Research Article Multiview Discriminative Geometry...
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Research Article Multiview Discriminative Geometry Preserving Projection for Image Classification Ziqiang Wang, Xia Sun, Lijun Sun, and Yuchun Huang School of Information Science and Engineering, Henan University of Technology, Zhengzhou 450001, China Correspondence should be addressed to Ziqiang Wang; [email protected] Received 19 December 2013; Accepted 22 January 2014; Published 9 March 2014 Academic Editors: X. Meng, Z. Zhou, and X. Zhu Copyright © 2014 Ziqiang Wang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In many image classification applications, it is common to extract multiple visual features from different views to describe an image. Since different visual features have their own specific statistical properties and discriminative powers for image classification, the conventional solution for multiple view data is to concatenate these feature vectors as a new feature vector. However, this simple concatenation strategy not only ignores the complementary nature of different views, but also ends up with “curse of dimensionality.” To address this problem, we propose a novel multiview subspace learning algorithm in this paper, named multiview discriminative geometry preserving projection (MDGPP) for feature extraction and classification. MDGPP can not only preserve the intraclass geometry and interclass discrimination information under a single view, but also explore the complementary property of different views to obtain a low-dimensional optimal consensus embedding by using an alternating-optimization-based iterative algorithm. Experimental results on face recognition and facial expression recognition demonstrate the effectiveness of the proposed algorithm. 1. Introduction Many computer vision and pattern recognition applications involve processing data in a high-dimensional space. Directly operating on such high-dimensional data is difficult due to the so-called “curse of dimensionality.” For computational time, storage, and classification performance considerations, dimensionality reduction (DR) techniques provide a means to solve this problem by generating a succinct and rep- resentative low-dimensional subspace of the original high- dimensional data space. Over the past two decades, many dimensionality reduction algorithms have been proposed and successfully applied to face recognition [1]. e most representative ones are principal component analysis (PCA) and linear discriminant analysis (LDA) [2]. PCA is an unsupervised dimensionality reduction method, which aims to project the high-dimensional data into a low-dimensional subspace spanned by the leading eigenvectors of a covariance matrix. LDA is supervised and its goal is to pursue a low-dimensional subspace by maximizing the ratio of between-class variance to within- class variance. Due to the utilization of label information, LDA usually outperforms PCA for classification tasks when sufficient labeled training data are available. While these two algorithms have attained reasonable good performance in pattern classification, they may fail to discover a highly nonlinear submanifold embedded in the high-dimensional ambient space as they seek only a compact Euclidean sub- space for data representation and classification [3]. Recently, there has been considerable interest in manifold learning algorithms for dimensionality reduction and feature extraction. e basic consideration of these algorithms is that the high-dimensional data may lie on an intrinsic nonlinear low-dimensional manifold. In order to detect the underlying manifold structure, nonlinear dimensionality reduction algo- rithms such as ISOMAP [4], locally linear embedding (LLE) [5], and Laplacian eigenmap (LE) [6] have been proposed. All of these algorithms are defined only on the training data, and the issue of how to map new testing data remains difficult. erefore, they cannot be applied to classification problem directly. To overcome the above so-called out-of- sample problem, He and Niyogi [7] developed the locality preserving projection (LPP), in which the linear projection function is adopted for mapping new data samples. As LPP is originally unsupervised, some recent attempts have exploited the discriminant information and derived many discriminant Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 924090, 11 pages http://dx.doi.org/10.1155/2014/924090
Transcript of Research Article Multiview Discriminative Geometry...
Research ArticleMultiview Discriminative Geometry PreservingProjection for Image Classification
Ziqiang Wang Xia Sun Lijun Sun and Yuchun Huang
School of Information Science and Engineering Henan University of Technology Zhengzhou 450001 China
Correspondence should be addressed to Ziqiang Wang zi-qiang-wanghotmailcom
Received 19 December 2013 Accepted 22 January 2014 Published 9 March 2014
Academic Editors X Meng Z Zhou and X Zhu
Copyright copy 2014 Ziqiang Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Inmany image classification applications it is common to extract multiple visual features from different views to describe an imageSince different visual features have their own specific statistical properties and discriminative powers for image classification theconventional solution for multiple view data is to concatenate these feature vectors as a new feature vector However this simpleconcatenation strategy not only ignores the complementary nature of different views but also ends upwith ldquocurse of dimensionalityrdquoTo address this problem we propose a novel multiview subspace learning algorithm in this paper named multiview discriminativegeometry preserving projection (MDGPP) for feature extraction and classification MDGPP can not only preserve the intraclassgeometry and interclass discrimination information under a single view but also explore the complementary property of differentviews to obtain a low-dimensional optimal consensus embedding by using an alternating-optimization-based iterative algorithmExperimental results on face recognition and facial expression recognition demonstrate the effectiveness of the proposed algorithm
1 Introduction
Many computer vision and pattern recognition applicationsinvolve processing data in a high-dimensional space Directlyoperating on such high-dimensional data is difficult due tothe so-called ldquocurse of dimensionalityrdquo For computationaltime storage and classification performance considerationsdimensionality reduction (DR) techniques provide a meansto solve this problem by generating a succinct and rep-resentative low-dimensional subspace of the original high-dimensional data space Over the past two decades manydimensionality reduction algorithms have been proposedand successfully applied to face recognition [1] The mostrepresentative ones are principal component analysis (PCA)and linear discriminant analysis (LDA) [2]
PCA is an unsupervised dimensionality reductionmethod which aims to project the high-dimensional datainto a low-dimensional subspace spanned by the leadingeigenvectors of a covariance matrix LDA is supervisedand its goal is to pursue a low-dimensional subspace bymaximizing the ratio of between-class variance to within-class variance Due to the utilization of label informationLDA usually outperforms PCA for classification tasks when
sufficient labeled training data are available While thesetwo algorithms have attained reasonable good performancein pattern classification they may fail to discover a highlynonlinear submanifold embedded in the high-dimensionalambient space as they seek only a compact Euclidean sub-space for data representation and classification [3]
Recently there has been considerable interest inmanifoldlearning algorithms for dimensionality reduction and featureextractionThe basic consideration of these algorithms is thatthe high-dimensional data may lie on an intrinsic nonlinearlow-dimensional manifold In order to detect the underlyingmanifold structure nonlinear dimensionality reduction algo-rithms such as ISOMAP [4] locally linear embedding (LLE)[5] and Laplacian eigenmap (LE) [6] have been proposedAll of these algorithms are defined only on the trainingdata and the issue of how to map new testing data remainsdifficult Therefore they cannot be applied to classificationproblem directly To overcome the above so-called out-of-sample problem He and Niyogi [7] developed the localitypreserving projection (LPP) in which the linear projectionfunction is adopted for mapping new data samples As LPP isoriginally unsupervised some recent attempts have exploitedthe discriminant information and derivedmany discriminant
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 924090 11 pageshttpdxdoiorg1011552014924090
2 The Scientific World Journal
manifold learning algorithms to enhance the classificationperformance The representative algorithms include localdiscriminant embedding (LDE) [8] locality sensitive dis-criminant analysis (LSDA) [9] margin Fisher analysis (MFA)[10] local Fisher discriminant analysis (LFDA) [11] anddiscriminative geometry preserving projection (DGPP) [12]Despite having different assumptions all these algorithmscan be unified into a general graph embedding framework(GEF) [10] with different constraints While these algorithmshave utilized both local geometry and the discriminativeinformation for dimensionality reduction and achieved rea-sonably good performance in different pattern classificationtasks they assume that the data are represented in a singlevector They can be regarded as single-view-based methodsand thus cannot handle data described by multiview featuresdirectly In many practical pattern classification applicationsdifferent views (visual features) have their own specificstatistical properties and each view represents the datapartially To address this problem the traditional solution formultiple viewdata is to simply concatenate vectors of differentviews into a new long vector and then apply dimensionalityreduction algorithms directly on the concatenated vectorHowever this concatenation ignores the diversity of multipleviews and thus cannot explore the complementary nature andspecific statistical properties of different views Recent studieshave provided convincing evidence of this fact [13ndash15]Henceit is more reasonable to assign different weights to differentviews (features) for feature extraction and classification Incomputer vision andmachine learning research many workshave shown that leveraging the complementary nature ofthe multiple views can better represent the data for featureextraction and classification [13ndash15] Therefore an efficientmanifold learning algorithm that can cope with multiviewdata and place proper weights on different views is of greatinterest and significance
Motivated by the above observations and reasons wepropose unifying different views under a discriminant man-ifold learning framework called multiview discriminativegeometry preserving projection (MDGPP) Under each viewwe can implement the discrimination and local geometrypreservations as those used in discriminative geometry pre-serving projection (DGPP) [12] Unifying different views insuch a multiview discriminant manifold learning frameworkis meaningful since data with different features can beappropriately integrated to further improve the classificationperformance Specifically we first implement the discrim-ination preservation by maximizing the average weightedpairwise distance between samples in different classes andsimultaneously minimizing the average weighted pairwisedistance between samples in the same class Meanwhile thelocal geometry preservation is implemented by minimizingthe reconstruction error of samples in the same class Thenwe learn a low-dimensional feature subspace by utilizing bothintraclass geometry and interclass discrimination informa-tion such that the complementary nature of different views(features) can be fully exploited when classification is per-formed in the derived feature subspace Experimental resultson face recognition and facial expression recognition are
presented to demonstrate the effectiveness of the proposedalgorithm
The remainder of the paper is organized as followsSection 2 reviews the related works Section 3 presents thedetails of the proposed MDGPP algorithm Experimentalresults on face recognition are presented in Section 4 and theconcluding remarks are provided in Section 5
2 Related Works
Multiview learning is one important topic in the machinelearning and pattern recognition communities In such asetting view weight information is introduced to measurethe importance of different features in characterizing dataand different weights reflect different contribution to thelearning process The aim of multiview learning is to exploitmore complementary information of different views ratherthan only a single view to further improve the learningperformance The traditional solution for multiview data isto concatenate all features into one vector and then conductmachine learning for such feature space However thissolution is not optimal as these features usually have differentphysical properties Simply concatenating them will ignorethe complementary nature and specific statistical propertiesof different views and thus causing performance degradationIn addition this simple concatenation will end up with thecurse of dimensionality problem for the subsequent learningtask
In order to perform multiview learning much effort hasbeen focused on multiview metric learning [14] multiviewclassification and retrieval [16] multiview clustering [15] andmultiview semisupervised learning [17] All these approachesdemonstrated that the learning performance can be signif-icantly enhanced if the complementary nature of differentviews is exploited and all views are appropriately integratedIt is very natural that multiview learning idea should alsobe considered in dimensionality reduction However mostof the existing dimensionality reduction algorithms aredesigned only for single view data and cannot cope withmultiview data directly To address this problem Long et al[18] first proposedmultiple view spectral embedding (MVSE)method MVSE performs a dimensionality reduction processon each view independently and then based on the obtainedlow-dimensionality representation it constructs a commonlow-dimensional embedding that is close to each represen-tation as much as possible Although MVSE allow selectingdifferent dimensionality reduction algorithms for each viewthe original multiview data are invisible to the final learningprocessThusMVSE cannot well explore the complementaryinformation of different views Xia et al [19] proposedmultiview spectral embedding (MSE) method to find low-dimensional and sufficiently smooth embedding based onthe patch alignment framework [20] However MSE ignoresthe flexibility of allowing shared information between subsetof different views owing to the global coordinate alignmentprocess To unify different views for dimensionality reductionunder a probabilistics Xie et al [21] extended the stochastic
The Scientific World Journal 3
neighbor embedding (SNE) to its multiview version and pro-posed multiview stochastic neighbor embedding (MSNE)Although MSNE operates on a probabilistic framework itis an unsupervised method and its classification abilitiesmay be limited since the class label information is notused in the learning process More recently inspired by therecent advances of sparse coding technique Han et al [22]proposed spectral sparse multiview embedding (SSMVE)method to deal with dimensionality reduction for multiviewdata Although SSMVE can impose sparsity constraint on theloading matrix of multiview dimensionality reduction it isunsupervised and does not explicitly consider the manifoldstructure onwhich the high dimensional data possibly resideIn the next section focusing on the manifold learningand pattern classification we propose a novel multiviewdiscriminative geometry preserving projection (MDGPP)for multiview dimensionality reduction which explicitlyconsiders the local manifold structure and discriminativeinformation as well as the complementary characteristics ofdifferent views in high-dimensional data
3 Multiview Discriminative GeometryPreserving Projection (MDGPP)
In this section we propose a new manifold learning algo-rithm called multiview discriminative geometry preservingprojection (MDGPP) which aims to find a unified low-dimensional and sufficiently smooth embedding over allviews simultaneously To better explain the algorithm detailsof the proposed MDGPP we introduce some importantnotations used in the remainder of this paper Capital letterssuch as 119883 denote data matrices and 119883
119894119895represents the (119894 119895)
entry of 119883 Lower case letters such as 119909 denote data vectorsand 119909
119894represents the 119894th data element of 119909 Superscript
(119894) such as 119883(119894) and 119909(119894) represents data from the 119894th viewrespectively Based on these notations MDGPP can bedescribed as follows according to the DGPP framework [12]
Given a multiview data set with 119899 data samplesand each with 119898 feature representations that is 119883 =
119883(119894)= [119909(119894)
1 119909(119894)
2 119909
(119894)
119899] isin 119877119901119894times119899119898
119894=1 wherein 119883
(119894) repre-sents the feature matrix for the 119894th view the aim of MDGPPis to find a projective matrix 119882 isin 119877
119901119894times119889 to map 119883(119894) into alow-dimensional representation 119884(119894) through 119884(119894) = 119882119879119883(119894)where 119889 denotes the dimension of low-dimensional featurerepresentation and satisfies 119889 lt 119901
119894(1 le 119894 le 119898) The
workflow of MDGPP can be simply described as followsFirst MDGPP builds a part optimization for a sample on asingle view by preserving both the intraclass geometry andinterclass discrimination Afterward all parts of optimizationfrom different views are unified as a whole via view weightcoefficientsThen an alternating-optimization-based iterativealgorithm is derived to obtain the optimal low-dimensionalembedding from multiple views
Given the 119894th view 119883(119894) = [119909(119894)1 119909(119894)
2 119909
(119894)
119899] isin 119877
119901119894times119899MDGPP first makes an attempt to preserve discrimina-tive information in the reduced low-dimensional space bymaximizing the average weighted pairwise distance betweensamples in different classes and simultaneously minimizing
the average weighted pairwise distance between samples inthe same class on the 119894th view Thus we have
argmax119910(119894)
119895
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
= argmax119910(119894)
119895
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
= argmax119910(119894)
119895
Tr(119899
sum
119895119905=1
ℎ(119894)
119895119905(119910(119894)
119895minus 119910(119894)
119905) (119910(119894)
119895minus 119910(119894)
119905)119879
)
= argmax119910(119894)
119895
Tr(119899
sum
119895119905=1
ℎ(119894)
119895119905(119910(119894)
119895(119910(119894)
119895)119879
minus 119910(119894)
119905(119910(119894)
119895)119879
minus119910(119894)
119895(119910(119894)
119905)119879
+ 119910(119894)
119905(119910(119894)
119905)119879
))
= argmax 2Tr(119884(119894)119863(119894)(119884(119894))119879
minus 119884(119894)119867(119894)(119884(119894))119879
)
= argmax 2Tr(119884(119894)119871(119894)(119884(119894))119879
)
= argmax 2Tr(119880119879119883(119894)119871(119894)(119883(119894))119879
119880)
(1)
where Tr( ) denotes the trace operation ofmatrix 119871(119894)(= 119863(119894)minus119867(119894)) is the graph Laplacian on the 119894th view119863(119894) is a diagonal
matrix with its element 119863(119894)119895119895= sum119905119867(119894)
119895119905on the 119894th view and
119867(119894)= [ℎ
(119894)
119895119905]119899
119895119905=1is the weighting matrix which encodes
both the distance weighting information and the class labelinformation on the 119894th view
ℎ(119894)
119895119905=
119888119895119905(1
119899minus1
119899119897
) if 119897119895= 119897119905= 119897
1
119899 if 119897
119895= 119897119905
(2)
where in 119897119895is the class label of sample 119909(119894)
119895 119899119897is the number of
samples belonging to the 119897th class and 119888119895119905is set as exp(minus119909(119894)
119895minus
119909(119894)
1199051205752) according to LPP [7] for locality preservation
Second we try to implement the local geometry preser-vation by assuming that each sample 119909(119894)
119895can be linearly
reconstructed by the samples 119909(119894)119905which share the same class
label with 119909(119894)119895on the 119894th viewThus we can obtain the recon-
struction coefficient 119908(119894)119895119905
by minimizing the reconstructionerror sum119899
119895=11205761198952 on the 119894th view that is
argmin119908119895119905
119899
sum
119895=1
10038171003817100381710038171003817120576119895
10038171003817100381710038171003817
2
= argmin119908119895119905
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119909(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119909(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2(3)
4 The Scientific World Journal
under the constraint
sum
119905119897119905=119897119895
119908(119894)
119895119905= 1 119908
(119894)
119895119905= 0 for 119897
119905= 119897119895 (4)
Then by solving (3) and (4) we have
119908(119894)
119895=
sum119901119862minus1
119895119901
sum119901119902119862minus1119901119902
(5)
where 119862119901119902= (119909(119894)
119895minus 119909(119894)
119901)119879
(119909(119894)
119895minus 119909(119894)
119902) denotes the local Gram
matrix and 119897119901= 119897119902= 119897119895
Once obtaining the reconstruction coefficient 119908(119894)119895119905
on the119894th view then MDGPP aims to reconstruct 119910(119894)
119895(= 119880
119879119909(119894)
119895)
from 119910(119894)119905(= 119880119879119909(119894)
119905) (where 119897
119905= 119897119895) with 119908(119894)
119895119905in the projected
low-dimensional space thus we have
argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
Tr(119899
sum
119895=1
(119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
times (119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
119879
)
= argmin 2Tr(119884(119894)119868(119894)(119884(119894))119879
minus 119884(119894)119882(119894)(119884(119894))119879
)
= argmin 2Tr(119884(119894) (119868(119894) minus119882(119894)) (119884(119894))119879
)
= argmin 2Tr(119880119879119883(119894) (119868(119894) minus119882(119894)) (119883(119894))119879
119880)
(6)
where 119868(119894) is an identity matrix defined on the 119894th view and119882(119894)= [119908(119894)
119895119905]119899
119895119905=1is the reconstruction coefficient matrix on
the 119894th viewAs a result by combining (1) and (6) together the part
optimization for119883(119894) is
argmax119910(119894)
119895
(
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
minus120582
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905 119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
)
= argmax Tr(119880119879119883(119894)119871(119894)(119883(119894))119879
119880
minus120582119880119879119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
119880)
= argmax Tr(119880119879 (119883(119894)119871(119894)(119883(119894))119879
minus120582119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
)119880)
= argmax Tr (119880119879119876(119894)119880) (7)
where 119876(119894) = 119883(119894)119871(119894)(119883(119894))119879
minus 120582119883(119894)(119868(119894)minus 119882(119894))(119883(119894))119879
and120582 is a tradeoff coefficient which is empirically set as 1 in thisexperiment
Based on the local manifold information encoded in119871(119894) and 119882(119894) (7) aims at finding a sufficiently smooth low-
dimensional embedding 119884(119894)(= 119880119879119883(119894)) by preserving the
interclass discrimination and intraclass geometry on the 119894thview
Because multiviews could provide complementary infor-mation in characterizing data from different viewpointsdifferent views certainly have different contributions to thelow-dimensional feature subspace In order to well discoverthe complementary information of data from different viewsa nonnegative weighted set120590 = [120590
1 1205902 120590
119898] is imposed on
each view independently Generally speaking the larger 120590119894is
the more the contribution of the view 119883(119894) is made to obtainthe low-dimensional feature subspace Hence by summingover all parts of optimization defined in (7) we can formulateMDGPP as the following optimization problem
argmax119880120590
119898
sum
119894=1
120590119894Tr (119880119879119876(119894)119880) (8)
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (9)
The solution to 120590 in (8) subject to (9) is 120590119896= 1
corresponding to the maximum Tr(119880119879119876(119894)119880) over differentviews and 120590
119896= 0 otherwise which means that only the
best view is finally selected by this method Consequentlythis solution cannot meet the demand for exploring thecomplementary characteristics of different views to get abetter low-dimensional embedding than that based on asingle view In order to avoid this problem we set 120590
119894larr 120590119903
119894
with 119903 gt 1 by following the trick utilized in [16ndash19] In thiscondition sum119898
119894=1120590119903
119894= 1 achieves its maximum when 120590
119894=
1119898 according to sum119898119894=1120590119894= 1 and 120590
119894ge 0 Similarly 120590
119894for
different views can be obtained by setting 119903 gt 1 thus eachviewmakes a specific contribution to obtaining the final low-dimensional embedding Consequently the new objectivefunction of MDGPP can be defined as follows
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (10)
The Scientific World Journal 5
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (11)
The above optimization problem is a nonlinearly con-strained nonconvex optimization problem so there is nodirect approach to find its global optimal solution In thispaper we derive an alternating-optimization-based iterativealgorithm to find a local optimal solution The alternatingoptimization iteratively updates the projection matrix 119880 andweight vector 120590 = [120590
1 1205902 120590
119898]
First we update 120590 by fixing 119880 The optimal problem (10)subject to (11) becomes
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (12)
subject to119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (13)
Following the standard Lagrange multiplier we constructthe following Lagrangian function by incorporating theconstraint (13) into (12)
119871 (120590 120582) =
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) minus 120582(
119898
sum
119894=1
120590119894minus 1) (14)
where the Lagrange multiplier 120582 satisfies 120582 ge 0Taking the partial derivation of the Lagrangian function
119871(120590 120582) with respect to 120590119894and 120582 and setting them to zeros we
have120597119871 (120590 120582)
120597120590119894
= 119903120590119903minus1
119894Tr (119880119879119876(119894)119880) minus 120582 = 0 (15)
120597119871 (120590 120582)
120597120582=
119898
sum
119894=1
120590119894minus 1 = 0 (16)
Hence according to (13) and (14) the weight coefficient120590119894can be calculated as
120590119894=
(1Tr (119880119879119876(119894)119880))1(119903minus1)
sum119898
119894=1(1Tr (119880119879119876(119894)119880))1(119903minus1)
(17)
Then we can make the following observations accordingto (15) If 119903 rarr infin then the values of different 120590
119894will be close
to each other If 119903 rarr 1 then only 120590119894= 1 corresponding
to the maximum Tr(119880119879119876(119894)119880) over different views and 120590119894=
0 otherwise Thus the choice of 119903 should respect to thecomplementary property of different views The effect of theparameter 119903 will be discussed in the later experiments
Second we update 119880 by fixing 120590 The optimal problem(10) subject to (11) can be equivalently transformed into thefollowing form
argmax119880
Tr (119880119879119876119880) (18)
subject to
119880119879119880 = 119868 (19)
where119876 = sum119898119894=1120590119903
119894119876(119894) Since119876(119894) defined in (7) is a symmetric
matrix 119876 is also a symmetric matrixObviously the solution of (18) subject to (19) can be
obtained by solving the following standard eigendecomposi-tion problem
119876119880 = 120582119880 (20)
Let the eigenvectors 1198801 1198802 119880
119889be solutions of (20)
ordered according to eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
Then the optimal projection matrix 119880 is given by 119880 =
[1198801 1198802 119880
119889] Now we discuss how to determine the
reduced feature dimension 119889 by using the Ky Fan theorem[23]
Ky FanTheorem Let119867 be a symmetric matrix with eigenval-ues 1205821ge 1205822ge sdot sdot sdot ge 120582
119899and the corresponding eigenvectors
119880 = [1198801 1198802 119880
119899] Then
1205821+ 1205822+ sdot sdot sdot + 120582
119896= arg max
119883119879119883=119868
Tr (119883119879119867119883) (21)
Moreover the optimal 119883lowast is given by 119883lowast = 119880 = [1198801 1198802
119880119896]119877 where 119877 is an arbitrary orthogonal matrix
From the above Ky Fan theorem we can make thefollowing observations The optimal solution to (18) subjectto (19) is composed of the largest 119889 eigenvectors of the matrix119876 and the optimal value of objective function (18) equals thesum of the largest 119889 eigenvalues of the matrix 119876 Thereforethe optimal reduced feature dimension 119889 is equivalent to thenumber of positive eigenvalues of the matrix 119876
Alternately updating 120590 and 119880 by solving (17) and (20)until convergence we can obtain the final optimal projectionmatrix 119880 for multiple views A simple initialization for 120590could be 120590 = [1119898 1119898] According to the afore-mentioned statement the proposed MDGPP algorithm issummarized as follows
Algorithm 1 (MDGPP algorithm)
Input Amultiview data set119883 = 119883(119894) isin 119877119901119894times119899119898
119894=1 the dimen-
sion of the reduced low-dimensional subspace119889 (119889 lt 119901119894 1 le
119894 le 119898) tuning parameter 119903 gt 1 iteration number 119879max andconvergence error 120576
Output Projection matrix 119880
Algorithm
Step 1 Simultaneously consider both intraclass geometry andinterclass discrimination information to calculate 119876(119894) foreach view according to (7)
Step 2 (initialization)
(1) Set 120590 = [1119898 1119898 1119898](2) obtain 1198800 by solving the eigendecomposition prob-
lem (20)
6 The Scientific World Journal
Step 3 (local optimization) For 119905 = 1 2 119879max
(1) calculate 120590 as shown in (17)(2) solve the eigenvalue equation in (20)(3) sort their eigenvectors 119880
1 1198802 119880
119889according to
their corresponding eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
and obtain 119880119905 = [1198801 1198802 119880
119889]
(4) if 119905 gt 2 and |119880119905 minus 119880119905minus1| lt 120576 then go to Step 4
Step 4 (output projection matrix) Output the final optimalprojection matrix 119880 = 119880119905
We now briefly analyze the computational complexityof the MDGPP algorithm which is dominated by threeparts One is for constructing the matrix 119876(119894) for differentviews As shown in (7) the computational complexity ofthis part is 119874((sum119898
119894=1119901119894) times 119899
2) In addition each iteration
involves computing view weight 120590 and solving a standardeigendecomposition problem the computational complexityof running two parts in each iteration is 119874((119898 + 119889) times 1198992)and 119874(1198993) respectively Therefore the total computationalcomplexity of MDGPP is 119874((sum119898
119894=1119901119894) times 1198992+ ((119898 + 119889) times 119899
2+
1198993) times 119879max) where 119879max denotes the iteration number and is
always set to less than five in all experiments
4 Experimental Results
In this section we evaluate the effectiveness of our proposedMDGPP algorithm for two image classification tasks includ-ing face recognition and facial expression recognition Twowidely used face databases including AR [24] and CMU PIE[25] are employed for face recognition evaluation and thewell-known Japanese female facial expression (JAFFE) [26]database is used for facial expression recognition evaluationWe also compare the proposed MDGPP algorithm withsome traditional single-view-based dimensionality reductionalgorithms such as PCA [2] LDA [2] LPP [3] MFA [10]DGPP [12] and the three latest multiview dimensionalityreduction algorithms includingMVSE [18]MSNE [21]MSE[19] and SSMVE [22] The nearest neighbor classifier withthe Euclidean distance was adopted for classification For afair comparison all the results reported here are based on thebest tuned parameters of all the compared algorithms
41 Data Sets and Experimental Settings We conducted facerecognition experiments on the widely used AR and CMUPIE face databases and facial expression recognition experi-ments on the well-known Japanese female facial expression(JAFFE) database
The AR database [24] contains over 4000 color imagescorresponding to 126 people (70 men and 56 women) whichinclude frontal view faces with different facial expressionsillumination conditions and occlusions (sun glasses andscarf) Each person has 26 different images taken in twosessions (separated by two weeks) In our experiments weused a subset of 800 face images from 100 persons (50 menand 50women)with eight face images of different expressionsand lighting conditions per person Figure 1 shows eight
Figure 1 Sample face images from the AR database
Figure 2 Sample face images from the CMU PIE database
sample images of one individual from the subset of the ARdatabase
The CMU PIE database [25] comprises more than 40000facial images of 68 people with different poses illuminationconditions and facial expressions In this experiment weselected a subset of the CMU PIE database which consistsof 3060 frontal face images with varying expression andillumination from 68 persons with 45 images from eachperson Figure 2 shows some sample images of one individualfrom the subset of the CMU PIE database
The Japanese female facial expression (JAFFE) database[26] contains 213 facial images of ten Japanese womenEach facial image shows one of seven expressions neutralhappiness sadness surprise anger disgust or fear Figure 3shows some facial images from the JAFFE database In thisexperiment following the general setting scheme of facialexpression recognition we discard all the neutral facialimages and only utilize the remainder 183 facial images whichinclude six basic facial expressions
For all the face images in the above three face databasesthe facial part of each image was manually aligned croppedand resized into 32 times 32 according to the eyersquos positions Foreach facial image we extract the commonly used four kindsof low-level visual features to represent four different viewsThese four features include color histogram (CH) [27] scale-invariant feature transform (SIFT) [28] Gabor [29] and localbinary pattern (LBP) [30] For the CH feature extraction weused 64 bins to encode a histogram feature for each facialimage according to [27] For the SIFT feature extractionwe densely sampled and calculated the SIFT descriptors of16 times 16 patches over a grid with spacing of 8 pixels accordingto [28] For the Gabor feature extraction following [29] weadopted 40 Gabor kernel functions from five scales and eightorientations For the LBP feature extraction we followed theparameter settings in [30] and utilized 256 bins to encodea histogram feature for each facial image For more detailson these four feature descriptors please refer to [27ndash30]Because these four features are complementary to each otherin representing facial images we empirically set the tuningparameter 119903 in MDGPP to be five
In this experiment each facial image set was partitionedinto the nonoverlap training and testing sets For eachdatabase we randomly selected 50 data as the training setand use the remaining 50 data as the testing set To reducestatistical variation for each random partition we repeatedthese trials independently ten times and reported the averagerecognition results
The Scientific World Journal 7
Figure 3 Sample facial images from the JAFFE database
42 Compared Algorithms We compared our proposedMDGPP algorithm with the following dimensionality reduc-tion algorithms
(1) PCA [2] PCA is an unsupervised dimensionalityreduction algorithm
(2) LDA [2] LDA is a supervised dimensionality reduc-tion algorithm We adopted a Tikhonov regular-ization term 120583119868 rather than PCA preprocessing toavoid the well-known small sample size (singularity)problem in LDA
(3) LPP [3] LPP is an unsupervised manifold learningalgorithm There is a nearest neighbor number 119896 tobe tuned in LPP and it was empirically set to befive in our experiments In addition the Tikhonovregularization was also adopted to avoid the smallsample size (singularity) problem in LPP
(4) MFA [10] MFA is a supervised manifold learningalgorithm There are two parameters (ie 119896
1nearest
neighbor number and 1198962nearest neighbor number)
to be tuned in MFA We empirically set 1198961= 5
and 1198962= 20 in our experiments Meanwhile the
Tikhonov regularizationwas also adopted to avoid thesmall sample size (singularity) problem in MFA
(5) DGPP [12] DGPP is a supervised manifold learningalgorithmThere is a tradeoff parameter 120582 to be tunedin DGPP and it was empirically set to be one in ourexperiments
(6) MVSE [18] MSVE is an initially proposed multiviewalgorithm for dimensionality reduction
(7) MSNE [21] MSNE is a probability-based unsuper-vised multiview algorithm We followed the param-eter setting in [21] and set the tradeoff coefficient 120582 tobe five in our experiments
(8) MSE [19] MSE is a supervised multiview algorithmThere are two parameters (ie the nearest neighbornumber 119896 and the tuning coefficient 119903) to be tunedin MSE We empirically set 119896 = 5 and 119903 = 5 in ourexperiments
(9) SSMVE [22] SSMVE is a sparse unsupervised mul-tiview algorithm We followed the parameter settingmethod in [22] and set the regularized parameter 120582 tobe one in our experiments
It is worth noting that since PCA LDA LPP MFAand DGPP are all single-view-based algorithms these fivealgorithms adopt the conventional feature concatenation-based strategy to cope with the multiview data
Table 1 Comparisons of recognition accuracy on the AR database
Algorithm Accuracy DimensionsPCA 815 120LDA 904 99LPP 908 100MFA 919 100DGPP 932 100MVSE 934 100MSNE 937 100SSMVE 941 100MSE 950 100MDGPP 988 100
Table 2 Comparisons of recognition accuracy on the CMU PIEdatabase
Algorithm Accuracy DimensionsPCA 568 80LDA 634 67LPP 652 68MFA 735 68DGPP 784 68MVSE 796 68MSNE 827 68SSMVE 845 68MSE 873 68MDGPP 952 68
Table 3 Comparisons of recognition accuracy on the JAFFEdatabase
Algorithm Accuracy DimensionsPCA 526 20LDA 658 9LPP 715 10MFA 832 10DGPP 863 10MVSE 878 10MSNE 892 10SSMVE 916 10MSE 934 10MDGPP 965 10
43 Experimental Results For each face image databasethe recognition performance of different algorithms wasevaluated on the testing data separately The conventionalnearest neighbor classifier with the Euclidean distance wasapplied to perform recognition in the subspace derived fromdifferent dimensionality reduction algorithms Tables 1 2and 3 report the recognition accuracies and the correspond-ing optimal dimensions obtained on the AR CMU PIE andJAFFE databases respectively Figures 4 5 and 6 illustratethe recognition accuracies versus the variation of reduceddimensions on the AR CMU PIE and JAFFE databases
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
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2 The Scientific World Journal
manifold learning algorithms to enhance the classificationperformance The representative algorithms include localdiscriminant embedding (LDE) [8] locality sensitive dis-criminant analysis (LSDA) [9] margin Fisher analysis (MFA)[10] local Fisher discriminant analysis (LFDA) [11] anddiscriminative geometry preserving projection (DGPP) [12]Despite having different assumptions all these algorithmscan be unified into a general graph embedding framework(GEF) [10] with different constraints While these algorithmshave utilized both local geometry and the discriminativeinformation for dimensionality reduction and achieved rea-sonably good performance in different pattern classificationtasks they assume that the data are represented in a singlevector They can be regarded as single-view-based methodsand thus cannot handle data described by multiview featuresdirectly In many practical pattern classification applicationsdifferent views (visual features) have their own specificstatistical properties and each view represents the datapartially To address this problem the traditional solution formultiple viewdata is to simply concatenate vectors of differentviews into a new long vector and then apply dimensionalityreduction algorithms directly on the concatenated vectorHowever this concatenation ignores the diversity of multipleviews and thus cannot explore the complementary nature andspecific statistical properties of different views Recent studieshave provided convincing evidence of this fact [13ndash15]Henceit is more reasonable to assign different weights to differentviews (features) for feature extraction and classification Incomputer vision andmachine learning research many workshave shown that leveraging the complementary nature ofthe multiple views can better represent the data for featureextraction and classification [13ndash15] Therefore an efficientmanifold learning algorithm that can cope with multiviewdata and place proper weights on different views is of greatinterest and significance
Motivated by the above observations and reasons wepropose unifying different views under a discriminant man-ifold learning framework called multiview discriminativegeometry preserving projection (MDGPP) Under each viewwe can implement the discrimination and local geometrypreservations as those used in discriminative geometry pre-serving projection (DGPP) [12] Unifying different views insuch a multiview discriminant manifold learning frameworkis meaningful since data with different features can beappropriately integrated to further improve the classificationperformance Specifically we first implement the discrim-ination preservation by maximizing the average weightedpairwise distance between samples in different classes andsimultaneously minimizing the average weighted pairwisedistance between samples in the same class Meanwhile thelocal geometry preservation is implemented by minimizingthe reconstruction error of samples in the same class Thenwe learn a low-dimensional feature subspace by utilizing bothintraclass geometry and interclass discrimination informa-tion such that the complementary nature of different views(features) can be fully exploited when classification is per-formed in the derived feature subspace Experimental resultson face recognition and facial expression recognition are
presented to demonstrate the effectiveness of the proposedalgorithm
The remainder of the paper is organized as followsSection 2 reviews the related works Section 3 presents thedetails of the proposed MDGPP algorithm Experimentalresults on face recognition are presented in Section 4 and theconcluding remarks are provided in Section 5
2 Related Works
Multiview learning is one important topic in the machinelearning and pattern recognition communities In such asetting view weight information is introduced to measurethe importance of different features in characterizing dataand different weights reflect different contribution to thelearning process The aim of multiview learning is to exploitmore complementary information of different views ratherthan only a single view to further improve the learningperformance The traditional solution for multiview data isto concatenate all features into one vector and then conductmachine learning for such feature space However thissolution is not optimal as these features usually have differentphysical properties Simply concatenating them will ignorethe complementary nature and specific statistical propertiesof different views and thus causing performance degradationIn addition this simple concatenation will end up with thecurse of dimensionality problem for the subsequent learningtask
In order to perform multiview learning much effort hasbeen focused on multiview metric learning [14] multiviewclassification and retrieval [16] multiview clustering [15] andmultiview semisupervised learning [17] All these approachesdemonstrated that the learning performance can be signif-icantly enhanced if the complementary nature of differentviews is exploited and all views are appropriately integratedIt is very natural that multiview learning idea should alsobe considered in dimensionality reduction However mostof the existing dimensionality reduction algorithms aredesigned only for single view data and cannot cope withmultiview data directly To address this problem Long et al[18] first proposedmultiple view spectral embedding (MVSE)method MVSE performs a dimensionality reduction processon each view independently and then based on the obtainedlow-dimensionality representation it constructs a commonlow-dimensional embedding that is close to each represen-tation as much as possible Although MVSE allow selectingdifferent dimensionality reduction algorithms for each viewthe original multiview data are invisible to the final learningprocessThusMVSE cannot well explore the complementaryinformation of different views Xia et al [19] proposedmultiview spectral embedding (MSE) method to find low-dimensional and sufficiently smooth embedding based onthe patch alignment framework [20] However MSE ignoresthe flexibility of allowing shared information between subsetof different views owing to the global coordinate alignmentprocess To unify different views for dimensionality reductionunder a probabilistics Xie et al [21] extended the stochastic
The Scientific World Journal 3
neighbor embedding (SNE) to its multiview version and pro-posed multiview stochastic neighbor embedding (MSNE)Although MSNE operates on a probabilistic framework itis an unsupervised method and its classification abilitiesmay be limited since the class label information is notused in the learning process More recently inspired by therecent advances of sparse coding technique Han et al [22]proposed spectral sparse multiview embedding (SSMVE)method to deal with dimensionality reduction for multiviewdata Although SSMVE can impose sparsity constraint on theloading matrix of multiview dimensionality reduction it isunsupervised and does not explicitly consider the manifoldstructure onwhich the high dimensional data possibly resideIn the next section focusing on the manifold learningand pattern classification we propose a novel multiviewdiscriminative geometry preserving projection (MDGPP)for multiview dimensionality reduction which explicitlyconsiders the local manifold structure and discriminativeinformation as well as the complementary characteristics ofdifferent views in high-dimensional data
3 Multiview Discriminative GeometryPreserving Projection (MDGPP)
In this section we propose a new manifold learning algo-rithm called multiview discriminative geometry preservingprojection (MDGPP) which aims to find a unified low-dimensional and sufficiently smooth embedding over allviews simultaneously To better explain the algorithm detailsof the proposed MDGPP we introduce some importantnotations used in the remainder of this paper Capital letterssuch as 119883 denote data matrices and 119883
119894119895represents the (119894 119895)
entry of 119883 Lower case letters such as 119909 denote data vectorsand 119909
119894represents the 119894th data element of 119909 Superscript
(119894) such as 119883(119894) and 119909(119894) represents data from the 119894th viewrespectively Based on these notations MDGPP can bedescribed as follows according to the DGPP framework [12]
Given a multiview data set with 119899 data samplesand each with 119898 feature representations that is 119883 =
119883(119894)= [119909(119894)
1 119909(119894)
2 119909
(119894)
119899] isin 119877119901119894times119899119898
119894=1 wherein 119883
(119894) repre-sents the feature matrix for the 119894th view the aim of MDGPPis to find a projective matrix 119882 isin 119877
119901119894times119889 to map 119883(119894) into alow-dimensional representation 119884(119894) through 119884(119894) = 119882119879119883(119894)where 119889 denotes the dimension of low-dimensional featurerepresentation and satisfies 119889 lt 119901
119894(1 le 119894 le 119898) The
workflow of MDGPP can be simply described as followsFirst MDGPP builds a part optimization for a sample on asingle view by preserving both the intraclass geometry andinterclass discrimination Afterward all parts of optimizationfrom different views are unified as a whole via view weightcoefficientsThen an alternating-optimization-based iterativealgorithm is derived to obtain the optimal low-dimensionalembedding from multiple views
Given the 119894th view 119883(119894) = [119909(119894)1 119909(119894)
2 119909
(119894)
119899] isin 119877
119901119894times119899MDGPP first makes an attempt to preserve discrimina-tive information in the reduced low-dimensional space bymaximizing the average weighted pairwise distance betweensamples in different classes and simultaneously minimizing
the average weighted pairwise distance between samples inthe same class on the 119894th view Thus we have
argmax119910(119894)
119895
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
= argmax119910(119894)
119895
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
= argmax119910(119894)
119895
Tr(119899
sum
119895119905=1
ℎ(119894)
119895119905(119910(119894)
119895minus 119910(119894)
119905) (119910(119894)
119895minus 119910(119894)
119905)119879
)
= argmax119910(119894)
119895
Tr(119899
sum
119895119905=1
ℎ(119894)
119895119905(119910(119894)
119895(119910(119894)
119895)119879
minus 119910(119894)
119905(119910(119894)
119895)119879
minus119910(119894)
119895(119910(119894)
119905)119879
+ 119910(119894)
119905(119910(119894)
119905)119879
))
= argmax 2Tr(119884(119894)119863(119894)(119884(119894))119879
minus 119884(119894)119867(119894)(119884(119894))119879
)
= argmax 2Tr(119884(119894)119871(119894)(119884(119894))119879
)
= argmax 2Tr(119880119879119883(119894)119871(119894)(119883(119894))119879
119880)
(1)
where Tr( ) denotes the trace operation ofmatrix 119871(119894)(= 119863(119894)minus119867(119894)) is the graph Laplacian on the 119894th view119863(119894) is a diagonal
matrix with its element 119863(119894)119895119895= sum119905119867(119894)
119895119905on the 119894th view and
119867(119894)= [ℎ
(119894)
119895119905]119899
119895119905=1is the weighting matrix which encodes
both the distance weighting information and the class labelinformation on the 119894th view
ℎ(119894)
119895119905=
119888119895119905(1
119899minus1
119899119897
) if 119897119895= 119897119905= 119897
1
119899 if 119897
119895= 119897119905
(2)
where in 119897119895is the class label of sample 119909(119894)
119895 119899119897is the number of
samples belonging to the 119897th class and 119888119895119905is set as exp(minus119909(119894)
119895minus
119909(119894)
1199051205752) according to LPP [7] for locality preservation
Second we try to implement the local geometry preser-vation by assuming that each sample 119909(119894)
119895can be linearly
reconstructed by the samples 119909(119894)119905which share the same class
label with 119909(119894)119895on the 119894th viewThus we can obtain the recon-
struction coefficient 119908(119894)119895119905
by minimizing the reconstructionerror sum119899
119895=11205761198952 on the 119894th view that is
argmin119908119895119905
119899
sum
119895=1
10038171003817100381710038171003817120576119895
10038171003817100381710038171003817
2
= argmin119908119895119905
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119909(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119909(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2(3)
4 The Scientific World Journal
under the constraint
sum
119905119897119905=119897119895
119908(119894)
119895119905= 1 119908
(119894)
119895119905= 0 for 119897
119905= 119897119895 (4)
Then by solving (3) and (4) we have
119908(119894)
119895=
sum119901119862minus1
119895119901
sum119901119902119862minus1119901119902
(5)
where 119862119901119902= (119909(119894)
119895minus 119909(119894)
119901)119879
(119909(119894)
119895minus 119909(119894)
119902) denotes the local Gram
matrix and 119897119901= 119897119902= 119897119895
Once obtaining the reconstruction coefficient 119908(119894)119895119905
on the119894th view then MDGPP aims to reconstruct 119910(119894)
119895(= 119880
119879119909(119894)
119895)
from 119910(119894)119905(= 119880119879119909(119894)
119905) (where 119897
119905= 119897119895) with 119908(119894)
119895119905in the projected
low-dimensional space thus we have
argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
Tr(119899
sum
119895=1
(119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
times (119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
119879
)
= argmin 2Tr(119884(119894)119868(119894)(119884(119894))119879
minus 119884(119894)119882(119894)(119884(119894))119879
)
= argmin 2Tr(119884(119894) (119868(119894) minus119882(119894)) (119884(119894))119879
)
= argmin 2Tr(119880119879119883(119894) (119868(119894) minus119882(119894)) (119883(119894))119879
119880)
(6)
where 119868(119894) is an identity matrix defined on the 119894th view and119882(119894)= [119908(119894)
119895119905]119899
119895119905=1is the reconstruction coefficient matrix on
the 119894th viewAs a result by combining (1) and (6) together the part
optimization for119883(119894) is
argmax119910(119894)
119895
(
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
minus120582
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905 119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
)
= argmax Tr(119880119879119883(119894)119871(119894)(119883(119894))119879
119880
minus120582119880119879119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
119880)
= argmax Tr(119880119879 (119883(119894)119871(119894)(119883(119894))119879
minus120582119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
)119880)
= argmax Tr (119880119879119876(119894)119880) (7)
where 119876(119894) = 119883(119894)119871(119894)(119883(119894))119879
minus 120582119883(119894)(119868(119894)minus 119882(119894))(119883(119894))119879
and120582 is a tradeoff coefficient which is empirically set as 1 in thisexperiment
Based on the local manifold information encoded in119871(119894) and 119882(119894) (7) aims at finding a sufficiently smooth low-
dimensional embedding 119884(119894)(= 119880119879119883(119894)) by preserving the
interclass discrimination and intraclass geometry on the 119894thview
Because multiviews could provide complementary infor-mation in characterizing data from different viewpointsdifferent views certainly have different contributions to thelow-dimensional feature subspace In order to well discoverthe complementary information of data from different viewsa nonnegative weighted set120590 = [120590
1 1205902 120590
119898] is imposed on
each view independently Generally speaking the larger 120590119894is
the more the contribution of the view 119883(119894) is made to obtainthe low-dimensional feature subspace Hence by summingover all parts of optimization defined in (7) we can formulateMDGPP as the following optimization problem
argmax119880120590
119898
sum
119894=1
120590119894Tr (119880119879119876(119894)119880) (8)
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (9)
The solution to 120590 in (8) subject to (9) is 120590119896= 1
corresponding to the maximum Tr(119880119879119876(119894)119880) over differentviews and 120590
119896= 0 otherwise which means that only the
best view is finally selected by this method Consequentlythis solution cannot meet the demand for exploring thecomplementary characteristics of different views to get abetter low-dimensional embedding than that based on asingle view In order to avoid this problem we set 120590
119894larr 120590119903
119894
with 119903 gt 1 by following the trick utilized in [16ndash19] In thiscondition sum119898
119894=1120590119903
119894= 1 achieves its maximum when 120590
119894=
1119898 according to sum119898119894=1120590119894= 1 and 120590
119894ge 0 Similarly 120590
119894for
different views can be obtained by setting 119903 gt 1 thus eachviewmakes a specific contribution to obtaining the final low-dimensional embedding Consequently the new objectivefunction of MDGPP can be defined as follows
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (10)
The Scientific World Journal 5
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (11)
The above optimization problem is a nonlinearly con-strained nonconvex optimization problem so there is nodirect approach to find its global optimal solution In thispaper we derive an alternating-optimization-based iterativealgorithm to find a local optimal solution The alternatingoptimization iteratively updates the projection matrix 119880 andweight vector 120590 = [120590
1 1205902 120590
119898]
First we update 120590 by fixing 119880 The optimal problem (10)subject to (11) becomes
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (12)
subject to119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (13)
Following the standard Lagrange multiplier we constructthe following Lagrangian function by incorporating theconstraint (13) into (12)
119871 (120590 120582) =
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) minus 120582(
119898
sum
119894=1
120590119894minus 1) (14)
where the Lagrange multiplier 120582 satisfies 120582 ge 0Taking the partial derivation of the Lagrangian function
119871(120590 120582) with respect to 120590119894and 120582 and setting them to zeros we
have120597119871 (120590 120582)
120597120590119894
= 119903120590119903minus1
119894Tr (119880119879119876(119894)119880) minus 120582 = 0 (15)
120597119871 (120590 120582)
120597120582=
119898
sum
119894=1
120590119894minus 1 = 0 (16)
Hence according to (13) and (14) the weight coefficient120590119894can be calculated as
120590119894=
(1Tr (119880119879119876(119894)119880))1(119903minus1)
sum119898
119894=1(1Tr (119880119879119876(119894)119880))1(119903minus1)
(17)
Then we can make the following observations accordingto (15) If 119903 rarr infin then the values of different 120590
119894will be close
to each other If 119903 rarr 1 then only 120590119894= 1 corresponding
to the maximum Tr(119880119879119876(119894)119880) over different views and 120590119894=
0 otherwise Thus the choice of 119903 should respect to thecomplementary property of different views The effect of theparameter 119903 will be discussed in the later experiments
Second we update 119880 by fixing 120590 The optimal problem(10) subject to (11) can be equivalently transformed into thefollowing form
argmax119880
Tr (119880119879119876119880) (18)
subject to
119880119879119880 = 119868 (19)
where119876 = sum119898119894=1120590119903
119894119876(119894) Since119876(119894) defined in (7) is a symmetric
matrix 119876 is also a symmetric matrixObviously the solution of (18) subject to (19) can be
obtained by solving the following standard eigendecomposi-tion problem
119876119880 = 120582119880 (20)
Let the eigenvectors 1198801 1198802 119880
119889be solutions of (20)
ordered according to eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
Then the optimal projection matrix 119880 is given by 119880 =
[1198801 1198802 119880
119889] Now we discuss how to determine the
reduced feature dimension 119889 by using the Ky Fan theorem[23]
Ky FanTheorem Let119867 be a symmetric matrix with eigenval-ues 1205821ge 1205822ge sdot sdot sdot ge 120582
119899and the corresponding eigenvectors
119880 = [1198801 1198802 119880
119899] Then
1205821+ 1205822+ sdot sdot sdot + 120582
119896= arg max
119883119879119883=119868
Tr (119883119879119867119883) (21)
Moreover the optimal 119883lowast is given by 119883lowast = 119880 = [1198801 1198802
119880119896]119877 where 119877 is an arbitrary orthogonal matrix
From the above Ky Fan theorem we can make thefollowing observations The optimal solution to (18) subjectto (19) is composed of the largest 119889 eigenvectors of the matrix119876 and the optimal value of objective function (18) equals thesum of the largest 119889 eigenvalues of the matrix 119876 Thereforethe optimal reduced feature dimension 119889 is equivalent to thenumber of positive eigenvalues of the matrix 119876
Alternately updating 120590 and 119880 by solving (17) and (20)until convergence we can obtain the final optimal projectionmatrix 119880 for multiple views A simple initialization for 120590could be 120590 = [1119898 1119898] According to the afore-mentioned statement the proposed MDGPP algorithm issummarized as follows
Algorithm 1 (MDGPP algorithm)
Input Amultiview data set119883 = 119883(119894) isin 119877119901119894times119899119898
119894=1 the dimen-
sion of the reduced low-dimensional subspace119889 (119889 lt 119901119894 1 le
119894 le 119898) tuning parameter 119903 gt 1 iteration number 119879max andconvergence error 120576
Output Projection matrix 119880
Algorithm
Step 1 Simultaneously consider both intraclass geometry andinterclass discrimination information to calculate 119876(119894) foreach view according to (7)
Step 2 (initialization)
(1) Set 120590 = [1119898 1119898 1119898](2) obtain 1198800 by solving the eigendecomposition prob-
lem (20)
6 The Scientific World Journal
Step 3 (local optimization) For 119905 = 1 2 119879max
(1) calculate 120590 as shown in (17)(2) solve the eigenvalue equation in (20)(3) sort their eigenvectors 119880
1 1198802 119880
119889according to
their corresponding eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
and obtain 119880119905 = [1198801 1198802 119880
119889]
(4) if 119905 gt 2 and |119880119905 minus 119880119905minus1| lt 120576 then go to Step 4
Step 4 (output projection matrix) Output the final optimalprojection matrix 119880 = 119880119905
We now briefly analyze the computational complexityof the MDGPP algorithm which is dominated by threeparts One is for constructing the matrix 119876(119894) for differentviews As shown in (7) the computational complexity ofthis part is 119874((sum119898
119894=1119901119894) times 119899
2) In addition each iteration
involves computing view weight 120590 and solving a standardeigendecomposition problem the computational complexityof running two parts in each iteration is 119874((119898 + 119889) times 1198992)and 119874(1198993) respectively Therefore the total computationalcomplexity of MDGPP is 119874((sum119898
119894=1119901119894) times 1198992+ ((119898 + 119889) times 119899
2+
1198993) times 119879max) where 119879max denotes the iteration number and is
always set to less than five in all experiments
4 Experimental Results
In this section we evaluate the effectiveness of our proposedMDGPP algorithm for two image classification tasks includ-ing face recognition and facial expression recognition Twowidely used face databases including AR [24] and CMU PIE[25] are employed for face recognition evaluation and thewell-known Japanese female facial expression (JAFFE) [26]database is used for facial expression recognition evaluationWe also compare the proposed MDGPP algorithm withsome traditional single-view-based dimensionality reductionalgorithms such as PCA [2] LDA [2] LPP [3] MFA [10]DGPP [12] and the three latest multiview dimensionalityreduction algorithms includingMVSE [18]MSNE [21]MSE[19] and SSMVE [22] The nearest neighbor classifier withthe Euclidean distance was adopted for classification For afair comparison all the results reported here are based on thebest tuned parameters of all the compared algorithms
41 Data Sets and Experimental Settings We conducted facerecognition experiments on the widely used AR and CMUPIE face databases and facial expression recognition experi-ments on the well-known Japanese female facial expression(JAFFE) database
The AR database [24] contains over 4000 color imagescorresponding to 126 people (70 men and 56 women) whichinclude frontal view faces with different facial expressionsillumination conditions and occlusions (sun glasses andscarf) Each person has 26 different images taken in twosessions (separated by two weeks) In our experiments weused a subset of 800 face images from 100 persons (50 menand 50women)with eight face images of different expressionsand lighting conditions per person Figure 1 shows eight
Figure 1 Sample face images from the AR database
Figure 2 Sample face images from the CMU PIE database
sample images of one individual from the subset of the ARdatabase
The CMU PIE database [25] comprises more than 40000facial images of 68 people with different poses illuminationconditions and facial expressions In this experiment weselected a subset of the CMU PIE database which consistsof 3060 frontal face images with varying expression andillumination from 68 persons with 45 images from eachperson Figure 2 shows some sample images of one individualfrom the subset of the CMU PIE database
The Japanese female facial expression (JAFFE) database[26] contains 213 facial images of ten Japanese womenEach facial image shows one of seven expressions neutralhappiness sadness surprise anger disgust or fear Figure 3shows some facial images from the JAFFE database In thisexperiment following the general setting scheme of facialexpression recognition we discard all the neutral facialimages and only utilize the remainder 183 facial images whichinclude six basic facial expressions
For all the face images in the above three face databasesthe facial part of each image was manually aligned croppedand resized into 32 times 32 according to the eyersquos positions Foreach facial image we extract the commonly used four kindsof low-level visual features to represent four different viewsThese four features include color histogram (CH) [27] scale-invariant feature transform (SIFT) [28] Gabor [29] and localbinary pattern (LBP) [30] For the CH feature extraction weused 64 bins to encode a histogram feature for each facialimage according to [27] For the SIFT feature extractionwe densely sampled and calculated the SIFT descriptors of16 times 16 patches over a grid with spacing of 8 pixels accordingto [28] For the Gabor feature extraction following [29] weadopted 40 Gabor kernel functions from five scales and eightorientations For the LBP feature extraction we followed theparameter settings in [30] and utilized 256 bins to encodea histogram feature for each facial image For more detailson these four feature descriptors please refer to [27ndash30]Because these four features are complementary to each otherin representing facial images we empirically set the tuningparameter 119903 in MDGPP to be five
In this experiment each facial image set was partitionedinto the nonoverlap training and testing sets For eachdatabase we randomly selected 50 data as the training setand use the remaining 50 data as the testing set To reducestatistical variation for each random partition we repeatedthese trials independently ten times and reported the averagerecognition results
The Scientific World Journal 7
Figure 3 Sample facial images from the JAFFE database
42 Compared Algorithms We compared our proposedMDGPP algorithm with the following dimensionality reduc-tion algorithms
(1) PCA [2] PCA is an unsupervised dimensionalityreduction algorithm
(2) LDA [2] LDA is a supervised dimensionality reduc-tion algorithm We adopted a Tikhonov regular-ization term 120583119868 rather than PCA preprocessing toavoid the well-known small sample size (singularity)problem in LDA
(3) LPP [3] LPP is an unsupervised manifold learningalgorithm There is a nearest neighbor number 119896 tobe tuned in LPP and it was empirically set to befive in our experiments In addition the Tikhonovregularization was also adopted to avoid the smallsample size (singularity) problem in LPP
(4) MFA [10] MFA is a supervised manifold learningalgorithm There are two parameters (ie 119896
1nearest
neighbor number and 1198962nearest neighbor number)
to be tuned in MFA We empirically set 1198961= 5
and 1198962= 20 in our experiments Meanwhile the
Tikhonov regularizationwas also adopted to avoid thesmall sample size (singularity) problem in MFA
(5) DGPP [12] DGPP is a supervised manifold learningalgorithmThere is a tradeoff parameter 120582 to be tunedin DGPP and it was empirically set to be one in ourexperiments
(6) MVSE [18] MSVE is an initially proposed multiviewalgorithm for dimensionality reduction
(7) MSNE [21] MSNE is a probability-based unsuper-vised multiview algorithm We followed the param-eter setting in [21] and set the tradeoff coefficient 120582 tobe five in our experiments
(8) MSE [19] MSE is a supervised multiview algorithmThere are two parameters (ie the nearest neighbornumber 119896 and the tuning coefficient 119903) to be tunedin MSE We empirically set 119896 = 5 and 119903 = 5 in ourexperiments
(9) SSMVE [22] SSMVE is a sparse unsupervised mul-tiview algorithm We followed the parameter settingmethod in [22] and set the regularized parameter 120582 tobe one in our experiments
It is worth noting that since PCA LDA LPP MFAand DGPP are all single-view-based algorithms these fivealgorithms adopt the conventional feature concatenation-based strategy to cope with the multiview data
Table 1 Comparisons of recognition accuracy on the AR database
Algorithm Accuracy DimensionsPCA 815 120LDA 904 99LPP 908 100MFA 919 100DGPP 932 100MVSE 934 100MSNE 937 100SSMVE 941 100MSE 950 100MDGPP 988 100
Table 2 Comparisons of recognition accuracy on the CMU PIEdatabase
Algorithm Accuracy DimensionsPCA 568 80LDA 634 67LPP 652 68MFA 735 68DGPP 784 68MVSE 796 68MSNE 827 68SSMVE 845 68MSE 873 68MDGPP 952 68
Table 3 Comparisons of recognition accuracy on the JAFFEdatabase
Algorithm Accuracy DimensionsPCA 526 20LDA 658 9LPP 715 10MFA 832 10DGPP 863 10MVSE 878 10MSNE 892 10SSMVE 916 10MSE 934 10MDGPP 965 10
43 Experimental Results For each face image databasethe recognition performance of different algorithms wasevaluated on the testing data separately The conventionalnearest neighbor classifier with the Euclidean distance wasapplied to perform recognition in the subspace derived fromdifferent dimensionality reduction algorithms Tables 1 2and 3 report the recognition accuracies and the correspond-ing optimal dimensions obtained on the AR CMU PIE andJAFFE databases respectively Figures 4 5 and 6 illustratethe recognition accuracies versus the variation of reduceddimensions on the AR CMU PIE and JAFFE databases
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
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Distributed Sensor Networks
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Applied Computational Intelligence and Soft Computing
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Electrical and Computer Engineering
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Advances in
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ArtificialNeural Systems
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
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The Scientific World Journal 3
neighbor embedding (SNE) to its multiview version and pro-posed multiview stochastic neighbor embedding (MSNE)Although MSNE operates on a probabilistic framework itis an unsupervised method and its classification abilitiesmay be limited since the class label information is notused in the learning process More recently inspired by therecent advances of sparse coding technique Han et al [22]proposed spectral sparse multiview embedding (SSMVE)method to deal with dimensionality reduction for multiviewdata Although SSMVE can impose sparsity constraint on theloading matrix of multiview dimensionality reduction it isunsupervised and does not explicitly consider the manifoldstructure onwhich the high dimensional data possibly resideIn the next section focusing on the manifold learningand pattern classification we propose a novel multiviewdiscriminative geometry preserving projection (MDGPP)for multiview dimensionality reduction which explicitlyconsiders the local manifold structure and discriminativeinformation as well as the complementary characteristics ofdifferent views in high-dimensional data
3 Multiview Discriminative GeometryPreserving Projection (MDGPP)
In this section we propose a new manifold learning algo-rithm called multiview discriminative geometry preservingprojection (MDGPP) which aims to find a unified low-dimensional and sufficiently smooth embedding over allviews simultaneously To better explain the algorithm detailsof the proposed MDGPP we introduce some importantnotations used in the remainder of this paper Capital letterssuch as 119883 denote data matrices and 119883
119894119895represents the (119894 119895)
entry of 119883 Lower case letters such as 119909 denote data vectorsand 119909
119894represents the 119894th data element of 119909 Superscript
(119894) such as 119883(119894) and 119909(119894) represents data from the 119894th viewrespectively Based on these notations MDGPP can bedescribed as follows according to the DGPP framework [12]
Given a multiview data set with 119899 data samplesand each with 119898 feature representations that is 119883 =
119883(119894)= [119909(119894)
1 119909(119894)
2 119909
(119894)
119899] isin 119877119901119894times119899119898
119894=1 wherein 119883
(119894) repre-sents the feature matrix for the 119894th view the aim of MDGPPis to find a projective matrix 119882 isin 119877
119901119894times119889 to map 119883(119894) into alow-dimensional representation 119884(119894) through 119884(119894) = 119882119879119883(119894)where 119889 denotes the dimension of low-dimensional featurerepresentation and satisfies 119889 lt 119901
119894(1 le 119894 le 119898) The
workflow of MDGPP can be simply described as followsFirst MDGPP builds a part optimization for a sample on asingle view by preserving both the intraclass geometry andinterclass discrimination Afterward all parts of optimizationfrom different views are unified as a whole via view weightcoefficientsThen an alternating-optimization-based iterativealgorithm is derived to obtain the optimal low-dimensionalembedding from multiple views
Given the 119894th view 119883(119894) = [119909(119894)1 119909(119894)
2 119909
(119894)
119899] isin 119877
119901119894times119899MDGPP first makes an attempt to preserve discrimina-tive information in the reduced low-dimensional space bymaximizing the average weighted pairwise distance betweensamples in different classes and simultaneously minimizing
the average weighted pairwise distance between samples inthe same class on the 119894th view Thus we have
argmax119910(119894)
119895
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
= argmax119910(119894)
119895
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
= argmax119910(119894)
119895
Tr(119899
sum
119895119905=1
ℎ(119894)
119895119905(119910(119894)
119895minus 119910(119894)
119905) (119910(119894)
119895minus 119910(119894)
119905)119879
)
= argmax119910(119894)
119895
Tr(119899
sum
119895119905=1
ℎ(119894)
119895119905(119910(119894)
119895(119910(119894)
119895)119879
minus 119910(119894)
119905(119910(119894)
119895)119879
minus119910(119894)
119895(119910(119894)
119905)119879
+ 119910(119894)
119905(119910(119894)
119905)119879
))
= argmax 2Tr(119884(119894)119863(119894)(119884(119894))119879
minus 119884(119894)119867(119894)(119884(119894))119879
)
= argmax 2Tr(119884(119894)119871(119894)(119884(119894))119879
)
= argmax 2Tr(119880119879119883(119894)119871(119894)(119883(119894))119879
119880)
(1)
where Tr( ) denotes the trace operation ofmatrix 119871(119894)(= 119863(119894)minus119867(119894)) is the graph Laplacian on the 119894th view119863(119894) is a diagonal
matrix with its element 119863(119894)119895119895= sum119905119867(119894)
119895119905on the 119894th view and
119867(119894)= [ℎ
(119894)
119895119905]119899
119895119905=1is the weighting matrix which encodes
both the distance weighting information and the class labelinformation on the 119894th view
ℎ(119894)
119895119905=
119888119895119905(1
119899minus1
119899119897
) if 119897119895= 119897119905= 119897
1
119899 if 119897
119895= 119897119905
(2)
where in 119897119895is the class label of sample 119909(119894)
119895 119899119897is the number of
samples belonging to the 119897th class and 119888119895119905is set as exp(minus119909(119894)
119895minus
119909(119894)
1199051205752) according to LPP [7] for locality preservation
Second we try to implement the local geometry preser-vation by assuming that each sample 119909(119894)
119895can be linearly
reconstructed by the samples 119909(119894)119905which share the same class
label with 119909(119894)119895on the 119894th viewThus we can obtain the recon-
struction coefficient 119908(119894)119895119905
by minimizing the reconstructionerror sum119899
119895=11205761198952 on the 119894th view that is
argmin119908119895119905
119899
sum
119895=1
10038171003817100381710038171003817120576119895
10038171003817100381710038171003817
2
= argmin119908119895119905
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119909(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119909(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2(3)
4 The Scientific World Journal
under the constraint
sum
119905119897119905=119897119895
119908(119894)
119895119905= 1 119908
(119894)
119895119905= 0 for 119897
119905= 119897119895 (4)
Then by solving (3) and (4) we have
119908(119894)
119895=
sum119901119862minus1
119895119901
sum119901119902119862minus1119901119902
(5)
where 119862119901119902= (119909(119894)
119895minus 119909(119894)
119901)119879
(119909(119894)
119895minus 119909(119894)
119902) denotes the local Gram
matrix and 119897119901= 119897119902= 119897119895
Once obtaining the reconstruction coefficient 119908(119894)119895119905
on the119894th view then MDGPP aims to reconstruct 119910(119894)
119895(= 119880
119879119909(119894)
119895)
from 119910(119894)119905(= 119880119879119909(119894)
119905) (where 119897
119905= 119897119895) with 119908(119894)
119895119905in the projected
low-dimensional space thus we have
argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
Tr(119899
sum
119895=1
(119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
times (119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
119879
)
= argmin 2Tr(119884(119894)119868(119894)(119884(119894))119879
minus 119884(119894)119882(119894)(119884(119894))119879
)
= argmin 2Tr(119884(119894) (119868(119894) minus119882(119894)) (119884(119894))119879
)
= argmin 2Tr(119880119879119883(119894) (119868(119894) minus119882(119894)) (119883(119894))119879
119880)
(6)
where 119868(119894) is an identity matrix defined on the 119894th view and119882(119894)= [119908(119894)
119895119905]119899
119895119905=1is the reconstruction coefficient matrix on
the 119894th viewAs a result by combining (1) and (6) together the part
optimization for119883(119894) is
argmax119910(119894)
119895
(
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
minus120582
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905 119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
)
= argmax Tr(119880119879119883(119894)119871(119894)(119883(119894))119879
119880
minus120582119880119879119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
119880)
= argmax Tr(119880119879 (119883(119894)119871(119894)(119883(119894))119879
minus120582119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
)119880)
= argmax Tr (119880119879119876(119894)119880) (7)
where 119876(119894) = 119883(119894)119871(119894)(119883(119894))119879
minus 120582119883(119894)(119868(119894)minus 119882(119894))(119883(119894))119879
and120582 is a tradeoff coefficient which is empirically set as 1 in thisexperiment
Based on the local manifold information encoded in119871(119894) and 119882(119894) (7) aims at finding a sufficiently smooth low-
dimensional embedding 119884(119894)(= 119880119879119883(119894)) by preserving the
interclass discrimination and intraclass geometry on the 119894thview
Because multiviews could provide complementary infor-mation in characterizing data from different viewpointsdifferent views certainly have different contributions to thelow-dimensional feature subspace In order to well discoverthe complementary information of data from different viewsa nonnegative weighted set120590 = [120590
1 1205902 120590
119898] is imposed on
each view independently Generally speaking the larger 120590119894is
the more the contribution of the view 119883(119894) is made to obtainthe low-dimensional feature subspace Hence by summingover all parts of optimization defined in (7) we can formulateMDGPP as the following optimization problem
argmax119880120590
119898
sum
119894=1
120590119894Tr (119880119879119876(119894)119880) (8)
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (9)
The solution to 120590 in (8) subject to (9) is 120590119896= 1
corresponding to the maximum Tr(119880119879119876(119894)119880) over differentviews and 120590
119896= 0 otherwise which means that only the
best view is finally selected by this method Consequentlythis solution cannot meet the demand for exploring thecomplementary characteristics of different views to get abetter low-dimensional embedding than that based on asingle view In order to avoid this problem we set 120590
119894larr 120590119903
119894
with 119903 gt 1 by following the trick utilized in [16ndash19] In thiscondition sum119898
119894=1120590119903
119894= 1 achieves its maximum when 120590
119894=
1119898 according to sum119898119894=1120590119894= 1 and 120590
119894ge 0 Similarly 120590
119894for
different views can be obtained by setting 119903 gt 1 thus eachviewmakes a specific contribution to obtaining the final low-dimensional embedding Consequently the new objectivefunction of MDGPP can be defined as follows
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (10)
The Scientific World Journal 5
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (11)
The above optimization problem is a nonlinearly con-strained nonconvex optimization problem so there is nodirect approach to find its global optimal solution In thispaper we derive an alternating-optimization-based iterativealgorithm to find a local optimal solution The alternatingoptimization iteratively updates the projection matrix 119880 andweight vector 120590 = [120590
1 1205902 120590
119898]
First we update 120590 by fixing 119880 The optimal problem (10)subject to (11) becomes
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (12)
subject to119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (13)
Following the standard Lagrange multiplier we constructthe following Lagrangian function by incorporating theconstraint (13) into (12)
119871 (120590 120582) =
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) minus 120582(
119898
sum
119894=1
120590119894minus 1) (14)
where the Lagrange multiplier 120582 satisfies 120582 ge 0Taking the partial derivation of the Lagrangian function
119871(120590 120582) with respect to 120590119894and 120582 and setting them to zeros we
have120597119871 (120590 120582)
120597120590119894
= 119903120590119903minus1
119894Tr (119880119879119876(119894)119880) minus 120582 = 0 (15)
120597119871 (120590 120582)
120597120582=
119898
sum
119894=1
120590119894minus 1 = 0 (16)
Hence according to (13) and (14) the weight coefficient120590119894can be calculated as
120590119894=
(1Tr (119880119879119876(119894)119880))1(119903minus1)
sum119898
119894=1(1Tr (119880119879119876(119894)119880))1(119903minus1)
(17)
Then we can make the following observations accordingto (15) If 119903 rarr infin then the values of different 120590
119894will be close
to each other If 119903 rarr 1 then only 120590119894= 1 corresponding
to the maximum Tr(119880119879119876(119894)119880) over different views and 120590119894=
0 otherwise Thus the choice of 119903 should respect to thecomplementary property of different views The effect of theparameter 119903 will be discussed in the later experiments
Second we update 119880 by fixing 120590 The optimal problem(10) subject to (11) can be equivalently transformed into thefollowing form
argmax119880
Tr (119880119879119876119880) (18)
subject to
119880119879119880 = 119868 (19)
where119876 = sum119898119894=1120590119903
119894119876(119894) Since119876(119894) defined in (7) is a symmetric
matrix 119876 is also a symmetric matrixObviously the solution of (18) subject to (19) can be
obtained by solving the following standard eigendecomposi-tion problem
119876119880 = 120582119880 (20)
Let the eigenvectors 1198801 1198802 119880
119889be solutions of (20)
ordered according to eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
Then the optimal projection matrix 119880 is given by 119880 =
[1198801 1198802 119880
119889] Now we discuss how to determine the
reduced feature dimension 119889 by using the Ky Fan theorem[23]
Ky FanTheorem Let119867 be a symmetric matrix with eigenval-ues 1205821ge 1205822ge sdot sdot sdot ge 120582
119899and the corresponding eigenvectors
119880 = [1198801 1198802 119880
119899] Then
1205821+ 1205822+ sdot sdot sdot + 120582
119896= arg max
119883119879119883=119868
Tr (119883119879119867119883) (21)
Moreover the optimal 119883lowast is given by 119883lowast = 119880 = [1198801 1198802
119880119896]119877 where 119877 is an arbitrary orthogonal matrix
From the above Ky Fan theorem we can make thefollowing observations The optimal solution to (18) subjectto (19) is composed of the largest 119889 eigenvectors of the matrix119876 and the optimal value of objective function (18) equals thesum of the largest 119889 eigenvalues of the matrix 119876 Thereforethe optimal reduced feature dimension 119889 is equivalent to thenumber of positive eigenvalues of the matrix 119876
Alternately updating 120590 and 119880 by solving (17) and (20)until convergence we can obtain the final optimal projectionmatrix 119880 for multiple views A simple initialization for 120590could be 120590 = [1119898 1119898] According to the afore-mentioned statement the proposed MDGPP algorithm issummarized as follows
Algorithm 1 (MDGPP algorithm)
Input Amultiview data set119883 = 119883(119894) isin 119877119901119894times119899119898
119894=1 the dimen-
sion of the reduced low-dimensional subspace119889 (119889 lt 119901119894 1 le
119894 le 119898) tuning parameter 119903 gt 1 iteration number 119879max andconvergence error 120576
Output Projection matrix 119880
Algorithm
Step 1 Simultaneously consider both intraclass geometry andinterclass discrimination information to calculate 119876(119894) foreach view according to (7)
Step 2 (initialization)
(1) Set 120590 = [1119898 1119898 1119898](2) obtain 1198800 by solving the eigendecomposition prob-
lem (20)
6 The Scientific World Journal
Step 3 (local optimization) For 119905 = 1 2 119879max
(1) calculate 120590 as shown in (17)(2) solve the eigenvalue equation in (20)(3) sort their eigenvectors 119880
1 1198802 119880
119889according to
their corresponding eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
and obtain 119880119905 = [1198801 1198802 119880
119889]
(4) if 119905 gt 2 and |119880119905 minus 119880119905minus1| lt 120576 then go to Step 4
Step 4 (output projection matrix) Output the final optimalprojection matrix 119880 = 119880119905
We now briefly analyze the computational complexityof the MDGPP algorithm which is dominated by threeparts One is for constructing the matrix 119876(119894) for differentviews As shown in (7) the computational complexity ofthis part is 119874((sum119898
119894=1119901119894) times 119899
2) In addition each iteration
involves computing view weight 120590 and solving a standardeigendecomposition problem the computational complexityof running two parts in each iteration is 119874((119898 + 119889) times 1198992)and 119874(1198993) respectively Therefore the total computationalcomplexity of MDGPP is 119874((sum119898
119894=1119901119894) times 1198992+ ((119898 + 119889) times 119899
2+
1198993) times 119879max) where 119879max denotes the iteration number and is
always set to less than five in all experiments
4 Experimental Results
In this section we evaluate the effectiveness of our proposedMDGPP algorithm for two image classification tasks includ-ing face recognition and facial expression recognition Twowidely used face databases including AR [24] and CMU PIE[25] are employed for face recognition evaluation and thewell-known Japanese female facial expression (JAFFE) [26]database is used for facial expression recognition evaluationWe also compare the proposed MDGPP algorithm withsome traditional single-view-based dimensionality reductionalgorithms such as PCA [2] LDA [2] LPP [3] MFA [10]DGPP [12] and the three latest multiview dimensionalityreduction algorithms includingMVSE [18]MSNE [21]MSE[19] and SSMVE [22] The nearest neighbor classifier withthe Euclidean distance was adopted for classification For afair comparison all the results reported here are based on thebest tuned parameters of all the compared algorithms
41 Data Sets and Experimental Settings We conducted facerecognition experiments on the widely used AR and CMUPIE face databases and facial expression recognition experi-ments on the well-known Japanese female facial expression(JAFFE) database
The AR database [24] contains over 4000 color imagescorresponding to 126 people (70 men and 56 women) whichinclude frontal view faces with different facial expressionsillumination conditions and occlusions (sun glasses andscarf) Each person has 26 different images taken in twosessions (separated by two weeks) In our experiments weused a subset of 800 face images from 100 persons (50 menand 50women)with eight face images of different expressionsand lighting conditions per person Figure 1 shows eight
Figure 1 Sample face images from the AR database
Figure 2 Sample face images from the CMU PIE database
sample images of one individual from the subset of the ARdatabase
The CMU PIE database [25] comprises more than 40000facial images of 68 people with different poses illuminationconditions and facial expressions In this experiment weselected a subset of the CMU PIE database which consistsof 3060 frontal face images with varying expression andillumination from 68 persons with 45 images from eachperson Figure 2 shows some sample images of one individualfrom the subset of the CMU PIE database
The Japanese female facial expression (JAFFE) database[26] contains 213 facial images of ten Japanese womenEach facial image shows one of seven expressions neutralhappiness sadness surprise anger disgust or fear Figure 3shows some facial images from the JAFFE database In thisexperiment following the general setting scheme of facialexpression recognition we discard all the neutral facialimages and only utilize the remainder 183 facial images whichinclude six basic facial expressions
For all the face images in the above three face databasesthe facial part of each image was manually aligned croppedand resized into 32 times 32 according to the eyersquos positions Foreach facial image we extract the commonly used four kindsof low-level visual features to represent four different viewsThese four features include color histogram (CH) [27] scale-invariant feature transform (SIFT) [28] Gabor [29] and localbinary pattern (LBP) [30] For the CH feature extraction weused 64 bins to encode a histogram feature for each facialimage according to [27] For the SIFT feature extractionwe densely sampled and calculated the SIFT descriptors of16 times 16 patches over a grid with spacing of 8 pixels accordingto [28] For the Gabor feature extraction following [29] weadopted 40 Gabor kernel functions from five scales and eightorientations For the LBP feature extraction we followed theparameter settings in [30] and utilized 256 bins to encodea histogram feature for each facial image For more detailson these four feature descriptors please refer to [27ndash30]Because these four features are complementary to each otherin representing facial images we empirically set the tuningparameter 119903 in MDGPP to be five
In this experiment each facial image set was partitionedinto the nonoverlap training and testing sets For eachdatabase we randomly selected 50 data as the training setand use the remaining 50 data as the testing set To reducestatistical variation for each random partition we repeatedthese trials independently ten times and reported the averagerecognition results
The Scientific World Journal 7
Figure 3 Sample facial images from the JAFFE database
42 Compared Algorithms We compared our proposedMDGPP algorithm with the following dimensionality reduc-tion algorithms
(1) PCA [2] PCA is an unsupervised dimensionalityreduction algorithm
(2) LDA [2] LDA is a supervised dimensionality reduc-tion algorithm We adopted a Tikhonov regular-ization term 120583119868 rather than PCA preprocessing toavoid the well-known small sample size (singularity)problem in LDA
(3) LPP [3] LPP is an unsupervised manifold learningalgorithm There is a nearest neighbor number 119896 tobe tuned in LPP and it was empirically set to befive in our experiments In addition the Tikhonovregularization was also adopted to avoid the smallsample size (singularity) problem in LPP
(4) MFA [10] MFA is a supervised manifold learningalgorithm There are two parameters (ie 119896
1nearest
neighbor number and 1198962nearest neighbor number)
to be tuned in MFA We empirically set 1198961= 5
and 1198962= 20 in our experiments Meanwhile the
Tikhonov regularizationwas also adopted to avoid thesmall sample size (singularity) problem in MFA
(5) DGPP [12] DGPP is a supervised manifold learningalgorithmThere is a tradeoff parameter 120582 to be tunedin DGPP and it was empirically set to be one in ourexperiments
(6) MVSE [18] MSVE is an initially proposed multiviewalgorithm for dimensionality reduction
(7) MSNE [21] MSNE is a probability-based unsuper-vised multiview algorithm We followed the param-eter setting in [21] and set the tradeoff coefficient 120582 tobe five in our experiments
(8) MSE [19] MSE is a supervised multiview algorithmThere are two parameters (ie the nearest neighbornumber 119896 and the tuning coefficient 119903) to be tunedin MSE We empirically set 119896 = 5 and 119903 = 5 in ourexperiments
(9) SSMVE [22] SSMVE is a sparse unsupervised mul-tiview algorithm We followed the parameter settingmethod in [22] and set the regularized parameter 120582 tobe one in our experiments
It is worth noting that since PCA LDA LPP MFAand DGPP are all single-view-based algorithms these fivealgorithms adopt the conventional feature concatenation-based strategy to cope with the multiview data
Table 1 Comparisons of recognition accuracy on the AR database
Algorithm Accuracy DimensionsPCA 815 120LDA 904 99LPP 908 100MFA 919 100DGPP 932 100MVSE 934 100MSNE 937 100SSMVE 941 100MSE 950 100MDGPP 988 100
Table 2 Comparisons of recognition accuracy on the CMU PIEdatabase
Algorithm Accuracy DimensionsPCA 568 80LDA 634 67LPP 652 68MFA 735 68DGPP 784 68MVSE 796 68MSNE 827 68SSMVE 845 68MSE 873 68MDGPP 952 68
Table 3 Comparisons of recognition accuracy on the JAFFEdatabase
Algorithm Accuracy DimensionsPCA 526 20LDA 658 9LPP 715 10MFA 832 10DGPP 863 10MVSE 878 10MSNE 892 10SSMVE 916 10MSE 934 10MDGPP 965 10
43 Experimental Results For each face image databasethe recognition performance of different algorithms wasevaluated on the testing data separately The conventionalnearest neighbor classifier with the Euclidean distance wasapplied to perform recognition in the subspace derived fromdifferent dimensionality reduction algorithms Tables 1 2and 3 report the recognition accuracies and the correspond-ing optimal dimensions obtained on the AR CMU PIE andJAFFE databases respectively Figures 4 5 and 6 illustratethe recognition accuracies versus the variation of reduceddimensions on the AR CMU PIE and JAFFE databases
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
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[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
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4 The Scientific World Journal
under the constraint
sum
119905119897119905=119897119895
119908(119894)
119895119905= 1 119908
(119894)
119895119905= 0 for 119897
119905= 119897119895 (4)
Then by solving (3) and (4) we have
119908(119894)
119895=
sum119901119862minus1
119895119901
sum119901119902119862minus1119901119902
(5)
where 119862119901119902= (119909(119894)
119895minus 119909(119894)
119901)119879
(119909(119894)
119895minus 119909(119894)
119902) denotes the local Gram
matrix and 119897119901= 119897119902= 119897119895
Once obtaining the reconstruction coefficient 119908(119894)119895119905
on the119894th view then MDGPP aims to reconstruct 119910(119894)
119895(= 119880
119879119909(119894)
119895)
from 119910(119894)119905(= 119880119879119909(119894)
119905) (where 119897
119905= 119897119895) with 119908(119894)
119895119905in the projected
low-dimensional space thus we have
argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
= argmin119910(119894)
119895
Tr(119899
sum
119895=1
(119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
times (119910(119894)
119895minus sum
119905119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905)
119879
)
= argmin 2Tr(119884(119894)119868(119894)(119884(119894))119879
minus 119884(119894)119882(119894)(119884(119894))119879
)
= argmin 2Tr(119884(119894) (119868(119894) minus119882(119894)) (119884(119894))119879
)
= argmin 2Tr(119880119879119883(119894) (119868(119894) minus119882(119894)) (119883(119894))119879
119880)
(6)
where 119868(119894) is an identity matrix defined on the 119894th view and119882(119894)= [119908(119894)
119895119905]119899
119895119905=1is the reconstruction coefficient matrix on
the 119894th viewAs a result by combining (1) and (6) together the part
optimization for119883(119894) is
argmax119910(119894)
119895
(
119899
sum
119895119905=1
ℎ(119894)
119895119905
10038171003817100381710038171003817119910(119894)
119895minus 119910(119894)
119905
10038171003817100381710038171003817
2
minus120582
119899
sum
119895=1
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
119910(119894)
119895minus sum
119905 119897119905=119897119895
119908(119894)
119895119905119910(119894)
119905
100381710038171003817100381710038171003817100381710038171003817100381710038171003817
2
)
= argmax Tr(119880119879119883(119894)119871(119894)(119883(119894))119879
119880
minus120582119880119879119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
119880)
= argmax Tr(119880119879 (119883(119894)119871(119894)(119883(119894))119879
minus120582119883(119894)(119868(119894)minus119882(119894)) (119883(119894))119879
)119880)
= argmax Tr (119880119879119876(119894)119880) (7)
where 119876(119894) = 119883(119894)119871(119894)(119883(119894))119879
minus 120582119883(119894)(119868(119894)minus 119882(119894))(119883(119894))119879
and120582 is a tradeoff coefficient which is empirically set as 1 in thisexperiment
Based on the local manifold information encoded in119871(119894) and 119882(119894) (7) aims at finding a sufficiently smooth low-
dimensional embedding 119884(119894)(= 119880119879119883(119894)) by preserving the
interclass discrimination and intraclass geometry on the 119894thview
Because multiviews could provide complementary infor-mation in characterizing data from different viewpointsdifferent views certainly have different contributions to thelow-dimensional feature subspace In order to well discoverthe complementary information of data from different viewsa nonnegative weighted set120590 = [120590
1 1205902 120590
119898] is imposed on
each view independently Generally speaking the larger 120590119894is
the more the contribution of the view 119883(119894) is made to obtainthe low-dimensional feature subspace Hence by summingover all parts of optimization defined in (7) we can formulateMDGPP as the following optimization problem
argmax119880120590
119898
sum
119894=1
120590119894Tr (119880119879119876(119894)119880) (8)
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (9)
The solution to 120590 in (8) subject to (9) is 120590119896= 1
corresponding to the maximum Tr(119880119879119876(119894)119880) over differentviews and 120590
119896= 0 otherwise which means that only the
best view is finally selected by this method Consequentlythis solution cannot meet the demand for exploring thecomplementary characteristics of different views to get abetter low-dimensional embedding than that based on asingle view In order to avoid this problem we set 120590
119894larr 120590119903
119894
with 119903 gt 1 by following the trick utilized in [16ndash19] In thiscondition sum119898
119894=1120590119903
119894= 1 achieves its maximum when 120590
119894=
1119898 according to sum119898119894=1120590119894= 1 and 120590
119894ge 0 Similarly 120590
119894for
different views can be obtained by setting 119903 gt 1 thus eachviewmakes a specific contribution to obtaining the final low-dimensional embedding Consequently the new objectivefunction of MDGPP can be defined as follows
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (10)
The Scientific World Journal 5
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (11)
The above optimization problem is a nonlinearly con-strained nonconvex optimization problem so there is nodirect approach to find its global optimal solution In thispaper we derive an alternating-optimization-based iterativealgorithm to find a local optimal solution The alternatingoptimization iteratively updates the projection matrix 119880 andweight vector 120590 = [120590
1 1205902 120590
119898]
First we update 120590 by fixing 119880 The optimal problem (10)subject to (11) becomes
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (12)
subject to119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (13)
Following the standard Lagrange multiplier we constructthe following Lagrangian function by incorporating theconstraint (13) into (12)
119871 (120590 120582) =
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) minus 120582(
119898
sum
119894=1
120590119894minus 1) (14)
where the Lagrange multiplier 120582 satisfies 120582 ge 0Taking the partial derivation of the Lagrangian function
119871(120590 120582) with respect to 120590119894and 120582 and setting them to zeros we
have120597119871 (120590 120582)
120597120590119894
= 119903120590119903minus1
119894Tr (119880119879119876(119894)119880) minus 120582 = 0 (15)
120597119871 (120590 120582)
120597120582=
119898
sum
119894=1
120590119894minus 1 = 0 (16)
Hence according to (13) and (14) the weight coefficient120590119894can be calculated as
120590119894=
(1Tr (119880119879119876(119894)119880))1(119903minus1)
sum119898
119894=1(1Tr (119880119879119876(119894)119880))1(119903minus1)
(17)
Then we can make the following observations accordingto (15) If 119903 rarr infin then the values of different 120590
119894will be close
to each other If 119903 rarr 1 then only 120590119894= 1 corresponding
to the maximum Tr(119880119879119876(119894)119880) over different views and 120590119894=
0 otherwise Thus the choice of 119903 should respect to thecomplementary property of different views The effect of theparameter 119903 will be discussed in the later experiments
Second we update 119880 by fixing 120590 The optimal problem(10) subject to (11) can be equivalently transformed into thefollowing form
argmax119880
Tr (119880119879119876119880) (18)
subject to
119880119879119880 = 119868 (19)
where119876 = sum119898119894=1120590119903
119894119876(119894) Since119876(119894) defined in (7) is a symmetric
matrix 119876 is also a symmetric matrixObviously the solution of (18) subject to (19) can be
obtained by solving the following standard eigendecomposi-tion problem
119876119880 = 120582119880 (20)
Let the eigenvectors 1198801 1198802 119880
119889be solutions of (20)
ordered according to eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
Then the optimal projection matrix 119880 is given by 119880 =
[1198801 1198802 119880
119889] Now we discuss how to determine the
reduced feature dimension 119889 by using the Ky Fan theorem[23]
Ky FanTheorem Let119867 be a symmetric matrix with eigenval-ues 1205821ge 1205822ge sdot sdot sdot ge 120582
119899and the corresponding eigenvectors
119880 = [1198801 1198802 119880
119899] Then
1205821+ 1205822+ sdot sdot sdot + 120582
119896= arg max
119883119879119883=119868
Tr (119883119879119867119883) (21)
Moreover the optimal 119883lowast is given by 119883lowast = 119880 = [1198801 1198802
119880119896]119877 where 119877 is an arbitrary orthogonal matrix
From the above Ky Fan theorem we can make thefollowing observations The optimal solution to (18) subjectto (19) is composed of the largest 119889 eigenvectors of the matrix119876 and the optimal value of objective function (18) equals thesum of the largest 119889 eigenvalues of the matrix 119876 Thereforethe optimal reduced feature dimension 119889 is equivalent to thenumber of positive eigenvalues of the matrix 119876
Alternately updating 120590 and 119880 by solving (17) and (20)until convergence we can obtain the final optimal projectionmatrix 119880 for multiple views A simple initialization for 120590could be 120590 = [1119898 1119898] According to the afore-mentioned statement the proposed MDGPP algorithm issummarized as follows
Algorithm 1 (MDGPP algorithm)
Input Amultiview data set119883 = 119883(119894) isin 119877119901119894times119899119898
119894=1 the dimen-
sion of the reduced low-dimensional subspace119889 (119889 lt 119901119894 1 le
119894 le 119898) tuning parameter 119903 gt 1 iteration number 119879max andconvergence error 120576
Output Projection matrix 119880
Algorithm
Step 1 Simultaneously consider both intraclass geometry andinterclass discrimination information to calculate 119876(119894) foreach view according to (7)
Step 2 (initialization)
(1) Set 120590 = [1119898 1119898 1119898](2) obtain 1198800 by solving the eigendecomposition prob-
lem (20)
6 The Scientific World Journal
Step 3 (local optimization) For 119905 = 1 2 119879max
(1) calculate 120590 as shown in (17)(2) solve the eigenvalue equation in (20)(3) sort their eigenvectors 119880
1 1198802 119880
119889according to
their corresponding eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
and obtain 119880119905 = [1198801 1198802 119880
119889]
(4) if 119905 gt 2 and |119880119905 minus 119880119905minus1| lt 120576 then go to Step 4
Step 4 (output projection matrix) Output the final optimalprojection matrix 119880 = 119880119905
We now briefly analyze the computational complexityof the MDGPP algorithm which is dominated by threeparts One is for constructing the matrix 119876(119894) for differentviews As shown in (7) the computational complexity ofthis part is 119874((sum119898
119894=1119901119894) times 119899
2) In addition each iteration
involves computing view weight 120590 and solving a standardeigendecomposition problem the computational complexityof running two parts in each iteration is 119874((119898 + 119889) times 1198992)and 119874(1198993) respectively Therefore the total computationalcomplexity of MDGPP is 119874((sum119898
119894=1119901119894) times 1198992+ ((119898 + 119889) times 119899
2+
1198993) times 119879max) where 119879max denotes the iteration number and is
always set to less than five in all experiments
4 Experimental Results
In this section we evaluate the effectiveness of our proposedMDGPP algorithm for two image classification tasks includ-ing face recognition and facial expression recognition Twowidely used face databases including AR [24] and CMU PIE[25] are employed for face recognition evaluation and thewell-known Japanese female facial expression (JAFFE) [26]database is used for facial expression recognition evaluationWe also compare the proposed MDGPP algorithm withsome traditional single-view-based dimensionality reductionalgorithms such as PCA [2] LDA [2] LPP [3] MFA [10]DGPP [12] and the three latest multiview dimensionalityreduction algorithms includingMVSE [18]MSNE [21]MSE[19] and SSMVE [22] The nearest neighbor classifier withthe Euclidean distance was adopted for classification For afair comparison all the results reported here are based on thebest tuned parameters of all the compared algorithms
41 Data Sets and Experimental Settings We conducted facerecognition experiments on the widely used AR and CMUPIE face databases and facial expression recognition experi-ments on the well-known Japanese female facial expression(JAFFE) database
The AR database [24] contains over 4000 color imagescorresponding to 126 people (70 men and 56 women) whichinclude frontal view faces with different facial expressionsillumination conditions and occlusions (sun glasses andscarf) Each person has 26 different images taken in twosessions (separated by two weeks) In our experiments weused a subset of 800 face images from 100 persons (50 menand 50women)with eight face images of different expressionsand lighting conditions per person Figure 1 shows eight
Figure 1 Sample face images from the AR database
Figure 2 Sample face images from the CMU PIE database
sample images of one individual from the subset of the ARdatabase
The CMU PIE database [25] comprises more than 40000facial images of 68 people with different poses illuminationconditions and facial expressions In this experiment weselected a subset of the CMU PIE database which consistsof 3060 frontal face images with varying expression andillumination from 68 persons with 45 images from eachperson Figure 2 shows some sample images of one individualfrom the subset of the CMU PIE database
The Japanese female facial expression (JAFFE) database[26] contains 213 facial images of ten Japanese womenEach facial image shows one of seven expressions neutralhappiness sadness surprise anger disgust or fear Figure 3shows some facial images from the JAFFE database In thisexperiment following the general setting scheme of facialexpression recognition we discard all the neutral facialimages and only utilize the remainder 183 facial images whichinclude six basic facial expressions
For all the face images in the above three face databasesthe facial part of each image was manually aligned croppedand resized into 32 times 32 according to the eyersquos positions Foreach facial image we extract the commonly used four kindsof low-level visual features to represent four different viewsThese four features include color histogram (CH) [27] scale-invariant feature transform (SIFT) [28] Gabor [29] and localbinary pattern (LBP) [30] For the CH feature extraction weused 64 bins to encode a histogram feature for each facialimage according to [27] For the SIFT feature extractionwe densely sampled and calculated the SIFT descriptors of16 times 16 patches over a grid with spacing of 8 pixels accordingto [28] For the Gabor feature extraction following [29] weadopted 40 Gabor kernel functions from five scales and eightorientations For the LBP feature extraction we followed theparameter settings in [30] and utilized 256 bins to encodea histogram feature for each facial image For more detailson these four feature descriptors please refer to [27ndash30]Because these four features are complementary to each otherin representing facial images we empirically set the tuningparameter 119903 in MDGPP to be five
In this experiment each facial image set was partitionedinto the nonoverlap training and testing sets For eachdatabase we randomly selected 50 data as the training setand use the remaining 50 data as the testing set To reducestatistical variation for each random partition we repeatedthese trials independently ten times and reported the averagerecognition results
The Scientific World Journal 7
Figure 3 Sample facial images from the JAFFE database
42 Compared Algorithms We compared our proposedMDGPP algorithm with the following dimensionality reduc-tion algorithms
(1) PCA [2] PCA is an unsupervised dimensionalityreduction algorithm
(2) LDA [2] LDA is a supervised dimensionality reduc-tion algorithm We adopted a Tikhonov regular-ization term 120583119868 rather than PCA preprocessing toavoid the well-known small sample size (singularity)problem in LDA
(3) LPP [3] LPP is an unsupervised manifold learningalgorithm There is a nearest neighbor number 119896 tobe tuned in LPP and it was empirically set to befive in our experiments In addition the Tikhonovregularization was also adopted to avoid the smallsample size (singularity) problem in LPP
(4) MFA [10] MFA is a supervised manifold learningalgorithm There are two parameters (ie 119896
1nearest
neighbor number and 1198962nearest neighbor number)
to be tuned in MFA We empirically set 1198961= 5
and 1198962= 20 in our experiments Meanwhile the
Tikhonov regularizationwas also adopted to avoid thesmall sample size (singularity) problem in MFA
(5) DGPP [12] DGPP is a supervised manifold learningalgorithmThere is a tradeoff parameter 120582 to be tunedin DGPP and it was empirically set to be one in ourexperiments
(6) MVSE [18] MSVE is an initially proposed multiviewalgorithm for dimensionality reduction
(7) MSNE [21] MSNE is a probability-based unsuper-vised multiview algorithm We followed the param-eter setting in [21] and set the tradeoff coefficient 120582 tobe five in our experiments
(8) MSE [19] MSE is a supervised multiview algorithmThere are two parameters (ie the nearest neighbornumber 119896 and the tuning coefficient 119903) to be tunedin MSE We empirically set 119896 = 5 and 119903 = 5 in ourexperiments
(9) SSMVE [22] SSMVE is a sparse unsupervised mul-tiview algorithm We followed the parameter settingmethod in [22] and set the regularized parameter 120582 tobe one in our experiments
It is worth noting that since PCA LDA LPP MFAand DGPP are all single-view-based algorithms these fivealgorithms adopt the conventional feature concatenation-based strategy to cope with the multiview data
Table 1 Comparisons of recognition accuracy on the AR database
Algorithm Accuracy DimensionsPCA 815 120LDA 904 99LPP 908 100MFA 919 100DGPP 932 100MVSE 934 100MSNE 937 100SSMVE 941 100MSE 950 100MDGPP 988 100
Table 2 Comparisons of recognition accuracy on the CMU PIEdatabase
Algorithm Accuracy DimensionsPCA 568 80LDA 634 67LPP 652 68MFA 735 68DGPP 784 68MVSE 796 68MSNE 827 68SSMVE 845 68MSE 873 68MDGPP 952 68
Table 3 Comparisons of recognition accuracy on the JAFFEdatabase
Algorithm Accuracy DimensionsPCA 526 20LDA 658 9LPP 715 10MFA 832 10DGPP 863 10MVSE 878 10MSNE 892 10SSMVE 916 10MSE 934 10MDGPP 965 10
43 Experimental Results For each face image databasethe recognition performance of different algorithms wasevaluated on the testing data separately The conventionalnearest neighbor classifier with the Euclidean distance wasapplied to perform recognition in the subspace derived fromdifferent dimensionality reduction algorithms Tables 1 2and 3 report the recognition accuracies and the correspond-ing optimal dimensions obtained on the AR CMU PIE andJAFFE databases respectively Figures 4 5 and 6 illustratethe recognition accuracies versus the variation of reduceddimensions on the AR CMU PIE and JAFFE databases
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Human-ComputerInteraction
Advances in
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The Scientific World Journal 5
subject to
119880119879119880 = 119868
119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (11)
The above optimization problem is a nonlinearly con-strained nonconvex optimization problem so there is nodirect approach to find its global optimal solution In thispaper we derive an alternating-optimization-based iterativealgorithm to find a local optimal solution The alternatingoptimization iteratively updates the projection matrix 119880 andweight vector 120590 = [120590
1 1205902 120590
119898]
First we update 120590 by fixing 119880 The optimal problem (10)subject to (11) becomes
argmax119880120590
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) (12)
subject to119898
sum
119894=1
120590119894= 1 120590
119894ge 0 (13)
Following the standard Lagrange multiplier we constructthe following Lagrangian function by incorporating theconstraint (13) into (12)
119871 (120590 120582) =
119898
sum
119894=1
120590119903
119894Tr (119880119879119876(119894)119880) minus 120582(
119898
sum
119894=1
120590119894minus 1) (14)
where the Lagrange multiplier 120582 satisfies 120582 ge 0Taking the partial derivation of the Lagrangian function
119871(120590 120582) with respect to 120590119894and 120582 and setting them to zeros we
have120597119871 (120590 120582)
120597120590119894
= 119903120590119903minus1
119894Tr (119880119879119876(119894)119880) minus 120582 = 0 (15)
120597119871 (120590 120582)
120597120582=
119898
sum
119894=1
120590119894minus 1 = 0 (16)
Hence according to (13) and (14) the weight coefficient120590119894can be calculated as
120590119894=
(1Tr (119880119879119876(119894)119880))1(119903minus1)
sum119898
119894=1(1Tr (119880119879119876(119894)119880))1(119903minus1)
(17)
Then we can make the following observations accordingto (15) If 119903 rarr infin then the values of different 120590
119894will be close
to each other If 119903 rarr 1 then only 120590119894= 1 corresponding
to the maximum Tr(119880119879119876(119894)119880) over different views and 120590119894=
0 otherwise Thus the choice of 119903 should respect to thecomplementary property of different views The effect of theparameter 119903 will be discussed in the later experiments
Second we update 119880 by fixing 120590 The optimal problem(10) subject to (11) can be equivalently transformed into thefollowing form
argmax119880
Tr (119880119879119876119880) (18)
subject to
119880119879119880 = 119868 (19)
where119876 = sum119898119894=1120590119903
119894119876(119894) Since119876(119894) defined in (7) is a symmetric
matrix 119876 is also a symmetric matrixObviously the solution of (18) subject to (19) can be
obtained by solving the following standard eigendecomposi-tion problem
119876119880 = 120582119880 (20)
Let the eigenvectors 1198801 1198802 119880
119889be solutions of (20)
ordered according to eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
Then the optimal projection matrix 119880 is given by 119880 =
[1198801 1198802 119880
119889] Now we discuss how to determine the
reduced feature dimension 119889 by using the Ky Fan theorem[23]
Ky FanTheorem Let119867 be a symmetric matrix with eigenval-ues 1205821ge 1205822ge sdot sdot sdot ge 120582
119899and the corresponding eigenvectors
119880 = [1198801 1198802 119880
119899] Then
1205821+ 1205822+ sdot sdot sdot + 120582
119896= arg max
119883119879119883=119868
Tr (119883119879119867119883) (21)
Moreover the optimal 119883lowast is given by 119883lowast = 119880 = [1198801 1198802
119880119896]119877 where 119877 is an arbitrary orthogonal matrix
From the above Ky Fan theorem we can make thefollowing observations The optimal solution to (18) subjectto (19) is composed of the largest 119889 eigenvectors of the matrix119876 and the optimal value of objective function (18) equals thesum of the largest 119889 eigenvalues of the matrix 119876 Thereforethe optimal reduced feature dimension 119889 is equivalent to thenumber of positive eigenvalues of the matrix 119876
Alternately updating 120590 and 119880 by solving (17) and (20)until convergence we can obtain the final optimal projectionmatrix 119880 for multiple views A simple initialization for 120590could be 120590 = [1119898 1119898] According to the afore-mentioned statement the proposed MDGPP algorithm issummarized as follows
Algorithm 1 (MDGPP algorithm)
Input Amultiview data set119883 = 119883(119894) isin 119877119901119894times119899119898
119894=1 the dimen-
sion of the reduced low-dimensional subspace119889 (119889 lt 119901119894 1 le
119894 le 119898) tuning parameter 119903 gt 1 iteration number 119879max andconvergence error 120576
Output Projection matrix 119880
Algorithm
Step 1 Simultaneously consider both intraclass geometry andinterclass discrimination information to calculate 119876(119894) foreach view according to (7)
Step 2 (initialization)
(1) Set 120590 = [1119898 1119898 1119898](2) obtain 1198800 by solving the eigendecomposition prob-
lem (20)
6 The Scientific World Journal
Step 3 (local optimization) For 119905 = 1 2 119879max
(1) calculate 120590 as shown in (17)(2) solve the eigenvalue equation in (20)(3) sort their eigenvectors 119880
1 1198802 119880
119889according to
their corresponding eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
and obtain 119880119905 = [1198801 1198802 119880
119889]
(4) if 119905 gt 2 and |119880119905 minus 119880119905minus1| lt 120576 then go to Step 4
Step 4 (output projection matrix) Output the final optimalprojection matrix 119880 = 119880119905
We now briefly analyze the computational complexityof the MDGPP algorithm which is dominated by threeparts One is for constructing the matrix 119876(119894) for differentviews As shown in (7) the computational complexity ofthis part is 119874((sum119898
119894=1119901119894) times 119899
2) In addition each iteration
involves computing view weight 120590 and solving a standardeigendecomposition problem the computational complexityof running two parts in each iteration is 119874((119898 + 119889) times 1198992)and 119874(1198993) respectively Therefore the total computationalcomplexity of MDGPP is 119874((sum119898
119894=1119901119894) times 1198992+ ((119898 + 119889) times 119899
2+
1198993) times 119879max) where 119879max denotes the iteration number and is
always set to less than five in all experiments
4 Experimental Results
In this section we evaluate the effectiveness of our proposedMDGPP algorithm for two image classification tasks includ-ing face recognition and facial expression recognition Twowidely used face databases including AR [24] and CMU PIE[25] are employed for face recognition evaluation and thewell-known Japanese female facial expression (JAFFE) [26]database is used for facial expression recognition evaluationWe also compare the proposed MDGPP algorithm withsome traditional single-view-based dimensionality reductionalgorithms such as PCA [2] LDA [2] LPP [3] MFA [10]DGPP [12] and the three latest multiview dimensionalityreduction algorithms includingMVSE [18]MSNE [21]MSE[19] and SSMVE [22] The nearest neighbor classifier withthe Euclidean distance was adopted for classification For afair comparison all the results reported here are based on thebest tuned parameters of all the compared algorithms
41 Data Sets and Experimental Settings We conducted facerecognition experiments on the widely used AR and CMUPIE face databases and facial expression recognition experi-ments on the well-known Japanese female facial expression(JAFFE) database
The AR database [24] contains over 4000 color imagescorresponding to 126 people (70 men and 56 women) whichinclude frontal view faces with different facial expressionsillumination conditions and occlusions (sun glasses andscarf) Each person has 26 different images taken in twosessions (separated by two weeks) In our experiments weused a subset of 800 face images from 100 persons (50 menand 50women)with eight face images of different expressionsand lighting conditions per person Figure 1 shows eight
Figure 1 Sample face images from the AR database
Figure 2 Sample face images from the CMU PIE database
sample images of one individual from the subset of the ARdatabase
The CMU PIE database [25] comprises more than 40000facial images of 68 people with different poses illuminationconditions and facial expressions In this experiment weselected a subset of the CMU PIE database which consistsof 3060 frontal face images with varying expression andillumination from 68 persons with 45 images from eachperson Figure 2 shows some sample images of one individualfrom the subset of the CMU PIE database
The Japanese female facial expression (JAFFE) database[26] contains 213 facial images of ten Japanese womenEach facial image shows one of seven expressions neutralhappiness sadness surprise anger disgust or fear Figure 3shows some facial images from the JAFFE database In thisexperiment following the general setting scheme of facialexpression recognition we discard all the neutral facialimages and only utilize the remainder 183 facial images whichinclude six basic facial expressions
For all the face images in the above three face databasesthe facial part of each image was manually aligned croppedand resized into 32 times 32 according to the eyersquos positions Foreach facial image we extract the commonly used four kindsof low-level visual features to represent four different viewsThese four features include color histogram (CH) [27] scale-invariant feature transform (SIFT) [28] Gabor [29] and localbinary pattern (LBP) [30] For the CH feature extraction weused 64 bins to encode a histogram feature for each facialimage according to [27] For the SIFT feature extractionwe densely sampled and calculated the SIFT descriptors of16 times 16 patches over a grid with spacing of 8 pixels accordingto [28] For the Gabor feature extraction following [29] weadopted 40 Gabor kernel functions from five scales and eightorientations For the LBP feature extraction we followed theparameter settings in [30] and utilized 256 bins to encodea histogram feature for each facial image For more detailson these four feature descriptors please refer to [27ndash30]Because these four features are complementary to each otherin representing facial images we empirically set the tuningparameter 119903 in MDGPP to be five
In this experiment each facial image set was partitionedinto the nonoverlap training and testing sets For eachdatabase we randomly selected 50 data as the training setand use the remaining 50 data as the testing set To reducestatistical variation for each random partition we repeatedthese trials independently ten times and reported the averagerecognition results
The Scientific World Journal 7
Figure 3 Sample facial images from the JAFFE database
42 Compared Algorithms We compared our proposedMDGPP algorithm with the following dimensionality reduc-tion algorithms
(1) PCA [2] PCA is an unsupervised dimensionalityreduction algorithm
(2) LDA [2] LDA is a supervised dimensionality reduc-tion algorithm We adopted a Tikhonov regular-ization term 120583119868 rather than PCA preprocessing toavoid the well-known small sample size (singularity)problem in LDA
(3) LPP [3] LPP is an unsupervised manifold learningalgorithm There is a nearest neighbor number 119896 tobe tuned in LPP and it was empirically set to befive in our experiments In addition the Tikhonovregularization was also adopted to avoid the smallsample size (singularity) problem in LPP
(4) MFA [10] MFA is a supervised manifold learningalgorithm There are two parameters (ie 119896
1nearest
neighbor number and 1198962nearest neighbor number)
to be tuned in MFA We empirically set 1198961= 5
and 1198962= 20 in our experiments Meanwhile the
Tikhonov regularizationwas also adopted to avoid thesmall sample size (singularity) problem in MFA
(5) DGPP [12] DGPP is a supervised manifold learningalgorithmThere is a tradeoff parameter 120582 to be tunedin DGPP and it was empirically set to be one in ourexperiments
(6) MVSE [18] MSVE is an initially proposed multiviewalgorithm for dimensionality reduction
(7) MSNE [21] MSNE is a probability-based unsuper-vised multiview algorithm We followed the param-eter setting in [21] and set the tradeoff coefficient 120582 tobe five in our experiments
(8) MSE [19] MSE is a supervised multiview algorithmThere are two parameters (ie the nearest neighbornumber 119896 and the tuning coefficient 119903) to be tunedin MSE We empirically set 119896 = 5 and 119903 = 5 in ourexperiments
(9) SSMVE [22] SSMVE is a sparse unsupervised mul-tiview algorithm We followed the parameter settingmethod in [22] and set the regularized parameter 120582 tobe one in our experiments
It is worth noting that since PCA LDA LPP MFAand DGPP are all single-view-based algorithms these fivealgorithms adopt the conventional feature concatenation-based strategy to cope with the multiview data
Table 1 Comparisons of recognition accuracy on the AR database
Algorithm Accuracy DimensionsPCA 815 120LDA 904 99LPP 908 100MFA 919 100DGPP 932 100MVSE 934 100MSNE 937 100SSMVE 941 100MSE 950 100MDGPP 988 100
Table 2 Comparisons of recognition accuracy on the CMU PIEdatabase
Algorithm Accuracy DimensionsPCA 568 80LDA 634 67LPP 652 68MFA 735 68DGPP 784 68MVSE 796 68MSNE 827 68SSMVE 845 68MSE 873 68MDGPP 952 68
Table 3 Comparisons of recognition accuracy on the JAFFEdatabase
Algorithm Accuracy DimensionsPCA 526 20LDA 658 9LPP 715 10MFA 832 10DGPP 863 10MVSE 878 10MSNE 892 10SSMVE 916 10MSE 934 10MDGPP 965 10
43 Experimental Results For each face image databasethe recognition performance of different algorithms wasevaluated on the testing data separately The conventionalnearest neighbor classifier with the Euclidean distance wasapplied to perform recognition in the subspace derived fromdifferent dimensionality reduction algorithms Tables 1 2and 3 report the recognition accuracies and the correspond-ing optimal dimensions obtained on the AR CMU PIE andJAFFE databases respectively Figures 4 5 and 6 illustratethe recognition accuracies versus the variation of reduceddimensions on the AR CMU PIE and JAFFE databases
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
6 The Scientific World Journal
Step 3 (local optimization) For 119905 = 1 2 119879max
(1) calculate 120590 as shown in (17)(2) solve the eigenvalue equation in (20)(3) sort their eigenvectors 119880
1 1198802 119880
119889according to
their corresponding eigenvalues 1205821gt 1205822gt sdot sdot sdot gt 120582
119889
and obtain 119880119905 = [1198801 1198802 119880
119889]
(4) if 119905 gt 2 and |119880119905 minus 119880119905minus1| lt 120576 then go to Step 4
Step 4 (output projection matrix) Output the final optimalprojection matrix 119880 = 119880119905
We now briefly analyze the computational complexityof the MDGPP algorithm which is dominated by threeparts One is for constructing the matrix 119876(119894) for differentviews As shown in (7) the computational complexity ofthis part is 119874((sum119898
119894=1119901119894) times 119899
2) In addition each iteration
involves computing view weight 120590 and solving a standardeigendecomposition problem the computational complexityof running two parts in each iteration is 119874((119898 + 119889) times 1198992)and 119874(1198993) respectively Therefore the total computationalcomplexity of MDGPP is 119874((sum119898
119894=1119901119894) times 1198992+ ((119898 + 119889) times 119899
2+
1198993) times 119879max) where 119879max denotes the iteration number and is
always set to less than five in all experiments
4 Experimental Results
In this section we evaluate the effectiveness of our proposedMDGPP algorithm for two image classification tasks includ-ing face recognition and facial expression recognition Twowidely used face databases including AR [24] and CMU PIE[25] are employed for face recognition evaluation and thewell-known Japanese female facial expression (JAFFE) [26]database is used for facial expression recognition evaluationWe also compare the proposed MDGPP algorithm withsome traditional single-view-based dimensionality reductionalgorithms such as PCA [2] LDA [2] LPP [3] MFA [10]DGPP [12] and the three latest multiview dimensionalityreduction algorithms includingMVSE [18]MSNE [21]MSE[19] and SSMVE [22] The nearest neighbor classifier withthe Euclidean distance was adopted for classification For afair comparison all the results reported here are based on thebest tuned parameters of all the compared algorithms
41 Data Sets and Experimental Settings We conducted facerecognition experiments on the widely used AR and CMUPIE face databases and facial expression recognition experi-ments on the well-known Japanese female facial expression(JAFFE) database
The AR database [24] contains over 4000 color imagescorresponding to 126 people (70 men and 56 women) whichinclude frontal view faces with different facial expressionsillumination conditions and occlusions (sun glasses andscarf) Each person has 26 different images taken in twosessions (separated by two weeks) In our experiments weused a subset of 800 face images from 100 persons (50 menand 50women)with eight face images of different expressionsand lighting conditions per person Figure 1 shows eight
Figure 1 Sample face images from the AR database
Figure 2 Sample face images from the CMU PIE database
sample images of one individual from the subset of the ARdatabase
The CMU PIE database [25] comprises more than 40000facial images of 68 people with different poses illuminationconditions and facial expressions In this experiment weselected a subset of the CMU PIE database which consistsof 3060 frontal face images with varying expression andillumination from 68 persons with 45 images from eachperson Figure 2 shows some sample images of one individualfrom the subset of the CMU PIE database
The Japanese female facial expression (JAFFE) database[26] contains 213 facial images of ten Japanese womenEach facial image shows one of seven expressions neutralhappiness sadness surprise anger disgust or fear Figure 3shows some facial images from the JAFFE database In thisexperiment following the general setting scheme of facialexpression recognition we discard all the neutral facialimages and only utilize the remainder 183 facial images whichinclude six basic facial expressions
For all the face images in the above three face databasesthe facial part of each image was manually aligned croppedand resized into 32 times 32 according to the eyersquos positions Foreach facial image we extract the commonly used four kindsof low-level visual features to represent four different viewsThese four features include color histogram (CH) [27] scale-invariant feature transform (SIFT) [28] Gabor [29] and localbinary pattern (LBP) [30] For the CH feature extraction weused 64 bins to encode a histogram feature for each facialimage according to [27] For the SIFT feature extractionwe densely sampled and calculated the SIFT descriptors of16 times 16 patches over a grid with spacing of 8 pixels accordingto [28] For the Gabor feature extraction following [29] weadopted 40 Gabor kernel functions from five scales and eightorientations For the LBP feature extraction we followed theparameter settings in [30] and utilized 256 bins to encodea histogram feature for each facial image For more detailson these four feature descriptors please refer to [27ndash30]Because these four features are complementary to each otherin representing facial images we empirically set the tuningparameter 119903 in MDGPP to be five
In this experiment each facial image set was partitionedinto the nonoverlap training and testing sets For eachdatabase we randomly selected 50 data as the training setand use the remaining 50 data as the testing set To reducestatistical variation for each random partition we repeatedthese trials independently ten times and reported the averagerecognition results
The Scientific World Journal 7
Figure 3 Sample facial images from the JAFFE database
42 Compared Algorithms We compared our proposedMDGPP algorithm with the following dimensionality reduc-tion algorithms
(1) PCA [2] PCA is an unsupervised dimensionalityreduction algorithm
(2) LDA [2] LDA is a supervised dimensionality reduc-tion algorithm We adopted a Tikhonov regular-ization term 120583119868 rather than PCA preprocessing toavoid the well-known small sample size (singularity)problem in LDA
(3) LPP [3] LPP is an unsupervised manifold learningalgorithm There is a nearest neighbor number 119896 tobe tuned in LPP and it was empirically set to befive in our experiments In addition the Tikhonovregularization was also adopted to avoid the smallsample size (singularity) problem in LPP
(4) MFA [10] MFA is a supervised manifold learningalgorithm There are two parameters (ie 119896
1nearest
neighbor number and 1198962nearest neighbor number)
to be tuned in MFA We empirically set 1198961= 5
and 1198962= 20 in our experiments Meanwhile the
Tikhonov regularizationwas also adopted to avoid thesmall sample size (singularity) problem in MFA
(5) DGPP [12] DGPP is a supervised manifold learningalgorithmThere is a tradeoff parameter 120582 to be tunedin DGPP and it was empirically set to be one in ourexperiments
(6) MVSE [18] MSVE is an initially proposed multiviewalgorithm for dimensionality reduction
(7) MSNE [21] MSNE is a probability-based unsuper-vised multiview algorithm We followed the param-eter setting in [21] and set the tradeoff coefficient 120582 tobe five in our experiments
(8) MSE [19] MSE is a supervised multiview algorithmThere are two parameters (ie the nearest neighbornumber 119896 and the tuning coefficient 119903) to be tunedin MSE We empirically set 119896 = 5 and 119903 = 5 in ourexperiments
(9) SSMVE [22] SSMVE is a sparse unsupervised mul-tiview algorithm We followed the parameter settingmethod in [22] and set the regularized parameter 120582 tobe one in our experiments
It is worth noting that since PCA LDA LPP MFAand DGPP are all single-view-based algorithms these fivealgorithms adopt the conventional feature concatenation-based strategy to cope with the multiview data
Table 1 Comparisons of recognition accuracy on the AR database
Algorithm Accuracy DimensionsPCA 815 120LDA 904 99LPP 908 100MFA 919 100DGPP 932 100MVSE 934 100MSNE 937 100SSMVE 941 100MSE 950 100MDGPP 988 100
Table 2 Comparisons of recognition accuracy on the CMU PIEdatabase
Algorithm Accuracy DimensionsPCA 568 80LDA 634 67LPP 652 68MFA 735 68DGPP 784 68MVSE 796 68MSNE 827 68SSMVE 845 68MSE 873 68MDGPP 952 68
Table 3 Comparisons of recognition accuracy on the JAFFEdatabase
Algorithm Accuracy DimensionsPCA 526 20LDA 658 9LPP 715 10MFA 832 10DGPP 863 10MVSE 878 10MSNE 892 10SSMVE 916 10MSE 934 10MDGPP 965 10
43 Experimental Results For each face image databasethe recognition performance of different algorithms wasevaluated on the testing data separately The conventionalnearest neighbor classifier with the Euclidean distance wasapplied to perform recognition in the subspace derived fromdifferent dimensionality reduction algorithms Tables 1 2and 3 report the recognition accuracies and the correspond-ing optimal dimensions obtained on the AR CMU PIE andJAFFE databases respectively Figures 4 5 and 6 illustratethe recognition accuracies versus the variation of reduceddimensions on the AR CMU PIE and JAFFE databases
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 7
Figure 3 Sample facial images from the JAFFE database
42 Compared Algorithms We compared our proposedMDGPP algorithm with the following dimensionality reduc-tion algorithms
(1) PCA [2] PCA is an unsupervised dimensionalityreduction algorithm
(2) LDA [2] LDA is a supervised dimensionality reduc-tion algorithm We adopted a Tikhonov regular-ization term 120583119868 rather than PCA preprocessing toavoid the well-known small sample size (singularity)problem in LDA
(3) LPP [3] LPP is an unsupervised manifold learningalgorithm There is a nearest neighbor number 119896 tobe tuned in LPP and it was empirically set to befive in our experiments In addition the Tikhonovregularization was also adopted to avoid the smallsample size (singularity) problem in LPP
(4) MFA [10] MFA is a supervised manifold learningalgorithm There are two parameters (ie 119896
1nearest
neighbor number and 1198962nearest neighbor number)
to be tuned in MFA We empirically set 1198961= 5
and 1198962= 20 in our experiments Meanwhile the
Tikhonov regularizationwas also adopted to avoid thesmall sample size (singularity) problem in MFA
(5) DGPP [12] DGPP is a supervised manifold learningalgorithmThere is a tradeoff parameter 120582 to be tunedin DGPP and it was empirically set to be one in ourexperiments
(6) MVSE [18] MSVE is an initially proposed multiviewalgorithm for dimensionality reduction
(7) MSNE [21] MSNE is a probability-based unsuper-vised multiview algorithm We followed the param-eter setting in [21] and set the tradeoff coefficient 120582 tobe five in our experiments
(8) MSE [19] MSE is a supervised multiview algorithmThere are two parameters (ie the nearest neighbornumber 119896 and the tuning coefficient 119903) to be tunedin MSE We empirically set 119896 = 5 and 119903 = 5 in ourexperiments
(9) SSMVE [22] SSMVE is a sparse unsupervised mul-tiview algorithm We followed the parameter settingmethod in [22] and set the regularized parameter 120582 tobe one in our experiments
It is worth noting that since PCA LDA LPP MFAand DGPP are all single-view-based algorithms these fivealgorithms adopt the conventional feature concatenation-based strategy to cope with the multiview data
Table 1 Comparisons of recognition accuracy on the AR database
Algorithm Accuracy DimensionsPCA 815 120LDA 904 99LPP 908 100MFA 919 100DGPP 932 100MVSE 934 100MSNE 937 100SSMVE 941 100MSE 950 100MDGPP 988 100
Table 2 Comparisons of recognition accuracy on the CMU PIEdatabase
Algorithm Accuracy DimensionsPCA 568 80LDA 634 67LPP 652 68MFA 735 68DGPP 784 68MVSE 796 68MSNE 827 68SSMVE 845 68MSE 873 68MDGPP 952 68
Table 3 Comparisons of recognition accuracy on the JAFFEdatabase
Algorithm Accuracy DimensionsPCA 526 20LDA 658 9LPP 715 10MFA 832 10DGPP 863 10MVSE 878 10MSNE 892 10SSMVE 916 10MSE 934 10MDGPP 965 10
43 Experimental Results For each face image databasethe recognition performance of different algorithms wasevaluated on the testing data separately The conventionalnearest neighbor classifier with the Euclidean distance wasapplied to perform recognition in the subspace derived fromdifferent dimensionality reduction algorithms Tables 1 2and 3 report the recognition accuracies and the correspond-ing optimal dimensions obtained on the AR CMU PIE andJAFFE databases respectively Figures 4 5 and 6 illustratethe recognition accuracies versus the variation of reduceddimensions on the AR CMU PIE and JAFFE databases
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
8 The Scientific World Journal
Dimension
Reco
gniti
on ac
cura
cy (
)
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80 100 120
Figure 4 Recognition accuracy versus reduced dimension on theAR database
Dimension
Reco
gniti
on ac
cura
cy (
)
30
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 20 40 60 80
Figure 5 Recognition accuracy versus reduced dimension on theCMU PIE database
respectively According to the above experimental results wecan make the following observations
(1) As can be seen from Tables 1 2 and 3 and Figures4 5 and 6 our proposed MDGPP algorithm con-sistently outperforms the conventional single-view-based algorithms (ie PCA LDA LPP MFA andDGPP) and the latest multiview algorithms (ieMVSE MSNE MSE and SSMVE) in all the experi-ments which implies that extracting a discriminative
Dimension
Reco
gniti
on ac
cura
cy (
)
40
50
60
70
80
90
100
PCALDALPPMFADGPP
MVSEMSNESSMVEMSEMDGPP
0 5 10 15 20
Figure 6 Recognition accuracy versus reduced dimension on theJAFFE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
90
92
94
96
98
100
2 4 6 8 10
Figure 7 Recognition accuracy of MDGPP versus different num-bers of iteration on the AR database
feature subspace by using both intraclass geometryand interclass discrimination and explicitly consid-ering the complementary information of differentfacial features can achieve the best recognition per-formance
(2) The multiview learning algorithms (ie MVSEMSNE MSE SSMVE and MDGPP ) perform muchbetter than single-view-based algorithms (ie PCALDA LPP MFA and DGPP) which demonstratesthat simple concatenation strategy cannot duly com-bine features frommultiple views and the recognitionperformance can be successfully improved by explor-ing the complementary characteristics of differentviews
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 9
Number of iteration
Reco
gniti
on ac
cura
cy (
)
91
92
93
94
95
96
2 4 6 8 10
Figure 8 Recognition accuracy of MDGPP versus different num-bers of iteration on the CMU PIE database
Number of iteration
Reco
gniti
on ac
cura
cy (
)
92
93
94
95
96
97
2 4 6 8 10
Figure 9 Recognition accuracy of MDGPP versus different num-bers of iteration on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9862
9864
9866
9868
9870
9872
9874
9876
9878
9880
9882
1 2 3 4 5 6
120582
Figure 10 Recognition accuracy of MDGPP versus varying param-eter 120582 on the AR database
Reco
gniti
on ac
cura
cy (
)
947
948
949
950
951
952
953
1 2 3 4 5 6
120582
Figure 11 Recognition accuracy of MDGPP versus varying param-eter 120582 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
950
955
960
965
970
975
980
1 2 3 4 5 6
120582
Figure 12 Recognition accuracy of MDGPP versus varying param-eter 120582 on the JAFFE database
Reco
gniti
on ac
cura
cy (
)
9850
9855
9860
9865
9870
9875
9880
9885
0 2 4 6 8 10
r
Figure 13 Recognition accuracy of MDGPP versus varying param-eter 119903 on the AR database
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
10 The Scientific World JournalRe
cogn
ition
accu
racy
()
9508
9510
9512
9514
9516
9518
9520
9522
0 2 4 6 8 10
r
Figure 14 Recognition accuracy of MDGPP versus varying param-eter 119903 on the CMU PIE database
Reco
gniti
on ac
cura
cy (
)
9638
9640
9642
9644
9646
9648
9650
9652
0 2 4 6 8 10
r
Figure 15 Recognition accuracy of MDGPP versus varying param-eter 119903 on the JAFFE database
(3) For the single-view-based algorithms the manifoldlearning algorithms (ie LPP MFA and DGPP)perform much better than the conventional dimen-sionality reduction algorithms (ie PCA and LDA)This observation confirms that the local manifoldstructure information is crucial for image classifi-cation Moreover the supervised manifold learningalgorithms (ie MFA and DGPP) perform muchbetter than the unsupervised manifold learning algo-rithm LPP which demonstrates that the utilizationof discriminant information is useful to improve theimage classification performance
(4) For themultiview learning algorithms the supervisedmultiview algorithms (ie MSE and MDGPP) out-perform the unsupervised multiview algorithms (ieMVSE MSNE and SSMVE) due to the utilization ofthe labeled facial images
(5) Although MVSE MSNE and SSMVE are all unsu-pervised multiview learning algorithms SSMVE per-forms much better than MVSE and MSNE Thepossible explanation is that the SSMVE algorithmadopts the sparse coding technique which is naturallydiscriminative in determining the appropriate combi-nation of different views
(6) Among the comparedmultiview learning algorithmsMVSE performs the worst The reason is that MVSEperforms a dimensionality reduction process on eachview independently Hence it cannot fully integratethe complementary information of different views toproduce a good low-dimensional embedding
(7) MDGPP can improve the recognition performance ofDGPP The reason is that MDGPP can make use ofmultiple facial feature representations in a commonlearned subspace such that some complementaryinformation can be explored for recognition task
44 Convergence Analysis Since our proposed MDGPP isan iteration algorithm we also evaluate its recognitionperformance with different numbers of iteration Figures 78 and 9 show the recognition accuracy of MDGPP versusdifferent numbers of iteration on the AR CMU PIE andJAFFE databases respectively As can be seen from thesefigures we can observe that our proposedMDGPP algorithmcan converge to a local optimal optimum value in less thanfive iterations
45 Parameter Analysis We investigate the parameter effectsof our proposed MDGPP algorithm tradeoff coefficient 120582and tuning parameter 119903 Since each parameter can affect therecognition performance we fix one parameter as used in theprevious experiments and test the effect of the remaining oneFigures 10 11 and 12 show the influence of the parameter120582 in the MDGPP algorithm on the AR CMU PIE andJAFFE databases respectively Figures 13 14 and 15 showthe influence of the parameter 119903 in the MDGPP algorithmon the AR CMU PIE and JAFFE databases respectivelyFrom Figure 10 to Figure 15 we can observe that MDGPPdemonstrates a stable recognition performance over a largerange of both 120582 and 119903 Therefore we can conclude that theperformance of MDGPP is not sensitive to the parameters 120582and 119903
5 Conclusion
In this paper we have proposed a new multiview learn-ing algorithm called multiview discriminative geometrypreserving projection (MDGPP) for feature extraction andclassification by exploring the complementary property ofdifferent views MDGPP can encode different features fromdifferent views in a physically meaningful subspace and learna low-dimensional and sufficiently smooth embedding overall views simultaneously with an alternating-optimization-based iterative algorithm Experimental results on three faceimage databases show that the proposed MDGPP algorithm
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World Journal 11
outperforms other multiview and single view learning algo-rithms
Conflict of Interests
The authors declare that they have no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation of China under Grants no 70701013 and 61174056the Natural Science Foundation of Henan Province underGrant no 102300410020 the National Science Founda-tion for Postdoctoral Scientists of China under Grant no2011M500035 and the Specialized Research Fund for theDoctoral ProgramofHigher Education of China underGrantno 20110023110002
References
[1] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003
[2] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces vs fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997
[3] X He S Yan Y Hu P Niyogi and H-J Zhang ldquoFacerecognition using Laplacianfacesrdquo IEEETransactions on PatternAnalysis and Machine Intelligence vol 27 no 3 pp 328ndash3402005
[4] J B Tenenbaum V De Silva and J C Langford ldquoA globalgeometric framework for nonlinear dimensionality reductionrdquoScience vol 290 no 5500 pp 2319ndash2323 2000
[5] S T Roweis and L K Saul ldquoNonlinear dimensionality reduc-tion by locally linear embeddingrdquo Science vol 290 no 5500pp 2323ndash2326 2000
[6] M Belkin and P Niyogi ldquoLaplacian eigenmaps for dimension-ality reduction and data representationrdquo Neural Computationvol 15 no 6 pp 1373ndash1396 2003
[7] X He and P Niyogi ldquoLocality preserving projectionsrdquoAdvancesin Neural Information Processing Systems vol 16 pp 585ndash5912003
[8] H-T Chen H-W Chang and T-L Liu ldquoLocal discrimi-nant embedding and its variantsrdquo in Proceedings of the IEEEComputer Society Conference on Computer Vision and PatternRecognition (CVPR rsquo05) pp 846ndash853 San Diego Calif USAJune 2005
[9] D Cai X He K Zhou J W Han and H J Bao ldquoLocalitysensitive discriminant analysisrdquo in Proceedings of the 20thInternational Joint Conference on Artificial Intelligence (IJCAIrsquo07) pp 708ndash713 Hyderabad India January 2007
[10] S YanD Xu B ZhangH-J ZhangQ Yang and S Lin ldquoGraphembedding and extensions a general framework for dimen-sionality reductionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 29 no 1 pp 40ndash51 2007
[11] M Sugiyama ldquoDimensionality reduction ofmultimodal labeleddata by local fisher discriminant analysisrdquo Journal of MachineLearning Research vol 8 pp 1027ndash1061 2007
[12] D Song and D Tao ldquoBiologically inspired feature manifold forscene classificationrdquo IEEE Transactions on Image Processing vol19 no 1 pp 174ndash184 2010
[13] J Lu and Y-P Tan ldquoA doubly weighted approach forappearance-based subspace learning methodsrdquo IEEE Transac-tions on Information Forensics and Security vol 5 no 1 pp 71ndash81 2010
[14] J Lu J Hu X Zhou Y Shang Y Tan -P and G WangldquoNeighborhood repulsed metric learning for kinship verifica-tionrdquo in Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo12) pp 2594ndash2601 Providence RI USA June 2012
[15] D Zhou and C Burges ldquoSpectral clustering and transductivelearning with multiple viewsrdquo in Proceedings of the 24th Inter-national Conference on Machine Learning (ICML rsquo07) pp 1159ndash1166 Corvalis Ore USA June 2007
[16] J Yu D Liu D Tao and H S Seah ldquoOn combining multiplefeatures for cartoon character retrieval and clip synthesisrdquo IEEETransactions on Systems Man and Cybernetics B vol 42 no 5pp 1413ndash1427 2012
[17] J YuMWang andD Tao ldquoSemisupervisedmultiview distancemetric learning for cartoon synthesisrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4636ndash4648 2012
[18] B Long P S Yu and Z Zhang ldquoA general model for multipleview unsupervised learningrdquo in Proceedings of the 8th SIAMInternational Conference on Data Mining pp 822ndash833 AtlantaGa USA April 2008
[19] T Xia D Tao TMei andY Zhang ldquoMultiview spectral embed-dingrdquo IEEE Transactions on Systems Man and Cybernetics Bvol 40 no 6 pp 1438ndash1446 2010
[20] T Zhang D Tao X Li and J Yang ldquoPatch alignment fordimensionality reductionrdquo IEEETransactions onKnowledge andData Engineering vol 21 no 9 pp 1299ndash1313 2009
[21] B Xie Y Mu D Tao and K Huang ldquoM-SNE multiviewstochastic neighbor embeddingrdquo IEEE Transactions on SystemsMan and Cybernetics B vol 41 no 4 pp 1088ndash1096 2011
[22] Y Han F Wu D Tao J Shao Y Zhuang and J Jiang ldquoSparseunsupervised dimensionality reduction for multiple view datardquoIEEE Transactions on Circuits and Systems for Video Technologyvol 22 no 10 pp 1485ndash1496 2012
[23] R BhatiaMatrix Analysis Springer New York NY USA 1997[24] A M Martinez and R Benavente AR Face Database CVC
Technology Report 1998[25] T Sim S Baker andM Bsat ldquoTheCMUPose illumination and
expression databaserdquo IEEETransactions on Pattern Analysis andMachine Intelligence vol 25 no 12 pp 1615ndash1618 2003
[26] M J Lyons ldquoAutomatic classification of single facial imagesrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 21 no 12 pp 1357ndash1362 1999
[27] L G Shapiro and G C Stockman Computer Vision Prentice-Hall Englewood Cliffs NJ USA 2003
[28] D G Lowe ldquoDistinctive image features from scale-invariantkeypointsrdquo International Journal of Computer Vision vol 60 no2 pp 91ndash110 2004
[29] T S Lee ldquoImage representation using 2d gabor waveletsrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol18 no 10 pp 959ndash971 1996
[30] T Ahonen A Hadid and M Pietikainen ldquoFace descriptionwith local binary patterns application to face recognitionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol28 no 12 pp 2037ndash2041 2006
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Submit your manuscripts athttpwwwhindawicom
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
Artificial Intelligence
HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation
httpwwwhindawicom Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
