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Research ArticleLaplace Graph Embedding Class Specific DictionaryLearning for Face Recognition
LiWang Yan-JiangWang and Bao-Di Liu
College of Information and Control Engineering China University of Petroleum (East China) Qingdao China
Correspondence should be addressed to Yan-Jiang Wang yjwangupceducn
Received 23 September 2017 Revised 2 December 2017 Accepted 6 December 2017 Published 7 February 2018
Academic Editor Tongliang Liu
Copyright copy 2018 Li Wang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The sparse representation based classification (SRC) method and collaborative representation based classification (CRC) methodhave attracted more and more attention in recent years due to their promising results and robustness However both SRC andCRC algorithms directly use the training samples as the dictionary which leads to a large fitting error In this paper we proposethe Laplace graph embedding class specific dictionary learning (LGECSDL) algorithm which trains a weight matrix and embedsa Laplace graph to reconstruct the dictionary Firstly it can increase the dimension of the dictionary matrix which can be usedto classify the small sample database Secondly it gives different dictionary atoms with different weights to improve classificationaccuracy Additionally in each class dictionary training process the LGECSDL algorithm introduces the Laplace graph embeddingmethod to the objective function in order to keep the local structure of each class and the proposed method is capable ofimproving the performance of face recognition according to the class specific dictionary learning and Laplace graph embeddingregularizer Moreover we also extend the proposed method to an arbitrary kernel space Extensive experimental results on severalface recognition benchmark databases demonstrate the superior performance of our proposed algorithm
1 Introduction
The sparse representation algorithm based on dictionarylearning (dictionary learning for sparse representation) isattracting more and more attention in computer vision dueto its impressive performance in many applications suchas image processing image ranking [1] human activityrecognition [2] and image classification [3 4] Different fromthe traditional subspace methods such as PCA the sparserepresentation algorithmallows the bases of a dictionary to bemuch larger than the dimension of the sample characteristicsso the sample can be fitted more effectively
We know that deep learning-basedmethods are currentlythe mainstream methods in image classification Taigman etal [5] proposed a DeepFace neural network for face recog-nition which has achieved human-level performance Dingand Tao [6] proposed a comprehensive framework basedon Convolutional Neural Networks to overcome challengesin video-based face recognition Florian Schroff et al [7]proposed a FaceNet which can learn the mapping from faceimages to a compact Euclidean space Sun et al [8] proposed
a DeepID2+ convolutional network which increases thedimension of hidden representations and adds supervision toearly convolutional layers Liu et al [9] proposed amultipatchdeep CNN and deep metric learning method to extractdiscriminative features for face recognition However thedepth learning method performed well when the samplesize was large and the effect was not satisfactory underthe condition of a small database Therefore we propose adictionary learning method based on Laplacian embeddingand sparse representationwhich can still achieve good resultsin the case of very small samples
The sparse representation based classifier has been widelyused in the field of face recognition Normally classifying thesamples involves two stages first obtain the sample featureand then the sample feature can be sent to the classifierfor classification In the process of feature extraction manysubspace methods are proposed The principal componentanalysis method was proposed to reduce a complex dataset to a lower dimension to reveal the hidden simplifieddata structures [10] The linear discriminant analysis (LDA)algorithm was proposed to find the projection hyperplane
HindawiJournal of Electrical and Computer EngineeringVolume 2018 Article ID 2179049 11 pageshttpsdoiorg10115520182179049
2 Journal of Electrical and Computer Engineering
that minimizes the interclass variance and maximizes thedistance between the projected means of the classes [11] Taoet al [12] proposed a general tensor discriminant analysismethod as a preprocessing step for the LDA algorithm toreduce the undersampling problem The locality preservingprojection algorithm was proposed to preserve the neighbor-hood structure of the data [13] In the procedure of classifi-cation Liu et al [14] proposed a new belief-based 119896-nearestneighbor classifier to make the classification result morerobust to misclassification errors Noh et al [15] proposed anearest neighbor algorithm to enhance the performance ofthe nearest neighbor classification by learning a local metricAlthough the 119896-nearest neighbor classifier and the nearestneighbor classifier have achieved good results on some datasets they did not select the most discriminatory feature ofthe sample to classify So subspace-based classifier designmethods were proposed to improve the classification effect
The sparse representation based classification algorithmuses training samples to construct an overcomplete dictio-nary and the test samples can be well represented as a sparselinear combination of elements from the dictionary [16]But the subsequent research shows that sparseness cannotextract the most discriminatory features of the samplesCollaborative representation based classification (CRC) wasproposed which uses the L2-norm constraint to reveal theinternal structure of the testing sample [17] Although theSRC and CRC methods have achieved superior performancein visual recognition both SRC and CRC algorithms directlyuse the training samples as the dictionary matrix The directuse of training samples to build dictionaries can lead to twodrawbacks first very few samples to build an overcompletedictionary which may result in low classification accuracyand second very redundant dictionary samples which pre-vent the original signals from being effectively expressedresulting in poor classifier performance
So dictionary learning methods are proposed to improvethe classification effect Discriminative dictionary learningapproaches can be divided into three types shared dictionarylearning class specific dictionary learning and hybrid dictio-nary learningThe shared dictionary learningmethod usuallyuses all training samples to obtain a classification dictionaryLu et al [18] proposed a locality weighted sparse representa-tion based classification (WSRC) method which utilizes bothdata locality and linearity to train a classification dictionaryYang et al [19] proposed a novel dictionary learning methodbased on the Fisher discrimination criterion to improve thepattern classification performance Yang et al [20] proposeda latent dictionary learning method to learn a discriminativedictionary and build its relationship to class labels adaptivelyJiang et al [21] proposed an algorithm to learn a singleovercomplete dictionary and an optimal linear classifier forface recognition Zhou et al [22] presented a dictionarylearning algorithm to exploit the visual correlation withina group of visually similar object categories for dictionarylearning where a commonly shared dictionary and multiplecategory-specific dictionaries are accordingly modeled Theclass specific dictionary learningmethod trained a dictionaryfor each class of samples Sun et al [23] learned a class specificsubdictionary for each class and a common subdictionary
shared by all classes to improve the classification perfor-mance Wang and Kong [24] proposed a method to explicitlylearn a class specific dictionary for each category whichcaptures the most discriminative features of this categoryand simultaneously learn a common pattern pool whoseatoms are shared by all the categories and only contribute torepresentation of the data rather than discrimination
The hybrid dictionary learning method is the combi-nation of the above two methods Rodriguez and Sapiro[25] proposed a new dictionary learning method whichuses a class-dependent supervised constraint and orthogonalconstraint this method learns the intraclass structure whileincreasing the interclass discrimination and expands thedifference between classes Gao et al [26] learned a category-specific dictionary for each category and a shared dictionaryfor all the categories and this method improves conventionalbasic-level object categorization Liu et al [27] proposed alocality sensitive dictionary learning algorithm with globalconsistency and smoothness constraint to overcome therestriction of linearity at a relatively low cost Although thehybrid dictionary learning method achieved good resultsthese methods usually operate in the original Euclideanspace which cannot capture nonlinear structures hiddenin data So many kernel-based classifiers are designed tosolve this problem Nguyen et al [28] presented a dictionarylearning method for sparse representation based on thekernel method Liu et al [29] proposed a multiple-viewself-explanatory sparse representation dictionary learningalgorithm (MSSR) to capture and combine various salientregions and structures from different kernel spaces and thismethod achieved superior performance in the field of facerecognition
As better effects had been achieved by MSSR algorithmthis algorithm neither took into consideration the details oftraining samples in the original sample space nor protectedthis powerful information conducive to classification in thedictionary space Therefore in this algorithm the Laplaceconstraint is added to the objective function to make theclosely similar samples in low dimensional space also veryclose in the high dimensional dictionary space
Motivated by this we proposed a Laplace graphembedding class specific dictionary learning algorithmand extended this method to arbitrary kernel space Themain contribution is listed in four aspects (1) We proposea Laplace embedding sparse representation algorithm Itcombines the advantages of SRCrsquos discriminant ability andmaintains the intrinsic local geometric feature of the samplefeatures by Laplace embedding (2) We propose a Laplaceembedding constraint dictionary learning algorithm toconstruct superior subspace and reduce the residual error(3) We extend this algorithm to arbitrary kernel space tofind the nonlinear structure of face images (4) Experimentalresults on several benchmark databases demonstrate thesuperior performance of our proposed algorithm
The rest of the paper is organized as follows Section 2overviews the three classical face recognition algorithmsSection 3 proposes our Laplace graph embedding class spe-cific dictionary learning algorithm with kernelsThe solutionto the minimization of the objective function is elaborated in
Journal of Electrical and Computer Engineering 3
Section 4 Then experimental results and analysis are shownin Section 5 Finally discussions and conclusions are drawnin Section 6
2 Overview of SRC and CRC
In this section we will briefly overview two classical facerecognition algorithms SRC and CRC
Suppose that there are 119862 classes in the training samplesand each class has 119873119888 elements 119873 = sum119862119888=1119873119888 where 119873represents the total number of training samples119883 representsall the training samples119883 = [1198831 1198832 119883119888 119883119873] where119883119888 isin 119877119863times119873119888 119863 represents the dimension of the samplefeatures 119883119888 represents the 119888th class of the training samplesSupposing that 119910 is a test sample and 119910 isin 119877119863times1 the sparserepresentation of sample 119910 can be expressed as
119878 = argmin1199041003817100381710038171003817119910 minus 119883119888119904100381710038171003817100381722 + 2120572 1199041 (1)
where 119904 is the sparse coding of sample 119910 in the 119888th dictionaryand 120572 is the regularization parameter in formula (1) which isused to control the sparsity and accuracy of the expression
The collaborative representation based classification algo-rithm applies L2-norm constraint on the object function theobjective of the CRC algorithm can be rewritten as follows
119878 = argmin1199041003817100381710038171003817119910 minus 119883119888119904100381710038171003817100381722 + 2120572 11990422 (2)
where 120573 is the regularization parameter to control theexpression accuracy of the object function
Both SRC and CRC methods directly use the trainingsamples as the dictionary And each base in the dictionary hasthe same contribution to the sample expression The testingsample 119910 can be encoded as
119910 asymp 119883119888119904 (3)
Here 119883119888 is the dictionary matrix composed of the 119888th classtraining samples and 119904 is the sparse coding of 119910
Directly using the training samples as the dictionaryleads to high residual error Liu et al [29] proposed asingle-view self-explanatory sparse representation dictionarylearning algorithm (SSSR) Supposing that 119888 represents theclass number of the training samples and 119883119888 means thecollection of sample characteristics of class 119888 the objectivefunction of the SSSR method can be formulated as
min119882119888119878119888
119891 (119882119888 119878119888) = 1003817100381710038171003817119883119888 minus 11988311988811988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119894=1
1003817100381710038171003817(119878119888119894 )10038171003817100381710038171st 1003817100381710038171003817119883119888 (119882119888)1198961003817100381710038171003817 le 1 forall119896 = 1 2 119870
(4)
where 119878119888 is the sparse codes of the 119888th class and 119878119888119894 representsthe 119894th column of 119878119888 The SSSR algorithm reconstructed thedictionary matrix119882119888 is the dictionary weight matrix119882119888 isin119877119873119888times119870 and 119873119888 is the number of the 119888th classes 119882119888 expandsthe original dictionary space into amore complete dictionaryspace when the identity matrix 119882119888 isin 119868119873119888times119873119888 appears theclass specific dictionary learning algorithm evolves into the
SRC method The existence of 119882119888 matrix makes dictionarylearning more flexible in the process of expression and thereconstruction error may be reduced as well
Meanwhile Liu et al [29] extended the SSSR algorithminto kernel spaces which can map the original samplefeatures into a high dimensional nonlinear space for bettermining of nonlinear relationships between samples Theobjective function of the multiple-view kernel-based classspecific dictionary learning algorithm (KCSDL) is shown asfollows
min119882119888119878119888
119891 (119882119888 119878119888)
= 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119894=1
1003817100381710038171003817(119878119888119894 )10038171003817100381710038171st 1003817100381710038171003817120601 (119883119888) (119882119888)1198961003817100381710038171003817 le 1 forall119896 = 1 2 119870
(5)
where 120601 119877119863 rarr 119877119905 means the kernel function it maps theoriginal feature space into a high dimensional kernel space
3 Our Proposed Approach
Although the above methods have achieved good results inthe field of face recognition there are still some deficienciesThe SSSR algorithm uses a reconstructed dictionary matrixto make sparse representation on samples however it doesnot take into account the fact that only the sparsity constrainton the target is not necessary to gain results for betterclassification
Motivated by this we have proposed the sparse represen-tation algorithm based on Laplace graph embedding whiletaking into account the sparse representation on the samplesthis algorithm mines the details implicit in the trainingsamples therefore the same sample is more concentrated inthe sparse expression space so as to reduce the fitting errorand improve the classification effect
The objective function of our proposed sparse represen-tation algorithm based on Laplace graph embedding nowbecomes
119891 (119878119888)= 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572
119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171+ 120573sum119894 =119895
10038171003817100381710038171003817119878119888119894 minus 11987811988811989510038171003817100381710038171003817 119901119894119895st 10038161003816100381610038161003816100381610038161003816120601 (119883119888) (119882119888)11989610038161003816100381610038161003816100381610038161003816 le 1 forall119896 = 1 2 119870
(6)
where 119883119888 means class 119888 training samples 119882119888 means dictio-nary weight matrix 119878119888 means the dictionary representationof class 119888 samples and 119878119888119899 represents the 119899th column of 1198781198884 Optimization of the Objective Function
In this section we focus on solving the optimization problemfor the proposed Laplace graph embedding class specific
4 Journal of Electrical and Computer Engineering
dictionary learning algorithm The dictionary weight matrix119882c and sparse representation matrix 119878119888 can be optimized byiterative approaches
When each element in the 119882119888 matrix is updated theremaining elements in 119882119888 matrix and 119878119888 matrix are fixedat this time the objective function is changed into an L2-norm constrained least-squares minimization subproblemSimilarly when each element in 119878119888 matrix is updated 119882119888matrix and the remaining elements in 119878119888 matrix are fixedThe objective function can be seen as an L1-norm constrainedleast-squares minimization subproblem
41 L1-Norm Regularized Minimization Subproblem Whenupdating the elements in 119878119888matrix the nonupdated elementsin 119878119888 and119882119888matrix will be fixed Here the objective functioncan be formulated as
119891 (119878119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171+ 120573sum119894 =119895
10038171003817100381710038171003817119878119888119894 minus 11987811988811989510038171003817100381710038171003817 119901119894119895(7)
where 119901119894119895 is the weight value which describes the neighboringdegree of 119909119888119894 and 119909119888119895 and119901119894119895 = 119890minus119909119888119894minus119909119888119895119905 119909119888119894 and 119909119888119895 are trainingsamples that belong to the 119888th class and 119905 is a constant whichcontrols the range of 119901119894119895 Formula (7) can be simplified as
119891 (119878119888)= trace 120601 (119883119888)119879 120601 (119883119888) minus 2120601 (119883119888)119879 120601 (119883119888)119882119888119878119888+ trace (119878119888)119879 (119882119888119879120601 (119883119888)119879 120601 (119883119888)119882119888) 119878119888+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573 trace 119878119888119871119878119888119879
= trace 120581 (119883119888 119883119888) minus 2119873119888sum119899=1
[120581 (119883119888 119883119888)119882119888]119899 119878119888119899
+ 119873119888sum119899=1
(119878119888119879)119899[119882119888119879120581 (119883119888 119883119888)119882119888] 119878119888119899
+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573119870sum119896=1
[119878119888119896119871119878119888119879119896 ]= trace 120581 (119883119888 119883119888)minus 2119873119888sum119899=1
119870sum119896=1
[120581 (119883119888 119883119888)119882119888]119899119896 119878119888119896119899
+ 119873119888sum119899=1
119870sum119896=1
[ 119870sum119897=1
(119878119888119879)119899119897(119882119888119879120581 (119883119888 119883119888)119882119888)
119897119896] 119878119888119896119899
+ 2120572 119870sum119896=1
119873119888sum119899=1
10038161003816100381610038161198781198881198961198991003816100381610038161003816
+ 120573 119870sum119896=1
[119873119888sum119899=1
(119873119888sum119897=1
119878119888119896119897119871 119897119899)(119878119879)119899119896]
(8)
where 119871 = 119863 minus 119875 119863119894119894 = sum119895 119875119894119895 and matrix 119875 is the weightmatrix expressing the sample neighboring distance 120581(119883119888 119883119888)means the kernel function of the sample and 120581(119883119888 119883119888)is calculated prior to dictionary updating 120581(119883119888 119883119888) =120601(119883119888)119879120601(119883119888)
In this algorithm each element in 119878119888 is updated sequen-tially when 119878119896119899 is updated the other elements in the 119878matrixare regarded as constants After ignoring the constant term offormula (8) formula (8) can be simplified as
119891 (119878119888119896119899) = (119882119888119879120581 (119883119888 119883119888)119882119888)119896119896(119878119888119896119899)2
+ 2 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899119878119888119896119899
minus 2 [120581 (119883119888 119883119888)119882119888]119899119896 119878c119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816 + 120573119871119899119899 (119878119888119896119899)2
+ 2120573 119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897119878119888119896119899 = [(119882119888119879120581 (119883119888 119883119888)119882119888)119896119896
+ 120573119871119899119899] (119878119888119896119899)2 minus 2[(120581 (119883119888 119883119888)119882119888)119899119896
minus 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899 minus 120573
119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897]sdot 119878119888119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816
(9)
According to the solving method in [29] it is easy toobtain the solution of the minimum value of 119891(119878119888119896119899) underthe current iteration condition
119878119888119896119899 = 11 + 120573119871119899119899 min (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 minus120572 + 11 + 120573119871119899119899
sdotmax (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 120572
(10)
where 119864 = 119882119888119879120581(119883119888 119883119888)119882119888 and 119878119888119896119899 = 119878119888119901119902 119901 = 119896 119902 =119899 0 119901 = 119896 119902 = 11989942 ℓ2 Norm Constrained Minimization Subproblem Whenupdating the dictionary matrix 119882119888 119878119888 and the nonupdatedelements in 119882119888 matrix will be fixed The objective functioncan be transformed into the following form
119891 (119882119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865st 1003817100381710038171003817120601 (119883119888)119882119888119896 100381710038171003817100381722 ⩽ 1 forall119896 = 1 2 119870 (11)
Journal of Electrical and Computer Engineering 5
The Lagrange multiplier method is used to optimize theabove problems then the objective function can be reducedto
119871 (119882119888 120582119896) = minus2 119870sum119896=1
[119878119888120581 (119883119888 119883119888)]119896119882119888119896+ 119870sum119896=1
119882119888119896 [120581 (119883119888 119883119888)119882119888119878119888119878119888119879]119896+ 120582119896 (1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]119896119896)
(12)
Here 120582119896 is a variable Meanwhile the algorithm usesKarush-Kuhn-Tucker (KKT) conditions to optimize theobjective function and Karush-Kuhn-Tucker (KKT) condi-tions meet the following three criteria
(1) 120597119871 (119882119888 120582119896)120597119882119888119896
= 0(2) 1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]
119896119896= 0
(3) 120582119896 gt 0(13)
Hence the solution to119882119888119896 becomes
119882119888119896= 119878119888119896 minus [119882119888119896119865]119896radic(119878119888119896119879 minus [119882119888119896119865]
119896)119879 120581 (119883119888 119883119888) (119878119888119896119879 minus [119882119888119896119865]119896)
(14)
where 119865 = 119878119888119878119888119879 and119882119888119896 = 119882119888119901 119901 = 119896 0 119901 = 1198965 Experimental Results
In this section we present experimental results on five bench-mark databases to illustrate the effectiveness of our methodWe compare the Laplace graph embedding class specificdictionary learning algorithm (LGECSDL) with some state-of-the-art methods In the following section we introducethe experimental environment setting database descriptionsand experimental results In the end we accordingly analyzethe experimental results
51 Experimental Settings In this section we evaluateour method on five benchmark databases The proposedLGECSDL algorithm is compared with another seven classi-cal face recognition algorithms nearest neighbor (NN) clas-sification collaborative representation based classification(CRC) [30] sparse representation based classification (SRC)[31] kernel-based probabilistic collaborative representationbased classifier (ProKCRC) [32] VGG19 [33] kernel-basedclass specific dictionary learning (KCSDL) algorithm [29]and SVM [34]
There are two parameters in the objective function ofthe LGECSDL algorithm that need to be specified 120572 is animportant parameter in the LGECSDL algorithm which is
Figure 1 Examples of the Extended YaleB database
used to adjust the trade-off between the reconstruction errorand the sparsity We increase 120572 from 2minus12 to 2minus1 in eachexperiment and find the best 120572 in our experiments120573 is another important factor in the LGECSDL algorithm120573 is used to control the trade-off between the reconstructionerror and the collaborative information We increase 120573 from2minus12 to 2minus1 and find the best 120573 in all of our experiments
We also evaluate the effect of different kernels for theLGECSDL algorithm Three different kernel functions areadopted as linear kernel (120581(119909 119910) = 119909119879119910) Hellinger kernel(120581(119909 119910) = sum119863119889=1radic119909119889119910119889) and polynomial kernel (120581(119909 119910) =(119901 + 119909119879119910)119902) Here in our experiments 119901 and 119902 are set to be 4and 2 respectively52 Database Descriptions There are five image databasesinvolved in our experiments The first one is the ExtendedYaleB database which contains 38 categories and 2414frontal-face images All the images are captured undervarying illumination conditions In our experiments theimage has been cropped and normalized to 32 times 32 pixelsFigure 1 shows several example images in the Extended YaleBdatabase
The second one is the AR database The AR databasecontains over 3000 images of 126 individuals images are shotunder different conditions of expression illumination andocclusion and each person has 26 images Figure 3 showssome examples in the AR database
The third database is the CMU-PIE database The CMU-PIE database consists of 41368 pieces of pictures whichare captured under different lighting conditions poses andexpressions The database contains 68 individuals in totaland each person has 43 different kinds of images with 13different poses We selected two types of images to carryout our experiment five near-frontal poses and all differentillumination conditions We chose 11554 images in total forour evaluation Each person has about 170 images Figure 5shows some example images in the CMU-PIE database
We also selected the Caltech101 database to verify theLGECSDL algorithm The Caltech101 database contains 9144images belonging to 101 categories each class has 31 to 800images We selected 5 images as training images in each classand the rest as test images Figure 7 shows some examples inthe Caltech101 database
The fifth database is Oxford-102 flower database thatcontains 8189 flower images belonging to 102 categories Eachimage contains 40 to 250 images and the minimum edge
6 Journal of Electrical and Computer Engineering
Table 1 Recognition rate on the Extended YaleB database ()
Methods Linear Hellinger PolynomialNN 3317 plusmn 145 NA NAVGG19 5379 plusmn 121 NA NASVM 6552 plusmn 277 7997 plusmn 257 6489 plusmn 243CRC 7807 plusmn 216 8723 plusmn 144 7633 plusmn 217SRC 7759 plusmn 159 8858 plusmn 156 7482 plusmn 235ProKCRC 7642 plusmn 213 8784 plusmn 189 7494 plusmn 203KCSDL 7855 plusmn 225 8898 plusmn 162 7968 plusmn 241LGECSDL 8018 plusmn 133 9193 plusmn 131 8105 plusmn 142
length of the image is greater than 500 pixelsTheOxford-102flower database contains pictures of flowers taken in differentillumination angle and occlusion environments and eachkind of flower image has a high degree of similarity In ourexperiments all the images aremanually cropped and resizedto 224 times 224 pixels Figure 9 shows several images in theOxford-102 flower database
53 Experiments on the Extended YaleB Database We ran-domly selected 5 images as the training samples in each cate-gory and 10 images as the testing samples In our experimentswe set the weight of the sparsity term 120572 as 2minus9 2minus7 and 2minus7for the linear kernel Hellinger kernel and polynomial kernelrespectivelyThe optimal 120573 is 2minus10 2minus8 and 2minus10 for the linearkernel Hellinger kernel and polynomial kernel respectivelyWe independently performed all the methods ten times andthen reported the average recognition rates Table 1 shows therecognition rates of all the algorithms using different kernelmethods
From Table 1 we can clearly see that LGECSDL achievesthe best recognition rates of 8018 9193 and 8105 inthe linear kernel Hellinger kernel and polynomial kernelspace respectively while KCSDL the second best methodarrives at 7855 8898 and 7968 Since illuminationvariations of images are relatively large in the Extended YaleBdatabase these experiment results validate the effectivenessand robustness of LGECSDL for image recognition withillumination variations VGG19 neural network in this exper-iment can only achieve the highest recognition rate of 5379Using a small database to train neural networks does not takeadvantage of neural networks We also verify the effect of120572 and 120573 on the LGECSDL algorithm and the experimentalresults are shown in Figure 2
From Figure 2 we can easily know that the LGECSDLalgorithmhas achieved better recognition results inHellingerkernel space With the parameter 120572 varied from 2minus13 to 2minus3the recognition rate increased gradually and then decreasedThe influence of parameter 120573 on the LGECSDL algorithm issimilar to that of the parameter 120572 The highest recognitionrate was achieved when 120572 = 2minus7 and 120573 = 2minus8 in Hellingerkernel space In the linear kernel space the recognition rateachieves the maximum value at 120572 = 2minus9 and 120573 = 2minus10 and inthe polynomial kernel space the maximum recognition ratewas obtained at 120572 = 2minus7 and 120573 = 2minus10
53
61
69
77
85
93
73
77
81
85
89
93Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 2 Parameter tuning on the Extended YaleB database
54 AR Database In this experiment we randomly selected5 images of each individual as training samples and the restfor testing Each image has been cropped to 32times 32 and pulledinto a column vector the image vectors have been performedby 1198972 normalization All the methods are independently runten times and the average recognition rates are reportedTherecognition rate of AR database is shown in Table 2
FromTable 2 we can clearly see that LGECSDL algorithmoutperforms the other methods in all kernel spaces TheLGECSDL algorithm achieves the best recognition rate of946 in the polynomial kernel space in the linear kernelspace and Hellinger kernel space the recognition rates are945 and 9413 respectively
Moreover we can also know fromTable 2 that the KCSDLalgorithm achieves the best recognition rate of 9112 inthe linear kernel space which is the highest one among theother methods From these experimental results we furtherconfirm the effectiveness and robustness of the LGECSDLalgorithm for image recognition with illumination variations
Journal of Electrical and Computer Engineering 7
Table 2 Recognition rate on the AR database ()
Methods Linear Hellinger PolynomialNN 3299 plusmn 197 NA NAVGG19 6580 plusmn 135 NA NASVM 8067 plusmn 132 8059 plusmn 126 8084 plusmn 157CRC 9198 plusmn 086 9218 plusmn 078 9243 plusmn 075SRC 8917 plusmn 114 8532 plusmn 112 8525 plusmn 141ProKCRC 8774 plusmn 143 9104 plusmn 076 8990 plusmn 121KCSDL 9112 plusmn 157 8977 plusmn 127 8902 plusmn 131LGECSDL 9450 plusmn 101 9413 plusmn 091 9460 plusmn 086
Table 3 Recognition rate on the CMU-PIE database ()
Methods Linear Hellinger PolynomialNN 3009 plusmn 167 NA NAVGG19 6180 plusmn 128 NA NASVM 6679 plusmn 263 6571 plusmn 254 6546 plusmn 276CRC 7289 plusmn 221 7519 plusmn 209 7319 plusmn 220SRC 7216 plusmn 209 7026 plusmn 223 6919 plusmn 217ProKCRC 6891 plusmn 197 7541 plusmn 186 7264 plusmn 205KCSDL 7446 plusmn 201 7478 plusmn 205 7349 plusmn 222LGECSDL 7912 plusmn 152 8103 plusmn 134 8005 plusmn 127
and expression changes We also verify the effect of 120572 and 120573on the AR database Figure 4 shows the experiment results
FromFigure 4 we can clearly see that the recognition ratereached the maximum value when 120572 is 2minus7 and 120573 is 2minus8 inHellinger kernel space and polynomial kernel space in thelinear kernel space the recognition rate achieves the highestvalue when 120572 is equal to 2minus9 and 120573 is 2minus8 With 120572 changedfrom 2minus13 to 2minus3 the recognition rate increased first and thendecreased The 120573 parameter shows a similar trend and when120573 is greater than 2minus8 the recognition rate decreases rapidly
55 CMU-PIE Database In this experiment we chose theCMU-PIE database to evaluate the performance of theLGECSDL algorithm Five images of each individual arerandomly selected for training and the remainder for testingWe also cropped each image to 32 times 32 and then pulled theminto a column vector Finally we normalized all the vectors by1198972 normalization We independently ran all the methods tentimes and then reported the average recognition rates Table 3gives the recognition rates of all the methods under differentkernel spaces
From Table 3 we can see that the LGECSDL algorithmalways achieves the highest recognition rates under all dif-ferent kernel spaces In the polynomial kernel space theLGECSDL algorithm outperforms KCSDL which achievesthe second highest recognition rate by more than 4improvement of recognition rate In Hellinger kernel spacethe LGECSDL achieves the best recognition rate of 8103and 6 points higher than the KCSDL algorithm In thelinear kernel space the face recognition rate of LGECSDL andKCSDL is 7912and 7446 respectively From these exper-imental results we confirm the effectiveness and robustness
Figure 3 Examples of the AR database
53
61
69
77
85
93
81
84
87
90
93
96Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 4 Parameter tuning on the AR database
of the LGECSDL algorithm We also evaluate the effect of120572 and 120573 on the CMU-PIE database Figure 6 shows theexperiment results
From Figure 6 we can see that when 120572 is 2minus7 and 120573 is 2minus8the face recognition rate reaches the highest value of 8103in Hellinger kernel space and when 120572 is 2minus7 and 120573 is 2minus10 thehighest face recognition rate is obtained in the polynomialkernel space We can also know from Figure 6 that when 120572 is
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
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2 Journal of Electrical and Computer Engineering
that minimizes the interclass variance and maximizes thedistance between the projected means of the classes [11] Taoet al [12] proposed a general tensor discriminant analysismethod as a preprocessing step for the LDA algorithm toreduce the undersampling problem The locality preservingprojection algorithm was proposed to preserve the neighbor-hood structure of the data [13] In the procedure of classifi-cation Liu et al [14] proposed a new belief-based 119896-nearestneighbor classifier to make the classification result morerobust to misclassification errors Noh et al [15] proposed anearest neighbor algorithm to enhance the performance ofthe nearest neighbor classification by learning a local metricAlthough the 119896-nearest neighbor classifier and the nearestneighbor classifier have achieved good results on some datasets they did not select the most discriminatory feature ofthe sample to classify So subspace-based classifier designmethods were proposed to improve the classification effect
The sparse representation based classification algorithmuses training samples to construct an overcomplete dictio-nary and the test samples can be well represented as a sparselinear combination of elements from the dictionary [16]But the subsequent research shows that sparseness cannotextract the most discriminatory features of the samplesCollaborative representation based classification (CRC) wasproposed which uses the L2-norm constraint to reveal theinternal structure of the testing sample [17] Although theSRC and CRC methods have achieved superior performancein visual recognition both SRC and CRC algorithms directlyuse the training samples as the dictionary matrix The directuse of training samples to build dictionaries can lead to twodrawbacks first very few samples to build an overcompletedictionary which may result in low classification accuracyand second very redundant dictionary samples which pre-vent the original signals from being effectively expressedresulting in poor classifier performance
So dictionary learning methods are proposed to improvethe classification effect Discriminative dictionary learningapproaches can be divided into three types shared dictionarylearning class specific dictionary learning and hybrid dictio-nary learningThe shared dictionary learningmethod usuallyuses all training samples to obtain a classification dictionaryLu et al [18] proposed a locality weighted sparse representa-tion based classification (WSRC) method which utilizes bothdata locality and linearity to train a classification dictionaryYang et al [19] proposed a novel dictionary learning methodbased on the Fisher discrimination criterion to improve thepattern classification performance Yang et al [20] proposeda latent dictionary learning method to learn a discriminativedictionary and build its relationship to class labels adaptivelyJiang et al [21] proposed an algorithm to learn a singleovercomplete dictionary and an optimal linear classifier forface recognition Zhou et al [22] presented a dictionarylearning algorithm to exploit the visual correlation withina group of visually similar object categories for dictionarylearning where a commonly shared dictionary and multiplecategory-specific dictionaries are accordingly modeled Theclass specific dictionary learningmethod trained a dictionaryfor each class of samples Sun et al [23] learned a class specificsubdictionary for each class and a common subdictionary
shared by all classes to improve the classification perfor-mance Wang and Kong [24] proposed a method to explicitlylearn a class specific dictionary for each category whichcaptures the most discriminative features of this categoryand simultaneously learn a common pattern pool whoseatoms are shared by all the categories and only contribute torepresentation of the data rather than discrimination
The hybrid dictionary learning method is the combi-nation of the above two methods Rodriguez and Sapiro[25] proposed a new dictionary learning method whichuses a class-dependent supervised constraint and orthogonalconstraint this method learns the intraclass structure whileincreasing the interclass discrimination and expands thedifference between classes Gao et al [26] learned a category-specific dictionary for each category and a shared dictionaryfor all the categories and this method improves conventionalbasic-level object categorization Liu et al [27] proposed alocality sensitive dictionary learning algorithm with globalconsistency and smoothness constraint to overcome therestriction of linearity at a relatively low cost Although thehybrid dictionary learning method achieved good resultsthese methods usually operate in the original Euclideanspace which cannot capture nonlinear structures hiddenin data So many kernel-based classifiers are designed tosolve this problem Nguyen et al [28] presented a dictionarylearning method for sparse representation based on thekernel method Liu et al [29] proposed a multiple-viewself-explanatory sparse representation dictionary learningalgorithm (MSSR) to capture and combine various salientregions and structures from different kernel spaces and thismethod achieved superior performance in the field of facerecognition
As better effects had been achieved by MSSR algorithmthis algorithm neither took into consideration the details oftraining samples in the original sample space nor protectedthis powerful information conducive to classification in thedictionary space Therefore in this algorithm the Laplaceconstraint is added to the objective function to make theclosely similar samples in low dimensional space also veryclose in the high dimensional dictionary space
Motivated by this we proposed a Laplace graphembedding class specific dictionary learning algorithmand extended this method to arbitrary kernel space Themain contribution is listed in four aspects (1) We proposea Laplace embedding sparse representation algorithm Itcombines the advantages of SRCrsquos discriminant ability andmaintains the intrinsic local geometric feature of the samplefeatures by Laplace embedding (2) We propose a Laplaceembedding constraint dictionary learning algorithm toconstruct superior subspace and reduce the residual error(3) We extend this algorithm to arbitrary kernel space tofind the nonlinear structure of face images (4) Experimentalresults on several benchmark databases demonstrate thesuperior performance of our proposed algorithm
The rest of the paper is organized as follows Section 2overviews the three classical face recognition algorithmsSection 3 proposes our Laplace graph embedding class spe-cific dictionary learning algorithm with kernelsThe solutionto the minimization of the objective function is elaborated in
Journal of Electrical and Computer Engineering 3
Section 4 Then experimental results and analysis are shownin Section 5 Finally discussions and conclusions are drawnin Section 6
2 Overview of SRC and CRC
In this section we will briefly overview two classical facerecognition algorithms SRC and CRC
Suppose that there are 119862 classes in the training samplesand each class has 119873119888 elements 119873 = sum119862119888=1119873119888 where 119873represents the total number of training samples119883 representsall the training samples119883 = [1198831 1198832 119883119888 119883119873] where119883119888 isin 119877119863times119873119888 119863 represents the dimension of the samplefeatures 119883119888 represents the 119888th class of the training samplesSupposing that 119910 is a test sample and 119910 isin 119877119863times1 the sparserepresentation of sample 119910 can be expressed as
119878 = argmin1199041003817100381710038171003817119910 minus 119883119888119904100381710038171003817100381722 + 2120572 1199041 (1)
where 119904 is the sparse coding of sample 119910 in the 119888th dictionaryand 120572 is the regularization parameter in formula (1) which isused to control the sparsity and accuracy of the expression
The collaborative representation based classification algo-rithm applies L2-norm constraint on the object function theobjective of the CRC algorithm can be rewritten as follows
119878 = argmin1199041003817100381710038171003817119910 minus 119883119888119904100381710038171003817100381722 + 2120572 11990422 (2)
where 120573 is the regularization parameter to control theexpression accuracy of the object function
Both SRC and CRC methods directly use the trainingsamples as the dictionary And each base in the dictionary hasthe same contribution to the sample expression The testingsample 119910 can be encoded as
119910 asymp 119883119888119904 (3)
Here 119883119888 is the dictionary matrix composed of the 119888th classtraining samples and 119904 is the sparse coding of 119910
Directly using the training samples as the dictionaryleads to high residual error Liu et al [29] proposed asingle-view self-explanatory sparse representation dictionarylearning algorithm (SSSR) Supposing that 119888 represents theclass number of the training samples and 119883119888 means thecollection of sample characteristics of class 119888 the objectivefunction of the SSSR method can be formulated as
min119882119888119878119888
119891 (119882119888 119878119888) = 1003817100381710038171003817119883119888 minus 11988311988811988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119894=1
1003817100381710038171003817(119878119888119894 )10038171003817100381710038171st 1003817100381710038171003817119883119888 (119882119888)1198961003817100381710038171003817 le 1 forall119896 = 1 2 119870
(4)
where 119878119888 is the sparse codes of the 119888th class and 119878119888119894 representsthe 119894th column of 119878119888 The SSSR algorithm reconstructed thedictionary matrix119882119888 is the dictionary weight matrix119882119888 isin119877119873119888times119870 and 119873119888 is the number of the 119888th classes 119882119888 expandsthe original dictionary space into amore complete dictionaryspace when the identity matrix 119882119888 isin 119868119873119888times119873119888 appears theclass specific dictionary learning algorithm evolves into the
SRC method The existence of 119882119888 matrix makes dictionarylearning more flexible in the process of expression and thereconstruction error may be reduced as well
Meanwhile Liu et al [29] extended the SSSR algorithminto kernel spaces which can map the original samplefeatures into a high dimensional nonlinear space for bettermining of nonlinear relationships between samples Theobjective function of the multiple-view kernel-based classspecific dictionary learning algorithm (KCSDL) is shown asfollows
min119882119888119878119888
119891 (119882119888 119878119888)
= 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119894=1
1003817100381710038171003817(119878119888119894 )10038171003817100381710038171st 1003817100381710038171003817120601 (119883119888) (119882119888)1198961003817100381710038171003817 le 1 forall119896 = 1 2 119870
(5)
where 120601 119877119863 rarr 119877119905 means the kernel function it maps theoriginal feature space into a high dimensional kernel space
3 Our Proposed Approach
Although the above methods have achieved good results inthe field of face recognition there are still some deficienciesThe SSSR algorithm uses a reconstructed dictionary matrixto make sparse representation on samples however it doesnot take into account the fact that only the sparsity constrainton the target is not necessary to gain results for betterclassification
Motivated by this we have proposed the sparse represen-tation algorithm based on Laplace graph embedding whiletaking into account the sparse representation on the samplesthis algorithm mines the details implicit in the trainingsamples therefore the same sample is more concentrated inthe sparse expression space so as to reduce the fitting errorand improve the classification effect
The objective function of our proposed sparse represen-tation algorithm based on Laplace graph embedding nowbecomes
119891 (119878119888)= 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572
119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171+ 120573sum119894 =119895
10038171003817100381710038171003817119878119888119894 minus 11987811988811989510038171003817100381710038171003817 119901119894119895st 10038161003816100381610038161003816100381610038161003816120601 (119883119888) (119882119888)11989610038161003816100381610038161003816100381610038161003816 le 1 forall119896 = 1 2 119870
(6)
where 119883119888 means class 119888 training samples 119882119888 means dictio-nary weight matrix 119878119888 means the dictionary representationof class 119888 samples and 119878119888119899 represents the 119899th column of 1198781198884 Optimization of the Objective Function
In this section we focus on solving the optimization problemfor the proposed Laplace graph embedding class specific
4 Journal of Electrical and Computer Engineering
dictionary learning algorithm The dictionary weight matrix119882c and sparse representation matrix 119878119888 can be optimized byiterative approaches
When each element in the 119882119888 matrix is updated theremaining elements in 119882119888 matrix and 119878119888 matrix are fixedat this time the objective function is changed into an L2-norm constrained least-squares minimization subproblemSimilarly when each element in 119878119888 matrix is updated 119882119888matrix and the remaining elements in 119878119888 matrix are fixedThe objective function can be seen as an L1-norm constrainedleast-squares minimization subproblem
41 L1-Norm Regularized Minimization Subproblem Whenupdating the elements in 119878119888matrix the nonupdated elementsin 119878119888 and119882119888matrix will be fixed Here the objective functioncan be formulated as
119891 (119878119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171+ 120573sum119894 =119895
10038171003817100381710038171003817119878119888119894 minus 11987811988811989510038171003817100381710038171003817 119901119894119895(7)
where 119901119894119895 is the weight value which describes the neighboringdegree of 119909119888119894 and 119909119888119895 and119901119894119895 = 119890minus119909119888119894minus119909119888119895119905 119909119888119894 and 119909119888119895 are trainingsamples that belong to the 119888th class and 119905 is a constant whichcontrols the range of 119901119894119895 Formula (7) can be simplified as
119891 (119878119888)= trace 120601 (119883119888)119879 120601 (119883119888) minus 2120601 (119883119888)119879 120601 (119883119888)119882119888119878119888+ trace (119878119888)119879 (119882119888119879120601 (119883119888)119879 120601 (119883119888)119882119888) 119878119888+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573 trace 119878119888119871119878119888119879
= trace 120581 (119883119888 119883119888) minus 2119873119888sum119899=1
[120581 (119883119888 119883119888)119882119888]119899 119878119888119899
+ 119873119888sum119899=1
(119878119888119879)119899[119882119888119879120581 (119883119888 119883119888)119882119888] 119878119888119899
+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573119870sum119896=1
[119878119888119896119871119878119888119879119896 ]= trace 120581 (119883119888 119883119888)minus 2119873119888sum119899=1
119870sum119896=1
[120581 (119883119888 119883119888)119882119888]119899119896 119878119888119896119899
+ 119873119888sum119899=1
119870sum119896=1
[ 119870sum119897=1
(119878119888119879)119899119897(119882119888119879120581 (119883119888 119883119888)119882119888)
119897119896] 119878119888119896119899
+ 2120572 119870sum119896=1
119873119888sum119899=1
10038161003816100381610038161198781198881198961198991003816100381610038161003816
+ 120573 119870sum119896=1
[119873119888sum119899=1
(119873119888sum119897=1
119878119888119896119897119871 119897119899)(119878119879)119899119896]
(8)
where 119871 = 119863 minus 119875 119863119894119894 = sum119895 119875119894119895 and matrix 119875 is the weightmatrix expressing the sample neighboring distance 120581(119883119888 119883119888)means the kernel function of the sample and 120581(119883119888 119883119888)is calculated prior to dictionary updating 120581(119883119888 119883119888) =120601(119883119888)119879120601(119883119888)
In this algorithm each element in 119878119888 is updated sequen-tially when 119878119896119899 is updated the other elements in the 119878matrixare regarded as constants After ignoring the constant term offormula (8) formula (8) can be simplified as
119891 (119878119888119896119899) = (119882119888119879120581 (119883119888 119883119888)119882119888)119896119896(119878119888119896119899)2
+ 2 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899119878119888119896119899
minus 2 [120581 (119883119888 119883119888)119882119888]119899119896 119878c119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816 + 120573119871119899119899 (119878119888119896119899)2
+ 2120573 119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897119878119888119896119899 = [(119882119888119879120581 (119883119888 119883119888)119882119888)119896119896
+ 120573119871119899119899] (119878119888119896119899)2 minus 2[(120581 (119883119888 119883119888)119882119888)119899119896
minus 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899 minus 120573
119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897]sdot 119878119888119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816
(9)
According to the solving method in [29] it is easy toobtain the solution of the minimum value of 119891(119878119888119896119899) underthe current iteration condition
119878119888119896119899 = 11 + 120573119871119899119899 min (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 minus120572 + 11 + 120573119871119899119899
sdotmax (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 120572
(10)
where 119864 = 119882119888119879120581(119883119888 119883119888)119882119888 and 119878119888119896119899 = 119878119888119901119902 119901 = 119896 119902 =119899 0 119901 = 119896 119902 = 11989942 ℓ2 Norm Constrained Minimization Subproblem Whenupdating the dictionary matrix 119882119888 119878119888 and the nonupdatedelements in 119882119888 matrix will be fixed The objective functioncan be transformed into the following form
119891 (119882119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865st 1003817100381710038171003817120601 (119883119888)119882119888119896 100381710038171003817100381722 ⩽ 1 forall119896 = 1 2 119870 (11)
Journal of Electrical and Computer Engineering 5
The Lagrange multiplier method is used to optimize theabove problems then the objective function can be reducedto
119871 (119882119888 120582119896) = minus2 119870sum119896=1
[119878119888120581 (119883119888 119883119888)]119896119882119888119896+ 119870sum119896=1
119882119888119896 [120581 (119883119888 119883119888)119882119888119878119888119878119888119879]119896+ 120582119896 (1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]119896119896)
(12)
Here 120582119896 is a variable Meanwhile the algorithm usesKarush-Kuhn-Tucker (KKT) conditions to optimize theobjective function and Karush-Kuhn-Tucker (KKT) condi-tions meet the following three criteria
(1) 120597119871 (119882119888 120582119896)120597119882119888119896
= 0(2) 1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]
119896119896= 0
(3) 120582119896 gt 0(13)
Hence the solution to119882119888119896 becomes
119882119888119896= 119878119888119896 minus [119882119888119896119865]119896radic(119878119888119896119879 minus [119882119888119896119865]
119896)119879 120581 (119883119888 119883119888) (119878119888119896119879 minus [119882119888119896119865]119896)
(14)
where 119865 = 119878119888119878119888119879 and119882119888119896 = 119882119888119901 119901 = 119896 0 119901 = 1198965 Experimental Results
In this section we present experimental results on five bench-mark databases to illustrate the effectiveness of our methodWe compare the Laplace graph embedding class specificdictionary learning algorithm (LGECSDL) with some state-of-the-art methods In the following section we introducethe experimental environment setting database descriptionsand experimental results In the end we accordingly analyzethe experimental results
51 Experimental Settings In this section we evaluateour method on five benchmark databases The proposedLGECSDL algorithm is compared with another seven classi-cal face recognition algorithms nearest neighbor (NN) clas-sification collaborative representation based classification(CRC) [30] sparse representation based classification (SRC)[31] kernel-based probabilistic collaborative representationbased classifier (ProKCRC) [32] VGG19 [33] kernel-basedclass specific dictionary learning (KCSDL) algorithm [29]and SVM [34]
There are two parameters in the objective function ofthe LGECSDL algorithm that need to be specified 120572 is animportant parameter in the LGECSDL algorithm which is
Figure 1 Examples of the Extended YaleB database
used to adjust the trade-off between the reconstruction errorand the sparsity We increase 120572 from 2minus12 to 2minus1 in eachexperiment and find the best 120572 in our experiments120573 is another important factor in the LGECSDL algorithm120573 is used to control the trade-off between the reconstructionerror and the collaborative information We increase 120573 from2minus12 to 2minus1 and find the best 120573 in all of our experiments
We also evaluate the effect of different kernels for theLGECSDL algorithm Three different kernel functions areadopted as linear kernel (120581(119909 119910) = 119909119879119910) Hellinger kernel(120581(119909 119910) = sum119863119889=1radic119909119889119910119889) and polynomial kernel (120581(119909 119910) =(119901 + 119909119879119910)119902) Here in our experiments 119901 and 119902 are set to be 4and 2 respectively52 Database Descriptions There are five image databasesinvolved in our experiments The first one is the ExtendedYaleB database which contains 38 categories and 2414frontal-face images All the images are captured undervarying illumination conditions In our experiments theimage has been cropped and normalized to 32 times 32 pixelsFigure 1 shows several example images in the Extended YaleBdatabase
The second one is the AR database The AR databasecontains over 3000 images of 126 individuals images are shotunder different conditions of expression illumination andocclusion and each person has 26 images Figure 3 showssome examples in the AR database
The third database is the CMU-PIE database The CMU-PIE database consists of 41368 pieces of pictures whichare captured under different lighting conditions poses andexpressions The database contains 68 individuals in totaland each person has 43 different kinds of images with 13different poses We selected two types of images to carryout our experiment five near-frontal poses and all differentillumination conditions We chose 11554 images in total forour evaluation Each person has about 170 images Figure 5shows some example images in the CMU-PIE database
We also selected the Caltech101 database to verify theLGECSDL algorithm The Caltech101 database contains 9144images belonging to 101 categories each class has 31 to 800images We selected 5 images as training images in each classand the rest as test images Figure 7 shows some examples inthe Caltech101 database
The fifth database is Oxford-102 flower database thatcontains 8189 flower images belonging to 102 categories Eachimage contains 40 to 250 images and the minimum edge
6 Journal of Electrical and Computer Engineering
Table 1 Recognition rate on the Extended YaleB database ()
Methods Linear Hellinger PolynomialNN 3317 plusmn 145 NA NAVGG19 5379 plusmn 121 NA NASVM 6552 plusmn 277 7997 plusmn 257 6489 plusmn 243CRC 7807 plusmn 216 8723 plusmn 144 7633 plusmn 217SRC 7759 plusmn 159 8858 plusmn 156 7482 plusmn 235ProKCRC 7642 plusmn 213 8784 plusmn 189 7494 plusmn 203KCSDL 7855 plusmn 225 8898 plusmn 162 7968 plusmn 241LGECSDL 8018 plusmn 133 9193 plusmn 131 8105 plusmn 142
length of the image is greater than 500 pixelsTheOxford-102flower database contains pictures of flowers taken in differentillumination angle and occlusion environments and eachkind of flower image has a high degree of similarity In ourexperiments all the images aremanually cropped and resizedto 224 times 224 pixels Figure 9 shows several images in theOxford-102 flower database
53 Experiments on the Extended YaleB Database We ran-domly selected 5 images as the training samples in each cate-gory and 10 images as the testing samples In our experimentswe set the weight of the sparsity term 120572 as 2minus9 2minus7 and 2minus7for the linear kernel Hellinger kernel and polynomial kernelrespectivelyThe optimal 120573 is 2minus10 2minus8 and 2minus10 for the linearkernel Hellinger kernel and polynomial kernel respectivelyWe independently performed all the methods ten times andthen reported the average recognition rates Table 1 shows therecognition rates of all the algorithms using different kernelmethods
From Table 1 we can clearly see that LGECSDL achievesthe best recognition rates of 8018 9193 and 8105 inthe linear kernel Hellinger kernel and polynomial kernelspace respectively while KCSDL the second best methodarrives at 7855 8898 and 7968 Since illuminationvariations of images are relatively large in the Extended YaleBdatabase these experiment results validate the effectivenessand robustness of LGECSDL for image recognition withillumination variations VGG19 neural network in this exper-iment can only achieve the highest recognition rate of 5379Using a small database to train neural networks does not takeadvantage of neural networks We also verify the effect of120572 and 120573 on the LGECSDL algorithm and the experimentalresults are shown in Figure 2
From Figure 2 we can easily know that the LGECSDLalgorithmhas achieved better recognition results inHellingerkernel space With the parameter 120572 varied from 2minus13 to 2minus3the recognition rate increased gradually and then decreasedThe influence of parameter 120573 on the LGECSDL algorithm issimilar to that of the parameter 120572 The highest recognitionrate was achieved when 120572 = 2minus7 and 120573 = 2minus8 in Hellingerkernel space In the linear kernel space the recognition rateachieves the maximum value at 120572 = 2minus9 and 120573 = 2minus10 and inthe polynomial kernel space the maximum recognition ratewas obtained at 120572 = 2minus7 and 120573 = 2minus10
53
61
69
77
85
93
73
77
81
85
89
93Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 2 Parameter tuning on the Extended YaleB database
54 AR Database In this experiment we randomly selected5 images of each individual as training samples and the restfor testing Each image has been cropped to 32times 32 and pulledinto a column vector the image vectors have been performedby 1198972 normalization All the methods are independently runten times and the average recognition rates are reportedTherecognition rate of AR database is shown in Table 2
FromTable 2 we can clearly see that LGECSDL algorithmoutperforms the other methods in all kernel spaces TheLGECSDL algorithm achieves the best recognition rate of946 in the polynomial kernel space in the linear kernelspace and Hellinger kernel space the recognition rates are945 and 9413 respectively
Moreover we can also know fromTable 2 that the KCSDLalgorithm achieves the best recognition rate of 9112 inthe linear kernel space which is the highest one among theother methods From these experimental results we furtherconfirm the effectiveness and robustness of the LGECSDLalgorithm for image recognition with illumination variations
Journal of Electrical and Computer Engineering 7
Table 2 Recognition rate on the AR database ()
Methods Linear Hellinger PolynomialNN 3299 plusmn 197 NA NAVGG19 6580 plusmn 135 NA NASVM 8067 plusmn 132 8059 plusmn 126 8084 plusmn 157CRC 9198 plusmn 086 9218 plusmn 078 9243 plusmn 075SRC 8917 plusmn 114 8532 plusmn 112 8525 plusmn 141ProKCRC 8774 plusmn 143 9104 plusmn 076 8990 plusmn 121KCSDL 9112 plusmn 157 8977 plusmn 127 8902 plusmn 131LGECSDL 9450 plusmn 101 9413 plusmn 091 9460 plusmn 086
Table 3 Recognition rate on the CMU-PIE database ()
Methods Linear Hellinger PolynomialNN 3009 plusmn 167 NA NAVGG19 6180 plusmn 128 NA NASVM 6679 plusmn 263 6571 plusmn 254 6546 plusmn 276CRC 7289 plusmn 221 7519 plusmn 209 7319 plusmn 220SRC 7216 plusmn 209 7026 plusmn 223 6919 plusmn 217ProKCRC 6891 plusmn 197 7541 plusmn 186 7264 plusmn 205KCSDL 7446 plusmn 201 7478 plusmn 205 7349 plusmn 222LGECSDL 7912 plusmn 152 8103 plusmn 134 8005 plusmn 127
and expression changes We also verify the effect of 120572 and 120573on the AR database Figure 4 shows the experiment results
FromFigure 4 we can clearly see that the recognition ratereached the maximum value when 120572 is 2minus7 and 120573 is 2minus8 inHellinger kernel space and polynomial kernel space in thelinear kernel space the recognition rate achieves the highestvalue when 120572 is equal to 2minus9 and 120573 is 2minus8 With 120572 changedfrom 2minus13 to 2minus3 the recognition rate increased first and thendecreased The 120573 parameter shows a similar trend and when120573 is greater than 2minus8 the recognition rate decreases rapidly
55 CMU-PIE Database In this experiment we chose theCMU-PIE database to evaluate the performance of theLGECSDL algorithm Five images of each individual arerandomly selected for training and the remainder for testingWe also cropped each image to 32 times 32 and then pulled theminto a column vector Finally we normalized all the vectors by1198972 normalization We independently ran all the methods tentimes and then reported the average recognition rates Table 3gives the recognition rates of all the methods under differentkernel spaces
From Table 3 we can see that the LGECSDL algorithmalways achieves the highest recognition rates under all dif-ferent kernel spaces In the polynomial kernel space theLGECSDL algorithm outperforms KCSDL which achievesthe second highest recognition rate by more than 4improvement of recognition rate In Hellinger kernel spacethe LGECSDL achieves the best recognition rate of 8103and 6 points higher than the KCSDL algorithm In thelinear kernel space the face recognition rate of LGECSDL andKCSDL is 7912and 7446 respectively From these exper-imental results we confirm the effectiveness and robustness
Figure 3 Examples of the AR database
53
61
69
77
85
93
81
84
87
90
93
96Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 4 Parameter tuning on the AR database
of the LGECSDL algorithm We also evaluate the effect of120572 and 120573 on the CMU-PIE database Figure 6 shows theexperiment results
From Figure 6 we can see that when 120572 is 2minus7 and 120573 is 2minus8the face recognition rate reaches the highest value of 8103in Hellinger kernel space and when 120572 is 2minus7 and 120573 is 2minus10 thehighest face recognition rate is obtained in the polynomialkernel space We can also know from Figure 6 that when 120572 is
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
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Journal of Electrical and Computer Engineering 3
Section 4 Then experimental results and analysis are shownin Section 5 Finally discussions and conclusions are drawnin Section 6
2 Overview of SRC and CRC
In this section we will briefly overview two classical facerecognition algorithms SRC and CRC
Suppose that there are 119862 classes in the training samplesand each class has 119873119888 elements 119873 = sum119862119888=1119873119888 where 119873represents the total number of training samples119883 representsall the training samples119883 = [1198831 1198832 119883119888 119883119873] where119883119888 isin 119877119863times119873119888 119863 represents the dimension of the samplefeatures 119883119888 represents the 119888th class of the training samplesSupposing that 119910 is a test sample and 119910 isin 119877119863times1 the sparserepresentation of sample 119910 can be expressed as
119878 = argmin1199041003817100381710038171003817119910 minus 119883119888119904100381710038171003817100381722 + 2120572 1199041 (1)
where 119904 is the sparse coding of sample 119910 in the 119888th dictionaryand 120572 is the regularization parameter in formula (1) which isused to control the sparsity and accuracy of the expression
The collaborative representation based classification algo-rithm applies L2-norm constraint on the object function theobjective of the CRC algorithm can be rewritten as follows
119878 = argmin1199041003817100381710038171003817119910 minus 119883119888119904100381710038171003817100381722 + 2120572 11990422 (2)
where 120573 is the regularization parameter to control theexpression accuracy of the object function
Both SRC and CRC methods directly use the trainingsamples as the dictionary And each base in the dictionary hasthe same contribution to the sample expression The testingsample 119910 can be encoded as
119910 asymp 119883119888119904 (3)
Here 119883119888 is the dictionary matrix composed of the 119888th classtraining samples and 119904 is the sparse coding of 119910
Directly using the training samples as the dictionaryleads to high residual error Liu et al [29] proposed asingle-view self-explanatory sparse representation dictionarylearning algorithm (SSSR) Supposing that 119888 represents theclass number of the training samples and 119883119888 means thecollection of sample characteristics of class 119888 the objectivefunction of the SSSR method can be formulated as
min119882119888119878119888
119891 (119882119888 119878119888) = 1003817100381710038171003817119883119888 minus 11988311988811988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119894=1
1003817100381710038171003817(119878119888119894 )10038171003817100381710038171st 1003817100381710038171003817119883119888 (119882119888)1198961003817100381710038171003817 le 1 forall119896 = 1 2 119870
(4)
where 119878119888 is the sparse codes of the 119888th class and 119878119888119894 representsthe 119894th column of 119878119888 The SSSR algorithm reconstructed thedictionary matrix119882119888 is the dictionary weight matrix119882119888 isin119877119873119888times119870 and 119873119888 is the number of the 119888th classes 119882119888 expandsthe original dictionary space into amore complete dictionaryspace when the identity matrix 119882119888 isin 119868119873119888times119873119888 appears theclass specific dictionary learning algorithm evolves into the
SRC method The existence of 119882119888 matrix makes dictionarylearning more flexible in the process of expression and thereconstruction error may be reduced as well
Meanwhile Liu et al [29] extended the SSSR algorithminto kernel spaces which can map the original samplefeatures into a high dimensional nonlinear space for bettermining of nonlinear relationships between samples Theobjective function of the multiple-view kernel-based classspecific dictionary learning algorithm (KCSDL) is shown asfollows
min119882119888119878119888
119891 (119882119888 119878119888)
= 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119894=1
1003817100381710038171003817(119878119888119894 )10038171003817100381710038171st 1003817100381710038171003817120601 (119883119888) (119882119888)1198961003817100381710038171003817 le 1 forall119896 = 1 2 119870
(5)
where 120601 119877119863 rarr 119877119905 means the kernel function it maps theoriginal feature space into a high dimensional kernel space
3 Our Proposed Approach
Although the above methods have achieved good results inthe field of face recognition there are still some deficienciesThe SSSR algorithm uses a reconstructed dictionary matrixto make sparse representation on samples however it doesnot take into account the fact that only the sparsity constrainton the target is not necessary to gain results for betterclassification
Motivated by this we have proposed the sparse represen-tation algorithm based on Laplace graph embedding whiletaking into account the sparse representation on the samplesthis algorithm mines the details implicit in the trainingsamples therefore the same sample is more concentrated inthe sparse expression space so as to reduce the fitting errorand improve the classification effect
The objective function of our proposed sparse represen-tation algorithm based on Laplace graph embedding nowbecomes
119891 (119878119888)= 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572
119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171+ 120573sum119894 =119895
10038171003817100381710038171003817119878119888119894 minus 11987811988811989510038171003817100381710038171003817 119901119894119895st 10038161003816100381610038161003816100381610038161003816120601 (119883119888) (119882119888)11989610038161003816100381610038161003816100381610038161003816 le 1 forall119896 = 1 2 119870
(6)
where 119883119888 means class 119888 training samples 119882119888 means dictio-nary weight matrix 119878119888 means the dictionary representationof class 119888 samples and 119878119888119899 represents the 119899th column of 1198781198884 Optimization of the Objective Function
In this section we focus on solving the optimization problemfor the proposed Laplace graph embedding class specific
4 Journal of Electrical and Computer Engineering
dictionary learning algorithm The dictionary weight matrix119882c and sparse representation matrix 119878119888 can be optimized byiterative approaches
When each element in the 119882119888 matrix is updated theremaining elements in 119882119888 matrix and 119878119888 matrix are fixedat this time the objective function is changed into an L2-norm constrained least-squares minimization subproblemSimilarly when each element in 119878119888 matrix is updated 119882119888matrix and the remaining elements in 119878119888 matrix are fixedThe objective function can be seen as an L1-norm constrainedleast-squares minimization subproblem
41 L1-Norm Regularized Minimization Subproblem Whenupdating the elements in 119878119888matrix the nonupdated elementsin 119878119888 and119882119888matrix will be fixed Here the objective functioncan be formulated as
119891 (119878119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171+ 120573sum119894 =119895
10038171003817100381710038171003817119878119888119894 minus 11987811988811989510038171003817100381710038171003817 119901119894119895(7)
where 119901119894119895 is the weight value which describes the neighboringdegree of 119909119888119894 and 119909119888119895 and119901119894119895 = 119890minus119909119888119894minus119909119888119895119905 119909119888119894 and 119909119888119895 are trainingsamples that belong to the 119888th class and 119905 is a constant whichcontrols the range of 119901119894119895 Formula (7) can be simplified as
119891 (119878119888)= trace 120601 (119883119888)119879 120601 (119883119888) minus 2120601 (119883119888)119879 120601 (119883119888)119882119888119878119888+ trace (119878119888)119879 (119882119888119879120601 (119883119888)119879 120601 (119883119888)119882119888) 119878119888+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573 trace 119878119888119871119878119888119879
= trace 120581 (119883119888 119883119888) minus 2119873119888sum119899=1
[120581 (119883119888 119883119888)119882119888]119899 119878119888119899
+ 119873119888sum119899=1
(119878119888119879)119899[119882119888119879120581 (119883119888 119883119888)119882119888] 119878119888119899
+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573119870sum119896=1
[119878119888119896119871119878119888119879119896 ]= trace 120581 (119883119888 119883119888)minus 2119873119888sum119899=1
119870sum119896=1
[120581 (119883119888 119883119888)119882119888]119899119896 119878119888119896119899
+ 119873119888sum119899=1
119870sum119896=1
[ 119870sum119897=1
(119878119888119879)119899119897(119882119888119879120581 (119883119888 119883119888)119882119888)
119897119896] 119878119888119896119899
+ 2120572 119870sum119896=1
119873119888sum119899=1
10038161003816100381610038161198781198881198961198991003816100381610038161003816
+ 120573 119870sum119896=1
[119873119888sum119899=1
(119873119888sum119897=1
119878119888119896119897119871 119897119899)(119878119879)119899119896]
(8)
where 119871 = 119863 minus 119875 119863119894119894 = sum119895 119875119894119895 and matrix 119875 is the weightmatrix expressing the sample neighboring distance 120581(119883119888 119883119888)means the kernel function of the sample and 120581(119883119888 119883119888)is calculated prior to dictionary updating 120581(119883119888 119883119888) =120601(119883119888)119879120601(119883119888)
In this algorithm each element in 119878119888 is updated sequen-tially when 119878119896119899 is updated the other elements in the 119878matrixare regarded as constants After ignoring the constant term offormula (8) formula (8) can be simplified as
119891 (119878119888119896119899) = (119882119888119879120581 (119883119888 119883119888)119882119888)119896119896(119878119888119896119899)2
+ 2 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899119878119888119896119899
minus 2 [120581 (119883119888 119883119888)119882119888]119899119896 119878c119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816 + 120573119871119899119899 (119878119888119896119899)2
+ 2120573 119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897119878119888119896119899 = [(119882119888119879120581 (119883119888 119883119888)119882119888)119896119896
+ 120573119871119899119899] (119878119888119896119899)2 minus 2[(120581 (119883119888 119883119888)119882119888)119899119896
minus 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899 minus 120573
119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897]sdot 119878119888119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816
(9)
According to the solving method in [29] it is easy toobtain the solution of the minimum value of 119891(119878119888119896119899) underthe current iteration condition
119878119888119896119899 = 11 + 120573119871119899119899 min (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 minus120572 + 11 + 120573119871119899119899
sdotmax (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 120572
(10)
where 119864 = 119882119888119879120581(119883119888 119883119888)119882119888 and 119878119888119896119899 = 119878119888119901119902 119901 = 119896 119902 =119899 0 119901 = 119896 119902 = 11989942 ℓ2 Norm Constrained Minimization Subproblem Whenupdating the dictionary matrix 119882119888 119878119888 and the nonupdatedelements in 119882119888 matrix will be fixed The objective functioncan be transformed into the following form
119891 (119882119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865st 1003817100381710038171003817120601 (119883119888)119882119888119896 100381710038171003817100381722 ⩽ 1 forall119896 = 1 2 119870 (11)
Journal of Electrical and Computer Engineering 5
The Lagrange multiplier method is used to optimize theabove problems then the objective function can be reducedto
119871 (119882119888 120582119896) = minus2 119870sum119896=1
[119878119888120581 (119883119888 119883119888)]119896119882119888119896+ 119870sum119896=1
119882119888119896 [120581 (119883119888 119883119888)119882119888119878119888119878119888119879]119896+ 120582119896 (1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]119896119896)
(12)
Here 120582119896 is a variable Meanwhile the algorithm usesKarush-Kuhn-Tucker (KKT) conditions to optimize theobjective function and Karush-Kuhn-Tucker (KKT) condi-tions meet the following three criteria
(1) 120597119871 (119882119888 120582119896)120597119882119888119896
= 0(2) 1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]
119896119896= 0
(3) 120582119896 gt 0(13)
Hence the solution to119882119888119896 becomes
119882119888119896= 119878119888119896 minus [119882119888119896119865]119896radic(119878119888119896119879 minus [119882119888119896119865]
119896)119879 120581 (119883119888 119883119888) (119878119888119896119879 minus [119882119888119896119865]119896)
(14)
where 119865 = 119878119888119878119888119879 and119882119888119896 = 119882119888119901 119901 = 119896 0 119901 = 1198965 Experimental Results
In this section we present experimental results on five bench-mark databases to illustrate the effectiveness of our methodWe compare the Laplace graph embedding class specificdictionary learning algorithm (LGECSDL) with some state-of-the-art methods In the following section we introducethe experimental environment setting database descriptionsand experimental results In the end we accordingly analyzethe experimental results
51 Experimental Settings In this section we evaluateour method on five benchmark databases The proposedLGECSDL algorithm is compared with another seven classi-cal face recognition algorithms nearest neighbor (NN) clas-sification collaborative representation based classification(CRC) [30] sparse representation based classification (SRC)[31] kernel-based probabilistic collaborative representationbased classifier (ProKCRC) [32] VGG19 [33] kernel-basedclass specific dictionary learning (KCSDL) algorithm [29]and SVM [34]
There are two parameters in the objective function ofthe LGECSDL algorithm that need to be specified 120572 is animportant parameter in the LGECSDL algorithm which is
Figure 1 Examples of the Extended YaleB database
used to adjust the trade-off between the reconstruction errorand the sparsity We increase 120572 from 2minus12 to 2minus1 in eachexperiment and find the best 120572 in our experiments120573 is another important factor in the LGECSDL algorithm120573 is used to control the trade-off between the reconstructionerror and the collaborative information We increase 120573 from2minus12 to 2minus1 and find the best 120573 in all of our experiments
We also evaluate the effect of different kernels for theLGECSDL algorithm Three different kernel functions areadopted as linear kernel (120581(119909 119910) = 119909119879119910) Hellinger kernel(120581(119909 119910) = sum119863119889=1radic119909119889119910119889) and polynomial kernel (120581(119909 119910) =(119901 + 119909119879119910)119902) Here in our experiments 119901 and 119902 are set to be 4and 2 respectively52 Database Descriptions There are five image databasesinvolved in our experiments The first one is the ExtendedYaleB database which contains 38 categories and 2414frontal-face images All the images are captured undervarying illumination conditions In our experiments theimage has been cropped and normalized to 32 times 32 pixelsFigure 1 shows several example images in the Extended YaleBdatabase
The second one is the AR database The AR databasecontains over 3000 images of 126 individuals images are shotunder different conditions of expression illumination andocclusion and each person has 26 images Figure 3 showssome examples in the AR database
The third database is the CMU-PIE database The CMU-PIE database consists of 41368 pieces of pictures whichare captured under different lighting conditions poses andexpressions The database contains 68 individuals in totaland each person has 43 different kinds of images with 13different poses We selected two types of images to carryout our experiment five near-frontal poses and all differentillumination conditions We chose 11554 images in total forour evaluation Each person has about 170 images Figure 5shows some example images in the CMU-PIE database
We also selected the Caltech101 database to verify theLGECSDL algorithm The Caltech101 database contains 9144images belonging to 101 categories each class has 31 to 800images We selected 5 images as training images in each classand the rest as test images Figure 7 shows some examples inthe Caltech101 database
The fifth database is Oxford-102 flower database thatcontains 8189 flower images belonging to 102 categories Eachimage contains 40 to 250 images and the minimum edge
6 Journal of Electrical and Computer Engineering
Table 1 Recognition rate on the Extended YaleB database ()
Methods Linear Hellinger PolynomialNN 3317 plusmn 145 NA NAVGG19 5379 plusmn 121 NA NASVM 6552 plusmn 277 7997 plusmn 257 6489 plusmn 243CRC 7807 plusmn 216 8723 plusmn 144 7633 plusmn 217SRC 7759 plusmn 159 8858 plusmn 156 7482 plusmn 235ProKCRC 7642 plusmn 213 8784 plusmn 189 7494 plusmn 203KCSDL 7855 plusmn 225 8898 plusmn 162 7968 plusmn 241LGECSDL 8018 plusmn 133 9193 plusmn 131 8105 plusmn 142
length of the image is greater than 500 pixelsTheOxford-102flower database contains pictures of flowers taken in differentillumination angle and occlusion environments and eachkind of flower image has a high degree of similarity In ourexperiments all the images aremanually cropped and resizedto 224 times 224 pixels Figure 9 shows several images in theOxford-102 flower database
53 Experiments on the Extended YaleB Database We ran-domly selected 5 images as the training samples in each cate-gory and 10 images as the testing samples In our experimentswe set the weight of the sparsity term 120572 as 2minus9 2minus7 and 2minus7for the linear kernel Hellinger kernel and polynomial kernelrespectivelyThe optimal 120573 is 2minus10 2minus8 and 2minus10 for the linearkernel Hellinger kernel and polynomial kernel respectivelyWe independently performed all the methods ten times andthen reported the average recognition rates Table 1 shows therecognition rates of all the algorithms using different kernelmethods
From Table 1 we can clearly see that LGECSDL achievesthe best recognition rates of 8018 9193 and 8105 inthe linear kernel Hellinger kernel and polynomial kernelspace respectively while KCSDL the second best methodarrives at 7855 8898 and 7968 Since illuminationvariations of images are relatively large in the Extended YaleBdatabase these experiment results validate the effectivenessand robustness of LGECSDL for image recognition withillumination variations VGG19 neural network in this exper-iment can only achieve the highest recognition rate of 5379Using a small database to train neural networks does not takeadvantage of neural networks We also verify the effect of120572 and 120573 on the LGECSDL algorithm and the experimentalresults are shown in Figure 2
From Figure 2 we can easily know that the LGECSDLalgorithmhas achieved better recognition results inHellingerkernel space With the parameter 120572 varied from 2minus13 to 2minus3the recognition rate increased gradually and then decreasedThe influence of parameter 120573 on the LGECSDL algorithm issimilar to that of the parameter 120572 The highest recognitionrate was achieved when 120572 = 2minus7 and 120573 = 2minus8 in Hellingerkernel space In the linear kernel space the recognition rateachieves the maximum value at 120572 = 2minus9 and 120573 = 2minus10 and inthe polynomial kernel space the maximum recognition ratewas obtained at 120572 = 2minus7 and 120573 = 2minus10
53
61
69
77
85
93
73
77
81
85
89
93Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 2 Parameter tuning on the Extended YaleB database
54 AR Database In this experiment we randomly selected5 images of each individual as training samples and the restfor testing Each image has been cropped to 32times 32 and pulledinto a column vector the image vectors have been performedby 1198972 normalization All the methods are independently runten times and the average recognition rates are reportedTherecognition rate of AR database is shown in Table 2
FromTable 2 we can clearly see that LGECSDL algorithmoutperforms the other methods in all kernel spaces TheLGECSDL algorithm achieves the best recognition rate of946 in the polynomial kernel space in the linear kernelspace and Hellinger kernel space the recognition rates are945 and 9413 respectively
Moreover we can also know fromTable 2 that the KCSDLalgorithm achieves the best recognition rate of 9112 inthe linear kernel space which is the highest one among theother methods From these experimental results we furtherconfirm the effectiveness and robustness of the LGECSDLalgorithm for image recognition with illumination variations
Journal of Electrical and Computer Engineering 7
Table 2 Recognition rate on the AR database ()
Methods Linear Hellinger PolynomialNN 3299 plusmn 197 NA NAVGG19 6580 plusmn 135 NA NASVM 8067 plusmn 132 8059 plusmn 126 8084 plusmn 157CRC 9198 plusmn 086 9218 plusmn 078 9243 plusmn 075SRC 8917 plusmn 114 8532 plusmn 112 8525 plusmn 141ProKCRC 8774 plusmn 143 9104 plusmn 076 8990 plusmn 121KCSDL 9112 plusmn 157 8977 plusmn 127 8902 plusmn 131LGECSDL 9450 plusmn 101 9413 plusmn 091 9460 plusmn 086
Table 3 Recognition rate on the CMU-PIE database ()
Methods Linear Hellinger PolynomialNN 3009 plusmn 167 NA NAVGG19 6180 plusmn 128 NA NASVM 6679 plusmn 263 6571 plusmn 254 6546 plusmn 276CRC 7289 plusmn 221 7519 plusmn 209 7319 plusmn 220SRC 7216 plusmn 209 7026 plusmn 223 6919 plusmn 217ProKCRC 6891 plusmn 197 7541 plusmn 186 7264 plusmn 205KCSDL 7446 plusmn 201 7478 plusmn 205 7349 plusmn 222LGECSDL 7912 plusmn 152 8103 plusmn 134 8005 plusmn 127
and expression changes We also verify the effect of 120572 and 120573on the AR database Figure 4 shows the experiment results
FromFigure 4 we can clearly see that the recognition ratereached the maximum value when 120572 is 2minus7 and 120573 is 2minus8 inHellinger kernel space and polynomial kernel space in thelinear kernel space the recognition rate achieves the highestvalue when 120572 is equal to 2minus9 and 120573 is 2minus8 With 120572 changedfrom 2minus13 to 2minus3 the recognition rate increased first and thendecreased The 120573 parameter shows a similar trend and when120573 is greater than 2minus8 the recognition rate decreases rapidly
55 CMU-PIE Database In this experiment we chose theCMU-PIE database to evaluate the performance of theLGECSDL algorithm Five images of each individual arerandomly selected for training and the remainder for testingWe also cropped each image to 32 times 32 and then pulled theminto a column vector Finally we normalized all the vectors by1198972 normalization We independently ran all the methods tentimes and then reported the average recognition rates Table 3gives the recognition rates of all the methods under differentkernel spaces
From Table 3 we can see that the LGECSDL algorithmalways achieves the highest recognition rates under all dif-ferent kernel spaces In the polynomial kernel space theLGECSDL algorithm outperforms KCSDL which achievesthe second highest recognition rate by more than 4improvement of recognition rate In Hellinger kernel spacethe LGECSDL achieves the best recognition rate of 8103and 6 points higher than the KCSDL algorithm In thelinear kernel space the face recognition rate of LGECSDL andKCSDL is 7912and 7446 respectively From these exper-imental results we confirm the effectiveness and robustness
Figure 3 Examples of the AR database
53
61
69
77
85
93
81
84
87
90
93
96Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 4 Parameter tuning on the AR database
of the LGECSDL algorithm We also evaluate the effect of120572 and 120573 on the CMU-PIE database Figure 6 shows theexperiment results
From Figure 6 we can see that when 120572 is 2minus7 and 120573 is 2minus8the face recognition rate reaches the highest value of 8103in Hellinger kernel space and when 120572 is 2minus7 and 120573 is 2minus10 thehighest face recognition rate is obtained in the polynomialkernel space We can also know from Figure 6 that when 120572 is
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
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4 Journal of Electrical and Computer Engineering
dictionary learning algorithm The dictionary weight matrix119882c and sparse representation matrix 119878119888 can be optimized byiterative approaches
When each element in the 119882119888 matrix is updated theremaining elements in 119882119888 matrix and 119878119888 matrix are fixedat this time the objective function is changed into an L2-norm constrained least-squares minimization subproblemSimilarly when each element in 119878119888 matrix is updated 119882119888matrix and the remaining elements in 119878119888 matrix are fixedThe objective function can be seen as an L1-norm constrainedleast-squares minimization subproblem
41 L1-Norm Regularized Minimization Subproblem Whenupdating the elements in 119878119888matrix the nonupdated elementsin 119878119888 and119882119888matrix will be fixed Here the objective functioncan be formulated as
119891 (119878119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865 + 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171+ 120573sum119894 =119895
10038171003817100381710038171003817119878119888119894 minus 11987811988811989510038171003817100381710038171003817 119901119894119895(7)
where 119901119894119895 is the weight value which describes the neighboringdegree of 119909119888119894 and 119909119888119895 and119901119894119895 = 119890minus119909119888119894minus119909119888119895119905 119909119888119894 and 119909119888119895 are trainingsamples that belong to the 119888th class and 119905 is a constant whichcontrols the range of 119901119894119895 Formula (7) can be simplified as
119891 (119878119888)= trace 120601 (119883119888)119879 120601 (119883119888) minus 2120601 (119883119888)119879 120601 (119883119888)119882119888119878119888+ trace (119878119888)119879 (119882119888119879120601 (119883119888)119879 120601 (119883119888)119882119888) 119878119888+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573 trace 119878119888119871119878119888119879
= trace 120581 (119883119888 119883119888) minus 2119873119888sum119899=1
[120581 (119883119888 119883119888)119882119888]119899 119878119888119899
+ 119873119888sum119899=1
(119878119888119879)119899[119882119888119879120581 (119883119888 119883119888)119882119888] 119878119888119899
+ 2120572119873119888sum119899=1
100381710038171003817100381711987811988811989910038171003817100381710038171 + 120573119870sum119896=1
[119878119888119896119871119878119888119879119896 ]= trace 120581 (119883119888 119883119888)minus 2119873119888sum119899=1
119870sum119896=1
[120581 (119883119888 119883119888)119882119888]119899119896 119878119888119896119899
+ 119873119888sum119899=1
119870sum119896=1
[ 119870sum119897=1
(119878119888119879)119899119897(119882119888119879120581 (119883119888 119883119888)119882119888)
119897119896] 119878119888119896119899
+ 2120572 119870sum119896=1
119873119888sum119899=1
10038161003816100381610038161198781198881198961198991003816100381610038161003816
+ 120573 119870sum119896=1
[119873119888sum119899=1
(119873119888sum119897=1
119878119888119896119897119871 119897119899)(119878119879)119899119896]
(8)
where 119871 = 119863 minus 119875 119863119894119894 = sum119895 119875119894119895 and matrix 119875 is the weightmatrix expressing the sample neighboring distance 120581(119883119888 119883119888)means the kernel function of the sample and 120581(119883119888 119883119888)is calculated prior to dictionary updating 120581(119883119888 119883119888) =120601(119883119888)119879120601(119883119888)
In this algorithm each element in 119878119888 is updated sequen-tially when 119878119896119899 is updated the other elements in the 119878matrixare regarded as constants After ignoring the constant term offormula (8) formula (8) can be simplified as
119891 (119878119888119896119899) = (119882119888119879120581 (119883119888 119883119888)119882119888)119896119896(119878119888119896119899)2
+ 2 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899119878119888119896119899
minus 2 [120581 (119883119888 119883119888)119882119888]119899119896 119878c119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816 + 120573119871119899119899 (119878119888119896119899)2
+ 2120573 119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897119878119888119896119899 = [(119882119888119879120581 (119883119888 119883119888)119882119888)119896119896
+ 120573119871119899119899] (119878119888119896119899)2 minus 2[(120581 (119883119888 119883119888)119882119888)119899119896
minus 119870sum119897=1119897 =119896
(119882119888119879120581 (119883119888 119883119888)119882119888)119896119897119878119888119897119899 minus 120573
119873sum119897=1119897 =119899
119871 119897119899119878119888119896119897]sdot 119878119888119896119899 + 2120572 10038161003816100381610038161198781198881198961198991003816100381610038161003816
(9)
According to the solving method in [29] it is easy toobtain the solution of the minimum value of 119891(119878119888119896119899) underthe current iteration condition
119878119888119896119899 = 11 + 120573119871119899119899 min (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 minus120572 + 11 + 120573119871119899119899
sdotmax (120581 (119883119888 119883119888)119882119888)119899119896 minus [119864119878119888119896119899]119896119899minus 120573 [119878119888119896119899119871]
119896119899 120572
(10)
where 119864 = 119882119888119879120581(119883119888 119883119888)119882119888 and 119878119888119896119899 = 119878119888119901119902 119901 = 119896 119902 =119899 0 119901 = 119896 119902 = 11989942 ℓ2 Norm Constrained Minimization Subproblem Whenupdating the dictionary matrix 119882119888 119878119888 and the nonupdatedelements in 119882119888 matrix will be fixed The objective functioncan be transformed into the following form
119891 (119882119888) = 1003817100381710038171003817120601 (119883119888) minus 120601 (119883119888)11988211988811987811988810038171003817100381710038172119865st 1003817100381710038171003817120601 (119883119888)119882119888119896 100381710038171003817100381722 ⩽ 1 forall119896 = 1 2 119870 (11)
Journal of Electrical and Computer Engineering 5
The Lagrange multiplier method is used to optimize theabove problems then the objective function can be reducedto
119871 (119882119888 120582119896) = minus2 119870sum119896=1
[119878119888120581 (119883119888 119883119888)]119896119882119888119896+ 119870sum119896=1
119882119888119896 [120581 (119883119888 119883119888)119882119888119878119888119878119888119879]119896+ 120582119896 (1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]119896119896)
(12)
Here 120582119896 is a variable Meanwhile the algorithm usesKarush-Kuhn-Tucker (KKT) conditions to optimize theobjective function and Karush-Kuhn-Tucker (KKT) condi-tions meet the following three criteria
(1) 120597119871 (119882119888 120582119896)120597119882119888119896
= 0(2) 1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]
119896119896= 0
(3) 120582119896 gt 0(13)
Hence the solution to119882119888119896 becomes
119882119888119896= 119878119888119896 minus [119882119888119896119865]119896radic(119878119888119896119879 minus [119882119888119896119865]
119896)119879 120581 (119883119888 119883119888) (119878119888119896119879 minus [119882119888119896119865]119896)
(14)
where 119865 = 119878119888119878119888119879 and119882119888119896 = 119882119888119901 119901 = 119896 0 119901 = 1198965 Experimental Results
In this section we present experimental results on five bench-mark databases to illustrate the effectiveness of our methodWe compare the Laplace graph embedding class specificdictionary learning algorithm (LGECSDL) with some state-of-the-art methods In the following section we introducethe experimental environment setting database descriptionsand experimental results In the end we accordingly analyzethe experimental results
51 Experimental Settings In this section we evaluateour method on five benchmark databases The proposedLGECSDL algorithm is compared with another seven classi-cal face recognition algorithms nearest neighbor (NN) clas-sification collaborative representation based classification(CRC) [30] sparse representation based classification (SRC)[31] kernel-based probabilistic collaborative representationbased classifier (ProKCRC) [32] VGG19 [33] kernel-basedclass specific dictionary learning (KCSDL) algorithm [29]and SVM [34]
There are two parameters in the objective function ofthe LGECSDL algorithm that need to be specified 120572 is animportant parameter in the LGECSDL algorithm which is
Figure 1 Examples of the Extended YaleB database
used to adjust the trade-off between the reconstruction errorand the sparsity We increase 120572 from 2minus12 to 2minus1 in eachexperiment and find the best 120572 in our experiments120573 is another important factor in the LGECSDL algorithm120573 is used to control the trade-off between the reconstructionerror and the collaborative information We increase 120573 from2minus12 to 2minus1 and find the best 120573 in all of our experiments
We also evaluate the effect of different kernels for theLGECSDL algorithm Three different kernel functions areadopted as linear kernel (120581(119909 119910) = 119909119879119910) Hellinger kernel(120581(119909 119910) = sum119863119889=1radic119909119889119910119889) and polynomial kernel (120581(119909 119910) =(119901 + 119909119879119910)119902) Here in our experiments 119901 and 119902 are set to be 4and 2 respectively52 Database Descriptions There are five image databasesinvolved in our experiments The first one is the ExtendedYaleB database which contains 38 categories and 2414frontal-face images All the images are captured undervarying illumination conditions In our experiments theimage has been cropped and normalized to 32 times 32 pixelsFigure 1 shows several example images in the Extended YaleBdatabase
The second one is the AR database The AR databasecontains over 3000 images of 126 individuals images are shotunder different conditions of expression illumination andocclusion and each person has 26 images Figure 3 showssome examples in the AR database
The third database is the CMU-PIE database The CMU-PIE database consists of 41368 pieces of pictures whichare captured under different lighting conditions poses andexpressions The database contains 68 individuals in totaland each person has 43 different kinds of images with 13different poses We selected two types of images to carryout our experiment five near-frontal poses and all differentillumination conditions We chose 11554 images in total forour evaluation Each person has about 170 images Figure 5shows some example images in the CMU-PIE database
We also selected the Caltech101 database to verify theLGECSDL algorithm The Caltech101 database contains 9144images belonging to 101 categories each class has 31 to 800images We selected 5 images as training images in each classand the rest as test images Figure 7 shows some examples inthe Caltech101 database
The fifth database is Oxford-102 flower database thatcontains 8189 flower images belonging to 102 categories Eachimage contains 40 to 250 images and the minimum edge
6 Journal of Electrical and Computer Engineering
Table 1 Recognition rate on the Extended YaleB database ()
Methods Linear Hellinger PolynomialNN 3317 plusmn 145 NA NAVGG19 5379 plusmn 121 NA NASVM 6552 plusmn 277 7997 plusmn 257 6489 plusmn 243CRC 7807 plusmn 216 8723 plusmn 144 7633 plusmn 217SRC 7759 plusmn 159 8858 plusmn 156 7482 plusmn 235ProKCRC 7642 plusmn 213 8784 plusmn 189 7494 plusmn 203KCSDL 7855 plusmn 225 8898 plusmn 162 7968 plusmn 241LGECSDL 8018 plusmn 133 9193 plusmn 131 8105 plusmn 142
length of the image is greater than 500 pixelsTheOxford-102flower database contains pictures of flowers taken in differentillumination angle and occlusion environments and eachkind of flower image has a high degree of similarity In ourexperiments all the images aremanually cropped and resizedto 224 times 224 pixels Figure 9 shows several images in theOxford-102 flower database
53 Experiments on the Extended YaleB Database We ran-domly selected 5 images as the training samples in each cate-gory and 10 images as the testing samples In our experimentswe set the weight of the sparsity term 120572 as 2minus9 2minus7 and 2minus7for the linear kernel Hellinger kernel and polynomial kernelrespectivelyThe optimal 120573 is 2minus10 2minus8 and 2minus10 for the linearkernel Hellinger kernel and polynomial kernel respectivelyWe independently performed all the methods ten times andthen reported the average recognition rates Table 1 shows therecognition rates of all the algorithms using different kernelmethods
From Table 1 we can clearly see that LGECSDL achievesthe best recognition rates of 8018 9193 and 8105 inthe linear kernel Hellinger kernel and polynomial kernelspace respectively while KCSDL the second best methodarrives at 7855 8898 and 7968 Since illuminationvariations of images are relatively large in the Extended YaleBdatabase these experiment results validate the effectivenessand robustness of LGECSDL for image recognition withillumination variations VGG19 neural network in this exper-iment can only achieve the highest recognition rate of 5379Using a small database to train neural networks does not takeadvantage of neural networks We also verify the effect of120572 and 120573 on the LGECSDL algorithm and the experimentalresults are shown in Figure 2
From Figure 2 we can easily know that the LGECSDLalgorithmhas achieved better recognition results inHellingerkernel space With the parameter 120572 varied from 2minus13 to 2minus3the recognition rate increased gradually and then decreasedThe influence of parameter 120573 on the LGECSDL algorithm issimilar to that of the parameter 120572 The highest recognitionrate was achieved when 120572 = 2minus7 and 120573 = 2minus8 in Hellingerkernel space In the linear kernel space the recognition rateachieves the maximum value at 120572 = 2minus9 and 120573 = 2minus10 and inthe polynomial kernel space the maximum recognition ratewas obtained at 120572 = 2minus7 and 120573 = 2minus10
53
61
69
77
85
93
73
77
81
85
89
93Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 2 Parameter tuning on the Extended YaleB database
54 AR Database In this experiment we randomly selected5 images of each individual as training samples and the restfor testing Each image has been cropped to 32times 32 and pulledinto a column vector the image vectors have been performedby 1198972 normalization All the methods are independently runten times and the average recognition rates are reportedTherecognition rate of AR database is shown in Table 2
FromTable 2 we can clearly see that LGECSDL algorithmoutperforms the other methods in all kernel spaces TheLGECSDL algorithm achieves the best recognition rate of946 in the polynomial kernel space in the linear kernelspace and Hellinger kernel space the recognition rates are945 and 9413 respectively
Moreover we can also know fromTable 2 that the KCSDLalgorithm achieves the best recognition rate of 9112 inthe linear kernel space which is the highest one among theother methods From these experimental results we furtherconfirm the effectiveness and robustness of the LGECSDLalgorithm for image recognition with illumination variations
Journal of Electrical and Computer Engineering 7
Table 2 Recognition rate on the AR database ()
Methods Linear Hellinger PolynomialNN 3299 plusmn 197 NA NAVGG19 6580 plusmn 135 NA NASVM 8067 plusmn 132 8059 plusmn 126 8084 plusmn 157CRC 9198 plusmn 086 9218 plusmn 078 9243 plusmn 075SRC 8917 plusmn 114 8532 plusmn 112 8525 plusmn 141ProKCRC 8774 plusmn 143 9104 plusmn 076 8990 plusmn 121KCSDL 9112 plusmn 157 8977 plusmn 127 8902 plusmn 131LGECSDL 9450 plusmn 101 9413 plusmn 091 9460 plusmn 086
Table 3 Recognition rate on the CMU-PIE database ()
Methods Linear Hellinger PolynomialNN 3009 plusmn 167 NA NAVGG19 6180 plusmn 128 NA NASVM 6679 plusmn 263 6571 plusmn 254 6546 plusmn 276CRC 7289 plusmn 221 7519 plusmn 209 7319 plusmn 220SRC 7216 plusmn 209 7026 plusmn 223 6919 plusmn 217ProKCRC 6891 plusmn 197 7541 plusmn 186 7264 plusmn 205KCSDL 7446 plusmn 201 7478 plusmn 205 7349 plusmn 222LGECSDL 7912 plusmn 152 8103 plusmn 134 8005 plusmn 127
and expression changes We also verify the effect of 120572 and 120573on the AR database Figure 4 shows the experiment results
FromFigure 4 we can clearly see that the recognition ratereached the maximum value when 120572 is 2minus7 and 120573 is 2minus8 inHellinger kernel space and polynomial kernel space in thelinear kernel space the recognition rate achieves the highestvalue when 120572 is equal to 2minus9 and 120573 is 2minus8 With 120572 changedfrom 2minus13 to 2minus3 the recognition rate increased first and thendecreased The 120573 parameter shows a similar trend and when120573 is greater than 2minus8 the recognition rate decreases rapidly
55 CMU-PIE Database In this experiment we chose theCMU-PIE database to evaluate the performance of theLGECSDL algorithm Five images of each individual arerandomly selected for training and the remainder for testingWe also cropped each image to 32 times 32 and then pulled theminto a column vector Finally we normalized all the vectors by1198972 normalization We independently ran all the methods tentimes and then reported the average recognition rates Table 3gives the recognition rates of all the methods under differentkernel spaces
From Table 3 we can see that the LGECSDL algorithmalways achieves the highest recognition rates under all dif-ferent kernel spaces In the polynomial kernel space theLGECSDL algorithm outperforms KCSDL which achievesthe second highest recognition rate by more than 4improvement of recognition rate In Hellinger kernel spacethe LGECSDL achieves the best recognition rate of 8103and 6 points higher than the KCSDL algorithm In thelinear kernel space the face recognition rate of LGECSDL andKCSDL is 7912and 7446 respectively From these exper-imental results we confirm the effectiveness and robustness
Figure 3 Examples of the AR database
53
61
69
77
85
93
81
84
87
90
93
96Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 4 Parameter tuning on the AR database
of the LGECSDL algorithm We also evaluate the effect of120572 and 120573 on the CMU-PIE database Figure 6 shows theexperiment results
From Figure 6 we can see that when 120572 is 2minus7 and 120573 is 2minus8the face recognition rate reaches the highest value of 8103in Hellinger kernel space and when 120572 is 2minus7 and 120573 is 2minus10 thehighest face recognition rate is obtained in the polynomialkernel space We can also know from Figure 6 that when 120572 is
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
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Journal of Electrical and Computer Engineering 5
The Lagrange multiplier method is used to optimize theabove problems then the objective function can be reducedto
119871 (119882119888 120582119896) = minus2 119870sum119896=1
[119878119888120581 (119883119888 119883119888)]119896119882119888119896+ 119870sum119896=1
119882119888119896 [120581 (119883119888 119883119888)119882119888119878119888119878119888119879]119896+ 120582119896 (1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]119896119896)
(12)
Here 120582119896 is a variable Meanwhile the algorithm usesKarush-Kuhn-Tucker (KKT) conditions to optimize theobjective function and Karush-Kuhn-Tucker (KKT) condi-tions meet the following three criteria
(1) 120597119871 (119882119888 120582119896)120597119882119888119896
= 0(2) 1 minus [119882119888119879120581 (119883119888 119883119888)119882119888]
119896119896= 0
(3) 120582119896 gt 0(13)
Hence the solution to119882119888119896 becomes
119882119888119896= 119878119888119896 minus [119882119888119896119865]119896radic(119878119888119896119879 minus [119882119888119896119865]
119896)119879 120581 (119883119888 119883119888) (119878119888119896119879 minus [119882119888119896119865]119896)
(14)
where 119865 = 119878119888119878119888119879 and119882119888119896 = 119882119888119901 119901 = 119896 0 119901 = 1198965 Experimental Results
In this section we present experimental results on five bench-mark databases to illustrate the effectiveness of our methodWe compare the Laplace graph embedding class specificdictionary learning algorithm (LGECSDL) with some state-of-the-art methods In the following section we introducethe experimental environment setting database descriptionsand experimental results In the end we accordingly analyzethe experimental results
51 Experimental Settings In this section we evaluateour method on five benchmark databases The proposedLGECSDL algorithm is compared with another seven classi-cal face recognition algorithms nearest neighbor (NN) clas-sification collaborative representation based classification(CRC) [30] sparse representation based classification (SRC)[31] kernel-based probabilistic collaborative representationbased classifier (ProKCRC) [32] VGG19 [33] kernel-basedclass specific dictionary learning (KCSDL) algorithm [29]and SVM [34]
There are two parameters in the objective function ofthe LGECSDL algorithm that need to be specified 120572 is animportant parameter in the LGECSDL algorithm which is
Figure 1 Examples of the Extended YaleB database
used to adjust the trade-off between the reconstruction errorand the sparsity We increase 120572 from 2minus12 to 2minus1 in eachexperiment and find the best 120572 in our experiments120573 is another important factor in the LGECSDL algorithm120573 is used to control the trade-off between the reconstructionerror and the collaborative information We increase 120573 from2minus12 to 2minus1 and find the best 120573 in all of our experiments
We also evaluate the effect of different kernels for theLGECSDL algorithm Three different kernel functions areadopted as linear kernel (120581(119909 119910) = 119909119879119910) Hellinger kernel(120581(119909 119910) = sum119863119889=1radic119909119889119910119889) and polynomial kernel (120581(119909 119910) =(119901 + 119909119879119910)119902) Here in our experiments 119901 and 119902 are set to be 4and 2 respectively52 Database Descriptions There are five image databasesinvolved in our experiments The first one is the ExtendedYaleB database which contains 38 categories and 2414frontal-face images All the images are captured undervarying illumination conditions In our experiments theimage has been cropped and normalized to 32 times 32 pixelsFigure 1 shows several example images in the Extended YaleBdatabase
The second one is the AR database The AR databasecontains over 3000 images of 126 individuals images are shotunder different conditions of expression illumination andocclusion and each person has 26 images Figure 3 showssome examples in the AR database
The third database is the CMU-PIE database The CMU-PIE database consists of 41368 pieces of pictures whichare captured under different lighting conditions poses andexpressions The database contains 68 individuals in totaland each person has 43 different kinds of images with 13different poses We selected two types of images to carryout our experiment five near-frontal poses and all differentillumination conditions We chose 11554 images in total forour evaluation Each person has about 170 images Figure 5shows some example images in the CMU-PIE database
We also selected the Caltech101 database to verify theLGECSDL algorithm The Caltech101 database contains 9144images belonging to 101 categories each class has 31 to 800images We selected 5 images as training images in each classand the rest as test images Figure 7 shows some examples inthe Caltech101 database
The fifth database is Oxford-102 flower database thatcontains 8189 flower images belonging to 102 categories Eachimage contains 40 to 250 images and the minimum edge
6 Journal of Electrical and Computer Engineering
Table 1 Recognition rate on the Extended YaleB database ()
Methods Linear Hellinger PolynomialNN 3317 plusmn 145 NA NAVGG19 5379 plusmn 121 NA NASVM 6552 plusmn 277 7997 plusmn 257 6489 plusmn 243CRC 7807 plusmn 216 8723 plusmn 144 7633 plusmn 217SRC 7759 plusmn 159 8858 plusmn 156 7482 plusmn 235ProKCRC 7642 plusmn 213 8784 plusmn 189 7494 plusmn 203KCSDL 7855 plusmn 225 8898 plusmn 162 7968 plusmn 241LGECSDL 8018 plusmn 133 9193 plusmn 131 8105 plusmn 142
length of the image is greater than 500 pixelsTheOxford-102flower database contains pictures of flowers taken in differentillumination angle and occlusion environments and eachkind of flower image has a high degree of similarity In ourexperiments all the images aremanually cropped and resizedto 224 times 224 pixels Figure 9 shows several images in theOxford-102 flower database
53 Experiments on the Extended YaleB Database We ran-domly selected 5 images as the training samples in each cate-gory and 10 images as the testing samples In our experimentswe set the weight of the sparsity term 120572 as 2minus9 2minus7 and 2minus7for the linear kernel Hellinger kernel and polynomial kernelrespectivelyThe optimal 120573 is 2minus10 2minus8 and 2minus10 for the linearkernel Hellinger kernel and polynomial kernel respectivelyWe independently performed all the methods ten times andthen reported the average recognition rates Table 1 shows therecognition rates of all the algorithms using different kernelmethods
From Table 1 we can clearly see that LGECSDL achievesthe best recognition rates of 8018 9193 and 8105 inthe linear kernel Hellinger kernel and polynomial kernelspace respectively while KCSDL the second best methodarrives at 7855 8898 and 7968 Since illuminationvariations of images are relatively large in the Extended YaleBdatabase these experiment results validate the effectivenessand robustness of LGECSDL for image recognition withillumination variations VGG19 neural network in this exper-iment can only achieve the highest recognition rate of 5379Using a small database to train neural networks does not takeadvantage of neural networks We also verify the effect of120572 and 120573 on the LGECSDL algorithm and the experimentalresults are shown in Figure 2
From Figure 2 we can easily know that the LGECSDLalgorithmhas achieved better recognition results inHellingerkernel space With the parameter 120572 varied from 2minus13 to 2minus3the recognition rate increased gradually and then decreasedThe influence of parameter 120573 on the LGECSDL algorithm issimilar to that of the parameter 120572 The highest recognitionrate was achieved when 120572 = 2minus7 and 120573 = 2minus8 in Hellingerkernel space In the linear kernel space the recognition rateachieves the maximum value at 120572 = 2minus9 and 120573 = 2minus10 and inthe polynomial kernel space the maximum recognition ratewas obtained at 120572 = 2minus7 and 120573 = 2minus10
53
61
69
77
85
93
73
77
81
85
89
93Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 2 Parameter tuning on the Extended YaleB database
54 AR Database In this experiment we randomly selected5 images of each individual as training samples and the restfor testing Each image has been cropped to 32times 32 and pulledinto a column vector the image vectors have been performedby 1198972 normalization All the methods are independently runten times and the average recognition rates are reportedTherecognition rate of AR database is shown in Table 2
FromTable 2 we can clearly see that LGECSDL algorithmoutperforms the other methods in all kernel spaces TheLGECSDL algorithm achieves the best recognition rate of946 in the polynomial kernel space in the linear kernelspace and Hellinger kernel space the recognition rates are945 and 9413 respectively
Moreover we can also know fromTable 2 that the KCSDLalgorithm achieves the best recognition rate of 9112 inthe linear kernel space which is the highest one among theother methods From these experimental results we furtherconfirm the effectiveness and robustness of the LGECSDLalgorithm for image recognition with illumination variations
Journal of Electrical and Computer Engineering 7
Table 2 Recognition rate on the AR database ()
Methods Linear Hellinger PolynomialNN 3299 plusmn 197 NA NAVGG19 6580 plusmn 135 NA NASVM 8067 plusmn 132 8059 plusmn 126 8084 plusmn 157CRC 9198 plusmn 086 9218 plusmn 078 9243 plusmn 075SRC 8917 plusmn 114 8532 plusmn 112 8525 plusmn 141ProKCRC 8774 plusmn 143 9104 plusmn 076 8990 plusmn 121KCSDL 9112 plusmn 157 8977 plusmn 127 8902 plusmn 131LGECSDL 9450 plusmn 101 9413 plusmn 091 9460 plusmn 086
Table 3 Recognition rate on the CMU-PIE database ()
Methods Linear Hellinger PolynomialNN 3009 plusmn 167 NA NAVGG19 6180 plusmn 128 NA NASVM 6679 plusmn 263 6571 plusmn 254 6546 plusmn 276CRC 7289 plusmn 221 7519 plusmn 209 7319 plusmn 220SRC 7216 plusmn 209 7026 plusmn 223 6919 plusmn 217ProKCRC 6891 plusmn 197 7541 plusmn 186 7264 plusmn 205KCSDL 7446 plusmn 201 7478 plusmn 205 7349 plusmn 222LGECSDL 7912 plusmn 152 8103 plusmn 134 8005 plusmn 127
and expression changes We also verify the effect of 120572 and 120573on the AR database Figure 4 shows the experiment results
FromFigure 4 we can clearly see that the recognition ratereached the maximum value when 120572 is 2minus7 and 120573 is 2minus8 inHellinger kernel space and polynomial kernel space in thelinear kernel space the recognition rate achieves the highestvalue when 120572 is equal to 2minus9 and 120573 is 2minus8 With 120572 changedfrom 2minus13 to 2minus3 the recognition rate increased first and thendecreased The 120573 parameter shows a similar trend and when120573 is greater than 2minus8 the recognition rate decreases rapidly
55 CMU-PIE Database In this experiment we chose theCMU-PIE database to evaluate the performance of theLGECSDL algorithm Five images of each individual arerandomly selected for training and the remainder for testingWe also cropped each image to 32 times 32 and then pulled theminto a column vector Finally we normalized all the vectors by1198972 normalization We independently ran all the methods tentimes and then reported the average recognition rates Table 3gives the recognition rates of all the methods under differentkernel spaces
From Table 3 we can see that the LGECSDL algorithmalways achieves the highest recognition rates under all dif-ferent kernel spaces In the polynomial kernel space theLGECSDL algorithm outperforms KCSDL which achievesthe second highest recognition rate by more than 4improvement of recognition rate In Hellinger kernel spacethe LGECSDL achieves the best recognition rate of 8103and 6 points higher than the KCSDL algorithm In thelinear kernel space the face recognition rate of LGECSDL andKCSDL is 7912and 7446 respectively From these exper-imental results we confirm the effectiveness and robustness
Figure 3 Examples of the AR database
53
61
69
77
85
93
81
84
87
90
93
96Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 4 Parameter tuning on the AR database
of the LGECSDL algorithm We also evaluate the effect of120572 and 120573 on the CMU-PIE database Figure 6 shows theexperiment results
From Figure 6 we can see that when 120572 is 2minus7 and 120573 is 2minus8the face recognition rate reaches the highest value of 8103in Hellinger kernel space and when 120572 is 2minus7 and 120573 is 2minus10 thehighest face recognition rate is obtained in the polynomialkernel space We can also know from Figure 6 that when 120572 is
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
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6 Journal of Electrical and Computer Engineering
Table 1 Recognition rate on the Extended YaleB database ()
Methods Linear Hellinger PolynomialNN 3317 plusmn 145 NA NAVGG19 5379 plusmn 121 NA NASVM 6552 plusmn 277 7997 plusmn 257 6489 plusmn 243CRC 7807 plusmn 216 8723 plusmn 144 7633 plusmn 217SRC 7759 plusmn 159 8858 plusmn 156 7482 plusmn 235ProKCRC 7642 plusmn 213 8784 plusmn 189 7494 plusmn 203KCSDL 7855 plusmn 225 8898 plusmn 162 7968 plusmn 241LGECSDL 8018 plusmn 133 9193 plusmn 131 8105 plusmn 142
length of the image is greater than 500 pixelsTheOxford-102flower database contains pictures of flowers taken in differentillumination angle and occlusion environments and eachkind of flower image has a high degree of similarity In ourexperiments all the images aremanually cropped and resizedto 224 times 224 pixels Figure 9 shows several images in theOxford-102 flower database
53 Experiments on the Extended YaleB Database We ran-domly selected 5 images as the training samples in each cate-gory and 10 images as the testing samples In our experimentswe set the weight of the sparsity term 120572 as 2minus9 2minus7 and 2minus7for the linear kernel Hellinger kernel and polynomial kernelrespectivelyThe optimal 120573 is 2minus10 2minus8 and 2minus10 for the linearkernel Hellinger kernel and polynomial kernel respectivelyWe independently performed all the methods ten times andthen reported the average recognition rates Table 1 shows therecognition rates of all the algorithms using different kernelmethods
From Table 1 we can clearly see that LGECSDL achievesthe best recognition rates of 8018 9193 and 8105 inthe linear kernel Hellinger kernel and polynomial kernelspace respectively while KCSDL the second best methodarrives at 7855 8898 and 7968 Since illuminationvariations of images are relatively large in the Extended YaleBdatabase these experiment results validate the effectivenessand robustness of LGECSDL for image recognition withillumination variations VGG19 neural network in this exper-iment can only achieve the highest recognition rate of 5379Using a small database to train neural networks does not takeadvantage of neural networks We also verify the effect of120572 and 120573 on the LGECSDL algorithm and the experimentalresults are shown in Figure 2
From Figure 2 we can easily know that the LGECSDLalgorithmhas achieved better recognition results inHellingerkernel space With the parameter 120572 varied from 2minus13 to 2minus3the recognition rate increased gradually and then decreasedThe influence of parameter 120573 on the LGECSDL algorithm issimilar to that of the parameter 120572 The highest recognitionrate was achieved when 120572 = 2minus7 and 120573 = 2minus8 in Hellingerkernel space In the linear kernel space the recognition rateachieves the maximum value at 120572 = 2minus9 and 120573 = 2minus10 and inthe polynomial kernel space the maximum recognition ratewas obtained at 120572 = 2minus7 and 120573 = 2minus10
53
61
69
77
85
93
73
77
81
85
89
93Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 2 Parameter tuning on the Extended YaleB database
54 AR Database In this experiment we randomly selected5 images of each individual as training samples and the restfor testing Each image has been cropped to 32times 32 and pulledinto a column vector the image vectors have been performedby 1198972 normalization All the methods are independently runten times and the average recognition rates are reportedTherecognition rate of AR database is shown in Table 2
FromTable 2 we can clearly see that LGECSDL algorithmoutperforms the other methods in all kernel spaces TheLGECSDL algorithm achieves the best recognition rate of946 in the polynomial kernel space in the linear kernelspace and Hellinger kernel space the recognition rates are945 and 9413 respectively
Moreover we can also know fromTable 2 that the KCSDLalgorithm achieves the best recognition rate of 9112 inthe linear kernel space which is the highest one among theother methods From these experimental results we furtherconfirm the effectiveness and robustness of the LGECSDLalgorithm for image recognition with illumination variations
Journal of Electrical and Computer Engineering 7
Table 2 Recognition rate on the AR database ()
Methods Linear Hellinger PolynomialNN 3299 plusmn 197 NA NAVGG19 6580 plusmn 135 NA NASVM 8067 plusmn 132 8059 plusmn 126 8084 plusmn 157CRC 9198 plusmn 086 9218 plusmn 078 9243 plusmn 075SRC 8917 plusmn 114 8532 plusmn 112 8525 plusmn 141ProKCRC 8774 plusmn 143 9104 plusmn 076 8990 plusmn 121KCSDL 9112 plusmn 157 8977 plusmn 127 8902 plusmn 131LGECSDL 9450 plusmn 101 9413 plusmn 091 9460 plusmn 086
Table 3 Recognition rate on the CMU-PIE database ()
Methods Linear Hellinger PolynomialNN 3009 plusmn 167 NA NAVGG19 6180 plusmn 128 NA NASVM 6679 plusmn 263 6571 plusmn 254 6546 plusmn 276CRC 7289 plusmn 221 7519 plusmn 209 7319 plusmn 220SRC 7216 plusmn 209 7026 plusmn 223 6919 plusmn 217ProKCRC 6891 plusmn 197 7541 plusmn 186 7264 plusmn 205KCSDL 7446 plusmn 201 7478 plusmn 205 7349 plusmn 222LGECSDL 7912 plusmn 152 8103 plusmn 134 8005 plusmn 127
and expression changes We also verify the effect of 120572 and 120573on the AR database Figure 4 shows the experiment results
FromFigure 4 we can clearly see that the recognition ratereached the maximum value when 120572 is 2minus7 and 120573 is 2minus8 inHellinger kernel space and polynomial kernel space in thelinear kernel space the recognition rate achieves the highestvalue when 120572 is equal to 2minus9 and 120573 is 2minus8 With 120572 changedfrom 2minus13 to 2minus3 the recognition rate increased first and thendecreased The 120573 parameter shows a similar trend and when120573 is greater than 2minus8 the recognition rate decreases rapidly
55 CMU-PIE Database In this experiment we chose theCMU-PIE database to evaluate the performance of theLGECSDL algorithm Five images of each individual arerandomly selected for training and the remainder for testingWe also cropped each image to 32 times 32 and then pulled theminto a column vector Finally we normalized all the vectors by1198972 normalization We independently ran all the methods tentimes and then reported the average recognition rates Table 3gives the recognition rates of all the methods under differentkernel spaces
From Table 3 we can see that the LGECSDL algorithmalways achieves the highest recognition rates under all dif-ferent kernel spaces In the polynomial kernel space theLGECSDL algorithm outperforms KCSDL which achievesthe second highest recognition rate by more than 4improvement of recognition rate In Hellinger kernel spacethe LGECSDL achieves the best recognition rate of 8103and 6 points higher than the KCSDL algorithm In thelinear kernel space the face recognition rate of LGECSDL andKCSDL is 7912and 7446 respectively From these exper-imental results we confirm the effectiveness and robustness
Figure 3 Examples of the AR database
53
61
69
77
85
93
81
84
87
90
93
96Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 4 Parameter tuning on the AR database
of the LGECSDL algorithm We also evaluate the effect of120572 and 120573 on the CMU-PIE database Figure 6 shows theexperiment results
From Figure 6 we can see that when 120572 is 2minus7 and 120573 is 2minus8the face recognition rate reaches the highest value of 8103in Hellinger kernel space and when 120572 is 2minus7 and 120573 is 2minus10 thehighest face recognition rate is obtained in the polynomialkernel space We can also know from Figure 6 that when 120572 is
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
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Submit your manuscripts atwwwhindawicom
Journal of Electrical and Computer Engineering 7
Table 2 Recognition rate on the AR database ()
Methods Linear Hellinger PolynomialNN 3299 plusmn 197 NA NAVGG19 6580 plusmn 135 NA NASVM 8067 plusmn 132 8059 plusmn 126 8084 plusmn 157CRC 9198 plusmn 086 9218 plusmn 078 9243 plusmn 075SRC 8917 plusmn 114 8532 plusmn 112 8525 plusmn 141ProKCRC 8774 plusmn 143 9104 plusmn 076 8990 plusmn 121KCSDL 9112 plusmn 157 8977 plusmn 127 8902 plusmn 131LGECSDL 9450 plusmn 101 9413 plusmn 091 9460 plusmn 086
Table 3 Recognition rate on the CMU-PIE database ()
Methods Linear Hellinger PolynomialNN 3009 plusmn 167 NA NAVGG19 6180 plusmn 128 NA NASVM 6679 plusmn 263 6571 plusmn 254 6546 plusmn 276CRC 7289 plusmn 221 7519 plusmn 209 7319 plusmn 220SRC 7216 plusmn 209 7026 plusmn 223 6919 plusmn 217ProKCRC 6891 plusmn 197 7541 plusmn 186 7264 plusmn 205KCSDL 7446 plusmn 201 7478 plusmn 205 7349 plusmn 222LGECSDL 7912 plusmn 152 8103 plusmn 134 8005 plusmn 127
and expression changes We also verify the effect of 120572 and 120573on the AR database Figure 4 shows the experiment results
FromFigure 4 we can clearly see that the recognition ratereached the maximum value when 120572 is 2minus7 and 120573 is 2minus8 inHellinger kernel space and polynomial kernel space in thelinear kernel space the recognition rate achieves the highestvalue when 120572 is equal to 2minus9 and 120573 is 2minus8 With 120572 changedfrom 2minus13 to 2minus3 the recognition rate increased first and thendecreased The 120573 parameter shows a similar trend and when120573 is greater than 2minus8 the recognition rate decreases rapidly
55 CMU-PIE Database In this experiment we chose theCMU-PIE database to evaluate the performance of theLGECSDL algorithm Five images of each individual arerandomly selected for training and the remainder for testingWe also cropped each image to 32 times 32 and then pulled theminto a column vector Finally we normalized all the vectors by1198972 normalization We independently ran all the methods tentimes and then reported the average recognition rates Table 3gives the recognition rates of all the methods under differentkernel spaces
From Table 3 we can see that the LGECSDL algorithmalways achieves the highest recognition rates under all dif-ferent kernel spaces In the polynomial kernel space theLGECSDL algorithm outperforms KCSDL which achievesthe second highest recognition rate by more than 4improvement of recognition rate In Hellinger kernel spacethe LGECSDL achieves the best recognition rate of 8103and 6 points higher than the KCSDL algorithm In thelinear kernel space the face recognition rate of LGECSDL andKCSDL is 7912and 7446 respectively From these exper-imental results we confirm the effectiveness and robustness
Figure 3 Examples of the AR database
53
61
69
77
85
93
81
84
87
90
93
96Tuning
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 4 Parameter tuning on the AR database
of the LGECSDL algorithm We also evaluate the effect of120572 and 120573 on the CMU-PIE database Figure 6 shows theexperiment results
From Figure 6 we can see that when 120572 is 2minus7 and 120573 is 2minus8the face recognition rate reaches the highest value of 8103in Hellinger kernel space and when 120572 is 2minus7 and 120573 is 2minus10 thehighest face recognition rate is obtained in the polynomialkernel space We can also know from Figure 6 that when 120572 is
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
8 Journal of Electrical and Computer Engineering
Figure 5 Examples of the CMU-PIE database
48
55
62
69
76
83
68
71
74
77
80
83
Tuning
Tuning
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
Tuning on linear spaceTuning on Hellinger spaceTuning on polynomial space
2minus14 2minus12 2minus10 2minus8 2minus6 2minus4
2minus13 2minus11 2minus9 2minus7 2minus5 2minus3
Figure 6 Parameter tuning on the CMU-PIE database
greater or less than the maximum value the recognition ratedecreases rapidly and parameter 120573 also has the same effect onthe face recognition rate
56 Caltech101 Database In this experiment we furtherevaluate the performance of the LGECSDL algorithm forimage recognition on the Caltech101 database Figure 7 shows
Figure 7 Examples of the Caltech101 database
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6863
80768204
80838157 8192 8223
8302 8288
70
80
Reco
gniti
on ra
te (
)
Figure 8 The best recognition rates of all the methods on Cal-tech101
some examples in the Caltech101 database We randomlysplit the Caltech101 database into two parts Part one whichcontains about 5 images of each subject is used as a trainingset and the other part is used as a testing set We use theVGG ILSVRC 19 layers model to obtain the features of eachimage Here we employ the second fully connected layeroutputs as the input features whose dimension size is 4096We independently ran all the methods ten times and thengave the average recognition rates of all the methods inFigure 8
LGECSDL-H represents the LGECSDL algorithm inHellinger kernel space and LGECSDL-P represents theLGECSDL algorithm in the polynomial kernel space simi-larly to the CSDL algorithm From Figure 8 we can easilysee that the LGECSDL-H algorithm achieves the highestrecognition rate and LGECSDL-P is the second one Moreconcretely LGECSDL-H and LGECSDL-P achieve the recog-nition rates of 8302 and 8288 respectively while CSDL-P the third best method arrives at 8223 Experimentalresults show that training the VGG19 network with a smalldatabase did not give the desired results VGG19 networkachieved the recognition rate of 6863
We also verified the computational time of each methodin Caltech101 database The experimental environment con-sists of the following Core i7 CPU (24GHz) 8GB memory
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Journal of Electrical and Computer Engineering 9
Table 4 Computational time of each method in Caltech101
Methods VGG19 SVM CRCTime 94mspic 8397mspic 105723mspicMethods ProKCRC SRC CSDL-HTime 12877mspic 11099mspic 10654mspicMethods CSDL-P LGECSDL-H LGECSDL-PTime 16668mspic 10950mspic 11225mspic
Figure 9 Examples of the Oxford-102 flower database
Windows 7 operating system and NVIDIA Quadro K2100Mcomputer graphics processor From Table 4 we can see thatthe VGG19 method achieves the best result This is mainlydue to the neural network GPU accelerated architecture thatsaves most of the computational time The second is SVMfollowed by CSDL-H algorithmThe LGECSDL-H algorithmneeds 10950milliseconds to classify each picture whereas theLGECSDL-P algorithm requires 11225 milliseconds
57 Oxford-102 Flower Database In this experiment wechose the Oxford-102 database to evaluate the performanceof the LGECSDL algorithm in the case of image recognitionwith precise image classification Five images of each indi-vidual are randomly selected for training and the rest of theimages are for testing The image features are obtained fromthe outputs of a pretrained VGG ILSVRC 19 layers networkwhich contains five convolutional and three fully connectedlayers Here we use the second fully connected layer outputsas the input features whose dimension size is 4096 Weindependently ran all the methods ten times and thenreported the average recognition rates The best recognitionrates of all the methods are presented in Figure 10
From Figure 10 we can clearly know that the LGECSDLachieves the best recognition rates of 7141 and 7085in polynomial kernel space and Hellinger kernel spacerespectively while CRC arrives at 697 which is thehighest one among those of the other methods LGECSDL-H and LGECSDL-P outperform VGG19 SVM SRC CRCProKCRC CSDL-H and CSDL-P by at least 12 improve-ment of recognition rateThe experimental results show thatin the small database experiment other methods have ahigher recognition rate than the VGG19 neural network Theclassical method has more advantages in the small samplebase experiment
62
64
66
68
70
72
Reco
gniti
on ra
te (
)
VGG
19
SVM
CRC
ProK
CRC
SRC
CSD
L-H
CSD
L-P
LGEC
SDL-
H
LGEC
SDL-
P
6314
679
697
6843 6836 6858 6852
70857141
Figure 10 The best recognition rates of all the methods on Oxford-102
We also verify the performance of the LGECSDL algo-rithmwith different values of 120572 or 120573 in different kernel spaceson the Oxford-102 database The performance with differentvalues of 120572 or 120573 is reported in Figures 11 and 12
From Figure 11 we can see that the LGECSDL algorithmachieves the maximum value when 120572 = 2minus4 and 120573 = 2minus4when 120572 is fixed the recognition rate increases firstly and thendecreases with the increase of 120573 Similarly the recognitionrate also increases firstly and then decreases with the increaseof 120572
From Figure 12 we can see that the LGECSDL algorithmin the polynomial kernel space achieves the maximumrecognition ratewhen120572 = 2minus5 and120573 = 2minus5 In the polynomialkernel space the influence of 120572 and 120573 on the algorithm issimilar to that in Hellinger kernel space
6 Conclusion
We present a novel Laplace graph embedding class specificdictionary learning algorithm with kernels The proposedLGECSDL algorithm improves the classical classificationalgorithm threefold First it concisely combines the dis-criminant ability (sparse representation) to enhance theinterpretability of face recognition Second it greatly reducesthe residual error according to Laplace constraint dictionarylearning Third it easily finds the nonlinear structure hiddenin face images by extending the LGECSDL algorithm toarbitrary kernel space Experimental results on several pub-licly available databases have demonstrated that LGECSDLcan provide superior performance to the traditional facerecognition approaches
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
10 Journal of Electrical and Computer Engineering
62
64
66
68
70
72
Tuning
202minus22minus42minus62minus82minus10
(a)
67
68
69
70
71
Tuning
2minus22minus42minus62minus82minus10
(b)
Figure 11 Parameter tuning on Oxford-102 database with Hellinger kernel (a) Tuning 120572 with 120573 = 2minus4 (b) Tuning 120573 with 120572 = 2minus4
67
68
69
70
71
72Tuning
2minus12minus32minus52minus72minus9
(a)
67
68
69
70
71
Tuning
2minus12minus32minus52minus72minus9
(b)
Figure 12 Parameter tuning on Oxford-102 database with polynomial kernel (a) Tuning 120572 with 120573 = 2minus5 (b) Tuning 120573 with 120572 = 2minus5
Acknowledgments
Thispaper is supported partly by theNationalNatural ScienceFoundation of China (Grants nos 61402535 and 61271407)the Natural Science Foundation for Youths of ShandongProvince China (Grant no ZR2014FQ001) the FundamentalResearch Funds for the Central Universities China Univer-sity of Petroleum (East China) (Grant no 16CX02060A) andthe International S and T Cooperation Program of China(Grant no 2015DFG12050)
References
[1] J Yu X Yang F Gao and D Tao ldquoDeep multimodal distancemetric learning using click constraints for image rankingrdquo IEEETransactions on Cybernetics vol 47 no 12 pp 4014ndash4024 2016
[2] W Liu Z-J Zha Y Wang K Lu and D Tao ldquo119901-laplacianregularized sparse coding for human activity recognitionrdquo IEEETransactions on Industrial Electronics vol 63 no 8 pp 5120ndash5129 2016
[3] T Liu and D Tao ldquoClassification with noisy labels by impor-tance reweightingrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 38 no 3 pp 447ndash461 2016
[4] D C Tao X Li X DWu and S J Maybank ldquoGeometric meanfor subspace selectionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 31 no 2 pp 260ndash274 2009
[5] Y TaigmanM YangM Ranzato and LWolf ldquoDeepFace clos-ing the gap to human-level performance in face verificationrdquoin Proceedings of the IEEE Conference on Computer Vision and
Pattern Recognition (CVPR rsquo14) pp 1701ndash1708 Columbus OhioUSA June 2014
[6] C Ding and D Tao ldquoTrunk-branch ensemble convolutionalneural networks for video-based face recognitionrdquo IEEE Trans-actions on Pattern Analysis and Machine Intelligence no 99 pp1-1 2017
[7] F Schroff D Kalenichenko and J Philbin ldquoFaceNet a unifiedembedding for face recognition and clusteringrdquo in Proceedingsof the IEEE Conference on Computer Vision and Pattern Recog-nition (CVPR rsquo15) pp 815ndash823 IEEE Boston Mass USA June2015
[8] Y Sun X Wang and X Tang ldquoDeeply learned face representa-tions are sparse selective and robustrdquo inProceedings of the IEEEConference on Computer Vision and Pattern Recognition (CVPRrsquo15) pp 2892ndash2900 Boston Ma USA June 2015
[9] J Liu Y Deng T Bai Z Wei and C Huang ldquoTargetingultimate accuracy face recognition via deep embeddingrdquo 2015httpsarxivorgabs150607310
[10] T Naeligs P B Brockhoff and O Tomic ldquoPrincipal componentanalysisrdquo in Statistics for Sensory and Consumer Science pp209ndash225 John Wiley amp Sons Hoboken NJ USA 2010
[11] P Xanthopoulos P M Pardalos and T B Trafalis ldquoLineardiscriminant analysisrdquo in Robust Data Mining SpringerBriefsin Optimization pp 27ndash33 Springer New York NY USA 2013
[12] D Tao X Li X Wu and S J Maybank ldquoGeneral tensordiscriminant analysis and Gabor features for gait recognitionrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 29 no 10 pp 1700ndash1715 2007
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Journal of Electrical and Computer Engineering 11
[13] X Niyogi ldquoLocality preserving projectionsrdquo inNeural Informa-tion Processing Systems vol 16 p 153 MIT Press CambridgeMass USA 2004
[14] Z-G Liu Q Pan and J Dezert ldquoA new belief-based K-nearestneighbor classification methodrdquo Pattern Recognition vol 46no 3 pp 834ndash844 2013
[15] Y Noh B Zhang and D D Lee ldquoGenerative local metriclearning for nearest neighbor classificationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 40 no 1 pp106ndash118 2018
[16] J Yang J Wright T S Huang and Y Ma ldquoImage super-resolution via sparse representationrdquo IEEE Transactions onImage Processing vol 19 no 11 pp 2861ndash2873 2010
[17] L Zhang M Yang F Xiangchu Y Ma and D Zhang ldquoCollab-orative representation based classification for face recognitionrdquo2012 httpsarxivorgabs12042358
[18] C-Y Lu HMin J Gui L Zhu and Y-K Lei ldquoFace recognitionviaweighted sparse representationrdquo Journal of Visual Communi-cation and Image Representation vol 24 no 2 pp 111ndash116 2013
[19] M Yang L Zhang X C Feng and D Zhang ldquoFisher dis-crimination dictionary learning for sparse representationrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 543ndash550 Barcelona Spain November2011
[20] M Yang D Dai L Shen and L Van Gool ldquoLatent dictionarylearning for sparse representation based classificationrdquo in Pro-ceedings of the 27th IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo14) pp 4138ndash4145 June 2014
[21] Z L Jiang Z Lin and L S Davis ldquoLearning a discriminativedictionary for sparse coding via label consistent K-SVDrdquo inPro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo11) pp 1697ndash1704 IEEE Providence RIUSA June 2011
[22] N Zhou Y Shen J Peng and J Fan ldquoLearning inter-relatedvisual dictionary for object recognitionrdquo in Proceedings of theIEEE Conference on Computer Vision and Pattern Recognition(CVPR rsquo12) pp 3490ndash3497 Providence RI USA June 2012
[23] Y Sun Q Liu J Tang and D Tao ldquoLearning discriminativedictionary for group sparse representationrdquo IEEE Transactionson Image Processing vol 23 no 9 pp 3816ndash3828 2014
[24] D Wang and S Kong ldquoA classification-oriented dictionarylearning model explicitly learning the particularity and com-monality across categoriesrdquo Pattern Recognition vol 47 no 2pp 885ndash898 2014
[25] F Rodriguez and G Sapiro ldquoSparse representations for imageclassification learning discriminative and reconstructive non-parametric dictionariesrdquo ADA513220 Defense Technical Infor-mation Center Fort Belvoir Va USA 2008
[26] S Gao I W Tsang and Y Ma ldquoLearning category-specificdictionary and shared dictionary for fine-grained image cate-gorizationrdquo IEEE Transactions on Image Processing vol 23 no2 pp 623ndash634 2014
[27] B-D Liu B Shen and X Li ldquoLocality sensitive dictionarylearning for image classificationrdquo in Proceedings of the IEEEInternational Conference on Image Processing (ICIP rsquo15) pp3807ndash3811 Quebec City Canada September 2015
[28] H V Nguyen V M Patel N M Nasrabadi and R ChellappaldquoKernel dictionary learningrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo12) pp 2021ndash2024 IEEE Kyoto Japan March 2012
[29] B-D Liu Y-X Wang B Shen Y-J Zhang and M HebertldquoSelf-explanatory sparse representation for image classifica-tionrdquo in European Conference on Computer Vision vol 8690of Lecture Notes in Computer Science pp 600ndash616 SpringerBerlin Germany 2014
[30] L Zhang M Yang and X Feng ldquoSparse representation orcollaborative representation Which helps face recognitionrdquo inProceedings of the IEEE International Conference on ComputerVision (ICCV rsquo11) pp 471ndash478 Barcelona Spain November2011
[31] JWright A Y Yang A Ganesh S S Sastry and YMa ldquoRobustface recognition via sparse representationrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 31 no 2 pp210ndash227 2009
[32] S Cai L Zhang W Zuo and X Feng ldquoA probabilistic collabo-rative representation based approach for pattern classificationrdquoin Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR rsquo16) pp 2950ndash2959 Las Vegas NevUSA July 2016
[33] K Simonyan and A Zisserman ldquoVery deep convolutional net-works for large-scale image recognitionrdquo 2014 httpsarxivorgabs14091556
[34] C Chang and C Lin ldquoLIBSVM a Library for support vectormachinesrdquo ACM Transactions on Intelligent Systems and Tech-nology vol 2 no 3 article 27 2011
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
International Journal of
AerospaceEngineeringHindawiwwwhindawicom Volume 2018
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Active and Passive Electronic Components
VLSI Design
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Shock and Vibration
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Control Scienceand Engineering
Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
SensorsJournal of
Hindawiwwwhindawicom Volume 2018
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
