Charles Darwin University

An MPCALDA Based Dimensionality Reduction Algorithm for Face Recognition

Huang Jun Su Kehua El-Den Jamal Hu Tao Li Junlong

Published inMathematical Problems in Engineering

DOI1011552014393265

Published 01012014

Document VersionPublishers PDF also known as Version of record

Link to publication

Citation for published version (APA)Huang J Su K El-Den J Hu T amp Li J (2014) An MPCALDA Based Dimensionality Reduction Algorithmfor Face Recognition Mathematical Problems in Engineering 2014 1-12 httpsdoiorg1011552014393265

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors andor other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights

bull Users may download and print one copy of any publication from the public portal for the purpose of private study or research bull You may not further distribute the material or use it for any profit-making activity or commercial gain bull You may freely distribute the URL identifying the publication in the public portal

Take down policyIf you believe that this document breaches copyright please contact us providing details and we will remove access to the work immediatelyand investigate your claim

Download date 11 Dec 2021

Research ArticleAn MPCALDA Based Dimensionality Reduction Algorithm forFace Recognition

Jun Huang1 Kehua Su2 Jamal El-Den3 Tao Hu1 and Junlong Li2

1 The State Key Laboratory of Information Engineering in Surveying Mapping and Remote Sensing Wuhan UniversityWuhan 430072 China

2 School of Computer Wuhan University Wuhan 430072 China3 School of Engineering and IT Charles Darwin University Darwin NT 0909 Australia

Correspondence should be addressed to Kehua Su skhemail163com

Received 10 January 2014 Revised 17 July 2014 Accepted 23 July 2014 Published 31 August 2014

Academic Editor Yi Chen

Copyright copy 2014 Jun Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We proposed a face recognition algorithm based on both the multilinear principal component analysis (MPCA) and lineardiscriminant analysis (LDA) Comparedwith current traditional existing face recognitionmethods our approach treats face imagesas multidimensional tensor in order to find the optimal tensor subspace for accomplishing dimension reduction The LDA is usedto project samples to a new discriminant feature space while theK nearest neighbor (KNN) is adopted for sample set classificationThe results of our study and the developed algorithm are validated with face databases ORL FERET and YALE and compared withPCA MPCA and PCA + LDA methods which demonstrates an improvement in face recognition accuracy

1 Introduction

Face recognition has become a topical and timely study focusin the fields of pattern recognition and computer vision for itswide application prospect [1 2] Feature extraction is the keyelement in face recognition Currently diverse recognitionmethods use different extraction strategies And one of themost popular algorithms is principal component analysisalgorithm (PCA) which aims to find the projected directionsalong with the minimum reconstructing error and then mapthe face dataset to a low-dimensional space spanned bythose directions corresponding to the top eigenvalues [34] Traditional PCA face recognition technology can reachaccuracy rate of 70ndash92 [5] However it is still not fullypractical

PCAhas certain limitationswhich result in bad adaptabil-ity in the image brightness and facial expression variety [6ndash9]Under either strong bright light or weak light environmentsthe information of the features of the face is deficient hencethe structural information from the feature points of the faceimage may hardly be captured using traditional algorithmslike PCA [10] In addition existing algorithms which are

based on capturing single expressions make it difficult andchallenging to capture the correct features of the same personif he changes his facial expressions Traditional PCA fails tosee the natural structure and correlation represented in dataset [3] which leads to potential additional loss of compactandor useful facial representations and will result in a higherreconstruction error rate [11]

There are many recognition proposals to address limita-tions of PCA presented above In [12] Bansal and Chawlaproposed normalized principal component analysis (NPCA)to improve the recognition rate They normalized images toremove the lightening variations by applying SVD insteadof eigenvalue decomposition Pereira et al [13] introduced anew techniquewhich can reduce face dimensions called class-modular image principal component analysis (CMIPCA) toextract local and global information to reduce illuminationeffects face expressions and head-pos changes resulting inspeed-up over PCA In [14] Tsai showed an application ofdimensionality reduction techniques such as PCA EM-PCAmultidimensional scaling and locally linear embedding toidentity emotion of facial animations But the application wasnot for realistic human faces

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 393265 12 pageshttpdxdoiorg1011552014393265

2 Mathematical Problems in Engineering

In our method we decided to complement some ofthese limitations of PCA by adopting the MPCA algorithmtogether with the LDA algorithm as the basis for the study[3 15] The MPCA algorithm disregards the traditionalmethod which is based on two-dimensional data and usesinstead vectors and integrates multiple face images intoa high-dimensional tensor and processes data in tensorspace The advantage of this approach lies in its ability topersistently structure facial information images and conse-quently increases the accuracy rate when spatial relationshipsbetween pixels are considered When the light brightnesschanges or facial expression changes spatial structural infor-mation between pixels becomes particularly important

LDA was adopted to further reduce the dimensions ofsamples processed by MPCA as it is capable of aggregatingthe samples in subspace and hence improving the facerecognition rate [16 17] We combine MPCA and LDA toform LDA subspace from which both MPCA features andLDA features can be extracted

The organization of this paper is as follows Our proposedalgorithm will be discussed in Section 2 Methodology ofthe approach is presented in Section 3 To demonstrate theeffectiveness of the proposed method experimental resultswill be shown in Section 4 Finally conclusions are drawn inSection 5

2 Principle of MPCA

In computer vision most of the objects are naturally con-sidered as 119899th-order tensors (119899 ge 2) [18] Take Figure 1as an example the image matrix in (i) is a 2nd-ordertensor and a movie clip while in (ii) it is a 3rd-ordertensor Traditional techniques for subspace dimensionalityreduction such as PCA could transform image matrix tovectors with high dimensionality in one mode only whichcannot meet the need of dimensionality reduction So suchtechniques are unable to handle multidimensional objectswell and get satisfactory results Therefore in order to reducedimensionality a reduction algorithm which can directlyoperate on a high-order tensor object is desirable Two-dimensional PCA (2DPCA) algorithm is proposed and devel-oped while researches are using dimensionality reductionsolutions which represent facial images as matrices (2nd-order tensors) instead of vectors [19ndash22] However 2DPCAcan only project images in single mode which results in baddimensionality of reduction [3 23] Thus a more efficientalgorithmMPCAhas been proposed to get better dimension-ality reduction

21 Tensor Notations and Definitions Multilinear principalcomponent analysis (MPCA) has been introduced in detailsin [3] which is used to solve the problem of gait recognitionBefore describing MPCA the notations will be shown in thispaper

Vector 120572 denotes 1st-order tensor Matrix119860 denotes 2nd-order tensor 119860

119894119895119896denotes 3rd-order tensor Higher-order

tensors are indicated by 11986011989411198942119894119899

Assume image matrix isindicated by119883 isin 119877

1198991times1198992 Tensor space is indicated by1198771198991times1198771198992

(1199061 1199062 119906

119899) indicates the orthonormal bases of vector

space1198771198991 and (V1 V2 V

1198992

) indicates the orthonormal basesof vector space 1198771198992 Vector 119906

119894V119879119895indicates orthonormal bases

of tensor space 1198771198991 otimes 1198771198992 Image matrix119883 equals

119883 = sum

119894119895

(119906119879

119894119883V119895) 119906119894V119879119895 (1)

Define two matrices 119880 = [1199061 1199062 119906

1198981

] isin 1198771198991times1198981 and

119881 = [V1 V2 V

1198981

] isin 1198771198992times1198982 Assume 119906 V indicate subspace

of space 1198771198991 1198771198992 formed by basis vectors 119906

1198941198981

119894= 1 and

V1198951198982

119895= 1 Then 119906 otimes V indicates subspace of tensor space

1198771198991otimes1198771198992 The result of 2nd-order tensor119883 isin 119877

1198991times1198992 projected

to 119906 otimes V is indicated by

119884 = 119880119879119883119881 isin 119877

1198981times1198982 (2)

Based on different objective functions transformationmatrices 119880 and 119881 can be obtained by iteration hencedimension reduction can be achieved

22 Principle of MPCA MPCA is developed based on thePCA algorithm Its advantage is that it operates on tensorreplacing the traditional algorithms which transform high-dimensional data into one-dimensional vector For exampleto process 100 face images with size 112 times 92 PCA treatsthem as a 100 times 10304 matrix while MPCA treat them as a100 times 112 times 92 tensor MPCA have the advantage of takinginto account correlation in the original data which is ignoredby PCA

Assume there are tensor sets of images 1198831 1198832 119883

119872

a tensor object is denoted by 119883119898

isin 1198771198681times1198682timessdotsdotsdottimes119868

119873 119868119899denotes

dimensionality of 119899-order tensor Each tensor can beunfolded as

119883 = 119878 times1119880(1)

times2119880(2)

times sdot sdot sdot times119873119880(119873)

(3)

Here 119880(119899) denotes orthogonal matrix So 1198832

119865= 119878

2

119865

[24] Decompose this matrix we can get

119883(119899)

= 119880(119899)

sdot 119878(119899)

sdot (119880(119899+1)

otimes 119880(119899+2)

otimes sdot sdot sdot otimes 119880(119873)

otimes119880(1)

otimes 119880(2)

otimes sdot sdot sdot otimes 119880(119899minus1)

)

119879

(4)

The key point of MPCA algorithm is to find a tensorsubspace which can catch the variety of tensor objects andextract features of object According to (4) projection oftensor samples onto tensor subspace is defined as

119884 = 119883 times1(1)119879

times2(2)119879

times sdot sdot sdot times119873(119873)119879

(5)

where119884 denotes tensor after projection119884 = 1198841 1198842 119884

119872

119884119898

isin 1198771198751times1198752timessdotsdotsdottimes119875

119873 Figure 2 depicts the processAs Figure 2 shows by projecting each mode of facial

tensor 119883 low-dimensional facial tensor which satisfies max-imum variance can be achieved

Mathematical Problems in Engineering 3

(i) Second-order tensor (ii) Third-order tensor

middot middotmiddot

Figure 1 2nd-order and 3rd-order tensor representations samples

1-modeprojection

Rows

1-mode vectorsX

I1 times I2 times I3

B(1)T

m1 times I1

m1 times I2 times I3

X times 1B(1)T

Figure 2 Illustration of the multilinear projection in the 1-mode vector space

Trainingimages Test images

PreprocessingMPCA

dimensionality reduction

Projection

LDA spaceClassifierinto

Figure 3 Flow chart of face recognition algorithm

Figure 4 Face image examples of two persons in ORL face database

4 Mathematical Problems in Engineering

The 1st group

k value

Erro

r rat

e022

02

018

016

014

012

01

008

006

004

0020 2 4 6 8 10

(a)

The 2nd group

Erro

r rat

e

04

035

03

025

02

015

01

0

005

k value1 2 3 4 5 6 7 8 9 10

(b)

The 3rd group

Erro

r rat

e

08

07

06

05

04

03

02

01

k value1 2 3 4 5 6 7 8 9 10

(c)

Figure 5 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In the 1st group 119896 equals 18 In the 2ndgroup 119896 equals 98 In the 3rd group 119896 equals 61

For tensor objects of image samples the variance beforeprojection is as follows

Ψ119883=

119872

sum

119898=1

10038171003817100381710038171003817119883119898minus 119883

10038171003817100381710038171003817

2

119865

119883 = (

1

119872

)

119872

sum

119898=1

119883119898

(6)

And the tensors after projection satisfy the followingequation

Ψ119884=

119872

sum

119898=1

10038171003817100381710038171003817119884119898minus 119884

10038171003817100381710038171003817

2

119865

119884 = (

1

119872

)

119872

sum

119898=1

119884119898

(7)

By combining (5) and (6) we can get the followingequation

120595119884=

119872

sum

119898=1

100381710038171003817100381710038171003817

119883119898times (1)119879

times (2)119879

times sdot sdot sdot times (119873)119879

minus119909 times (1)119879

times (2)119879

times sdot sdot sdot times (119873)119879100381710038171003817100381710038171003817

2

119865

(8)

The MPCA algorithm equals to the resolving optimiza-tion problem

(1)

119899 = 1 2 119873 = arg max120595119884 (9)

Mathematical Problems in Engineering 5

The 1st groupEr

ror r

ate

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

The 2nd group

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value1 2 3 4 5 6 7 8 9 10

(b)

The 3rd group

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(c)

Figure 6 Recognition error rate of MPCA against different 119896 values

In (9) by using alternating-least-square method (ALS)we are able to calculate local optimization procedure Whensolving the 119899th projection matrix

(119899) other matriceswere set constant tensor 119883 is projected to tensor space(1198771 119877

119899minus1 119877119899+1

119877119873) where

(119899)

119895+1= 119883 times

(1)119879

119895+1times sdot sdot sdot times

(119899minus1)

119879

119895+1times (119899+1)

119879

119895+1times (119873)119879

119895+1

Column of (119899) can be obtained from orthogonal basis ofprojection subspace Sample 119883

119898in (8) is projected to lower

dimensional tensor 119884119898

isin 1198771198751times1198752timessdotsdotsdottimes119875

119899minus1times119868119899times119875119899+1sdotsdotsdottimes119875119873 119884(119899)119898

119899th-mode unfoldingmatrix of119884

119898 is inputted to get PCA It equals

to

argmax119872

sum

119898=1

100381710038171003817100381710038171003817

(119899)119879

119884(119899)

119898minus (119899)119879

119884

(119899)100381710038171003817100381710038171003817

(10)

23 MPCA Algorithm MPCA have managed to handlemultidimensional objects According to the above sectionspseudocode for the computation of theMPCA algorithm canbe concluded [25] as shown in Figure 3

Step 1 Input sample images and center them as 119909119899

isin

1198771198751times1198752 119899 = 1 119873

Step 2 Obtain the total scatter matrixrsquos eigendecomposition

Step 3 Calculate the eigenvectors and their correspondingmost significant eigenvalues and the result is output as (119899)

Step 4 (i) Get 119910119899= (1)119879

times 119909119899times (2)

119899 = 1 119873

(ii) Calculate 1205951198840

= sum119872

119899=11199101198992

119865

(iii) For 119896 = 1 119870

6 Mathematical Problems in Engineering

The 1st group

Erro

r rat

e08

07

06

05

04

03

02

01

0

k value0 5 10 15

(a)

The 2nd group

Erro

r rat

e

08

07

06

05

04

03

02

01

0

k value0 5 10 15

(b)

The 3rd group

Erro

r rat

e

07

06

05

04

03

02

01

0

k value0 5 10 15

(c)

Figure 7 Recognition error rate of PCA + LDA against different LDA dimension reduction values

(a) calculate the total scatter matrixrsquos eigenvectors andtheir correspondingmost significant eigenvalues andthe result is output as (119895) for 119895 = 1 2

(b) get 120595119884119896

and 119910119899 119899 = 1 119873

(c) if 120595119884119896

minus 120595119884119896minus1

lt 119895 then break the loop and go toStep 5

Step 5 Finally calculate the feature matrix see the followingequation

119910119899= (1)119879

times 119909119898times (2)

119899 = 1 119873 (11)

24 LDA Algorithm LDA (linear discriminant analysis)projects image onto a lower-dimensional vector space toachieve maximum discrimination as follows

Mathematical Problems in Engineering 7

The 1st groupEr

ror r

ate

016

014

012

01

008

006

004

002

00 5 10 15 20

LDA dimensionality reduction

(a)

The 2nd group

0 5 10 15 20

LDA dimensionality reduction

Erro

r rat

e

025

02

015

01

005

0

(b)

The 3rd group

0 5 10 15 20 25 30 35 40

LDA dimensionality reduction

Erro

r rat

e

01

009

008

007

006

005

004

003

002

001

0

(c)

Figure 8 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

Step 1 Compute the average sample values for different kindsof facial images in the original space Total number is denotedby 119888119883

119894119895denotes the 119895th object of the 119894th class of samples

119898119894=

1

119899119894

1198991

sum

119895=1

119883119894119895 119883

119894119895isin 119877119889 119894 = 1 2 119888

119898 =

119888

sum

119894=1

119901119894119898119894

(12)

Step 2 Compute covariance matrix of each class

119888119894=

1

119899119894

119899119894

sum

119895=1

(119883119894119895minus 119898119894) sdot (119883119894119895minus 119898119894)119879 (13)

Step 3 Compute within-class and between-class scattermatrices

119862119887=

119888

sum

119894=1

119901119894(119898119894minus 119898) sdot (119898

119894minus 119898)119879

119862119908=

119888

sum

119894=1

119888119894

(14)

Step 4 Compute eigenvectors of matrix 119862minus1

119908119862119887to get pro-

jection vectors Then dimensionality reduction data can beobtained by projection [26 27]

After dimensionality reduction using MPCA the matri-ces are arranged in columns into vectors as inputs to theLDA algorithm By using MPCA algorithm to reduce thedimension of the image we not only solved the problem of

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

100

The 1st group The 2nd group The 3rd group

PCAMPCA

()

PCA + LDAMPCA + LDA

Figure 9 Histogram of recognition results in experiments

singular matrix but also retained structure information in theimages and thus improve the recognition rate

25 KNN Algorithm 119870-nearest neighbor (KNN) algorithm[28 29] is adopted for sample set classification here and theconcrete steps are as follows

Step 1 Select different parameters of119870 valueStep 2Adopt the method of cross-validation on training faceimages for 119896 = 1 119870

Step 3 Make the cross-validation error classification rateminimization and get its corresponding parameter 119896

Step 4 Construct a prediction model with 119896

3 Process of the Recognition Algorithm

31 Preprocessing Image preprocessing and normalizationare vital for face recognition systems as images are oftenaffected by image quality illumination face rotation facialexpression [8 30] and so forth In order to offset abovefactors it is necessary for us to carry out face normalizationbefore facial feature extraction

Our data is preprocessing normalized images with aresolution of 80times 80 In our research histogram equalizationwas applied (see (15))

1199101015840=

119910 minusmin (119910)

max (119910) minusmin (119910)

(15)

32 Dimensionality Reduction Using MPCA and FeatureMatrix Extraction Using LDA MPCA reduces dimensionsof input face images and generates feature projection matrix[30] that are then taken as input samples to LDA MPCAand LDA combination were used to construct LDA subspacefrom which both MPCA features and LDA features can beextracted

The detailed steps have been described in Sections 22 and23

33 Face Recognition Using L2 Distance Measure We usedresultant output acquired above as input samples for trainingand applied aforesaid techniques to get the feature matrixThen we carried out a similarity measure on image samplesIn our researchwe choose L2 distance formeasures (see (16))

119889 (119886 119887) = radic

119867

sum

ℎ=1

[119886 (ℎ) minus 119887(ℎ)]2 (16)

KNN classifier [31] is adopted for sample set classificationhere while the procedure and details are introduced inSection 24

The overall approach of face recognition proposed in thispaper is shown in Figure 3

4 Experiments

We evaluated the performance of our algorithm based onMPCA + LDA in this research and compared with thePCA MPCA and PCA + LDA algorithm by performingexperiments on ORL databases [32] In order to examine theability of our method we also try it on other classical facedatabases such as FERET and YALE

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

(i) Second-order tensor (ii) Third-order tensor

middot middotmiddot

Figure 1 2nd-order and 3rd-order tensor representations samples

1-modeprojection

Rows

1-mode vectorsX

I1 times I2 times I3

B(1)T

m1 times I1

m1 times I2 times I3

X times 1B(1)T

Figure 2 Illustration of the multilinear projection in the 1-mode vector space

Trainingimages Test images

PreprocessingMPCA

dimensionality reduction

Projection

LDA spaceClassifierinto

Figure 3 Flow chart of face recognition algorithm

Figure 4 Face image examples of two persons in ORL face database

4 Mathematical Problems in Engineering

The 1st group

k value

Erro

r rat

e022

02

018

016

014

012

01

008

006

004

0020 2 4 6 8 10

(a)

The 2nd group

Erro

r rat

e

04

035

03

025

02

015

01

0

005

k value1 2 3 4 5 6 7 8 9 10

(b)

The 3rd group

Erro

r rat

e

08

07

06

05

04

03

02

01

k value1 2 3 4 5 6 7 8 9 10

(c)

Figure 5 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In the 1st group 119896 equals 18 In the 2ndgroup 119896 equals 98 In the 3rd group 119896 equals 61

For tensor objects of image samples the variance beforeprojection is as follows

Ψ119883=

119872

sum

119898=1

10038171003817100381710038171003817119883119898minus 119883

10038171003817100381710038171003817

2

119865

119883 = (

1

119872

)

119872

sum

119898=1

119883119898

(6)

And the tensors after projection satisfy the followingequation

Ψ119884=

119872

sum

119898=1

10038171003817100381710038171003817119884119898minus 119884

10038171003817100381710038171003817

2

119865

119884 = (

1

119872

)

119872

sum

119898=1

119884119898

(7)

By combining (5) and (6) we can get the followingequation

120595119884=

119872

sum

119898=1

100381710038171003817100381710038171003817

119883119898times (1)119879

times (2)119879

times sdot sdot sdot times (119873)119879

minus119909 times (1)119879

times (2)119879

times sdot sdot sdot times (119873)119879100381710038171003817100381710038171003817

2

119865

(8)

The MPCA algorithm equals to the resolving optimiza-tion problem

(1)

119899 = 1 2 119873 = arg max120595119884 (9)

Mathematical Problems in Engineering 5

The 1st groupEr

ror r

ate

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

The 2nd group

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value1 2 3 4 5 6 7 8 9 10

(b)

The 3rd group

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(c)

Figure 6 Recognition error rate of MPCA against different 119896 values

In (9) by using alternating-least-square method (ALS)we are able to calculate local optimization procedure Whensolving the 119899th projection matrix

(119899) other matriceswere set constant tensor 119883 is projected to tensor space(1198771 119877

119899minus1 119877119899+1

119877119873) where

(119899)

119895+1= 119883 times

(1)119879

119895+1times sdot sdot sdot times

(119899minus1)

119879

119895+1times (119899+1)

119879

119895+1times (119873)119879

119895+1

Column of (119899) can be obtained from orthogonal basis ofprojection subspace Sample 119883

119898in (8) is projected to lower

dimensional tensor 119884119898

isin 1198771198751times1198752timessdotsdotsdottimes119875

119899minus1times119868119899times119875119899+1sdotsdotsdottimes119875119873 119884(119899)119898

119899th-mode unfoldingmatrix of119884

119898 is inputted to get PCA It equals

to

argmax119872

sum

119898=1

100381710038171003817100381710038171003817

(119899)119879

119884(119899)

119898minus (119899)119879

119884

(119899)100381710038171003817100381710038171003817

(10)

23 MPCA Algorithm MPCA have managed to handlemultidimensional objects According to the above sectionspseudocode for the computation of theMPCA algorithm canbe concluded [25] as shown in Figure 3

Step 1 Input sample images and center them as 119909119899

isin

1198771198751times1198752 119899 = 1 119873

Step 2 Obtain the total scatter matrixrsquos eigendecomposition

Step 3 Calculate the eigenvectors and their correspondingmost significant eigenvalues and the result is output as (119899)

Step 4 (i) Get 119910119899= (1)119879

times 119909119899times (2)

119899 = 1 119873

(ii) Calculate 1205951198840

= sum119872

119899=11199101198992

119865

(iii) For 119896 = 1 119870

6 Mathematical Problems in Engineering

The 1st group

Erro

r rat

e08

07

06

05

04

03

02

01

0

k value0 5 10 15

(a)

The 2nd group

Erro

r rat

e

08

07

06

05

04

03

02

01

0

k value0 5 10 15

(b)

The 3rd group

Erro

r rat

e

07

06

05

04

03

02

01

0

k value0 5 10 15

(c)

Figure 7 Recognition error rate of PCA + LDA against different LDA dimension reduction values

(a) calculate the total scatter matrixrsquos eigenvectors andtheir correspondingmost significant eigenvalues andthe result is output as (119895) for 119895 = 1 2

(b) get 120595119884119896

and 119910119899 119899 = 1 119873

(c) if 120595119884119896

minus 120595119884119896minus1

lt 119895 then break the loop and go toStep 5

Step 5 Finally calculate the feature matrix see the followingequation

119910119899= (1)119879

times 119909119898times (2)

119899 = 1 119873 (11)

24 LDA Algorithm LDA (linear discriminant analysis)projects image onto a lower-dimensional vector space toachieve maximum discrimination as follows

Mathematical Problems in Engineering 7

The 1st groupEr

ror r

ate

016

014

012

01

008

006

004

002

00 5 10 15 20

LDA dimensionality reduction

(a)

The 2nd group

0 5 10 15 20

LDA dimensionality reduction

Erro

r rat

e

025

02

015

01

005

0

(b)

The 3rd group

0 5 10 15 20 25 30 35 40

LDA dimensionality reduction

Erro

r rat

e

01

009

008

007

006

005

004

003

002

001

0

(c)

Figure 8 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

Step 1 Compute the average sample values for different kindsof facial images in the original space Total number is denotedby 119888119883

119894119895denotes the 119895th object of the 119894th class of samples

119898119894=

1

119899119894

1198991

sum

119895=1

119883119894119895 119883

119894119895isin 119877119889 119894 = 1 2 119888

119898 =

119888

sum

119894=1

119901119894119898119894

(12)

Step 2 Compute covariance matrix of each class

119888119894=

1

119899119894

119899119894

sum

119895=1

(119883119894119895minus 119898119894) sdot (119883119894119895minus 119898119894)119879 (13)

Step 3 Compute within-class and between-class scattermatrices

119862119887=

119888

sum

119894=1

119901119894(119898119894minus 119898) sdot (119898

119894minus 119898)119879

119862119908=

119888

sum

119894=1

119888119894

(14)

Step 4 Compute eigenvectors of matrix 119862minus1

119908119862119887to get pro-

jection vectors Then dimensionality reduction data can beobtained by projection [26 27]

After dimensionality reduction using MPCA the matri-ces are arranged in columns into vectors as inputs to theLDA algorithm By using MPCA algorithm to reduce thedimension of the image we not only solved the problem of

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

100

The 1st group The 2nd group The 3rd group

PCAMPCA

()

PCA + LDAMPCA + LDA

Figure 9 Histogram of recognition results in experiments

singular matrix but also retained structure information in theimages and thus improve the recognition rate

25 KNN Algorithm 119870-nearest neighbor (KNN) algorithm[28 29] is adopted for sample set classification here and theconcrete steps are as follows

Step 1 Select different parameters of119870 valueStep 2Adopt the method of cross-validation on training faceimages for 119896 = 1 119870

Step 3 Make the cross-validation error classification rateminimization and get its corresponding parameter 119896

Step 4 Construct a prediction model with 119896

3 Process of the Recognition Algorithm

31 Preprocessing Image preprocessing and normalizationare vital for face recognition systems as images are oftenaffected by image quality illumination face rotation facialexpression [8 30] and so forth In order to offset abovefactors it is necessary for us to carry out face normalizationbefore facial feature extraction

Our data is preprocessing normalized images with aresolution of 80times 80 In our research histogram equalizationwas applied (see (15))

1199101015840=

119910 minusmin (119910)

max (119910) minusmin (119910)

(15)

32 Dimensionality Reduction Using MPCA and FeatureMatrix Extraction Using LDA MPCA reduces dimensionsof input face images and generates feature projection matrix[30] that are then taken as input samples to LDA MPCAand LDA combination were used to construct LDA subspacefrom which both MPCA features and LDA features can beextracted

The detailed steps have been described in Sections 22 and23

33 Face Recognition Using L2 Distance Measure We usedresultant output acquired above as input samples for trainingand applied aforesaid techniques to get the feature matrixThen we carried out a similarity measure on image samplesIn our researchwe choose L2 distance formeasures (see (16))

119889 (119886 119887) = radic

119867

sum

ℎ=1

[119886 (ℎ) minus 119887(ℎ)]2 (16)

KNN classifier [31] is adopted for sample set classificationhere while the procedure and details are introduced inSection 24

The overall approach of face recognition proposed in thispaper is shown in Figure 3

4 Experiments

We evaluated the performance of our algorithm based onMPCA + LDA in this research and compared with thePCA MPCA and PCA + LDA algorithm by performingexperiments on ORL databases [32] In order to examine theability of our method we also try it on other classical facedatabases such as FERET and YALE

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

The 1st group

k value

Erro

r rat

e022

02

018

016

014

012

01

008

006

004

0020 2 4 6 8 10

(a)

The 2nd group

Erro

r rat

e

04

035

03

025

02

015

01

0

005

k value1 2 3 4 5 6 7 8 9 10

(b)

The 3rd group

Erro

r rat

e

08

07

06

05

04

03

02

01

k value1 2 3 4 5 6 7 8 9 10

(c)

Figure 5 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In the 1st group 119896 equals 18 In the 2ndgroup 119896 equals 98 In the 3rd group 119896 equals 61

For tensor objects of image samples the variance beforeprojection is as follows

Ψ119883=

119872

sum

119898=1

10038171003817100381710038171003817119883119898minus 119883

10038171003817100381710038171003817

2

119865

119883 = (

1

119872

)

119872

sum

119898=1

119883119898

(6)

And the tensors after projection satisfy the followingequation

Ψ119884=

119872

sum

119898=1

10038171003817100381710038171003817119884119898minus 119884

10038171003817100381710038171003817

2

119865

119884 = (

1

119872

)

119872

sum

119898=1

119884119898

(7)

By combining (5) and (6) we can get the followingequation

120595119884=

119872

sum

119898=1

100381710038171003817100381710038171003817

119883119898times (1)119879

times (2)119879

times sdot sdot sdot times (119873)119879

minus119909 times (1)119879

times (2)119879

times sdot sdot sdot times (119873)119879100381710038171003817100381710038171003817

2

119865

(8)

The MPCA algorithm equals to the resolving optimiza-tion problem

(1)

119899 = 1 2 119873 = arg max120595119884 (9)

Mathematical Problems in Engineering 5

The 1st groupEr

ror r

ate

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

The 2nd group

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value1 2 3 4 5 6 7 8 9 10

(b)

The 3rd group

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(c)

Figure 6 Recognition error rate of MPCA against different 119896 values

In (9) by using alternating-least-square method (ALS)we are able to calculate local optimization procedure Whensolving the 119899th projection matrix

(119899) other matriceswere set constant tensor 119883 is projected to tensor space(1198771 119877

119899minus1 119877119899+1

119877119873) where

(119899)

119895+1= 119883 times

(1)119879

119895+1times sdot sdot sdot times

(119899minus1)

119879

119895+1times (119899+1)

119879

119895+1times (119873)119879

119895+1

Column of (119899) can be obtained from orthogonal basis ofprojection subspace Sample 119883

119898in (8) is projected to lower

dimensional tensor 119884119898

isin 1198771198751times1198752timessdotsdotsdottimes119875

119899minus1times119868119899times119875119899+1sdotsdotsdottimes119875119873 119884(119899)119898

119899th-mode unfoldingmatrix of119884

119898 is inputted to get PCA It equals

to

argmax119872

sum

119898=1

100381710038171003817100381710038171003817

(119899)119879

119884(119899)

119898minus (119899)119879

119884

(119899)100381710038171003817100381710038171003817

(10)

23 MPCA Algorithm MPCA have managed to handlemultidimensional objects According to the above sectionspseudocode for the computation of theMPCA algorithm canbe concluded [25] as shown in Figure 3

Step 1 Input sample images and center them as 119909119899

isin

1198771198751times1198752 119899 = 1 119873

Step 2 Obtain the total scatter matrixrsquos eigendecomposition

Step 3 Calculate the eigenvectors and their correspondingmost significant eigenvalues and the result is output as (119899)

Step 4 (i) Get 119910119899= (1)119879

times 119909119899times (2)

119899 = 1 119873

(ii) Calculate 1205951198840

= sum119872

119899=11199101198992

119865

(iii) For 119896 = 1 119870

6 Mathematical Problems in Engineering

The 1st group

Erro

r rat

e08

07

06

05

04

03

02

01

0

k value0 5 10 15

(a)

The 2nd group

Erro

r rat

e

08

07

06

05

04

03

02

01

0

k value0 5 10 15

(b)

The 3rd group

Erro

r rat

e

07

06

05

04

03

02

01

0

k value0 5 10 15

(c)

Figure 7 Recognition error rate of PCA + LDA against different LDA dimension reduction values

(a) calculate the total scatter matrixrsquos eigenvectors andtheir correspondingmost significant eigenvalues andthe result is output as (119895) for 119895 = 1 2

(b) get 120595119884119896

and 119910119899 119899 = 1 119873

(c) if 120595119884119896

minus 120595119884119896minus1

lt 119895 then break the loop and go toStep 5

Step 5 Finally calculate the feature matrix see the followingequation

119910119899= (1)119879

times 119909119898times (2)

119899 = 1 119873 (11)

24 LDA Algorithm LDA (linear discriminant analysis)projects image onto a lower-dimensional vector space toachieve maximum discrimination as follows

Mathematical Problems in Engineering 7

The 1st groupEr

ror r

ate

016

014

012

01

008

006

004

002

00 5 10 15 20

LDA dimensionality reduction

(a)

The 2nd group

0 5 10 15 20

LDA dimensionality reduction

Erro

r rat

e

025

02

015

01

005

0

(b)

The 3rd group

0 5 10 15 20 25 30 35 40

LDA dimensionality reduction

Erro

r rat

e

01

009

008

007

006

005

004

003

002

001

0

(c)

Figure 8 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

Step 1 Compute the average sample values for different kindsof facial images in the original space Total number is denotedby 119888119883

119894119895denotes the 119895th object of the 119894th class of samples

119898119894=

1

119899119894

1198991

sum

119895=1

119883119894119895 119883

119894119895isin 119877119889 119894 = 1 2 119888

119898 =

119888

sum

119894=1

119901119894119898119894

(12)

Step 2 Compute covariance matrix of each class

119888119894=

1

119899119894

119899119894

sum

119895=1

(119883119894119895minus 119898119894) sdot (119883119894119895minus 119898119894)119879 (13)

Step 3 Compute within-class and between-class scattermatrices

119862119887=

119888

sum

119894=1

119901119894(119898119894minus 119898) sdot (119898

119894minus 119898)119879

119862119908=

119888

sum

119894=1

119888119894

(14)

Step 4 Compute eigenvectors of matrix 119862minus1

119908119862119887to get pro-

jection vectors Then dimensionality reduction data can beobtained by projection [26 27]

After dimensionality reduction using MPCA the matri-ces are arranged in columns into vectors as inputs to theLDA algorithm By using MPCA algorithm to reduce thedimension of the image we not only solved the problem of

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

100

The 1st group The 2nd group The 3rd group

PCAMPCA

()

PCA + LDAMPCA + LDA

Figure 9 Histogram of recognition results in experiments

singular matrix but also retained structure information in theimages and thus improve the recognition rate

25 KNN Algorithm 119870-nearest neighbor (KNN) algorithm[28 29] is adopted for sample set classification here and theconcrete steps are as follows

Step 1 Select different parameters of119870 valueStep 2Adopt the method of cross-validation on training faceimages for 119896 = 1 119870

Step 3 Make the cross-validation error classification rateminimization and get its corresponding parameter 119896

Step 4 Construct a prediction model with 119896

3 Process of the Recognition Algorithm

31 Preprocessing Image preprocessing and normalizationare vital for face recognition systems as images are oftenaffected by image quality illumination face rotation facialexpression [8 30] and so forth In order to offset abovefactors it is necessary for us to carry out face normalizationbefore facial feature extraction

Our data is preprocessing normalized images with aresolution of 80times 80 In our research histogram equalizationwas applied (see (15))

1199101015840=

119910 minusmin (119910)

max (119910) minusmin (119910)

(15)

32 Dimensionality Reduction Using MPCA and FeatureMatrix Extraction Using LDA MPCA reduces dimensionsof input face images and generates feature projection matrix[30] that are then taken as input samples to LDA MPCAand LDA combination were used to construct LDA subspacefrom which both MPCA features and LDA features can beextracted

The detailed steps have been described in Sections 22 and23

33 Face Recognition Using L2 Distance Measure We usedresultant output acquired above as input samples for trainingand applied aforesaid techniques to get the feature matrixThen we carried out a similarity measure on image samplesIn our researchwe choose L2 distance formeasures (see (16))

119889 (119886 119887) = radic

119867

sum

ℎ=1

[119886 (ℎ) minus 119887(ℎ)]2 (16)

KNN classifier [31] is adopted for sample set classificationhere while the procedure and details are introduced inSection 24

The overall approach of face recognition proposed in thispaper is shown in Figure 3

4 Experiments

We evaluated the performance of our algorithm based onMPCA + LDA in this research and compared with thePCA MPCA and PCA + LDA algorithm by performingexperiments on ORL databases [32] In order to examine theability of our method we also try it on other classical facedatabases such as FERET and YALE

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

The 1st groupEr

ror r

ate

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

The 2nd group

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value1 2 3 4 5 6 7 8 9 10

(b)

The 3rd group

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(c)

Figure 6 Recognition error rate of MPCA against different 119896 values

In (9) by using alternating-least-square method (ALS)we are able to calculate local optimization procedure Whensolving the 119899th projection matrix

(119899) other matriceswere set constant tensor 119883 is projected to tensor space(1198771 119877

119899minus1 119877119899+1

119877119873) where

(119899)

119895+1= 119883 times

(1)119879

119895+1times sdot sdot sdot times

(119899minus1)

119879

119895+1times (119899+1)

119879

119895+1times (119873)119879

119895+1

Column of (119899) can be obtained from orthogonal basis ofprojection subspace Sample 119883

119898in (8) is projected to lower

dimensional tensor 119884119898

isin 1198771198751times1198752timessdotsdotsdottimes119875

119899minus1times119868119899times119875119899+1sdotsdotsdottimes119875119873 119884(119899)119898

119899th-mode unfoldingmatrix of119884

119898 is inputted to get PCA It equals

to

argmax119872

sum

119898=1

100381710038171003817100381710038171003817

(119899)119879

119884(119899)

119898minus (119899)119879

119884

(119899)100381710038171003817100381710038171003817

(10)

23 MPCA Algorithm MPCA have managed to handlemultidimensional objects According to the above sectionspseudocode for the computation of theMPCA algorithm canbe concluded [25] as shown in Figure 3

Step 1 Input sample images and center them as 119909119899

isin

1198771198751times1198752 119899 = 1 119873

Step 2 Obtain the total scatter matrixrsquos eigendecomposition

Step 3 Calculate the eigenvectors and their correspondingmost significant eigenvalues and the result is output as (119899)

Step 4 (i) Get 119910119899= (1)119879

times 119909119899times (2)

119899 = 1 119873

(ii) Calculate 1205951198840

= sum119872

119899=11199101198992

119865

(iii) For 119896 = 1 119870

6 Mathematical Problems in Engineering

The 1st group

Erro

r rat

e08

07

06

05

04

03

02

01

0

k value0 5 10 15

(a)

The 2nd group

Erro

r rat

e

08

07

06

05

04

03

02

01

0

k value0 5 10 15

(b)

The 3rd group

Erro

r rat

e

07

06

05

04

03

02

01

0

k value0 5 10 15

(c)

Figure 7 Recognition error rate of PCA + LDA against different LDA dimension reduction values

(a) calculate the total scatter matrixrsquos eigenvectors andtheir correspondingmost significant eigenvalues andthe result is output as (119895) for 119895 = 1 2

(b) get 120595119884119896

and 119910119899 119899 = 1 119873

(c) if 120595119884119896

minus 120595119884119896minus1

lt 119895 then break the loop and go toStep 5

Step 5 Finally calculate the feature matrix see the followingequation

119910119899= (1)119879

times 119909119898times (2)

119899 = 1 119873 (11)

24 LDA Algorithm LDA (linear discriminant analysis)projects image onto a lower-dimensional vector space toachieve maximum discrimination as follows

Mathematical Problems in Engineering 7

The 1st groupEr

ror r

ate

016

014

012

01

008

006

004

002

00 5 10 15 20

LDA dimensionality reduction

(a)

The 2nd group

0 5 10 15 20

LDA dimensionality reduction

Erro

r rat

e

025

02

015

01

005

0

(b)

The 3rd group

0 5 10 15 20 25 30 35 40

LDA dimensionality reduction

Erro

r rat

e

01

009

008

007

006

005

004

003

002

001

0

(c)

Figure 8 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

Step 1 Compute the average sample values for different kindsof facial images in the original space Total number is denotedby 119888119883

119894119895denotes the 119895th object of the 119894th class of samples

119898119894=

1

119899119894

1198991

sum

119895=1

119883119894119895 119883

119894119895isin 119877119889 119894 = 1 2 119888

119898 =

119888

sum

119894=1

119901119894119898119894

(12)

Step 2 Compute covariance matrix of each class

119888119894=

1

119899119894

119899119894

sum

119895=1

(119883119894119895minus 119898119894) sdot (119883119894119895minus 119898119894)119879 (13)

Step 3 Compute within-class and between-class scattermatrices

119862119887=

119888

sum

119894=1

119901119894(119898119894minus 119898) sdot (119898

119894minus 119898)119879

119862119908=

119888

sum

119894=1

119888119894

(14)

Step 4 Compute eigenvectors of matrix 119862minus1

119908119862119887to get pro-

jection vectors Then dimensionality reduction data can beobtained by projection [26 27]

After dimensionality reduction using MPCA the matri-ces are arranged in columns into vectors as inputs to theLDA algorithm By using MPCA algorithm to reduce thedimension of the image we not only solved the problem of

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

100

The 1st group The 2nd group The 3rd group

PCAMPCA

()

PCA + LDAMPCA + LDA

Figure 9 Histogram of recognition results in experiments

singular matrix but also retained structure information in theimages and thus improve the recognition rate

25 KNN Algorithm 119870-nearest neighbor (KNN) algorithm[28 29] is adopted for sample set classification here and theconcrete steps are as follows

Step 1 Select different parameters of119870 valueStep 2Adopt the method of cross-validation on training faceimages for 119896 = 1 119870

Step 3 Make the cross-validation error classification rateminimization and get its corresponding parameter 119896

Step 4 Construct a prediction model with 119896

3 Process of the Recognition Algorithm

31 Preprocessing Image preprocessing and normalizationare vital for face recognition systems as images are oftenaffected by image quality illumination face rotation facialexpression [8 30] and so forth In order to offset abovefactors it is necessary for us to carry out face normalizationbefore facial feature extraction

Our data is preprocessing normalized images with aresolution of 80times 80 In our research histogram equalizationwas applied (see (15))

1199101015840=

119910 minusmin (119910)

max (119910) minusmin (119910)

(15)

32 Dimensionality Reduction Using MPCA and FeatureMatrix Extraction Using LDA MPCA reduces dimensionsof input face images and generates feature projection matrix[30] that are then taken as input samples to LDA MPCAand LDA combination were used to construct LDA subspacefrom which both MPCA features and LDA features can beextracted

The detailed steps have been described in Sections 22 and23

33 Face Recognition Using L2 Distance Measure We usedresultant output acquired above as input samples for trainingand applied aforesaid techniques to get the feature matrixThen we carried out a similarity measure on image samplesIn our researchwe choose L2 distance formeasures (see (16))

119889 (119886 119887) = radic

119867

sum

ℎ=1

[119886 (ℎ) minus 119887(ℎ)]2 (16)

KNN classifier [31] is adopted for sample set classificationhere while the procedure and details are introduced inSection 24

The overall approach of face recognition proposed in thispaper is shown in Figure 3

4 Experiments

We evaluated the performance of our algorithm based onMPCA + LDA in this research and compared with thePCA MPCA and PCA + LDA algorithm by performingexperiments on ORL databases [32] In order to examine theability of our method we also try it on other classical facedatabases such as FERET and YALE

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The 1st group

Erro

r rat

e08

07

06

05

04

03

02

01

0

k value0 5 10 15

(a)

The 2nd group

Erro

r rat

e

08

07

06

05

04

03

02

01

0

k value0 5 10 15

(b)

The 3rd group

Erro

r rat

e

07

06

05

04

03

02

01

0

k value0 5 10 15

(c)

Figure 7 Recognition error rate of PCA + LDA against different LDA dimension reduction values

(a) calculate the total scatter matrixrsquos eigenvectors andtheir correspondingmost significant eigenvalues andthe result is output as (119895) for 119895 = 1 2

(b) get 120595119884119896

and 119910119899 119899 = 1 119873

(c) if 120595119884119896

minus 120595119884119896minus1

lt 119895 then break the loop and go toStep 5

Step 5 Finally calculate the feature matrix see the followingequation

119910119899= (1)119879

times 119909119898times (2)

119899 = 1 119873 (11)

24 LDA Algorithm LDA (linear discriminant analysis)projects image onto a lower-dimensional vector space toachieve maximum discrimination as follows

Mathematical Problems in Engineering 7

The 1st groupEr

ror r

ate

016

014

012

01

008

006

004

002

00 5 10 15 20

LDA dimensionality reduction

(a)

The 2nd group

0 5 10 15 20

LDA dimensionality reduction

Erro

r rat

e

025

02

015

01

005

0

(b)

The 3rd group

0 5 10 15 20 25 30 35 40

LDA dimensionality reduction

Erro

r rat

e

01

009

008

007

006

005

004

003

002

001

0

(c)

Figure 8 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

Step 1 Compute the average sample values for different kindsof facial images in the original space Total number is denotedby 119888119883

119894119895denotes the 119895th object of the 119894th class of samples

119898119894=

1

119899119894

1198991

sum

119895=1

119883119894119895 119883

119894119895isin 119877119889 119894 = 1 2 119888

119898 =

119888

sum

119894=1

119901119894119898119894

(12)

Step 2 Compute covariance matrix of each class

119888119894=

1

119899119894

119899119894

sum

119895=1

(119883119894119895minus 119898119894) sdot (119883119894119895minus 119898119894)119879 (13)

Step 3 Compute within-class and between-class scattermatrices

119862119887=

119888

sum

119894=1

119901119894(119898119894minus 119898) sdot (119898

119894minus 119898)119879

119862119908=

119888

sum

119894=1

119888119894

(14)

Step 4 Compute eigenvectors of matrix 119862minus1

119908119862119887to get pro-

jection vectors Then dimensionality reduction data can beobtained by projection [26 27]

After dimensionality reduction using MPCA the matri-ces are arranged in columns into vectors as inputs to theLDA algorithm By using MPCA algorithm to reduce thedimension of the image we not only solved the problem of

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

100

The 1st group The 2nd group The 3rd group

PCAMPCA

()

PCA + LDAMPCA + LDA

Figure 9 Histogram of recognition results in experiments

singular matrix but also retained structure information in theimages and thus improve the recognition rate

25 KNN Algorithm 119870-nearest neighbor (KNN) algorithm[28 29] is adopted for sample set classification here and theconcrete steps are as follows

Step 1 Select different parameters of119870 valueStep 2Adopt the method of cross-validation on training faceimages for 119896 = 1 119870

Step 3 Make the cross-validation error classification rateminimization and get its corresponding parameter 119896

Step 4 Construct a prediction model with 119896

3 Process of the Recognition Algorithm

31 Preprocessing Image preprocessing and normalizationare vital for face recognition systems as images are oftenaffected by image quality illumination face rotation facialexpression [8 30] and so forth In order to offset abovefactors it is necessary for us to carry out face normalizationbefore facial feature extraction

Our data is preprocessing normalized images with aresolution of 80times 80 In our research histogram equalizationwas applied (see (15))

1199101015840=

119910 minusmin (119910)

max (119910) minusmin (119910)

(15)

32 Dimensionality Reduction Using MPCA and FeatureMatrix Extraction Using LDA MPCA reduces dimensionsof input face images and generates feature projection matrix[30] that are then taken as input samples to LDA MPCAand LDA combination were used to construct LDA subspacefrom which both MPCA features and LDA features can beextracted

The detailed steps have been described in Sections 22 and23

33 Face Recognition Using L2 Distance Measure We usedresultant output acquired above as input samples for trainingand applied aforesaid techniques to get the feature matrixThen we carried out a similarity measure on image samplesIn our researchwe choose L2 distance formeasures (see (16))

119889 (119886 119887) = radic

119867

sum

ℎ=1

[119886 (ℎ) minus 119887(ℎ)]2 (16)

KNN classifier [31] is adopted for sample set classificationhere while the procedure and details are introduced inSection 24

The overall approach of face recognition proposed in thispaper is shown in Figure 3

4 Experiments

We evaluated the performance of our algorithm based onMPCA + LDA in this research and compared with thePCA MPCA and PCA + LDA algorithm by performingexperiments on ORL databases [32] In order to examine theability of our method we also try it on other classical facedatabases such as FERET and YALE

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

The 1st groupEr

ror r

ate

016

014

012

01

008

006

004

002

00 5 10 15 20

LDA dimensionality reduction

(a)

The 2nd group

0 5 10 15 20

LDA dimensionality reduction

Erro

r rat

e

025

02

015

01

005

0

(b)

The 3rd group

0 5 10 15 20 25 30 35 40

LDA dimensionality reduction

Erro

r rat

e

01

009

008

007

006

005

004

003

002

001

0

(c)

Figure 8 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

Step 1 Compute the average sample values for different kindsof facial images in the original space Total number is denotedby 119888119883

119894119895denotes the 119895th object of the 119894th class of samples

119898119894=

1

119899119894

1198991

sum

119895=1

119883119894119895 119883

119894119895isin 119877119889 119894 = 1 2 119888

119898 =

119888

sum

119894=1

119901119894119898119894

(12)

Step 2 Compute covariance matrix of each class

119888119894=

1

119899119894

119899119894

sum

119895=1

(119883119894119895minus 119898119894) sdot (119883119894119895minus 119898119894)119879 (13)

Step 3 Compute within-class and between-class scattermatrices

119862119887=

119888

sum

119894=1

119901119894(119898119894minus 119898) sdot (119898

119894minus 119898)119879

119862119908=

119888

sum

119894=1

119888119894

(14)

Step 4 Compute eigenvectors of matrix 119862minus1

119908119862119887to get pro-

jection vectors Then dimensionality reduction data can beobtained by projection [26 27]

After dimensionality reduction using MPCA the matri-ces are arranged in columns into vectors as inputs to theLDA algorithm By using MPCA algorithm to reduce thedimension of the image we not only solved the problem of

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

100

The 1st group The 2nd group The 3rd group

PCAMPCA

()

PCA + LDAMPCA + LDA

Figure 9 Histogram of recognition results in experiments

singular matrix but also retained structure information in theimages and thus improve the recognition rate

25 KNN Algorithm 119870-nearest neighbor (KNN) algorithm[28 29] is adopted for sample set classification here and theconcrete steps are as follows

Step 1 Select different parameters of119870 valueStep 2Adopt the method of cross-validation on training faceimages for 119896 = 1 119870

Step 3 Make the cross-validation error classification rateminimization and get its corresponding parameter 119896

Step 4 Construct a prediction model with 119896

3 Process of the Recognition Algorithm

31 Preprocessing Image preprocessing and normalizationare vital for face recognition systems as images are oftenaffected by image quality illumination face rotation facialexpression [8 30] and so forth In order to offset abovefactors it is necessary for us to carry out face normalizationbefore facial feature extraction

Our data is preprocessing normalized images with aresolution of 80times 80 In our research histogram equalizationwas applied (see (15))

1199101015840=

119910 minusmin (119910)

max (119910) minusmin (119910)

(15)

32 Dimensionality Reduction Using MPCA and FeatureMatrix Extraction Using LDA MPCA reduces dimensionsof input face images and generates feature projection matrix[30] that are then taken as input samples to LDA MPCAand LDA combination were used to construct LDA subspacefrom which both MPCA features and LDA features can beextracted

The detailed steps have been described in Sections 22 and23

33 Face Recognition Using L2 Distance Measure We usedresultant output acquired above as input samples for trainingand applied aforesaid techniques to get the feature matrixThen we carried out a similarity measure on image samplesIn our researchwe choose L2 distance formeasures (see (16))

119889 (119886 119887) = radic

119867

sum

ℎ=1

[119886 (ℎ) minus 119887(ℎ)]2 (16)

KNN classifier [31] is adopted for sample set classificationhere while the procedure and details are introduced inSection 24

The overall approach of face recognition proposed in thispaper is shown in Figure 3

4 Experiments

We evaluated the performance of our algorithm based onMPCA + LDA in this research and compared with thePCA MPCA and PCA + LDA algorithm by performingexperiments on ORL databases [32] In order to examine theability of our method we also try it on other classical facedatabases such as FERET and YALE

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

10

20

30

40

50

60

70

80

90

100

The 1st group The 2nd group The 3rd group

PCAMPCA

()

PCA + LDAMPCA + LDA

Figure 9 Histogram of recognition results in experiments

singular matrix but also retained structure information in theimages and thus improve the recognition rate

25 KNN Algorithm 119870-nearest neighbor (KNN) algorithm[28 29] is adopted for sample set classification here and theconcrete steps are as follows

Step 1 Select different parameters of119870 valueStep 2Adopt the method of cross-validation on training faceimages for 119896 = 1 119870

Step 3 Make the cross-validation error classification rateminimization and get its corresponding parameter 119896

Step 4 Construct a prediction model with 119896

3 Process of the Recognition Algorithm

31 Preprocessing Image preprocessing and normalizationare vital for face recognition systems as images are oftenaffected by image quality illumination face rotation facialexpression [8 30] and so forth In order to offset abovefactors it is necessary for us to carry out face normalizationbefore facial feature extraction

Our data is preprocessing normalized images with aresolution of 80times 80 In our research histogram equalizationwas applied (see (15))

1199101015840=

119910 minusmin (119910)

max (119910) minusmin (119910)

(15)

32 Dimensionality Reduction Using MPCA and FeatureMatrix Extraction Using LDA MPCA reduces dimensionsof input face images and generates feature projection matrix[30] that are then taken as input samples to LDA MPCAand LDA combination were used to construct LDA subspacefrom which both MPCA features and LDA features can beextracted

The detailed steps have been described in Sections 22 and23

33 Face Recognition Using L2 Distance Measure We usedresultant output acquired above as input samples for trainingand applied aforesaid techniques to get the feature matrixThen we carried out a similarity measure on image samplesIn our researchwe choose L2 distance formeasures (see (16))

119889 (119886 119887) = radic

119867

sum

ℎ=1

[119886 (ℎ) minus 119887(ℎ)]2 (16)

KNN classifier [31] is adopted for sample set classificationhere while the procedure and details are introduced inSection 24

The overall approach of face recognition proposed in thispaper is shown in Figure 3

4 Experiments

We evaluated the performance of our algorithm based onMPCA + LDA in this research and compared with thePCA MPCA and PCA + LDA algorithm by performingexperiments on ORL databases [32] In order to examine theability of our method we also try it on other classical facedatabases such as FERET and YALE

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Figure 10 Face image examples of two persons in FERET face database

Figure 11 Face image examples of two persons in YALE face database

The experiments were conducted with three groups Wechoose part of images in each group for training while therest for testing As the probabilities for each kind of facialsamples are the same then 119875

119894that equals 1 is set in LDA

algorithmInitially we tested how different parameters affect the

recognition error rate and how classification result is affectedby dimensionality using the MPCA dimensionality afterusing the LDAand 119896 value of KNNalgorithm LDAalgorithmrequires dimension reduction not greater than the totalnumber of samplesminus 1 so 1 le LDAdimension reductionle 19 There are 10 samples in each category so 1 le 119896 le 10Other parameters of MPCA are set to the optimized values

41 Experiments on the ORL TheORL face database containsa total of 400 images of 40 individuals (each individual has 10gray scale images) [33] Some photos are taken in differentperiods and some are taken with the various countenancesand the facial details Each image is of a resolution of 256 greylevels per pixel [34] Figure 4 shows image examples of twopersons before preprocessing

Now images have been divided into three differentgroupsWith the first group we select the first 5 images of thefirst 20 persons as training data and the last 5 images of thefirst 20 persons as test samples for face identification Withthe second group we select the first 5 images of the rest 20persons as training data and the last 5 images of the rest 20persons as testing samples With the third group we selectthe first 5 images of 40 persons as training data and the last 5images of 40 persons as testing samples

Recognition error rate of PCA is shown in Figure 5

Judging from the figure when 119896 equals 1 the recognitionerror rate reaches minimal value PCA recognition accuracyreaches 58ndash82

Error rate of MPCA under different 119896 value is shown inFigure 6 As shown in the figure when 119896 equals 1 error rateisminimalMPCA recognition accuracy reaches 75ndash85 inthe experiments

When 119896 equals 8 error rate of PCA + LDA algorithmreaches minimal value 7 How different LDA dimensionreduction affects recognition accuracy is shown in Figure 7

We can see from Figures 5 6 and 7 that dimension afterLDA increases as the number of samples also increasesWhenapplying PCA + LDA algorithm we use MPCA to decreasedimension of facial samples to 11 and then use LDA For the1st group reduce dimension to 7 For the 2nd group reducedimension to 8 For the 3rd group reduce dimension to 10We can conclude that the LDA algorithm is not satisfied withmultidimensional objects Accuracy of PCA + LDA reaches86ndash88

MPCA + LDA algorithm only produces higher errorrate of 10ndash25 when 119896 equals 10 In other situations therecognition error rate is very low When 119896 equals 8 the errorrate of different LDA dimensionality reduction is shown inFigure 8 Algorithm recognition accuracy rate reaches a highvalue

Result of the experiments on ORL database is shown inFigure 9 Take recognition accuracy of four algorithms forcomparison the combination of MPCA and LDA does resultin better recognition performance than traditional methods

42 Experiments onMore Face Databases We choose FERETand YALE for our experiments Implement steps are similar

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

k value1 2 3 4 5 6 7 8 9 10

(a)

A group in YALE

Erro

r rat

e

04

035

03

025

06

065

07

075

055

05

045

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 12 Recognition error rate of PCA against different 119896 values (best LDA dimension reduction) In one group of FERET 119896 equals 27 Inone group of YALE 119896 equals 63

A group in FERET

Erro

r rat

e

04

035

03

025

02

015

01

005

0

k value20 4 6 8 10

(a)

025

02

015

01

005

0

A group in YALEEr

ror r

ate

k value1 2 3 4 5 6 7 8 9 10

(b)

Figure 13 Recognition error rate of MPCA against different 119896 values

to those in above section so we just simplify steps and focuson the results

FERET face database consists of a total of 1400 imagesof 200 individuals (each person has 7 different images)Figure 10 shows image examples of two persons beforepreprocessing

YALE face database contains 165 images of 15 individualsFigure 11 shows image examples of two persons before pre-processing [35]

The performance of PCA MPCA PCA plus LDA andMPCA + LDA techniques is tested by varying the numberof eigenvectors We have chosen one group of result in eachdatabase for comparison

PCA performed worse on YALE than on FERET becauseof the poor adaptability for the image brightness and facialexpression which is shown in Figure 12

Though in Figure 13MPCA performedmuch well on facerecognition in YALE database the process takes longer timethan with PCA

Figures 14 and 15 show that both PCA + LDA andMPCA + LDA can turn to high accuracy and low error ratein recognition However PCA + LDA effectively sees onlythe Euclidean structure while MPCA + LDA successes todiscover the underlying structure [36]

Compared against all the other algorithms although withsimple preprocessing we can learn that MPCA + LDA hasachieved best overall performance in both FERET and YALEdatabases

5 Conclusions

This paper presents an algorithm for face recognition basedon MPCA and LDA As opposed to other traditional meth-ods our proposed algorithm treats data as multidimensionaltensor and fully considers the spatial relationshipThe advan-tage of our approach is of great relevance to applications andis capable of recognizing face dataset under different lighting

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

A group in FERET

Erro

r rat

e

1

09

08

07

06

05

04

03

02

01

k value0 5 10 15

(a)

08

06

04

02

0

A group in YALE

Erro

r rat

ek value

0 5 10 15

(b)

Figure 14 Recognition error rate of PCA + LDA against different LDA dimension reduction values

A group in FERET

0 10 20

LDA dimensionality reduction

Erro

r rat

e

008

006

004

002

(a)

0

A group in YALE

010 20

LDA dimensionality reduction

Erro

r rat

e

01

005

(b)

Figure 15 Error rate of MPCA + LDA algorithm against different LDA dimension reduction values

conditions and with various facial expressions LDA algo-rithmprojects the data to a new space and has exact clusteringresult in our experiments Compared with traditional facerecognition algorithms our proposed algorithm is not onlya boost in recognition accuracy but also an unclogging ofdimensionality bottlenecks and an efficient resolution of thesmall sample size problem Future work of our research willinclude applying this approach on larger face databases suchas on the CMUMulti-PIE NISTrsquos FRGC and MBGC

Conflict of Interests

The authors declared that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to show an appreciation of reviewersrsquoinsightful and constructive comments and would like tothank everyone for their hardwork on this researchTheworkwas supported by a Grant from the PhD Programs Founda-tion of Ministry of Education of China (no 20120141120006)Hubei Planning Project of Research and Development (no2011BAB035) and Wuhan Planning Project of Science andTechnology (no 2013010501010146)

References

[1] F Song H Liu D Zhang and J Yang ldquoA highly scalableincremental facial feature extractionmethodrdquoNeurocomputingvol 71 no 10-12 pp 1883ndash1888 2008

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[2] W Zhao R Chellappa P J Phillips and A Rosenfeld ldquoFacerecognition a literature surveyrdquo ACM Computing Surveys vol35 no 4 pp 399ndash458 2003

[3] H Lu K N Plataniotis and A N Venetsanopoulos ldquoMPCAmultilinear principal component analysis of tensor objectsrdquoIEEE Transactions on Neural Networks vol 19 no 1 pp 18ndash392008

[4] S Fernandes and J Bala ldquoPerformance analysis of PCA-basedand LDA-based algorithms for face recognitionrdquo InternationalJournal of Signal Processing Systems vol 1 no 1 pp 1ndash6 2013

[5] B A Draper K Baek M S Bartlett and J R BeveridgeldquoRecognizing faces with PCA and ICArdquo Computer Vision andImage Understanding vol 91 no 1-2 pp 115ndash137 2003

[6] K Choudhary and N Goel ldquoA review on face recognitiontechniquesrdquo in Proceedings of the International Conference onCommunication and Electronics System Design InternationalSociety for Optics and Photonics 2013

[7] R Gottumukkal and V K Asari ldquoAn improved face recognitiontechnique based on modular PCA approachrdquo Pattern Recogni-tion Letters vol 25 no 4 pp 429ndash436 2004

[8] J Li B Zhao and H Zhang ldquoFace recognition based on PCAand LDA combination feature extractionrdquo in Proceedings ofthe 1st International Conference on Information Science andEngineering (ICISE rsquo09) pp 1240ndash1243 IEEE December 2009

[9] P Viola and M J Jones ldquoRobust real-time face detectionrdquoInternational Journal of Computer Vision vol 57 no 2 pp 137ndash154 2004

[10] X R L Y Tang Liang ldquoAn face recognition technique basedon discriminative common vector in PCA transform spacerdquoJournal of Wuhan University vol 34 no 4 2009

[11] H Lu K N Plataniotis and A N Venetsanopoulos ldquoAsurvey of multilinear subspace learning for tensor datardquo PatternRecognition vol 44 no 7 pp 1540ndash1551 2011

[12] A K Bansal and P Chawla ldquoPerformance evaluation of facerecognition using PCA and N-PCArdquo International Journal ofComputer Applications vol 76 no 8 pp 14ndash20 2013

[13] J F Pereira R M Barreto G D C Cavalcanti and T I RenldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSP rsquo11) 2011

[14] F S Tsai ldquoDimensionality reduction for computer facial anima-tionrdquo Expert Systems with Applications vol 39 no 5 pp 4965ndash4971 2012

[15] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncorre-lated multilinear discriminant analysis with regularization andaggregation for tensor object recognitionrdquo IEEETransactions onNeural Networks vol 20 no 1 pp 103ndash123 2009

[16] W Zhao R Chellappa and N Nandhakumar ldquoEmpiricalperformance analysis of linear discriminant classifierrdquo in Pro-ceedings of the IEEE Computer Society Conference on ComputerVision and Pattern Recognition pp 164ndash169 IEEE June 1998

[17] Y Xie ldquoLDA algorithm and its application to face recognitionrdquoComputer Engineering and Applications vol 46 no 19 pp 189ndash192 2010

[18] S Yan D Xu Q Yang L Zhang and H Zhang ldquoMultilineardiscriminant analysis for face recognitionrdquo IEEE Transactionson Image Processing vol 16 no 1 pp 212ndash220 2007

[19] D Zhang and Z Zhou ldquo(2D)2 PCA two-directional two-dimensional PCA for efficient face representation and recogni-tionrdquo Neurocomputing vol 69 no 1ndash3 pp 224ndash231 2005

[20] D Zhang X You P Wang S N Yanushkevich and Y YTang ldquoFacial biometrics using nontensor product wavelet and2d discriminant techniquesrdquo International Journal of PatternRecognition and Artificial Intelligence vol 23 no 3 pp 521ndash5432009

[21] J Yang D Zhang and A F Frangi ldquoTwo-dimensionalPCA a new approach to appearance-based face representationand recognitionrdquo IEEE Transactions on Pattern Analysis andMachine Intelligence vol 26 no 1 pp 131ndash137 2004

[22] Y Li H Xie and Y Zhou ldquoStudy of eyebrow recognition basedon 2 DPCArdquo Journal ofWuhan University vol 57 no 6 pp 517ndash522 2011

[23] H Lu K N Plataniotis and A N Venetsanopoulos ldquoUncor-related multilinear principal component analysis through suc-cessive variance maximizationrdquo in Proceedings of the 25thInternational Conference onMachine Learning pp 616ndash623 July2008

[24] L De Lathauwer B De Moor and J Vandewalle ldquoOn the bestrank-1 and rank-(119877

11198772 119877119899) approximation of higher-order

tensorsrdquo SIAM Journal onMatrix Analysis and Applications vol21 no 4 pp 1324ndash1342 2000

[25] C Chen S Zhang and Y Chen ldquoFace recognition based onMPCArdquo in Proceedings of the 2nd International Conference onIndustrial Mechatronics and Automation (ICIMA rsquo10) pp 322ndash325 Wuhan China May 2010

[26] P N Belhumeur J P Hespanha and D J Kriegman ldquoEigen-faces versus fisherfaces recognition using class specific linearprojectionrdquo IEEE Transactions on Pattern Analysis andMachineIntelligence vol 19 no 7 pp 711ndash720 1997

[27] X Su Q Zeng and X Wang ldquoSeveral combination methods offace recognition based on PCA and LDArdquo Computer Engineer-ing and Design vol 33 no 9 pp 3574ndash3578 2012

[28] G-F Lu Y J Wang and J Zou ldquoImproved complete neigh-bourhood preserving embedding for face recognitionrdquo IETComputer Vision vol 7 no 1 pp 71ndash79 2013

[29] E Nasibov and C Kandemir-Cavas ldquoEfficiency analysis ofKNNandminimumdistance-based classifiers in enzyme familypredictionrdquo Computational Biology and Chemistry vol 33 no6 pp 461ndash464 2009

[30] J Shermina ldquoFace recognition system using multilinear princi-pal component analysis and locality preserving projectionrdquo inProceedings of the IEEE GCC Conference and Exhibition (GCCrsquo11) pp 283ndash286 IEEE February 2011

[31] Y Liaw M Leou and C Wu ldquoFast exact k nearest neighborssearch using an orthogonal search treerdquo Pattern Recognitionvol 43 no 6 pp 2351ndash2358 2010

[32] F S Samaria and A C Harter ldquoParameterisation of a stochasticmodel for human face identificationrdquo in Proceedings of the 2ndIEEEWorkshop onApplications of Computer Vision pp 138ndash142Sarasota Fla USA December 1994

[33] Y Jin and Q Ruan ldquoOrthogonal locality sensitive discriminantanalysis for face recognitionrdquo Journal of Information Science andEngineering vol 25 no 2 pp 419ndash433 2009

[34] P P Paul and M Gavrilova ldquoMultimodal cancelable biomet-ricsrdquo in Proceedings of the IEEE 11th International Conference onCognitive Informatics amp Cognitive Computing (ICCI rsquo12) 2012

[35] P Punitha and D S Guru ldquoSymbolic image indexing andretrieval by spatial similarity an approach based on B-treerdquoPattern Recognition vol 41 no 6 pp 2068ndash2085 2008

[36] X He S Yan Y Hu P Niyogi and H Zhang ldquoFace recognitionusing Laplacianfacesrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 27 no 3 pp 328ndash340 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of