rate versus dimensionality reduction for the Eigenface, Fisherface, Laplacianface, TSA and baseline methods. [sent-229, score-0.233]

89 Here, for a given d, we show its performance in the (d×d)dimensional tensor subspace. [sent-231, score-0.363]

90 The best result obtained in the optimal subspace and the running time (millisecond) of computing the eigenvectors for each method are shown in Table 2. [sent-233, score-0.17]

91 4 Conclusions and Future Work Tensor based face analysis (representation and recognition) is introduced in this paper in order to detect the underlying nonlinear face manifold structure in the manner of tensor subspace learning. [sent-236, score-1.03]

92 The manifold structure is approximated by the adjacency graph computed from the data points. [sent-237, score-0.187]

93 The optimal tensor subspace respecting the graph structure is then obtained by solving an optimization problem. [sent-238, score-0.637]

94 Most of traditional appearance based face recognition methods (i. [sent-240, score-0.251]

95 Eigenface, Fisherface, and Laplacianface) consider an image as a vector in high dimensional space. [sent-242, score-0.079]

96 In our work, an image is naturally represented as a matrix, or the second order tensor. [sent-244, score-0.06]

97 Therefore, if the face manifold is highly nonlinear, it may fail to discover the intrinsic geometrical structure. [sent-248, score-0.347]

98 Also, in our algorithm, the adjacency graph is induced from the local geometry and class information. [sent-250, score-0.095]

99 It remains unclear how to deﬁne the optimal graph structure in the sense of discrimination. [sent-252, score-0.124]

100 Yang, “Two-dimensional PCA: a new approach to appearance-based face representation and recognition,”IEEE. [sent-313, score-0.219]

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