- Email: [email protected]
L1-norm plus L2-norm sparse parameter for image recognition
Optik 126 (2015) 4078–4082
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
L1-norm plus L2-norm sparse parameter for image recognition Qingxiang Feng a , Qi Zhu b,∗ , Lin-Lin Tang a , Jeng-Shyang Pan a a b
Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 27 August 2014 Accepted 25 August 2015 Keywords: Sparse representation classification Nearest neighbor classifier Simple-fast representation-based classifier Image recognition
a b s t r a c t In this paper, a new classifier based on both sparse representation classification (SRC) and simple-fast representation-based classifier (SFR), called L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier, is proposed for image recognition. The SRC classifier only utilizes the distance between test image vector and the L1-norm sparse representation for classification. The SFR classifier uses the test image vector and the nearest image vector of class subspace to classify the test sample. However, the L1L2-SP classifier firstly computes the L1-norm sparse parameter and L1-norm sparse representation, which are used to compose a novel robust global space. Then the L1L2-SP utilizes the novel global space to get the L2-norm sparse parameter. In the last, L1L2-SP uses the sum of L1-norm sparse parameter and L2-norm sparse parameter for classification. A mass number of experiments on coil100 object database, eth80 object database, Oxford flower 17 database and FERET face database are used to assess the proposed classifier. The experimental results prove that the proposed approach achieves better recognition rate than the SRC classifier, SFR classifier, and several other classifiers. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction A typical face recognition system contains two stages: feature extraction and classification. For the feature extraction stage, there are a lots of methods, such as PCA [1–3], LDA [4,5], ICA [6] and laplacianfaces [7,8]. For the classification, there are also lots of methods. Nearest neighbor (NN) [9] is the one of the important classifier in pattern recognition area. However, the number of training-samples is usually very small, which makes high recognition rate hard to achieve. So nearest feature line (NFL) [10] was proposed for face recognition by Li et al. in 1999. NFL tries to reinforce the representational capacity of a sample set of limited size by using the line passing through each pair of the samples belonging to the same class. The authors of ref. [10] explain that a feature line provides information about the possible linear variants of two sample points. After the NFL, some other improved methods are proposed, such as Feng et al. proposed the nearest feature center (NFC) [11] classifier, Gao et al. proposed the center-based nearest neighbor (CNN) [12] classifier, and Han et al. proposed the shortest feature line segment (SFLS) [13] classifier. Recently, researcher proposed the sparse representation classification (SRC) [14,15]. SRC uses the combination of all the training samples and the test sample to solve the L1-norm minimum
∗ Corresponding author. Tel.: +86 13951981021. E-mail address: [email protected] (Q. Zhu). http://dx.doi.org/10.1016/j.ijleo.2015.08.172 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
problem. Then SRC gains the sparse representation by using the L1norm sparse parameter. After the SRC, some improved classifiers are proposed [16–26]. These improved classifiers could be classified into two kinds. The first kind utilizes the novel representations of each class, such as Zhang et al. proposed the collaborative representation classification (CRC) [16], which uses the collaborative representation instead of the sparse representation. Xu et al. proposed the two-phase test sample sparse representation (TPSR) [17] classifier, which utilizes the first phase to choose some training samples and gains the novel sparse representation in the second phase. Xu et al. proposed simple-fast representation based (SFR) classifier [18], which uses the nearest training sample to form the global model and solve the linear regression problem for classification. Yang et al. proposed regularized robust coding (RRC) [19] and relaxed collaborative representation (RCR) [20]. The second kind uses the sparse representation to make the subspace discriminant analysis learning. E.g., ref. [24] uses the discriminant projection to make feature selection, ref. [25] utilizes the tensor discriminant to make feature selection, and ref. [26] uses the sparse eigenface to make feature selection. Motivated by the SRC and SFR classifiers, the L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier is proposed for face recognition in this paper. The SRC classifier only utilizes the distance between test image vector and the L1-norm sparse representation for classification. The SFR classifier uses the test image vector and the nearest image vector of class subspace to classify the test sample. However, the L1L2-SP classifier firstly computes
Q. Feng et al. / Optik 126 (2015) 4078–4082
4079
the L1-norm sparse parameter and L1-norm sparse representation, which will be used to compose a novel robust global space. Then the L1L2-SP utilizes the novel global space to get the L2-norm sparse parameter. In the last, L1L2-SP will use the sum of L1-norm sparse parameter and L2-norm sparse parameter for classification. A mass number of experiments on coil100 object database, eth80 object database, Oxford flower 17 database and FERET face database are used to assess the proposed classifier. The experimental results prove that the proposed approach achieves better recognition rate than the SRC classifier, SFR classifier, RRC classifiers, RCR classifier, CRC classifier, TPSR classifier and CFSR classifier.
The rule of the SFR in favor of the class with the minimum distance as
2. Reviews of SRC and SFR classifiers
3.1. L1-norm plus L2-norm sparse parameter (L1L2-SP)
Let Y = {yic , c = 1, 2, . . ., M, i = 1, 2, . . ., Nc } ⊂ RD denote the prototype set, where yic is the ith prototype belonging to cth class, M is the number of classes, and Nc is the number of prototypes belonging to the cth class. Let each training image be with a × b pixels, each image is transformed to column vector such as yic ∈
Suppose that each training image is an order a × b pixels. Each gallery image is transformed to column vector such that yic ∈
Ra×b → xic ∈ Rq×1 , where q = a × b.
2.1. Sparse representation-based classification (SRC) classifier Develops a class-specific model Xc by stacking the qdimensional image vectors c
X =
[ x1c
x2c
...
c xN c
]∈R
q×Nc
.
(1)
Suppose that we have M classes of subjects, we can collect the entire class-specific model Xc defined in (1) to form the complete data model as X = [ X1
X2
...
XM ] ∈ Rq×MNc .
(2)
For the spare representation-based classification (SRC), we first normalize the columns of X to become unit vector. Then, we can solve the L1-norm minimization problem as: g = arg ming ||g||1 subject to Xg = x.
c min dSFR (x),
(10)
3. The proposed classifier Motivated by the SRC classifier and SFR classifier, the proposed classifier called L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier is given in this section.
Ra×b → xic ∈ Rq×1 , where q = a × b. The L1L2-SP firstly constitutes a class-model Xc by stacking the q-dimensional column vectors.
X c = [ x1c
x2c
c .... xN ] ∈ Rq×Nc , c
(11)
Suppose that we have M classes of subjects, we can collect the entire class-specific model Xc defined in (11) to form the all-classes-model as X = [ X1
X2
...
XM ] ∈ Rq×MNc .
(12)
For the spare representation-based classification (SRC), we first normalize the columns of X to become unit vector. Then, we can solve the L1-norm minimization problem as: g = arg ming ||g||1 subject to Xg = x.
(13)
Let ˇ ∈ RM×1 be the sum of L1-norm sparse parameter of each class, which can be calculated as ˇc =
(3)
Compute the regularized residuals rc as
c = 1, 2, . . ., M.
c∗
Nc
gic ,
c = 1, 2, . . ., M.
(14)
i=1
And the SRC classification rule in favor of the class with the minimum distance can be expressed by
After gaining the L1-norm sparse parameter, we need to compute the sparse representation. If x belongs to the cth class, it may be represented as a sparse combination of the training images from the same class. Let xc be the sparse representation vector of x on the cth class, which can be computed as
minr c , c = 1, 2, . . ., M.
x ≈ xc = X c g c ,
c
c c
r = ||x − X gˆ ||.
(4)
(5)
c∗
By using the M sparse representation vectors, we can constitute the global-model G, which can be computed by
2.2. Simple-fast representation based (SFR) classifier Give a test image vector x, the simple-fast representation based (SFR) classifier consists of two processes. In the first process, SFR chooses the nearest training sample from each class and uses the M nearest training samples to constitute the novel global model as S = [ s1
s2
...
sM ] ∈ Rq×M .
(6)
Then SFR solve the L2 -norm minimization problem as r = (S T S)
−1 T
S x,
r ∈ RM×1 .
(7)
c as Compute the predicted vector xSFR c xSFR
c c
=s r ,
c xSFR
∈R
M×1
.
G = [ x1
x2
....
(8)
(9)
xM ] ∈ Rq×M .
(16)
Let ∈ RM×1 be the vector of L2-norm sparse parameter of the novel global-model, which can be calculated as = (GT G)
−1
GT x.
(17)
Let s ∈ RM×1 be the score vector of each class, which can be computed as sc = ˇ c + c ,
The SFR classifier computes the distance measure between the predicted vector xc and the original test vector x as c c (x) = ||x − xSFR ||. dSFR
(15)
c = 1, 2, . . ., M.
(18)
And the rule of the L1L2-SRC classifier is in favor of the class with the max value as max sc , c∗
c = 1, 2, . . ., M.
(19)
4080
Q. Feng et al. / Optik 126 (2015) 4078–4082
Fig. 2. Some object images of coil100 object database.
Table 1 The recognition rate of several classifiers on coil100 object database with ‘first N’ scheme.
Fig. 1. The relationship of L1L2-SP, SRC and SFR.
3.2. Classification procedure of L1L2-SP classifier Algorithm: L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier. Inputs: The entire training sample xic , c = 1, 2, . . ., M, i = 1, 2, . . ., Nc and a test image vector x ∈ Rq×1 . Output: Class of x. 1. Constitute the all-classes-model by formula (11) and (12). 2. Use the all-classes-model to solve the L1-norm minimum problem and gain the sparse parameter by formula (13). 3. Compute the sum of L1-norm sparse parameter of each class by formula (14). 4. Utilize the sparse parameter to gain M sparse representations by formula (15). 5. Constitute the novel global-model by using M sparse representations in formula (16). 6. Calculate L2-norm sparse parameter of the novel global-model by formula (17). 7. Compute the score vector of each class by formula (18). 8. The classification rule of L1L2-SRC is in favor of formula (19).
Classifier
4
5
6
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
64.87% 70.02% 59.88% 56.00% 66.29% 69.43% 58.25% 72.29%
67.43% 71.57% 61.29% 57.43% 71.00% 75.00% 60.57% 78.17%
73.00% 76.83% 65.83% 63.00% 81.40% 85.00% 66.83% 87.20%
68.43% 72.81% 62.33% 58.81% 72.90% 76.48% 61.88% 79.22%
the effectiveness of the new classifier. It is noted that the source codes of CRC classifier, RCR classifier, and RRC classifier are from the URL (http://www4.comp.polyu.edu.hk/∼cslzhang/papers.htm) and the source codes of CFSR classifier, TPSR classifier are from the URL (http://www.yongxu.org/lunwen.html). 4.1. Object recognition on coil100 object database
The SRC only uses the L1-norm sparse representation to classify the query sample. However, the L1L2-SP uses the L1-norm sparse representation to constitute the new robust-global-model. Then the L1L2-SP utilizes the novel global space to get the L2-norm sparse parameter. Finally, L1L2-SP will use the sum of L1-norm sparse parameter and L2-norm sparse parameter for classification. The L1L2-SP classifier can be seen as viewed sparse representation. The SFR classifier utilizes the nearest sample vector of class subspace to form the novel global-model. What is more, the SFR classifier uses the novel global-model to compute the distance between the test sample and the predicted vector for classification. So, the mainly difference between the L1L2-SP classifier and the SFR classifier contains two points. The first point is that L1L2-SP classifier uses the L1-norm sparse representation vector instead of the nearest sample vector. The second point is that L1L2-SP classifier uses the L1-norm plus L2-norm sparse parameter instead of the novel predicted vector. The detailed relationship of L1L2-SP, SRC and SFR is described as Fig. 1.
The Coil-100 data set [27] was widely used as an objectrecognition benchmark. In this data set, there are 100 objects and each object has 72 different views (images) that are taken every 5◦ around an axis passing through the object. Each image is a 128 × 128 color one with R, G, B channels. We use only a limited number of views per objects for experiments. In our experiments, 12 different views per object (0◦ , 30◦ , 60◦ , 90◦ , 120◦ , 150◦ , 180◦ , 210◦ , 240◦ , 270◦ , 300◦ and 330◦ ) were used, shown in Fig. 3. So the subset of Coil-100 data set contains 1200 images, and all images in subset of Coil-100 database were manually cropped into a 32 × 32 color one with R, G, B channels. Some object images of coil100 object database are shown in Fig. 2. In the first experiment, we utilize the “first N” scheme on the coil100 object database. The result is shown in Table 1. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 17.34%, 2.74%, 6.32%, 20.41%, 16.89%, 6.41% and 10.79%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set.
4. Experimental results
4.2. Object recognition on eth80 object database
The classification performance of L1L2-SP classifier is compared with those of SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier in this section. In our experiments, four famous databases and “first N” scheme are taken for comparisons: the first N images of each class are used as the prototype samples. The remaining images of test database are used as test samples. The recognition rate (RR) is used to assess
In the eth80-cropped-close128 object database [28,29], all images are cropped, so that they contain only the object without any border area. In addition, they are rescaled to a size of 128 × 128 pixels. Again, the scale is left the same for all images of the same object. This dataset is useful when no derivatives need to be calculated. In our experimental, all images are resized to 32 × 32 gray images. Some images of eth80 object database are shown in Fig. 3.
3.3. The relationship of L1L2-SP, SRC and SFR
Q. Feng et al. / Optik 126 (2015) 4078–4082
4081
Table 3 The recognition rate of several classifiers on Oxford 17 subjects database with ‘first N’ scheme.
Fig. 3. Some sampled images of eth80 object database.
Classifier
4
5
6
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
28.87% 28.56% 22.14% 18.81% 24.54% 26.94% 25.70% 29.58%
30.59% 28.63% 22.28% 20.78% 25.41% 29.02% 25.80% 31.45%
32.03% 29.41% 24.09% 21.30% 28.14% 30.29% 28.46% 32.27%
30.50% 28.87% 22.84% 20.30% 26.03% 28.75% 26.65% 31.10%
Table 2 The recognition rate of several classifiers on eth80 object database with ‘first N’ scheme. Classifier
4
5
6
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
10.51% 12.57% 8.18% 8.95% 12.80% 14.97% 8.34% 15.85%
11.25% 13.61% 10.28% 9.93% 14.41% 16.22% 9.97% 17.16%
11.11% 15.43% 11.54% 10.71% 14.11% 17.21% 9.46% 18.14%
10.96% 13.87% 10.00% 9.86% 13.77% 16.13% 9.26% 17.05%
Fig. 5. Some face images of FERET face database. Table 4 The recognition rate of several classifiers on FERET face database with ‘first N’ scheme. Classifier
3
4
5
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
46.88% 54.25% 42.88% 45.13% 48.63% 56.63% 42.88% 57.88%
55.17% 65.17% 53.67% 51.00% 54.33% 72.50% 48.50% 74.17%
64.75% 75.25% 70.25% 59.75% 68.00% 79.00% 62.25% 81.50%
55.60% 64.89% 55.60% 51.96% 56.99% 69.38% 51.21% 71.18%
In the third experiment, we utilize the “first N” scheme on the oxford flower 17 database. The result is shown in Table 3. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 4.45%, 2.35%, 5.07%, 10.80%, 8.26%, 2.23% and 0.60%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set. 4.4. Face recognition on FERET face database
Fig. 4. Some flower images of Oxford 17 subjects database.
In the second experiment, we utilize the “first N” scheme on the eth80 object database. The result is shown in Table 2. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 7.79%, 0.92%, 3.28%, 7.19%, 7.05%, 3.18% and 6.09%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set. 4.3. Flower recognition on Oxford flower database Oxford flower 17 database [30,31] has created a 17 category flower dataset with 80 images for each class. The flowers chosen are some common flowers in the UK. The images have large scale, pose and light variations and there are also classes with large variations of images within the class and close similarity to other classes. Fig. 4 shows some selected flower images of Oxford flower 17 database.
We selected a subset of 1400 face images of 200 subjects from the FERET face database [32]. Each subject has 7 face images of different poses (whose names are marked with two-character strings: “ba,” “bj,” “bk,” “be,” “bf,” “bd,” and “bg” in the database). Each image is downsampled to a 40 × 40 gray image. Some face images of FERET face database are shown in Fig. 5. In the fourth experiment, we utilize the “first N” scheme on the FERET database. The result is shown in Table 4. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 19.97%, 1.80%, 14.19%, 19.22%, 15.58%, 6.29% and 15.58%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set. 5. Conclusion In this paper, a novel classifier called L1-norm plus L2-norm sparse parameter (L1L2-SP) is proposed for image recognition. L1L2-SP classifier uses the L1-norm sparse representation of each class to form the new robust global space. Then L1L2-SP utilizes the
4082
Q. Feng et al. / Optik 126 (2015) 4078–4082
robust global space to get the sum of L2-norm sparse parameter of each class. Finally, L1L2-SP computes the score of each class by using the L1-norm sparse parameter and L2-norm sparse parameter. The proposed classifier achieves the better recognition rate than SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier. The experimental results on four databases confirm the effectiveness of the proposed algorithm. Acknowledgements This article is partly supported by Nature Science Foundation of China under grants (nos. 61375021, 61501230), China Postdoctoral Science Foundation funded project (no. 2015M570446), Jiangsu Planned Projects for Postdoctoral Research Funds (no. 1402047B), Open Project Program of Key Laboratory of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education (no. JYB201503), and Natural Science Foundation of Jiangsu Province (nos. BK20131365, BK20150751). References [1] M. Turk, A. Pentland, Eigenfaces for recognition, J. Cogn. Neurosci. 3 (1) (1991) 71–86. [2] Q. Zhu, Y. Xu, Multi-directional two-dimensional PCA with matching score level fusion for face recognition, Neural Comput. Appl. 23 (1) (2013) 169–174. [3] P. Belhumeur, J. Hespanha, D. Kriegman, Eigenfaces vs. fisherfaces: recognition using class specific linear projection, IEEE Trans. Pattern Anal. Mach. Intell. 19 (7) (1997) 711–720. [4] Yong Xu, Qi Zhu, Zizhu Fan, Yaowu Wang, Jeng-Shyang Pan, From the idea of “sparse representation” to a representation-based transformation method for feature extraction, Neurocomputing 113 (2013) 168–176. [5] C.-Y. Chang, C.-W. Chang, C.-Y. Hsieh, Applications of block linear discriminant analysis for face recognition, J. Inform. Hiding Multimed. Sign. Process. 2 (3) (2011) 259–269. [6] M.S. Bartlett, J.R. Movellan, T.J. Sejnowski, Face recognition by independent component analysis, IEEE Trans. Neural Netw. 13 (6) (2002) 1450–1464. [7] X. He, S. Yan, Y. Hu, P. Niyogi, H.J. Zhang, Face recognition using laplacianfaces, IEEE Trans. Pattern Anal. Mach. Intell. 27 (3) (2005) 1–13. [8] Yong Xu, Aini Zhong, Jian Yang, David Zhang, LPP solution schemes for use with face recognition, Pattern Recognit. 43 (12) (2010) 4165–4176. [9] T.M. Cover, P.E. Hart, Nearest neighbor pattern classification, IEEE Trans. Inform. Theory 13 (1) (1967) 21–27. [10] S.Z. Li, J. Lu, Face recognition using the nearest feature line method, IEEE Trans. Neural Netw. 10 (2) (1999) 439–443. [11] Q. Feng, J.S. Pan, L. Yan, Nearest feature centre classifier for face recognition, Electron. Lett. 48 (August (18)) (2012) 1120–1122.
[12] Q.B. Gao, Z.Z. Wang, Center-based nearest neighbor classifier, Pattern Recognit. 40 (January (1)) (2006) 346–349. [13] D.Q. Han, C.Z. Han, Y. Yang, A novel classifier based on shortest feature line segment, Pattern Recognit. Lett. 32 (3) (2011) 485–493. [14] J. Wright, Y. Ma, J. Mairal, G. Sapiro, T.S. Huang, S. Yan, Sparse representation for computer vision and pattern recognition, Proc. IEEE 98 (June (6)) (2010) 1031–1044. [15] J. Wright, A.Y. Yang, A. Ganesh, S.S. Sastry, Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Anal. Mach. Intell. 31 (February (2)) (2009) 210–227. [16] L. Zhang, M. Yang, X. Feng, Sparse representation or collaborative representation: which helps face recognition? Proc. IEEE Int’l Conf. Comput. Vis. (November) (2011) 471–478. [17] Y. Xu, D. Zhang, J. Yang, J.-Y. Yang, A two-phase test sample sparse representation method for use with face recognition, IEEE Trans. Circuits Syst. Video Technol. 21 (9) (2011) 1255–1262. [18] Y. Xu, Q. Zhu, A simple and fast representation-based face recognition method, Neural Comput. Appl. 22 (7) (2013) 1543–1549. [19] M. Yang, L. Zhang, J. Yang, D. Zhang, Regularized robust coding for face recognition, IEEE Trans. Image Process. 22 (5) (2013) 1753–1766. [20] M. Yang, L. Zhang, D. Zhang, S. Wang, Relaxed collaborative representation for pattern classification, in: CVPR, 2012. [21] Y. Xu, Q. Zhu, Z. Fan, D. Zhang, J. Mi, Z. Lai, Using the idea of the sparse representation to perform coarse to fine face recognition, Inform. Sci. 238 (2013) 138–148. [22] Y. Xu, Q. Zhu, Z. Fan, M. Qiu, Y. Chen, H. Liu, Coarse to fine K nearest neighbor classifier, Pattern Recognit. Lett. 34 (2013) 980–986. [23] Y. Xu, X. Zhu, Z. Li, G. Liu, Y. Lu, H. Liu, Using the original and ‘symmetrical face’ training samples to perform representation based two-step face recognition, Pattern Recognit. vol.46 (2013) 1151–1158. [24] J. Yang, D. Chu, L. Zhang, Y. Xu, J. Yang, Sparse representation classifier steered discriminative projection with applications to face recognition, IEEE Trans. Neural Netw. Learn. Syst. 24 (July (7)) (2013) 1023–1035. [25] Z. Lai, Y. Xu, J. Yang, J. Tang, D. Zhang, Sparse tensor discriminant analysis, IEEE Trans. Image Process. 22 (10) (2013) 3904–3905. [26] Z. Lai, W.K. Wong, Z. Jin, J. Yang, Y. Xu, Sparse approximation to the eigensubspace for discrimination, IEEE Trans. Neural Netw. Learn. Syst. 23 (12) (2012) 1948–1960. [27] S.A. Nene, S.K. Nayar, H. Murase, Columbia object image library (COIL-100), Technical Report CUCS-006-96, Columbia University, 1996. [28] http://www.d2.mpi-inf.mpg.de/Datasets/ETH80. [29] B. Leibe, B. Schiele, Analyzing appearance and contour based methods for object categorization, in: Proc. of International Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003. [30] http://www.robots.ox.ac.uk/∼vgg/data/flowers/17/index.html. [31] M.-E. Nilsback, A.A. Zisserman, Visual vocabulary for flower classification, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2006. [32] P.J. Phillips, H. Moon, P.J. Rauss, S. Rizvi, The FERET evaluation methodology for face recognition algorithms, IEEE Trans. Pattern Anal. Mach. Intell. 22 (October (10)) (2000).
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
L1-norm plus L2-norm sparse parameter for image recognition Qingxiang Feng a , Qi Zhu b,∗ , Lin-Lin Tang a , Jeng-Shyang Pan a a b
Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
a r t i c l e
i n f o
Article history: Received 27 August 2014 Accepted 25 August 2015 Keywords: Sparse representation classification Nearest neighbor classifier Simple-fast representation-based classifier Image recognition
a b s t r a c t In this paper, a new classifier based on both sparse representation classification (SRC) and simple-fast representation-based classifier (SFR), called L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier, is proposed for image recognition. The SRC classifier only utilizes the distance between test image vector and the L1-norm sparse representation for classification. The SFR classifier uses the test image vector and the nearest image vector of class subspace to classify the test sample. However, the L1L2-SP classifier firstly computes the L1-norm sparse parameter and L1-norm sparse representation, which are used to compose a novel robust global space. Then the L1L2-SP utilizes the novel global space to get the L2-norm sparse parameter. In the last, L1L2-SP uses the sum of L1-norm sparse parameter and L2-norm sparse parameter for classification. A mass number of experiments on coil100 object database, eth80 object database, Oxford flower 17 database and FERET face database are used to assess the proposed classifier. The experimental results prove that the proposed approach achieves better recognition rate than the SRC classifier, SFR classifier, and several other classifiers. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction A typical face recognition system contains two stages: feature extraction and classification. For the feature extraction stage, there are a lots of methods, such as PCA [1–3], LDA [4,5], ICA [6] and laplacianfaces [7,8]. For the classification, there are also lots of methods. Nearest neighbor (NN) [9] is the one of the important classifier in pattern recognition area. However, the number of training-samples is usually very small, which makes high recognition rate hard to achieve. So nearest feature line (NFL) [10] was proposed for face recognition by Li et al. in 1999. NFL tries to reinforce the representational capacity of a sample set of limited size by using the line passing through each pair of the samples belonging to the same class. The authors of ref. [10] explain that a feature line provides information about the possible linear variants of two sample points. After the NFL, some other improved methods are proposed, such as Feng et al. proposed the nearest feature center (NFC) [11] classifier, Gao et al. proposed the center-based nearest neighbor (CNN) [12] classifier, and Han et al. proposed the shortest feature line segment (SFLS) [13] classifier. Recently, researcher proposed the sparse representation classification (SRC) [14,15]. SRC uses the combination of all the training samples and the test sample to solve the L1-norm minimum
∗ Corresponding author. Tel.: +86 13951981021. E-mail address: [email protected] (Q. Zhu). http://dx.doi.org/10.1016/j.ijleo.2015.08.172 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
problem. Then SRC gains the sparse representation by using the L1norm sparse parameter. After the SRC, some improved classifiers are proposed [16–26]. These improved classifiers could be classified into two kinds. The first kind utilizes the novel representations of each class, such as Zhang et al. proposed the collaborative representation classification (CRC) [16], which uses the collaborative representation instead of the sparse representation. Xu et al. proposed the two-phase test sample sparse representation (TPSR) [17] classifier, which utilizes the first phase to choose some training samples and gains the novel sparse representation in the second phase. Xu et al. proposed simple-fast representation based (SFR) classifier [18], which uses the nearest training sample to form the global model and solve the linear regression problem for classification. Yang et al. proposed regularized robust coding (RRC) [19] and relaxed collaborative representation (RCR) [20]. The second kind uses the sparse representation to make the subspace discriminant analysis learning. E.g., ref. [24] uses the discriminant projection to make feature selection, ref. [25] utilizes the tensor discriminant to make feature selection, and ref. [26] uses the sparse eigenface to make feature selection. Motivated by the SRC and SFR classifiers, the L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier is proposed for face recognition in this paper. The SRC classifier only utilizes the distance between test image vector and the L1-norm sparse representation for classification. The SFR classifier uses the test image vector and the nearest image vector of class subspace to classify the test sample. However, the L1L2-SP classifier firstly computes
Q. Feng et al. / Optik 126 (2015) 4078–4082
4079
the L1-norm sparse parameter and L1-norm sparse representation, which will be used to compose a novel robust global space. Then the L1L2-SP utilizes the novel global space to get the L2-norm sparse parameter. In the last, L1L2-SP will use the sum of L1-norm sparse parameter and L2-norm sparse parameter for classification. A mass number of experiments on coil100 object database, eth80 object database, Oxford flower 17 database and FERET face database are used to assess the proposed classifier. The experimental results prove that the proposed approach achieves better recognition rate than the SRC classifier, SFR classifier, RRC classifiers, RCR classifier, CRC classifier, TPSR classifier and CFSR classifier.
The rule of the SFR in favor of the class with the minimum distance as
2. Reviews of SRC and SFR classifiers
3.1. L1-norm plus L2-norm sparse parameter (L1L2-SP)
Let Y = {yic , c = 1, 2, . . ., M, i = 1, 2, . . ., Nc } ⊂ RD denote the prototype set, where yic is the ith prototype belonging to cth class, M is the number of classes, and Nc is the number of prototypes belonging to the cth class. Let each training image be with a × b pixels, each image is transformed to column vector such as yic ∈
Suppose that each training image is an order a × b pixels. Each gallery image is transformed to column vector such that yic ∈
Ra×b → xic ∈ Rq×1 , where q = a × b.
2.1. Sparse representation-based classification (SRC) classifier Develops a class-specific model Xc by stacking the qdimensional image vectors c
X =
[ x1c
x2c
...
c xN c
]∈R
q×Nc
.
(1)
Suppose that we have M classes of subjects, we can collect the entire class-specific model Xc defined in (1) to form the complete data model as X = [ X1
X2
...
XM ] ∈ Rq×MNc .
(2)
For the spare representation-based classification (SRC), we first normalize the columns of X to become unit vector. Then, we can solve the L1-norm minimization problem as: g = arg ming ||g||1 subject to Xg = x.
c min dSFR (x),
(10)
3. The proposed classifier Motivated by the SRC classifier and SFR classifier, the proposed classifier called L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier is given in this section.
Ra×b → xic ∈ Rq×1 , where q = a × b. The L1L2-SP firstly constitutes a class-model Xc by stacking the q-dimensional column vectors.
X c = [ x1c
x2c
c .... xN ] ∈ Rq×Nc , c
(11)
Suppose that we have M classes of subjects, we can collect the entire class-specific model Xc defined in (11) to form the all-classes-model as X = [ X1
X2
...
XM ] ∈ Rq×MNc .
(12)
For the spare representation-based classification (SRC), we first normalize the columns of X to become unit vector. Then, we can solve the L1-norm minimization problem as: g = arg ming ||g||1 subject to Xg = x.
(13)
Let ˇ ∈ RM×1 be the sum of L1-norm sparse parameter of each class, which can be calculated as ˇc =
(3)
Compute the regularized residuals rc as
c = 1, 2, . . ., M.
c∗
Nc
gic ,
c = 1, 2, . . ., M.
(14)
i=1
And the SRC classification rule in favor of the class with the minimum distance can be expressed by
After gaining the L1-norm sparse parameter, we need to compute the sparse representation. If x belongs to the cth class, it may be represented as a sparse combination of the training images from the same class. Let xc be the sparse representation vector of x on the cth class, which can be computed as
minr c , c = 1, 2, . . ., M.
x ≈ xc = X c g c ,
c
c c
r = ||x − X gˆ ||.
(4)
(5)
c∗
By using the M sparse representation vectors, we can constitute the global-model G, which can be computed by
2.2. Simple-fast representation based (SFR) classifier Give a test image vector x, the simple-fast representation based (SFR) classifier consists of two processes. In the first process, SFR chooses the nearest training sample from each class and uses the M nearest training samples to constitute the novel global model as S = [ s1
s2
...
sM ] ∈ Rq×M .
(6)
Then SFR solve the L2 -norm minimization problem as r = (S T S)
−1 T
S x,
r ∈ RM×1 .
(7)
c as Compute the predicted vector xSFR c xSFR
c c
=s r ,
c xSFR
∈R
M×1
.
G = [ x1
x2
....
(8)
(9)
xM ] ∈ Rq×M .
(16)
Let ∈ RM×1 be the vector of L2-norm sparse parameter of the novel global-model, which can be calculated as = (GT G)
−1
GT x.
(17)
Let s ∈ RM×1 be the score vector of each class, which can be computed as sc = ˇ c + c ,
The SFR classifier computes the distance measure between the predicted vector xc and the original test vector x as c c (x) = ||x − xSFR ||. dSFR
(15)
c = 1, 2, . . ., M.
(18)
And the rule of the L1L2-SRC classifier is in favor of the class with the max value as max sc , c∗
c = 1, 2, . . ., M.
(19)
4080
Q. Feng et al. / Optik 126 (2015) 4078–4082
Fig. 2. Some object images of coil100 object database.
Table 1 The recognition rate of several classifiers on coil100 object database with ‘first N’ scheme.
Fig. 1. The relationship of L1L2-SP, SRC and SFR.
3.2. Classification procedure of L1L2-SP classifier Algorithm: L1-norm plus L2-norm sparse parameter (L1L2-SP) classifier. Inputs: The entire training sample xic , c = 1, 2, . . ., M, i = 1, 2, . . ., Nc and a test image vector x ∈ Rq×1 . Output: Class of x. 1. Constitute the all-classes-model by formula (11) and (12). 2. Use the all-classes-model to solve the L1-norm minimum problem and gain the sparse parameter by formula (13). 3. Compute the sum of L1-norm sparse parameter of each class by formula (14). 4. Utilize the sparse parameter to gain M sparse representations by formula (15). 5. Constitute the novel global-model by using M sparse representations in formula (16). 6. Calculate L2-norm sparse parameter of the novel global-model by formula (17). 7. Compute the score vector of each class by formula (18). 8. The classification rule of L1L2-SRC is in favor of formula (19).
Classifier
4
5
6
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
64.87% 70.02% 59.88% 56.00% 66.29% 69.43% 58.25% 72.29%
67.43% 71.57% 61.29% 57.43% 71.00% 75.00% 60.57% 78.17%
73.00% 76.83% 65.83% 63.00% 81.40% 85.00% 66.83% 87.20%
68.43% 72.81% 62.33% 58.81% 72.90% 76.48% 61.88% 79.22%
the effectiveness of the new classifier. It is noted that the source codes of CRC classifier, RCR classifier, and RRC classifier are from the URL (http://www4.comp.polyu.edu.hk/∼cslzhang/papers.htm) and the source codes of CFSR classifier, TPSR classifier are from the URL (http://www.yongxu.org/lunwen.html). 4.1. Object recognition on coil100 object database
The SRC only uses the L1-norm sparse representation to classify the query sample. However, the L1L2-SP uses the L1-norm sparse representation to constitute the new robust-global-model. Then the L1L2-SP utilizes the novel global space to get the L2-norm sparse parameter. Finally, L1L2-SP will use the sum of L1-norm sparse parameter and L2-norm sparse parameter for classification. The L1L2-SP classifier can be seen as viewed sparse representation. The SFR classifier utilizes the nearest sample vector of class subspace to form the novel global-model. What is more, the SFR classifier uses the novel global-model to compute the distance between the test sample and the predicted vector for classification. So, the mainly difference between the L1L2-SP classifier and the SFR classifier contains two points. The first point is that L1L2-SP classifier uses the L1-norm sparse representation vector instead of the nearest sample vector. The second point is that L1L2-SP classifier uses the L1-norm plus L2-norm sparse parameter instead of the novel predicted vector. The detailed relationship of L1L2-SP, SRC and SFR is described as Fig. 1.
The Coil-100 data set [27] was widely used as an objectrecognition benchmark. In this data set, there are 100 objects and each object has 72 different views (images) that are taken every 5◦ around an axis passing through the object. Each image is a 128 × 128 color one with R, G, B channels. We use only a limited number of views per objects for experiments. In our experiments, 12 different views per object (0◦ , 30◦ , 60◦ , 90◦ , 120◦ , 150◦ , 180◦ , 210◦ , 240◦ , 270◦ , 300◦ and 330◦ ) were used, shown in Fig. 3. So the subset of Coil-100 data set contains 1200 images, and all images in subset of Coil-100 database were manually cropped into a 32 × 32 color one with R, G, B channels. Some object images of coil100 object database are shown in Fig. 2. In the first experiment, we utilize the “first N” scheme on the coil100 object database. The result is shown in Table 1. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 17.34%, 2.74%, 6.32%, 20.41%, 16.89%, 6.41% and 10.79%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set.
4. Experimental results
4.2. Object recognition on eth80 object database
The classification performance of L1L2-SP classifier is compared with those of SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier in this section. In our experiments, four famous databases and “first N” scheme are taken for comparisons: the first N images of each class are used as the prototype samples. The remaining images of test database are used as test samples. The recognition rate (RR) is used to assess
In the eth80-cropped-close128 object database [28,29], all images are cropped, so that they contain only the object without any border area. In addition, they are rescaled to a size of 128 × 128 pixels. Again, the scale is left the same for all images of the same object. This dataset is useful when no derivatives need to be calculated. In our experimental, all images are resized to 32 × 32 gray images. Some images of eth80 object database are shown in Fig. 3.
3.3. The relationship of L1L2-SP, SRC and SFR
Q. Feng et al. / Optik 126 (2015) 4078–4082
4081
Table 3 The recognition rate of several classifiers on Oxford 17 subjects database with ‘first N’ scheme.
Fig. 3. Some sampled images of eth80 object database.
Classifier
4
5
6
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
28.87% 28.56% 22.14% 18.81% 24.54% 26.94% 25.70% 29.58%
30.59% 28.63% 22.28% 20.78% 25.41% 29.02% 25.80% 31.45%
32.03% 29.41% 24.09% 21.30% 28.14% 30.29% 28.46% 32.27%
30.50% 28.87% 22.84% 20.30% 26.03% 28.75% 26.65% 31.10%
Table 2 The recognition rate of several classifiers on eth80 object database with ‘first N’ scheme. Classifier
4
5
6
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
10.51% 12.57% 8.18% 8.95% 12.80% 14.97% 8.34% 15.85%
11.25% 13.61% 10.28% 9.93% 14.41% 16.22% 9.97% 17.16%
11.11% 15.43% 11.54% 10.71% 14.11% 17.21% 9.46% 18.14%
10.96% 13.87% 10.00% 9.86% 13.77% 16.13% 9.26% 17.05%
Fig. 5. Some face images of FERET face database. Table 4 The recognition rate of several classifiers on FERET face database with ‘first N’ scheme. Classifier
3
4
5
ARR
TPSR CFSR RRC RCR CRC SRC SFR L1L2-SP
46.88% 54.25% 42.88% 45.13% 48.63% 56.63% 42.88% 57.88%
55.17% 65.17% 53.67% 51.00% 54.33% 72.50% 48.50% 74.17%
64.75% 75.25% 70.25% 59.75% 68.00% 79.00% 62.25% 81.50%
55.60% 64.89% 55.60% 51.96% 56.99% 69.38% 51.21% 71.18%
In the third experiment, we utilize the “first N” scheme on the oxford flower 17 database. The result is shown in Table 3. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 4.45%, 2.35%, 5.07%, 10.80%, 8.26%, 2.23% and 0.60%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set. 4.4. Face recognition on FERET face database
Fig. 4. Some flower images of Oxford 17 subjects database.
In the second experiment, we utilize the “first N” scheme on the eth80 object database. The result is shown in Table 2. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 7.79%, 0.92%, 3.28%, 7.19%, 7.05%, 3.18% and 6.09%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set. 4.3. Flower recognition on Oxford flower database Oxford flower 17 database [30,31] has created a 17 category flower dataset with 80 images for each class. The flowers chosen are some common flowers in the UK. The images have large scale, pose and light variations and there are also classes with large variations of images within the class and close similarity to other classes. Fig. 4 shows some selected flower images of Oxford flower 17 database.
We selected a subset of 1400 face images of 200 subjects from the FERET face database [32]. Each subject has 7 face images of different poses (whose names are marked with two-character strings: “ba,” “bj,” “bk,” “be,” “bf,” “bd,” and “bg” in the database). Each image is downsampled to a 40 × 40 gray image. Some face images of FERET face database are shown in Fig. 5. In the fourth experiment, we utilize the “first N” scheme on the FERET database. The result is shown in Table 4. Contrasted to the SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier, the average recognition rate (ARR) of L1L2-SP classifier surpasses the ARR of them with 19.97%, 1.80%, 14.19%, 19.22%, 15.58%, 6.29% and 15.58%, respectively, when the first 4, 5, 6 samples of each class is used as prototype set. 5. Conclusion In this paper, a novel classifier called L1-norm plus L2-norm sparse parameter (L1L2-SP) is proposed for image recognition. L1L2-SP classifier uses the L1-norm sparse representation of each class to form the new robust global space. Then L1L2-SP utilizes the
4082
Q. Feng et al. / Optik 126 (2015) 4078–4082
robust global space to get the sum of L2-norm sparse parameter of each class. Finally, L1L2-SP computes the score of each class by using the L1-norm sparse parameter and L2-norm sparse parameter. The proposed classifier achieves the better recognition rate than SFR classifier, SRC classifier, CRC classifier, RCR classifier, RRC classifier, CFSR classifier and TPSR classifier. The experimental results on four databases confirm the effectiveness of the proposed algorithm. Acknowledgements This article is partly supported by Nature Science Foundation of China under grants (nos. 61375021, 61501230), China Postdoctoral Science Foundation funded project (no. 2015M570446), Jiangsu Planned Projects for Postdoctoral Research Funds (no. 1402047B), Open Project Program of Key Laboratory of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education (no. JYB201503), and Natural Science Foundation of Jiangsu Province (nos. BK20131365, BK20150751). References [1] M. Turk, A. Pentland, Eigenfaces for recognition, J. Cogn. Neurosci. 3 (1) (1991) 71–86. [2] Q. Zhu, Y. Xu, Multi-directional two-dimensional PCA with matching score level fusion for face recognition, Neural Comput. Appl. 23 (1) (2013) 169–174. [3] P. Belhumeur, J. Hespanha, D. Kriegman, Eigenfaces vs. fisherfaces: recognition using class specific linear projection, IEEE Trans. Pattern Anal. Mach. Intell. 19 (7) (1997) 711–720. [4] Yong Xu, Qi Zhu, Zizhu Fan, Yaowu Wang, Jeng-Shyang Pan, From the idea of “sparse representation” to a representation-based transformation method for feature extraction, Neurocomputing 113 (2013) 168–176. [5] C.-Y. Chang, C.-W. Chang, C.-Y. Hsieh, Applications of block linear discriminant analysis for face recognition, J. Inform. Hiding Multimed. Sign. Process. 2 (3) (2011) 259–269. [6] M.S. Bartlett, J.R. Movellan, T.J. Sejnowski, Face recognition by independent component analysis, IEEE Trans. Neural Netw. 13 (6) (2002) 1450–1464. [7] X. He, S. Yan, Y. Hu, P. Niyogi, H.J. Zhang, Face recognition using laplacianfaces, IEEE Trans. Pattern Anal. Mach. Intell. 27 (3) (2005) 1–13. [8] Yong Xu, Aini Zhong, Jian Yang, David Zhang, LPP solution schemes for use with face recognition, Pattern Recognit. 43 (12) (2010) 4165–4176. [9] T.M. Cover, P.E. Hart, Nearest neighbor pattern classification, IEEE Trans. Inform. Theory 13 (1) (1967) 21–27. [10] S.Z. Li, J. Lu, Face recognition using the nearest feature line method, IEEE Trans. Neural Netw. 10 (2) (1999) 439–443. [11] Q. Feng, J.S. Pan, L. Yan, Nearest feature centre classifier for face recognition, Electron. Lett. 48 (August (18)) (2012) 1120–1122.
[12] Q.B. Gao, Z.Z. Wang, Center-based nearest neighbor classifier, Pattern Recognit. 40 (January (1)) (2006) 346–349. [13] D.Q. Han, C.Z. Han, Y. Yang, A novel classifier based on shortest feature line segment, Pattern Recognit. Lett. 32 (3) (2011) 485–493. [14] J. Wright, Y. Ma, J. Mairal, G. Sapiro, T.S. Huang, S. Yan, Sparse representation for computer vision and pattern recognition, Proc. IEEE 98 (June (6)) (2010) 1031–1044. [15] J. Wright, A.Y. Yang, A. Ganesh, S.S. Sastry, Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Anal. Mach. Intell. 31 (February (2)) (2009) 210–227. [16] L. Zhang, M. Yang, X. Feng, Sparse representation or collaborative representation: which helps face recognition? Proc. IEEE Int’l Conf. Comput. Vis. (November) (2011) 471–478. [17] Y. Xu, D. Zhang, J. Yang, J.-Y. Yang, A two-phase test sample sparse representation method for use with face recognition, IEEE Trans. Circuits Syst. Video Technol. 21 (9) (2011) 1255–1262. [18] Y. Xu, Q. Zhu, A simple and fast representation-based face recognition method, Neural Comput. Appl. 22 (7) (2013) 1543–1549. [19] M. Yang, L. Zhang, J. Yang, D. Zhang, Regularized robust coding for face recognition, IEEE Trans. Image Process. 22 (5) (2013) 1753–1766. [20] M. Yang, L. Zhang, D. Zhang, S. Wang, Relaxed collaborative representation for pattern classification, in: CVPR, 2012. [21] Y. Xu, Q. Zhu, Z. Fan, D. Zhang, J. Mi, Z. Lai, Using the idea of the sparse representation to perform coarse to fine face recognition, Inform. Sci. 238 (2013) 138–148. [22] Y. Xu, Q. Zhu, Z. Fan, M. Qiu, Y. Chen, H. Liu, Coarse to fine K nearest neighbor classifier, Pattern Recognit. Lett. 34 (2013) 980–986. [23] Y. Xu, X. Zhu, Z. Li, G. Liu, Y. Lu, H. Liu, Using the original and ‘symmetrical face’ training samples to perform representation based two-step face recognition, Pattern Recognit. vol.46 (2013) 1151–1158. [24] J. Yang, D. Chu, L. Zhang, Y. Xu, J. Yang, Sparse representation classifier steered discriminative projection with applications to face recognition, IEEE Trans. Neural Netw. Learn. Syst. 24 (July (7)) (2013) 1023–1035. [25] Z. Lai, Y. Xu, J. Yang, J. Tang, D. Zhang, Sparse tensor discriminant analysis, IEEE Trans. Image Process. 22 (10) (2013) 3904–3905. [26] Z. Lai, W.K. Wong, Z. Jin, J. Yang, Y. Xu, Sparse approximation to the eigensubspace for discrimination, IEEE Trans. Neural Netw. Learn. Syst. 23 (12) (2012) 1948–1960. [27] S.A. Nene, S.K. Nayar, H. Murase, Columbia object image library (COIL-100), Technical Report CUCS-006-96, Columbia University, 1996. [28] http://www.d2.mpi-inf.mpg.de/Datasets/ETH80. [29] B. Leibe, B. Schiele, Analyzing appearance and contour based methods for object categorization, in: Proc. of International Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003. [30] http://www.robots.ox.ac.uk/∼vgg/data/flowers/17/index.html. [31] M.-E. Nilsback, A.A. Zisserman, Visual vocabulary for flower classification, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2006. [32] P.J. Phillips, H. Moon, P.J. Rauss, S. Rizvi, The FERET evaluation methodology for face recognition algorithms, IEEE Trans. Pattern Anal. Mach. Intell. 22 (October (10)) (2000).