Dual-Space Linear Discriminant Analysis for Face Recognition

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1 Dual-Space Linear Discriminant nalysis for Face Recognition Xiaogang Wang and Xiaoou ang Department of Information Engineering he Chinese University of Hong Kong {xgang1, bstract Linear Discriminant nalysis (LD) is a popular feature extraction technique for face recognition. Hoever, it often suffers from the small sample size problem hen dealing ith the high dimensional face data. Some approaches have been proposed to overcome this problem, but they are often unstable and have to discard some discriminative information. In this paper, a dual-space LD approach for face recognition is proposed to take full advantage of the discriminative information in the face space. ased on a probabilistic visual model, the eigenvalue spectrum in the null space of ithin-class scatter matrix is estimated, and discriminant analysis is simultaneously applied in the principal and null subspaces of the ithin-class scatter matrix. he to sets of discriminative features are then combined for recognition. It outperforms existing LD approaches. 1. Introduction LD is a popular face recognition approach. It determines a set of proection vectors maximizing the beteen-class scatter matrix ( S b ) hile minimizing the ithin-class scatter matrix ( S ) in the proective feature space. Hoever, LD often suffers from the small sample size problem hen dealing ith the high dimensional face data. When there are not enough training samples, S may become singular, and it is difficult to compute the LD vectors. Several approaches have been proposed to address this problem. In a to-stage C+LD approach [1], the data dimensionality is first reduced by rincipal Component nalysis (C), and LD is performed in the reduced C subspace, in hich S is non-singular. Hoever, Chen et al. [] suggested that the null space spanned by the eigenvectors of S ith zero eigenvalues contains the most discriminative information. LD method in the null space of S as proposed. It chooses the proection vectors maximizing S b ith the constraint that S is zero. ut this approach discards the discriminative information outside the null space of S. Yu. et al. [3] proposed a direct LD algorithm. It first removes the null space of S b, and assumes that no discriminative information exists in this space. Unfortunately, e can sho that this assumption is incorrect. We ill demonstrate that the optimal discriminant vectors do not necessarily lie in the subspace spanned by the class centers. common problem ith all these proposed LD approaches is that they all lose some discriminative information in the high dimensional face space. In this paper, using the probabilistic visual model [4], the eigenvalue spectrum in the null space of S is estimated. We then apply discriminant analysis in both the principal and null subspaces of S. he to parts of discriminative features are combined in recognition. his dual-space LD approach successfully resolves the small sample size problem. Compared ith conventional LD approaches, it is more stable and makes use of all the discriminative information in both the principal and null space. he experiments on FERE database clearly demonstrate its efficacy.. Linear Discriminant nalysis.1. LD LD method tries to find a set of proection vectors W maximizing the ratio of determinant of S b to S, W S W W arg max b (1) W SW Let the training set contain L classes and each class X i has n i samples. S and S b are defined as, L S xkmixkmi, () i1 x kxi L Sb ni mi mm i m, (3) i1 here m is the center of the hole training set, m i is the center for the class X i, and x k is the sample belonging to class X i. W can be computed from the eigenvectors of S 1 Sb [7]. Hoever, hen the small sample size problem occurs, S becomes singular and it is difficult to 1 compute S. o avoid the singularity of S, a to- roceedings of the 4 IEEE Computer Society Conference on Computer Vision and attern Recognition (CVR 4) /4 $. 4 IEEE

2 stage C+LD approach is used in [1]. C is first used to proect the high dimensional face data into a lo dimensional feature space. hen LD is performed in the reduced C subspace, in hich S is non-singular. Fukunnaga [7] has proved that W can also be computed from simultaneous diagonalization of S and S b. First S is hitened by 1/ 1/ S I, (4) here, and are the eigenvector matrix and eigenvalue matrix of S. o avoid singularity and overfitting for noise, only the eigenvectors ith non-zero and non-trivial eigenvalues are selected in the enhanced LD model proposed by Liu et. al. [6]. Second, apply C on class centers of the transformed data. o do this, 1/ the class centers are proected onto, and the beteen-class scatter matrix is transformed to K, 1/ 1/ Sb K b. (5) fter computing the eigenvector matrix and eigenvalue matrix of K b, the overall proection vectors of LD can be defined as 1/ W. (6).. LD in the Null Space of S he LD approaches described above are all performed in the principal subspace of S, in hich W S W. Hoever, the null space of S, in hich W S W, also contains much discriminative information, since it is possible to find some proection vectors W satisfying W S W and W S b W, thus the Fisher criteria in Eq. (1) definitely reaches its maximum value. LD in the null space of S as proposed by Chen et. al. []. First, the null space of S is computed as, V SV ( V V I ). (7) hen S b is proected to the null space of S, S ~ b V SbV. (8) Choose the eigenvectors U of S ~ b ith the largest eigenvalues, ~ U S b U. (9) he LD transformation matrix is defined as W VU. s the rank of S increases, the null space of S becomes small, and much discriminative information outside the null space is discarded []. b.3. Direct LD Yu et. al. [3] proposed a direct LD method. S b is first diagonalized, and the null space of S b is removed, Y S b Y D b, (1) here Y are eigenvectors and D b are the corresponding non-zero eigenvalues of S b. S is transformed to 1/ 1/ Db Y SYDb K. (11) K is diagonalized by eigenanalysis, K U U D. (1) he LD transformation matrix for classification is defined as, 1/ 1/ YD b UD W. (13) In direct LD, the null space of S b is first removed. It is assumed that the null space of S b contains no discriminative information. his assumption is not true. In direct LD, proection vectors are restricted in the subspace spanned by class centers. Hoever the optimal discriminant vectors do not necessarily lie in the subspace spanned by class centers. his point can be clearly illustrated in Figure 1. For a binary classification problem, using direct LD, the derived discriminant proection vector is constrained to the line passing through the to class centers. ut according to the Fisher criteria, the optimal discriminant vector should be line..4. Discussion ll these proposed LD approaches lose some discriminative information in the face space. his point can be further summarized in Figure, here is the principal subspace of S, and is the principal subspace of S b. Since the total scatter matrix S t is equal to the summarization of S and S b [7], S S S, (14) t Class 1 m1 m b Class Figure 1. Using direct LD, the discriminant vector is constrained to the line passing through the to class centers m 1 and m, but according to the Fisher criteria, the optimal discriminant proection should be line roceedings of the 4 IEEE Computer Society Conference on Computer Vision and attern Recognition (CVR 4) /4 $. 4 IEEE

3 the face space is composed of and. When as shon in Figure (a), LD in the principal subspace of S, contains all the discriminative information in the face space. When as shon in Figure (b), direct LD, orking in subspace, can keep all the discriminative information. When = as shon in Figure (c), LD in the null space of S can keep all the discriminative information. Finally hen and are only partially overlapped as shon in Figure (d), some discriminative information ill definitely be lost using any of the existing LD approaches. Furthermore, conventional LD approaches suffer from the overfitting problem. roection vectors are tuned to the training set ith the existence of noise. s suggested in [8], an eigenvector ill be very sensitive to small perturbation if its eigenvalue is close to another eigenvalue of the same matrix, so the eigenvectors of S ith very small eigenvalues are unstable. In Eq. (6) and (13), LD in the principal subspace of S and direct LD all need to be hitened using the inverse of eigenvalues of S. Since the trivial eigenvalues sensitive to noise are not ell estimated because of the small sample size problem, they can substantially change the proection vectors. If data vector is hitened on noisy eigenvectors, overfitting ill happen. For LD in the null space of S, the rank of S, r S, is bounded by minm L, N, here M is the total training sample number, L is the class number, and N is the dimensionality of the face data. r S is almost equal to this bound because of the existence of noise. s shon by experiments in [], hen the training sample number is large, the null space of S becomes small, so much discriminative information outside it ill be lost. 3. Dual-Space LD 3.1. robabilistic model In LD, the main difficulty for the small sample size problem is that S, especially the eigenvalue spectrum in the null space of S, is not ell estimated. In order to estimate the eigenvalue spectrum in the null space of S, e use the probabilistic visual model proposed in [4]. In classification, the likelihood of an input vector x belonging to class X is often estimated ith a Gaussian density, 1 1 x X exp N / 1 x m x m, (15) here is the covariance matrix for X, and N is the dimensionality of the face vector. When there are not enough samples in each class, is replaced by S [5], 1 1 x X exp N / S 1 x m S x m. (16) It is called Mahalanobis likelihood [5], characterized by a Mahalanobis distance, x d ~ 1 x S ~ x, (17) here ~ x x m. S can be diagonalized as, S 1 here, N, (18) 1, N are the eigenvector matrix of S. he Mahalanobis distance can be estimated as the sum of to independent parts, distance-in-feature-space (DIFS) and distance-from-feature space (DFFS), K i corresponding to the principal subspace F i 1, spanned by the K eigenvectors ith the largest eigenvalues, and its orthogonal complement i N i K F 1, K ~ x y dx i, (19) i1 i here y i is the component proecting ~ x to the ith eigenvector, and i is the corresponding eigenvalue in F. ~ x is the C reconstruction error of ~ x in F. he eigenvalues in F are not ell estimated and have zero values. s suggested in [4], they are simply the estimated noise spectrum, and tend to be the flattest portion of eigenvalue spectrum. So they can be estimated by a single value, 1 N * i N K ik 1. () he unknon * i in F are estimated by fitting a nonlinear function to the available portion of the eigenvalue spectrum in F. No the ithin-class scatter matrix can be estimated as, roceedings of the 4 IEEE Computer Society Conference on Computer Vision and attern Recognition (CVR 4) /4 $. 4 IEEE

4 (a) Figure. is the principal subspace of case (a),, LD in the principal space of (b) S, is the principal subspace of S b, and is the hole face space. In S can keep all the discriminative information. In case (b),, direct LD can keep all the discriminative information. In case (c), =, LD in the null space of S can keep all the discriminative information. In case (d), and are only partially intersect, and so discriminative information in data space ill definitely be lost in conventional LD approaches. (c) (d) Sˆ 1 Ŝ is no non-singular. K (1) he relationship beteen the Mahalanobis distance and LD implies that LD can be simultaneously applied to the principal and null subspaces of S, and the to parts of discriminative features can be combined to make full use of the discriminative information in the face space. ased on this observation, e develop a dualspace LD algorithm: t training stage, 1. Compute S and S b from the training set. 3.. Discriminant nalysis in Dual Space oth LD and the Mahalanobis distance in (19) evaluate the distance from the input pattern to the class center after proecting to the subspace. Recall that LD in the principal subspace of S can be divided into to steps. In the first hitening step, face data is proected 1/ onto and normalized by. Most of the ithinclass variation concentrates on several largest eigenvectors in. Since represents the energy distribution of ithin-class variation, the hitening process effectively reduces the ithin-class variation. his process is essentially the same as the Mahalanobis distance in F. hen, C is applied on the hitened class centers to find the dominant direction that best separates the class centers. his process further reduces the noise and compacts the discriminative features onto a small number of principal components. It is also easy to see that the null space of S is S essentially identical to F. LD in the null space of extracts the discriminative features in F. F has removed most of the ithin-class variation, and LD further separates the class centers by C.. pply C to S, and compute the principal subspace F, ith K eigenvectors V 1,, K, and its complementary subspace F. Estimate the average eigenvalue in F. 3. ll of the class centers are proected to F and normalized by the K eigenvalues. S b is transformed to 1/ 1/ V SbV Kb, () here is the eigenvalue matrix for F. pply C to K b, and compute l eigenvectors ith the largest eigenvalues. he l discriminative vectors in F are defined as W V 1/. (3) 4. roect all the class centers to F and compute the reconstruction difference as r here m, VV I VV (4) m L 1, is the class centers matrix. In fact, r is the proection of into F. In F, S b is transformed to roceedings of the 4 IEEE Computer Society Conference on Computer Vision and attern Recognition (CVR 4) /4 $. 4 IEEE

5 C Kb I VV S I VV. (5) b C Compute l C eigenvectors C of K b ith the largest eigenvalues. he l C discriminative vectors in the F are defined as WC I VV C. (6) t the recognition stage, 1. ll the face class centers m in the gallery and the probe face data x t are proected to the discriminant vectors in F and F to get, a W m C a WC m at W xt C at WC xt, (7). (8), (9). (3). Class is found to minimize the distance measure C d a a a a /. (31) t his dual-space LD algorithm has several advantages over conventional LD algorithms. First, it takes advantage of all the discriminative information in the full face space hile other LD approaches all lose some discriminative information one ay or the other. In the principal and null subspaces of S, LD vectors can be computed using different criterions, C W SbW W arg max W S W, (3) W S W W C arg maxwc SbWC. (33) WC S WC oth of the to sets of features contain discriminative information for recognition. Hoever, the to subspaces have different metric scales. he principal subspace of S has been hitened, so proection vectors in W are not orthornormal. It is not suitable, at least not optimal, to combine the distances in the to subspaces directly. In dual-space LD, the null space of S is also hitened by the average eigenvalues. In Eq.(34), t a a t and C a C a t / are computed under the same metric scale measure, and the distances in the to subspaces are similarly hitened by the eigenvalue spectrum of S. he second advantage of the ne algorithm is that it is more stable and insensitive to noise than existing LD approaches. Since eigenvectors of S ith very small eigenvalues are unstable and sensitive to small perturbation, e avoid computing these unstable eigenvectors by grouping them into F. In addition, the eigenvalue spectrum of S is better estimated, thus it avoids hitening ith very small eigenvalues. Finally, this approach can be vieed as an improvement to the Mahalanobis likelihood computed from the subspace estimation. It is more effective for classification and more efficient in computation. esides effective reducing the ithin-class variation like the Mahalanobis distance, it further distances class centers, and removes some noise disturbance by compacting the discriminative features. It is also much faster than the Mahalanobis distance. Computing the reconstruction error x in Eq. () is expensive. Its computational cost is comparable to the correlation beteen the to original high dimensional face data vectors. Our approach only needs to compute the distances beteen vectors of l dimensions. l C 4. Experiment In this section, e apply the dual-space LD to face recognition and compare ith conventional approaches. by experiments on the data sets from the FERE face database [9]. he high dimensional image intensity vector is used as input pattern for classification. ll the 1195 people from the FERE Fa/Fb data set are used in the experiment. here are to face images for each person. 495 people are used for training, and the remaining 7 people are used for testing. For each testing people, one face image is in the gallery and the other is for probe. First, e compare the dual-space LD ith the Mahalanobis distance estimated by the probabilistic visual model. Figure 3 reports their op 1 recognition accuracies ith different feature numbers. he feature number for the dual-space LD is the summation of discriminant feature numbers in F and F. he feature number for the Mahalanobis distance is the dimensionality (K) of F. hree distance measures, DIFS, DFFS and DIFS+DFFS, for the Mahalanobis distance, are evaluated. Notice that only for DIFS. the feature number relates to computation cost. Even for a small K, the computation cost of DFFS and DIFS+DFFS are very high, since they need to compute the reconstruction error roceedings of the 4 IEEE Computer Society Conference on Computer Vision and attern Recognition (CVR 4) /4 $. 4 IEEE

6 the unified subspace analysis [1] and face sketch recognition [11], and further compare ith the random sampling LD, hich combine the to LD subspaces at the decision level [1]. cknoledgement he ork described in this paper as fully supported by grants from the Research Grants Council of the Hong Kong Special dministrative Region (roect no. CUHK 419/1E and CUHK 44/3E). Figure 3. Recognition accuracy comparison beteen the Dual-Space LD and the Mahalanobis distances. Figure 4 ccumulative matching scores for the dualspace LD, Mahanobis distance, Fisherface, LD in null space, and direct LD on the FERE database. x. Dual-space LD outperforms DIFS significantly at the same computational cost. It achieves over 96% recognition accuracy. he best performance for the three Mahalanobis distance measures is about 93%. his is over 4% reduction in recognition error rate. he dual-space LD also outperforms conventional LD approaches. Figure 4 reports the accumulative matching scores comparing the Dual-Space LD ith Fisherface, hich uses to stage C+LD, LD in the null space of S, and direct LD. he novel method has reduced 5% error rate than conventional approaches. 5. Conclusion In this paper, a dual-space LD approach for high dimensional data classification is proposed. Compared ith existing LD approaches, it is more stable and makes use of all the discriminative information in the face space. Experiments on the FERE face database have shon that the method is much more effective that existing LD methods. In future study, e ill investigate the application of this dual-space approach to Reference [1]. N. elhumeur, J. Hespanda, and D. Kiregeman, Eigenfaces vs. Fisherfaces: Recognition Using Class Specific Linear roection, IEEE rans. on MI, Vol. 19, No. 7, pp , July [] L. Chen, H. Liao, M. Ko, J. Lin, and G. Yu, Ne LD- ased Face Recognition System Which can Solve the Small Sample Size roblem, Journal of attern Recognition, Vol. 33, No. 1, pp , Oct.. [3] H. Yu and J. Yang, Direct LD lgorithm for High- Dimensional Data - ith pplication to Face Recognition, attern Recognition, Vol. 34, pp. 67-7, 1. [4]. Moghaddam and. entland, robabilistic Visual Learning for Obect Representation, IEEE rans. on MI, Vol. 19, No. 7, pp , July, [5] W. Hang and J. Weng, "Hierarchical Discriminant Regression," IEEE rans. on MI, Vol., No. 11, pp , Nov.. [6] C. Liu and H. Wechsler, Enhanced Fisher Linear Discriminant Models for Face Recognition, roceedings of ICR, Vol., pp , [7] K. Fukunnaga, Introduction to Statistical attern Recognition, cademic ress, second edition, [8] G. W. Steart, Introduction to Matrix Computations, cademic ress, Ne York, [9]. J. hillips, H. Moon, S.. Rizvi, and. J. Rauss, "he FERE Evaluation," in Face Recognition: From heory to pplications, H. Wechsler,.J. hillips, V. ruce, F.F. Soulie, and.s. Huang, Eds., erlin: Springer-Verlag, [1] X. Wang and X. ang, Unified Subspace nalysis for Face Recognition, in roceedings of ICCV, pp , Nice, France, Oct. 3. [11] X. ang, and X. Wang, Face Sketch Synthesis and Recognition, in roceedings of ICCV, pp , Nice, France, Oct. 3. [1] X. Wang and X. ang, Random Sampling LD for Face Recognition, in roceedings of CVR, Washington D.C., US, 4. roceedings of the 4 IEEE Computer Society Conference on Computer Vision and attern Recognition (CVR 4) /4 $. 4 IEEE