Research Article PEM-PCA: A Parallel Expectation ...downloads.hindawi.com/journals/tswj/2014/468176.pdf · and/or verify a human face is facial recognition systems. Facial recognition

Research ArticlePEM-PCA A Parallel Expectation-Maximization PCA FaceRecognition Architecture

Kanokmon Rujirakul1 Chakchai So-In1 and Banchar Arnonkijpanich2

1 Applied Network Technology (ANT) Laboratory Department of Computer Science Faculty of Science Khon Kaen UniversityKhon Kaen Thailand

2Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen Thailand

Correspondence should be addressed to Chakchai So-In chaksokkuacth

Received 24 November 2013 Accepted 30 January 2014 Published 15 April 2014

Academic Editors J Shu and F Yu

Copyright copy 2014 Kanokmon Rujirakul et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Principal component analysis or PCA has been traditionally used as one of the feature extraction techniques in face recognitionsystems yielding high accuracy when requiring a small number of features However the covariance matrix and eigenvaluedecomposition stages cause high computational complexity especially for a large database Thus this research presents analternative approach utilizing an Expectation-Maximization algorithm to reduce the determinant matrix manipulation resulting inthe reduction of the stagesrsquo complexity To improve the computational time a novel parallel architecture was employed to utilize thebenefits of parallelization ofmatrix computation during feature extraction and classification stages including parallel preprocessingand their combinations so-called a Parallel Expectation-Maximization PCA architecture Comparing to a traditional PCA and itsderivatives the results indicate lower complexity with an insignificant difference in recognition precision leading to high speed facerecognition systems that is the speed-up over nine and three times over PCA and Parallel PCA

1 Introduction

Face recognition has recently brought the extensive attentionto the society for both research and commercial especiallywhen several applications have been practically adopted inseveral areas for example human biometrics pattern recog-nitions and computer visions within various practical usagessuch as access controls human identifications roboticscrowd surveillances and criminal forensics [1] In fact one ofthe computational techniques used to automatically identifyandor verify a human face is facial recognition systemsFacial recognition systems focus on the unanimated imageof a humanrsquos face The image can be retrieved either froma digital picture snapshot or a video frame Generally thereare two subsystems face detection [2] and face recognition[3 4] The first approach is used to identify the face positionbefore specifically distinguishing the face identity in therecognition stage There are a number of researches pro-posed to achieve high precision of the first subsystem [5ndash7]However although some approaches have been introduced

to resolve the second subsystem several issues which areamong research community interests still remain Thus inthis research the focus is on the latter

Consider the face recognition stage There are manyapproaches used to enhance a recognition precision oneof them is to compare the properly selected facial featuresrsquoimages to their facial database [8] In general a commonimage-based face recognition method with feature-basedextraction can be divided into two categories appearance-based andmodel-based approaches [9] Each of which has itsown distinctive characteristic for instance the first schemewhich applies the concept of transformed face data to a facespace analysis problem without using the human knowl-edge is designed to support the images with low resolutionandor poor quality For the second approach each standingpoint of the actual face structure including face variationis formed as the face feature model normally required aninteraction with the human Aside from a specific facemodelthat is expression and position human images the firstapproach yields high accuracy widely used in the traditional

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 468176 16 pageshttpdxdoiorg1011552014468176

2 The Scientific World Journal

Acquisition Preprocessing Feature extractor Classifier

Face image

Normalized new face image

Normalized face image

Training sets

Feature vectorClassified as

ldquoknownrdquo or ldquounknownrdquo

Figure 1 Face recognition system

image-based face recognitionThus many proposals adoptedthe appearance-based approaches for example PrincipalComponent Analysis (PCA) Independent Component Anal-ysis (ICA) Linear Discriminant Analysis (LDA) KernelPrincipal Component Analysis (KPCA) and ISOMAP [9]

Consider those approaches however PCA and its deriva-tives are commonly used due to several distinctive featuresFor example PCA is an optimal linear scheme in terms ofmean squared error for compressing a set of high dimen-sional vectors into a set of lower dimensional vectors Inaddition the model parameters used in PCA can be directlycomputed from the data without additional processing stepsMoreover compression and decompression operation com-plexity is given to the model parameters in other wordsPCA only requires matrix multiplication Most importantlyPCA requires less number of features which provide thenonreduction of precision quality and these advantages resultin high recognition accuracy even with a small data set [4]Although PCA expresses several distinctive outcomes one ofthe drawbacks is over a high complexity due to larger matrixoperation A large amount of memory is also required sincethe memory requirement grows with the image quality highresolution and the number of training images used for imagematching or classification [10 11]

With PCA derivations for years there are many attemptsto overcome the drawbacks of a traditional PCA to enhancethe performance of PCA recognition scheme for examplea symmetrical PCA and two-dimensional PCA [9] Sometechniques were involved in finding alternative approachesto lessen the computational complexity of PCA for exampleapplying QR decomposition instead of singular value decom-position (SVD) [11] Nevertheless a few of them focus onthe feasibility to absorb the high computational stage matrixmanipulation and to optimize the computational stagesespecially utilizing the parallelism to enhance the speed-upof PCA More details will be discussed in the related worksection

With a high computational task one of the probableapproaches to lessen the serial constraint is the usage of par-allelism concepts Recently with a rapid increase of computerchips and advances in integrated circuit technology resultsin affordable multicore CPUs and enhanced parallelizationtechniques [12] which have risen for a specific computa-tional task becoming a practical solution As a result thisresearch has also investigated a possibility to improve the facerecognition system and so our contribution lies in two foldsfirst using Expectation-Maximization (EM) instead of PCAcovariance matrix computation for face feature extractions

and second a novel parallel architecture for enhancingthe speed-up on matrix computational operations over ourfirst optimization called EM-PCA including the enhance-ment over face classification parallelizationThe contributionforms novel parallel face recognition architecture so-called aParallel Expectation-Maximization PCA architecture (PEM-PCA)

This research paper is organized as follows In Section 2background of face recognition systems is briefly revisitedincluding PCA and its limitation Then in Section 3 closelyrelated works of a traditional PCA PCA derivationsrsquo opti-mizations and a parallel system for PCA face recognitionsare comparatively surveyed Section 4 presents the parallelarchitecture proposal including the first PCA optimiza-tion (EM-PCA) and the second parallel face recognitionarchitecture in three substages (PEM-PCA) After that inSection 5 our comparative performance of various proposalsis discussed Finally the conclusions and future work aredrawn in Section 6

2 Background

In general a face recognition system consists of four com-ponents shown in Figure 1 These components are acqui-sition preprocessing feature extractor and classifier [13]Additionally in order to recognize the face identity thereare two main processes training and testing For trainingprocess given an image acquisition step from various inputsfor example captured photos scanned images and videoframes (acquisition) face images are fed as inputs intothe preprocessing step for example histogram equalizationbackground removal pose adjustment image size normal-ization gray-scale transformation median filtering high-pass filtering translation and rotational normalizations andillumination normalization in order to generate normalizedface imagesThen the system extracts themain features of theimages (feature extractor) resulting in feature vectors storedin a training set

The testing process is similar to the training process butwith fewer stepsThe testing image will be processed in orderto generate proper normalized face images which are readyfor face image classifier (classifier) in order to figure out theleast feature matching distance between testing and trainedfeatures Normally there are several techniques of featureextractor as well as classifier that is PCA ICA and LDA asface image matching for example Euclidian distance (ED)support vector machine (SVM) and K-nearest neighbor [1]It should be noted that PCA and ED are commonly used due

The Scientific World Journal 3

Mean estimation

Eigen selection

Mean subtraction

Covariance calculationand eigenvalue decomposition

Trained image projection

Mean subtraction

Test image projection

Image identification

PCA image identificationPCA Eigenspace Generation

Eigen projection

Trained image

Figure 2 PCA Eigenspace Generation and PCA image identification

to distinctive characteristics that is low complexity yieldinghigh classification speed [14 15]

Specifically consider PCA [16] one of the well-knownapproaches for face recognition systems PCA applies a linetransformation technique over a sample image to reducethe set of large imagesrsquo variances and then projects theirvariances into coordinate axis In other words the goal ofPCA is to decrease the high dimensional image data spaceinto low dimensional feature space [17] Figure 2 shows anoverview of PCA face recognition system Here there aretwo main components PCA Eigenspace Generation andPCA image identification Compared to Figure 1 these twocomponents correspond to two modules feature extractorand classifier Note that normally before processing the PCAface recognition stages it requires a preprocessing step asshown in Figure 1 and one of the necessities is to load theinput images into an initial matrix each of which will beconverted from RGB to grayscale and then reshaped into acommon size This matrix is also generally normalized inorder to reduce the data size leading to the reduction of timecomplexity [13]

(1) PCA Eigenspace Generation there are five submod-ules in PCA feature extraction computation as fol-lows (1) estimating themean vector of trained images(2) centering the input data around the mean vectorby finding the difference between the input imageand imagesrsquo mean (3) performing the covariancematrix calculation and then applying SVD over thecovariance matrix to obtain the eigenvectors and theeigenvalues (4) sorting the eigenvector in descendingorder and then selecting nonzero eigenvalues andfinally (5) projecting training images by calculating

the dot product between the trained image and theordered eigenvectors

(2) PCA image identification there are three submod-ules as follows (1) subtracting the testing image bymean vector (2) performing Eigenspace projectionby executing dot-product computation (3) projectingthe testing image and making a comparison betweentraining and testing images to retrieve the closetdistance

As discussed previously applying PCA for face recog-nition incurs several advantages however there are somelimitations for instance PCA involves a complex math-ematical procedure due to a transformation from a largenumber of correlated variables to a smaller number ofuncorrelated ones Thus in particular a high resolutionimage in addition to a large number of images produceshigh computational complexity especially during the matrixmanipulation that is multiplication transpose and divisionamong high dimensional vectors [18]

3 Related Work

There are many face recognition proposals employing PCA[9] Shamna et al and Zhao et al indicated that most of themused PCA for different purposes and obtained several dis-tinctive features for example less memory requirement andsimple computation complexity [5ndash7 15 19ndash21]The results ofthe survey brought about various PCA derivations includingincreasing the recognition rate For example in 2010 Gumuset al [22] applied a hybrid approach over PCA andwavelets toextract feature resulting in higher recognition rate Recentlysome researches have also improved the recognition rate


for instance Bansal and Chawla [23] in 2013 have proposednormalized principal component analysis (NPCA) whichnormalized images to remove the lightening variations andbackground effects by applying SVD instead of eigenvaluedecomposition

Furthermore in the same year Yue [24] proposed touse a radial basis function to construct a kernel matrix bycomputing the distance of two different vectors calculatedby the parameter of 2-norm exponential and then applyinga cosine distance to calculate the matching distance leadingto higher recognition rate over a traditional PCA SimilarlyMin et al [17] introduced a two-dimensional concept forPCA (2DPCA) for face feature extraction to maintain therecognition rate but with lower computational recognitiontime It should be noted that only a few proposals investigateda computational time complexity

Consider computational time complexity In 2011 Chenet al [25] proposed a local facial feature framework for stillimage and video-based face recognition leveraging FeatureAveraging (FA) Mutual Subspace Method (MSM) Manifoldto Manifold Distance (MMD) and Affine Hull Method(AHM) resulting in high speed processing especially forCCTV surveillance but with the limitation of lower recog-nition rate that is only 90 Similarly however Pereiraet al [26] proposed a technique to reduce face dimensionscalled class-modular image principal component analysis(CMIPCA) to extract local and global information to reduceillumination effects face expressions and head-pos changesresulting in speed-up over PCA Recently W Wang and WWang [16] have also applied K-L transformation for featureextraction to speed-up recognition rate but still maintainingrecognition precision

Specifically consider recognition stage complexityRoweis [28] generally discussed a possibility to useExpectation-Maximization or EM algorithm for PCAto resolve the covariance computation time of PCA-basedproblem in generalThis technique does not require a samplecovariance computation and so a complexity is substantiallyreduced when compared to a traditional PCA In additionAhn and Oh [29] proposed a constrained EM algorithmto enhance the performance of PCA to resolve the actualprincipal components extracting the problem using a coupleprobability model derived from single-standard factoranalysis models with isotropic noise structure

Moreover in 2010 Chan and Tsai [30] applied EMalgorithm over PCA to identity emotion of facial animationsbut not for realistic human faces Two years later Tsai [31]showed an application of dimensionality reduction tech-niques for computer facial animation in various techniquesfor example PCA EM-PCA Multidimensional Scaling andLocally Linear Embedding In addition in 2012 Reel etal [32] used EM-PCA to recognize medical images forthe purpose of computational complexity reduction A fewmonths later they [33] also proposed a similar techniqueinvolving an initial known transformation of predefinedaxesrsquo translation and rotation and these techniques resultedin a reduction of inherent dimensionality problem leadingto lower computational complexity Aiming at distinctiveadvantages of EM note that the EM derivation is our

promising technique to hybrid with PCA for human facerecognition

In order to overcome the major limitations of singlecore processing one of the promising approaches to speedup a computation is parallelism Several parallel architec-tures including parallel algorithms and machines have beeninvestigated Most of parallel face recognition systems onlyapplied computer hardware architectures for example eachindividual computer system is used to run each individualimage or a subset of face images Additionally recently in2013 Cavalcanti et al [34] proposed a novel method called(weighted) Modular Image PCA by dividing a single imageinto different modules to individually recognize human faceto reduce computational complexity

Previously in 2003 Jiang et al [35] proposed a dis-tributed parallel system for face recognition by dividingtrained face databases into five subdatabases feeding into eachindividual computer hardware system and then performedan individual face recognition algorithm individually inparallel over TCPIP socket communication and after thatthe recognition feedbacks are sent back for making a finaldecision at the master Similarly Liu and Su [36] modifieda distributed system to support parallel retrieval virtualmachines by allowing multivirtual machines for each slave torun individual face recognition algorithms

To improve recognition accuracy in 2005 Meng et al[1] applied a parallel matching algorithm with a multimodalpart face technique This technique uses five main faceparts on each face that is bare face based on a principalcomponent analysis and then used for recognition process ineach individual computer hardware system enabling MMXtechnology to speed up the matching process In addition in2007 Huiyuan et al [18] proposed a division of eigenblock inequal block size and then performed PCA face recognition indistributed mannersThis process can enhance the speed-uphowever the accuracy is still under consideration

Due to the advances of multicore-processors within asingle computer system Wang et al [37] proposed a parallelface analysis platform which basically used two-level parallelcomputing architecture The first level is assigned to eachindividual core for recognition purpose by dividing testingand training images in parallel and the second level is onlyto perform the final decision from the results of recognitionprocesses In 2010Numaan and Sibi [38] also discussed a pos-sibility of parallelizing and optimizing PCA with eigenfaceshowever there is a limitation of memory size

Notice that all of the approaches discussed above canachieve sorts of highest degree of parallelisms by onlyperforming an individual face recognition algorithm eitherin multivirtual machines or multicore-processing with thekey limitation on the number of CPU cores and so ingeneral these approaches do not utilize the parallelism ineach face recognition stage in order to achieve higher degreeof parallelisms and these are our main focus in this researchto propose a parallel architecture utilizing the parallelism offace recognition stage

Aside from the recognition stage the other two stagespreprocessing and classification phases are also importantFor example consider the first phase Zhang [20] introduced


Preprocess Feature extraction (EM-PCA)

(EM-PCA)

(EM-PCA)

Preprocess Feature extraction


Preprocess Eigen projection



Image identification(Euclidean distance)



Eigen projection(training image)



Figure 3 A Parallel EM-PCA Architecture (PEM-PCA)

wavelet transform discrete cosine transform and color nor-malization as preprocessing methods in face recognition toachieve better recognition precision Recently Kavitha et al[39] have also introduced color space transformation (CST)with fractional Fourier transform (FFT) and local binarypattern (BP) to increase recognition rate both of which lackincluding the investigation of stage parallelism to furtherspeed-up

Consider the classification stage Many approaches areintroduced for example ED Manhattan distance Maha-lanobis distance nearest neighbor and SVM [4 9 40] Forexample in 2013 I Melnykov and VMelnykov [41] proposedthe use of Mahalanobis distance for K-mean algorithm toimprove the performance when covariance matrices are notproperly initialized but with the increase of computationalcomplexity Moreover soft computing-based approachesthat is neural networks are also used to compute the match-ing distance for instance Yue [24] used nearest neighbormethods to compute the matching distance to improve theclassification precision Zhou et al [14] also proposed acombination of PCA and LDA for image reconstructionto increase recognition rate and then be classified withSVM Similarly Gumus et al [22] also applied SVM duringclassification steps resulting in higher recognition rate buthigher computational time

It should be noted that most of the face recognitionsystems have applied ED for face classification [2ndash4 9 34]to achieve simplification and to yield acceptable classificationprecision To emphasize a feasibility of this technique in2003 Draper et al [15] compared Manhattan distance EDand Mahalanobis distance over PCA and then found outthat ED have the highest accuracy rate In addition recentlyin 2013 Moon and Pan [42] compared Manhattan distanceED cosine similarity and Mahalanobis distance over LDAface recognition in which the results lead to an outstandingperformance of ED In addition to further speed up thecomputational time Li et al [43] proposed ED for matrix

calculation on a large dataset using multiprocessing whichapplied a traditional parallel matrix operation

To sum up as discussed above PCA yields high facerecognition precision together with several derivationshowever our proposals investigated enclosing Expectation-Maximization (EM) algorithm into PCA to reduce a com-putational time complexity during covariance calculationTo further enhance the speed-up of the recognition ratealthoughmany proposals focus on the parallelism utilizationour proposal deals with individual stage parallelisms duringmatrixmanipulation of our first enhancement by rearrangingthematrixmanipulation including determinant and orthogo-nalization processes Last but not least the optimization overparallel classification technique was also investigated Thesethree combinations lead to a parallel architecture for facerecognition called Parallel Expectation-Maximization PCAarchitecture (PEM-PCA)

4 Parallel Expectation-Maximization PCAFace Recognition Architecture (PEM-PCA)

An overall architecture of Parallel Expectation-MaximizationPCA (PEM-PCA) generally consists of three parts paral-lel face preprocessing parallel face feature extraction andparallel face classification To illustrate the detailed systemFigure 3 shows both training and testing processes of PEM-PCA in comparison to a traditional face recognition systemas shown in Figure 1 excluding an acquisition stage Inthis architecture both training and testing processes will beperformed in parallel In general initially an input imageis divided into pixels and then executed in parallel that isone pixel per thread During a high matrix computationcomplexity a manipulation process is performed in parallelby computing one element of a result matrix per thread Itis observed that the parallel efficacy depends upon a numberof cores and relationships between thread and core usages


Gray-scale conversion

Image size normalization



Figure 4 Parallel preprocessing

Mean estimation Orthogonalization

Mean subtraction

EM stage derivation

Data projection

PCA Eigenspace Generation

Trained image

Figure 5 EM-PCA face recognition stage

For example if there are twenty processes on eight coresbasically it performs the first eight processes followed by thelatter once completed iteratively

41 Parallel Face Preprocessing In general preprocessingis one of the major stages used to reduce computationalcomplexity as well as increase recognition precision fornoise reduction Although there are several preprocessingtechniques briefly stated in related work section here anecessary parallel method is considered to aid algorithmefficiency that is gray-scale conversion Regarding our previ-ous experiment the recognition precision between color andgray-scale images is not significantly different but with theincrease of computational time complexity [27] Thus in ourproposed architecture our focus is to parallelly perform gray-scale conversions once the input images are shaped into thesame size that is 180times200 pixels in both training and testingprocesses as shown in Figure 4 Note that other probablepreprocessing techniques can also be fit in this architecturewith paralleled modification

42 Parallel Face Feature Extraction As stated in relatedworks several enhanced proposals over PCA for reducinga number of dimensions have been introduced howeversome issues are to resolve for example outlier and noisereductions leading to lower accuracy and computationalcomplexity Thus here to lessen the limitation our firstproposal is to apply Expectation-Maximization (EM) tofigure out themaximum likelihood to estimate proper param-eters derived from the covariance computational step [28]called EM-PCA Then to speed up the recognition rate

several simplified matrix manipulation techniques are to beproposed called Parallel EM-PCA (PEM-PCA)

421 Expectation-Maximization PCA Face Recognition (EM-PCA) To enhance face recognition rate Figure 5 illustratesour hybrid EM-PCA face recognition scheme when applyingEM steps [44] instead of the covariance computational stageover a traditional PCA face recognition stated in the secondmodule of the first component as shown in Figure 2 Herethere are four substages EM-Step derivation orthogonal-ization data projection and PCA Eigenspace GenerationBefore moving to the these stages there are two moresubstages for preprocessing mean (vector) estimation 120583including centralized input data acquisition 119883 and meansubtraction used to compute the difference between the inputdata and mean Notice that with EM-PCA only a few eigen-vectors and eigenvalues are required to be extracted from alarge amount of high dimensional data set In addition thecovariance structure of an observed p-dimensional variablecan be captured with the operation less than 119901(119901 + 1)2

dimensions compared to 1199012 of a traditional PCA with full

covariance matrix computation

(1) EM-Step derivation given eigenvector matrix U thisprocess is used to estimate input parameters U forthe next stage (orthogonalization) Here there aretwo steps called E-Step and M-Step illustrated in(1) and (2) respectively Note that EM-Step will berepeatedly performed until the change epsilon (120576)of the difference between variances is equal or lessthan a specific threshold value It should be also noted


Input Matrix x

Output Matrix eigenvectors (119880) eigenvalues (120582)(1) Estimate the mean vector 120583 = (1119873)sum

119873

119894=1119909119894

(2) Center the input data around the mean119883 = 119883 minus 1205831times119873

(3) Set elements of 119880 isin R119872times119896 to random values(4) repeat(5) E-step 119860 = (119880

119879119880)minus1

119880119879119883

(6) M-step 119880 = 119883119860119879(119860119860119879)minus1

(7) until the change of 1205902= (1(119873 minus 119896)) (VAR(119883 ) minus VAR(119860)) le threshold

(8) Orthogonalize 119880

(9) Project input data on 119880 119860 = 119880119879119883

(10) Perform PCA on 119860 Obtain 1198801015840 and 120582

1015840

(11) Rotate 119880 for 119880 1198801015840= 119880119880

1015840

(12) Determine the eigenvalues 120582 = 1205821015840

Algorithm 1 Expectation-Maximization algorithm

that before entering this step the eigenvector will berandomly selected as the input

119860 = (119880119879119880)minus1

119880119879119883 (1)

119880 = 119883119860119879(119860119860119879)minus1

(2)

(2) Orthogonalization at this stage Gram-SchmidtOrthonormalization [32] was performed In generalthis process started by normalizing the first vectorand then iteratively transformed the remainingvectors into weighted normalized vectors Notethat the detailed description corresponding to ourproposed parallel algorithm will be discussed in nextsection

(3) Data projection at this stage the input vector isprojected into a transpose of M-Step matrix manip-ulation and then it is multiplied to the result frommean subtraction step

(4) PCA Eigenspace Generation the final stage is per-formed as a final round using a traditional PCAmean estimation mean subtraction covariance com-putation and SVD eigenvalue selection and trainedimage projection respectively as previously shown inFigure 2

Consider algorithm complexity As stated in Algorithm 1EM-PCA complexity is in order of 119874(119896119899119901) versus 119874(1198991199012)for covariance computation used in the traditional PCAface recognition where 119896 is the number of leading trainedeigenvectors (extracted component) It should be noted thatthe degree of accuracy of EM-PCA will be also based on theselection criteria of number of eigenvectors and epsilon (120576)with a time complexity trade-off

422 Parallel Expectation-Maximization PCA Face Recog-nition (PEM-PCA) To further enhance the speed-up we

propose PEM-PCA which introduces the parallelism ineach computational stage During EM-PCA face recognitionbased on our observation there are four different fundamen-tal matrix manipulations multiplication (matrixconstant)division (constant) transpose and subtraction (matrix)each of which can utilize the parallelism depending on itsdistinctive characteristic In addition to these four three extraprocesses determinant cofactor and orthogonalization canalso be performed in parallel in our architecture

For example for parallel matrix multiplication with4 times 4 dimension here a number of threads can be dividedinto four parts in each dimension (119860

1119895times 1198611198961) or sixteen

threads (119860119894119895times 119861119896119898

) to achieve high degree of parallelismNevertheless there is a trade-off over task distribution andself-contained computational complexity considering a com-munication process cost Similarly to achieve parallel matrixcomputation in each process of EM-PCA face recognitionthe degree of parallelism is based on the dimension of matrixversus a number of computational threads supported Ingeneral each computational step is parameterized by aniteration value and applies with a work-stealing algorithmand so the runtime process may reassign a set of iterationsto other available threads if any

As discussed previously our proposal is to hybridthose two techniques over a traditional PCA (Expectation-Maximization and parallelism) Essentially each distinctivematrix manipulation is to investigate the degree of paral-lelism that is during subtraction multiplication divisiontranspose determinant cofactor and orthogonal stated inAlgorithm 1 where the parallel algorithm structure is gen-erated as proposed by Blelloch and Maggs [45]

(1) Parallel Subtraction The subtraction of dimensionalmatrix can be performed in parallel to calculate the differenceof each element at the same position stated in (3) and thealgorithm of matrix subtraction is illustrated in Algorithm 2

119888 [119894 119895] = 119886 [119894 119895] minus 119887 [119894 119895] (3)


Input Matrix left Matrix right

Output Matrix result

(1) Matrix Subtraction

(2) parallel for i from 0 to rows

(3) for j from 0 to cols

(4) Subtraction 119903119890119904119906119897119905[119894 119895] = 119897119890119891119905[119894 119895] minus 119903119894119892ℎ119905[119894 119895]

(5) endfor(6) endfor(7)(8) Matrix Multiplication

(9) for i from 0 to leftRows

(10) parallel for j from 0 to rightColumns

(11) for k from 0 to leftColumns

(12) Calculate sum 119904119906119898 += 119897119890119891119905[119894 119896] times 119903119894119892ℎ119905[119896 119895](13) endfor(14) 119903119890119904119906119897119905[119894 119895] = 119904119906119898(15) endfor(16) endfor(17)(18) Matrix Transpose

(19) parallel for i from 0 to rows(20) for j from 0 to cols

(21) Transpose by convert row to column 119903119890119904119906119897119905[119895 119894] = 119894119899119901119906119905[119894 119895](22) endfor(23) endfor(24) return result

Algorithm 2 Parallel matrix manipulation with matrix computation

(2) Parallel Multiplication There are two types of matrixmultiplication used in our proposal either matrix multi-plication by matrix or by constant number Consider thefirst multiplication with matrix The value of each elementin result matrix can be computed by employing (4) (5)and (6) respectively These equations are independent andthus the computation can be performed in parallel as shownin Algorithm 2 The row dimensions of right matrix or themultiplier must be equal to the column dimension of the leftone The row dimension of result matrix is equal to the leftrow dimension while the column dimension of result matrixis equal to the multiplier column dimension Consider

119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)

Consider the later multiplication with constant valuesThe procedure is multiplying constant number to everyelement of the input matrix The result matrix dimensionwill be equal to the input matrix dimension Each elementvalue in the result matrix can be computed from (7) These

calculations are independent and so the parallelism can beachieved as shown in Algorithm 3 Consider

119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)

(3) Parallel Division The matrix division calculation can bealso performed in parallel because each calculation step isindependent The result at one position can be retrieved bydividing the same position of input matrix by constant valuesstated in (8) leading to an equalized dimensional matrixand the algorithm of parallel matrix division manipulation isillustrated in Algorithm 3 Consider

119888 [119894 119895] =119886 [119894 119895]

119887 (8)

(4) Parallel Transpose The matrix transpose procedure is toconvert the rows into columns For instance the 1st row ofinput matrix becomes the 1st column of result matrix and the2nd row of input matrix becomes the 2nd column of resultmatrix as shown in (9)

Example 3 times 3 dimension matrix transpose

[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)

Since the transformation from row to column is indepen-dent of each other the parallelism can be utilized by using


Input Matrix input Double constant


(1) Matrix Multiplication



(4) Multiplication 119903119890119904119906119897119905[119894 119895] = 119894119899119901119906119905[119894 119895] times 119888119900119899119904119905119886119899119888119890 (5) endfor(6) endfor(7)(8) Matrix Division



(11) Division 119903119890119904119906119897119905[119894 119895] = 119894119899119901119906119905[119894 119895]119888119900119899119904119905119886119899119888119890 (12) endfor(13) endfor(14) return result

Algorithm 3 Parallel matrix multiplication with a constant number computation

(10)The parallel matrix transpose manipulation algorithm isexplained in Algorithm 2 Consider

119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)

(5) Parallel Transpose Multiplication For the matrix withsymmetric and square property the multiplication of thismatrix with its transpose can be performed in parallel Dueto the nature of computational matrix in EM-PCA thesquare matrix one of which is the matrix multiplicationwith its transpose 119860 times 119860

119879 and the calculation processcould be optimized as stated in Algorithm 4 as followsfirst for diagonal computational elements traditionally thecomputation can be derived as stated in (11) However theoriginal matrix could be reused by removing the transpose

process for example accessing and reserving memory butperforming the square of the original matrix instead

Diag [119894 119895] = 119860[119894 119895]2

(11)

Second since the matrix is symmetry each element inthe upper-triangular matrix is the same as that in the lower-triangular matrix (see (13)) leading to lower computationalcomplexity by half (see (12)) Consider

Upper = Lower =119899

sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)

Optimized matrix multiplication with its transpose(upperlower and diagonal) Consider

[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)

(6) Parallel Determinant and Cofactor The cofactor matrix of119860 is the 119899 times 119899matrix C whose (119894 119895) entry is the (119894 119895) cofactorof 119860 (see (14))

119862119894119895= (minus1)

119894+119895119860119894119895 (14)

The calculation of each position can be computed at thesame time and so the parallelism can be utilized as shown in(15)

Example 3 times 3 dimension matrix cofactor calculation

cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+ ((a22 times a33) minus (a23 times a32)) minus ((11988621times 11988633) minus (119886

23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))

minus ((11988612times 11988633) minus (119886

13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input


(1) for i from 0 to input Rows

(2) for j from 0 to input Columns

(3) Calculate Diagonal Value result [ii] += (input[ij]times input[ij])

(4) endfor(5) endfor(6) c=0 b=0


(8) repeat(9) Calculate Non-Diagonal Value result [ic] += (input[ib]times input[cb])

(10) b++

(11) if (bgt=input Columns)

(12) c++

(13) b=0

(14) Copy lower-diagonal value to upper-diagonal value(15) result [c-1i]= result [ic-1]

(16) endif(17) until (((clti)ampamp(blt input Columns)))

(18) endfor(19) c=0

(20) b=0

Algorithm 4 Parallel optimized matrix multiplication with its transpose algorithm [27]

It should be noted that the determinant is a function of asquare matrix reduced into a single number Finding thedeterminant of an n-square matrix for 119899 gt 2 can be doneby recursively deleting rows and columns to create successivesmaller matrices until they are all in 2 times 2 dimensions

Given the 119899 times 119899matrix (119886119894119895) the determinant of 119860 can be

written as the sum of the cofactors of any rows or columnsof the matrix multiplied by the entries that generated themIn other words the cofactor expansion along the jth columngives

det (119860) = 11988611198951198621119895+ 11988621198951198622119895+ 11988631198951198623119895+ sdot sdot sdot + 119886

119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)

The cofactor expansion along the ith row gives


119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)

From (16) and (17) it is noticeable that their summationcan be also in parallel As shown in (18) the determinantvalue at each position can be computed at the same timebecause the calculation was independent Both determinantand cofactor can be calculated concurrently as shown inAlgorithm5 and here the calculation is performed iterativelyuntil the matrix size equals 2 times 2matrix

Example 3times3 dimensionmatrix determinant calculation

determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

+a11 times ((a22 times a33) minus (a23 times a32)) minus11988612times ((11988621times 11988633) minus (119886

23times 11988631)) +119886

13times ((11988621times 11988632) minus (119886

22times 11988631))

minus11988611times ((11988612times 11988633) minus (119886

13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))

+11988611times ((11988612times 11988623) minus (119886

13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)

(7) Parallel Orthogonal Gram-Schmidt orthonormalizationis a method for converting a set of vectors into a set oforthonormal vectors It basically begins by normalizingthe first vector and iteratively calculating a weightvector of the remaining vectors and then normalizing

them The matrix can be normalized by first powerevery matrix component by two then summarizes theresult in each column and finally divides each matrixcomponent in each column by the square root of thesummary Notice that the orthogonalization processes


Input Matrix input

Output Double detVal Matrix cofactor

(1) if matrixSize=2 then(2) detVal = (119894119899119901119906119905 [0 0] times 119894119899119901119906119905 [1 1]) minus (119894119899119901119906119905[0 1] times 119894119899119901119906119905[1 0])(3) result [119894 119895] = (minus1)

119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))

(4) return detVal cofactor

(5) else(6) parallel for i from 0 to dimension

(7) for j from 0 to columns

(8) detVal = detVal + 119894119899119901119906119905 [0 119894] times 119889119890119905119890119903119898119894119899119886119899119905(119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119894119899119901119906119905 0 119894)) times (minus1)119894+119895

(9) result [119894 119895] = (minus1)119894+119895

times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))(10) endfor(11) endfor(12) return detVal

(13) endif

Algorithm 5 Parallel matrix determinant and cofactor computation

1 middot 4 times 1

1 2 3 4

2 middot 4 times 2 3 middot 4 times 3 4 middot 5 times 4

Normalize w5

Calculate weight of 5th vector

5 = 5 minus 1 minus 2 minus 3 minus 4

u

u

u

u

u

u

u

u

w

Figure 6 Example weight calculation of the 5th vector

proposed in most of the previous works [29] performedmodified Gram Schmidt however in this researcha traditional algorithm classical Gram Schmidt [46]was selected in order to gain higher degree of parallelisms asshown in Figure 6 and Algorithm 6

Algorithms 2 to 6 illustrate our proposed matrix manip-ulation calculation and in general parallel for was applied[44] in NET C to achieve process concurrency Each of thecomputational tasks running over the algorithmis parameter-ized by an iteration value It should be noted that during theactual implementation to achieve the process concurrencythere is a possibility to generate out-of-bound index Inthis case we applied a delegate function to obscure thisproblem over resource allocation In this proposal NET Cwas implemented instead of MATLAB due to the limitationof computational matrix for huge dimensions over memorymanagement Most importantly in addition this implemen-tation is suitable for online-processing face recognition

43 Parallel Face Classification With our proposed paral-lel architecture face recognition for testing purposes ourparallel classification is based on ED to parallelly figure outthe closet distance of face [9] for simplicity purposes whileremaining the recognition precision especially during thematrix computation Traditionally one of the optimizationtechniques proposed in this research is based on Li et al [43]by utilizing the input matrix characteristic (one-dimensional

scale) leading to complexity reduction of (o1198993) to (o1198992) asshown in Algorithm 7 Furthermore to utilize the parallelismfor our parallel face recognition architecture Figure 7 showsthe feasibility to utilize the degree of parallelisms over ouroptimized ED in each EDrsquos matrixes The matrixes will besimultaneously computed in parallel for example with fourprocesses the computation at 119889

11 11988912 11988913 and 119889

21will be

performed in parallel It should be noted that other probableclassification techniques can also be fit in this architecturewith paralleled modification

5 Performance Evaluation

To investigate the performance of our parallel architecturein this section this research comparatively evaluates thesystem into three main scenarios in order to illustrate theoptimal number of eigenvectors and epsilon values over EM-based approaches to state the recognition computationalcomplexity and precision and to evaluate the performanceof degree of parallelization

51 Experimental Setup In general our testbed is based ona standard configuration on Windows 7 Ultimate operatingsystems (64 bits) CPU Intel(R) Core (TM) i-3770K 8-Cores350GHz (8MB L3 Cache) 8192 times 2MB DDR3-SDAM and500GB 5400 RPM Disk


Input Matrix a

Output Matrix q

(1) for j from 1 to n

(2) V119895= 119886119895

(3) parallel for i from 1 to (j-1)

(4) 119903119894119895= 119902119894times V119895

(5) Rewrite Vectors V119895= V119895minus 119903119894119895119902119894

(6) endfor(7) Normalize vector 119903

119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172

(8) Rewrite Remaining Vectors 119902119895= V119895119903119895119895

(9) endfor

Algorithm 6 Proposed parallel matrix orthogonalization computation

Input Matrix A Matrix B


(1) for row from 1 to A Rows

(2) sum = 00

(3) for col from 1 to A Cols

(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2

(5) endfor(6) result[row] = sum

(7) endfor(8) return result

Algorithm 7 Matrix Operation for Generalized Euclidean Distance

The performance evaluation process was implementedin NET C programming environment in order to emulatethe real-world application and illustrate the computationaltime complexity Public face database from FACE94 andFACE95 [47] was used for testing purposes A set of colorswas selected as 24-bits RGB PNG images ranging from100 to 500 images of 180 times 200 pixels In all evaluationsthere are two main metrics average (computational time andaccuracy) and standard deviation for varied random loads(a number of trained images) over five trails [9] Note thateach evaluation a classic statistical method was selected thatis simple random selection [48] by randomly selecting ninetesting images four images within the training image datasetand five from the outside

Three main scenarios are as follows first to illustratethe optimal number of eigenvectors and epsilon values forface recognition using EM-PCA and PEM-PCA in practicethe number of eigenvectors was varied by factor of 001003 005 007 and 009 respectively In addition variousepsilon values 10119864 minus 119909 in that 119909 is in range of 1 3 5 7and 9 respectively were applied in order to figure out theoptimal value yielding the performance improvement overcomputational time and in this scenario the number oftraining images was limited to 500 [47]

Second the selected optimal number of epsilons andeigenvectors are based on the outstanding performance inthe first scenario Then for scalability purposes in terms ofnumber of images to illustrate the outstanding performanceof our PEM-PCA the evaluation was to comparatively

perform the recognition process over two matrixes Here atraditional PCA our first enhancement - EM-PCA ParallelPCA (P-PCA) and finally our PEM-PCA Note that theselected proposal P-PCA is based on one of the promisingprevious works [35] in which divided the training dataset intomodules regarding the number of CPU cores that is witheight cores the dataset will be divided into eight sub-datasetand then perform the recognition in parallel

Finally to evaluate the performance of degree of par-allelization especially of PEM-PCA including P-PCA andEM-PCA the comparative performance over number ofcores were in range of 1 2 4 and 8 cores respectively Thecomputational time and accuracy were measured with 500training images and 10119864 minus 1 in epsilon and 0009 in numberof eigens

52 Experimental Results Consider the first scenario (EM-based approaches) Generally Figure 8 illustrates that byincreasing a number of eigenvectors the results indicatedhigher computational time-consuming operations Howeverwhile increasing of epsilon values leads to lower com-putational time complexity Similarly consider the secondscenario Figure 9 shows the increasing trend of accuracywhen the eigenvector number is increased Nevertheless thevariation of epsilon value has insignificantly affected therecognition accuracy

Second to explicitly show the performance improvementof PEM-PCA Figure 10 illustrates the computational speed


A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43

b11 b12 b13d11 = radic(a11 minus b11)

2

d12 = radic(a12 minus b12)2


d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43

Figure 7 Parallel matrix operation computation for generalized Euclidean distance

001003005

007009

19520

20521

21522

225

Epsilon valueEigen number

Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9

Figure 8 Effect of epsilon values and Eigen numbers on computa-tional time over EM-PCA and PEM-PCA

60

70

80

90

100

Eigen numberEpsilon value

Accu

racy

()

001003005

007009

e1e3

e5e7

e9

Figure 9 Effect of epsilon values and Eigen numbers on recognitionaccuracy over EM-PCA and PEM-PCA

and especially with 500 images PEM-PCA outperforms theother three approaches PCA P-PCA andEM-PCA by factorof nine three and two respectively In addition by rangingthe number of training images the computational speed is inan increasing order For example with 100 200 300 and 400images the speed-up of PEM-PCA over the three approachesis in order of 3 2 2 4 2 2 5 2 2 7 2 2 respectively Note thatthe speed-up of the enhancement also depends on the degreeof parallelisms and the number of cores as well as the numberof working individual threads

Moreover consider the recognition precision Figure 11shows that the recognition accuracy of PEM-PCA is insignif-icantly different from the other three It should be noted that

100 200 300 400 5000

100

200

300

400

PCA EM-PCAPEM-PCAP-PCA

Number of trained images

Com

puta

tiona

l tim

e (s)

Figure 10 Computation time over number of trained images (PCAP-PCA EM-PCA and PEM-PCA)

this behavior of PEM-PCA and EM-PCA is due to the ran-dom characteristic when choosing the random eigenvectors

Finally to illustrate the scalability when increasing thenumber of cores Figure 12 shows that EM-PCA speed-up hasno significant effect by the increase of cores and especiallywith low number of cores it incurs higher performance thanP-PCA but not over PEM-PCA Whenever the number ofcores is accordingly increased although P-PCA performanceshows the degree of parallelism effect PEM-PCA still hasoutstanding performance over the others that is in orderof three and two of P-PCA and EM-PCA respectively ateight cores In addition similarly P-PCA EM-PCA andPEM-PCA produced insignificant different face recognitionprecision as shown in Figure 13 Finally in order to justifyour algorithm implementation Figure 14 shows comparativeresults of Eigenface decomposition among PCA EM-PCA P-PCA and PEM-PCA

6 Conclusions and Future Work

Although a traditional PCA can improve the recognitionaccuracy for a face recognition system however there existsa limitation over PCA Therefore in this research several


0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500

Figure 11 Percentage of recognition accuracy over number oftrained images (PCA P-PCA EM-PCA and PEM-PCA)

1 core 2 cores 4 cores 8 cores0

100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)

Figure 12 Computation time over number of CPU cores (P-PCAEM-PCA and PEM-PCA)

issues were evaluated and investigated especially in termsof the computational time complexity during the covariancematrix computation stage In addition one of the possibilitiesto enhance PCA Expectation-Maximization (EM) PCA facerecognition was proposed to enhance the time complexitywhen the recognition accuracy remains insignificant differ-ence Plus due to the advance of parallelism novel face recog-nition architecture was proposed by applying parallelism forlarge matrix manipulation including parallel preprocessingparallel recognition and parallel classification all of whichrefer to Parallel Expectation-Maximization PCA or PEM-PCA

Based on our justified parallel algorithm implementationPEM-PCA outperforms the others namely a traditionalPCA our first enhancement EM-PCA and Parallel PCA bynine two and three respectively It should be noted that the

0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()

1 core 2 cores 4 cores 8 coresNumber of CPU cores

Figure 13 Percentage of recognition accuracy over number of CPUcores (P-PCA EM-PCA and PEM-PCA)

(a)

(b)

(c)

(d)

Figure 14 Eigenface decomposition ((a) traditional PCA (b) P-PCA (c) EM-PCA and (d) PEM-PCA)

results also depend on the number of training images withinsignificant difference for recognition precision Althoughthe proposed technique can achieve a high degree of speed-upover PCAmore investigation including intensive evaluationscan be performed for example improving preprocessingstages enhancing high degree of parallelism aside fromfocusing only on matrix manipulation reducing sensitivityoutliers and testing a large number of various imagesIn addition to show the efficiency of parallelism usagesautonomously separating recognition task can be performed


over message passing interface on each individual machineTo also fully complete the process of face recognition systemthe other aspects of the system that is face detection can alsobe further investigated They are left for future work

Conflict of Interests

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

References

[1] K Meng G-D Su C-C Li B Fu and J Zhou ldquoA highperformance face recognition system based on a huge facedatabaserdquo in Proceedings of the International Conference onMachine Learning andCybernetics (ICMLC rsquo05) vol 8 pp 5159ndash5164 Guangzhou China August 2005

[2] X Lu ldquoImage analysis for face recognitionrdquo Personal Notespp36 2003 httpwwwface-recorginteresting-papersgene-ralimana4facrcg lupdf

[3] L-H Chan S-H Salleh C-M Ting and A K Ariff ldquoFaceidentification and verification using PCA and LDArdquo in Proceed-ings of the International Symposium on Information Technology(ITSim rsquo08) vol 8 pp 1ndash6 Kuala Lumpur Malaysia August2008

[4] K Delac M Grgic and S Grgic ldquoIndependent comparativestudy of PCA ICA and LDA on the FERET data setrdquo Interna-tional Journal of Imaging Systems and Technology vol 15 no 5pp 252ndash260 2005

[5] N Rathore D Chaubey and N Rajput ldquoA survey on facedetection and recognitionrdquo International Journal of ComputerArchitecture and Mobility vol 1 no 5 2013

[6] D Dandotiya R Gupta S Dhakad and Y Tayal ldquoA surveypaper on biometric based face detection techniquesrdquo Interna-tional Journal of Software and Web Sciences vol 2 no 4 pp67ndash76 2013

[7] E Hjelmas and B K Low ldquoFace detection a surveyrdquo ComputerVision and Image Understanding vol 83 no 3 pp 236ndash2742001

[8] L Smith ldquoA tutorial on principal components analysisrdquo 2002httpwwwcsotagoacnzcosc453

[9] R Jafri and H R Arabnia ldquoSurvey of face recognition tech-niquesrdquo Journal of Information Processing Systems vol 5 no 2pp 41ndash68 2009

[10] M Turk and A Pentland ldquoEigenfaces for recognitionrdquo Journalof Cognitive Neuroscience vol 3 no 1 pp 71ndash86 1991

[11] J Gao L Fan and L Xu ldquoSolving the face recognition problemusing QR factorizationrdquo Journal of WSEAS Transactions onMathematics vol 11 no 1 pp 728ndash737 2012

[12] T Goodall S Gibson and M C Smith ldquoParallelizing principalcomponent analysis for robust facial recognition using CUDArdquoin Proceedings of the Symposium on Application Acceleratorsin High Performance Computing (SAAHPC rsquo12) pp 121ndash124Chicago Ill USA 2012

[13] V P Kshirsagar M R Baviskar and M E Gaikwad ldquoFacerecognition using Eigenfacesrdquo in Proceedings of the 3rd Inter-national Conference on Computer Research and Development(ICCRD rsquo11) vol 2 pp 302ndash306 Shanghai China March 2011

[14] C Zhou LWang Q Zhang andXWei ldquoFace recognition baseon PCA image reconstruction and LDArdquoOptik vol 124 no 22pp 5599ndash5603 2013

[15] 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

[16] W Wang and W Wang ldquoFace feature optimization based onPCArdquo Journal of Convergence Information Technology vol 8 no8 pp 693ndash700 2013

[17] L Min L Bo and W Bin ldquoComparison of face recognitionbased on PCA and 2DPCArdquo Journal of Advances in InformationSciences and Service Sciences vol 5 pp 545ndash553 2013

[18] W Huiyuan W Xiaojuan and L Qing ldquoEigenblock approachfor face recognitionrdquo International Journal of ComputationalIntelligence Research vol 3 no 1 pp 72ndash77 2007

[19] P Shamna A Paul and C Tripti ldquoAn exploratory survey onvarious face recognition methods using component analysisrdquoInternational Journal of Advanced Research in Computer andCommunication Engineering vol 2 no 5 pp 2081ndash2086 2013

[20] H Zhang ldquoImage preprocessing methods in face recognitionrdquoin Proceedings of the Symposium on Photonics and Optoelectron-ics (SOPO rsquo11) pp 1ndash4 Wuhan China May 2011

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

[22] E Gumus N Kilic A Sertbas and O N Ucan ldquoEvaluationof face recognition techniques using PCA wavelets and SVMrdquoExpert Systems with Applications vol 37 no 9 pp 6404ndash64082010

[23] 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

[24] X Yue ldquoHuman face recognition based on improved PCAalgorithmrdquo Journal ofMultimedia vol 8 no 4 pp 351ndash357 2013

[25] S Chen S Mau M T Harandi C Sanderson A Bigdeliand B C Lovell ldquoFace recognition from still images to videosequences a local-feature-based frameworkrdquoEurasip Journal onImage and Video Processing vol 2011 Article ID 790598 pp 1ndash14 2011

[26] J F Pereira R M Barreto G D C Cavalcanti and I R TsangldquoA robust feature extraction algorithm based on class-modularimage principal component analysis for face verificationrdquo inProceedings of the 36th IEEE International Conference on Acous-tics Speech and Signal Processing (ICASSP rsquo11) pp 1469ndash1472Prague Czech Republic May 2011

[27] K Rujirakul C So-In B Arnonkijpanich K Sunat and SPoolsanguan ldquoPFP-PCA parallel fixed point PCA face recog-nitionrdquo in Proceedings of the 4th International Conference onIntelligent Systems Modelling amp Simulation (ISMS rsquo13) pp 409ndash414 Bangkok Thailand January 2013

[28] S Roweis ldquoEM algorithms for PCA and SPCArdquo in Proceedingsof the Conference on Advances in Neural Information ProcessingSystems (NIPS rsquo98) pp 626ndash632 1998

[29] J-H Ahn and J-H Oh ldquoA constrained EM algorithm forprincipal component analysisrdquoNeural Computation vol 15 no1 pp 57ndash65 2003

[30] C S Chan and F S Tsai ldquoComputer animation of facialemotionsrdquo in Proceedings of the 10th International Conference onCyberworlds (CW rsquo10) pp 425ndash429 Singapore October 2010

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

[32] P S Reel L SDooley andPWong ldquoEfficient image registrationusing fast principal component analysisrdquo in Proceedings of the


19th IEEE International Conference on Image Processing (ICIPrsquo12) pp 1661ndash1664 Orlando Fla USA October 2012

[33] P S Reel L S Dooley and PWong ldquoA newmutual informationbased similarity measure for medical image registrationrdquo inProceedings of the IET Conference on Image Processing pp 1ndash6London UK July 2012

[34] G D C Cavalcanti T I Ren and J F Pereira ldquoWeighted mod-ular image principal component analysis for face recognitionrdquoExpert Systems with Applications vol 40 no 12 pp 4971ndash49772013

[35] C Jiang G Su and X Liu ldquoA distributed parallel systemfor face recognitionrdquo in Proceedings of the 4th InternationalConference on Parallel and Distributed Computing Applicationsand Technologies (PDCAT rsquo03) pp 797ndash800 August 2003

[36] X Liu and G Su ldquoA Cluster-based parallel face recognitionsystemrdquo in Proceedings of the International Conference onParallel and Distributed Processing Techniques and ApplicationsampConference on Real-Time Computing Systems andApplications(PDPTA rsquo06) vol 2 Las Vegas Nev USA June 2006

[37] L Wang K-Y Liu T Zhang Q-L Wang and Y Ma ldquoParallelface analysis platformrdquo in Proceedings of the IEEE InternationalConference on Multimedia and Expo (ICME rsquo10) pp 268ndash269Singapore July 2010

[38] A Numaan and A Sibi ldquoCUDA accelerated face recognitionrdquo2010 httpnestsoftwarecomnestwhitepapersRealTimeFace Recognition Using Cudapdf

[39] S Kavitha V Paul and T Gananasekaran ldquoColor face recog-nition using texture features and fractional fourier transformsrdquoInternational Journal of Scientific amp Engineering Research vol 4no 8 2013

[40] Q Hu D Yu and Z Xie ldquoNeighborhood classifiersrdquo ExpertSystems with Applications vol 34 no 2 pp 866ndash876 2008

[41] I Melnykov and V Melnykov ldquoOn K-means algorithm withthe use of Mahalanobis distancesrdquo Journal of Statistics andProbability Letters vol 84 no 1 pp 88ndash95 2013

[42] H M Moon and S B Pan ldquoThe LDA-based face recognitionat a distance using multiple distance imagerdquo in Proceedingsof the 7th International Conference on Innovation Mobile andInternet Services in Ubiquitous Computing (IMIS rsquo13) pp 249ndash255 Taichung Taiwan July 2013

[43] Q Li V Kecman and R Salman ldquoA chunking method foreuclidean distance matrix calculation on large dataset usingmulti-GPUrdquo in Proceedings of the 9th International Conferenceon Machine Learning and Applications (ICMLA rsquo10) pp 208ndash213 Washington DC USA December 2010

[44] D Skocaj Robust Subspace Approaches to Visual Learningand Recognition [PhD thesis] University of Ljubljana 2003httpeprintsfriuni-ljsi511ds phdpdf

[45] G E Blelloch and B M Maggs Parallel algorithms [Masterrsquosthesis] Carnegie Mellon University Pittsburgh Pa USA 1996

[46] P Blomgren Numerical Matrix Analysis San Diego State Uni-versity San Diego Calif USA 2012

[47] D L Spacek ldquoFace recognition datardquo University of EssexUK Computer Vision Science Research Projects 2012httpcswwwessexacukmvallfacesindexhtml

[48] S Gupta and J Shabbir ldquoVariance estimation in simple randomsampling using auxiliary informationrdquo Hacettepe Journal ofMathematics and Statistics vol 37 no 1 pp 57ndash67 2008

Submit your manuscripts athttpwwwhindawicom

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



Acquisition Preprocessing Feature extractor Classifier

Face image

Normalized new face image

Normalized face image

Training sets

Feature vectorClassified as

ldquoknownrdquo or ldquounknownrdquo

Figure 1 Face recognition system

image-based face recognitionThus many proposals adoptedthe appearance-based approaches for example PrincipalComponent Analysis (PCA) Independent Component Anal-ysis (ICA) Linear Discriminant Analysis (LDA) KernelPrincipal Component Analysis (KPCA) and ISOMAP [9]

Consider those approaches however PCA and its deriva-tives are commonly used due to several distinctive featuresFor example PCA is an optimal linear scheme in terms ofmean squared error for compressing a set of high dimen-sional vectors into a set of lower dimensional vectors Inaddition the model parameters used in PCA can be directlycomputed from the data without additional processing stepsMoreover compression and decompression operation com-plexity is given to the model parameters in other wordsPCA only requires matrix multiplication Most importantlyPCA requires less number of features which provide thenonreduction of precision quality and these advantages resultin high recognition accuracy even with a small data set [4]Although PCA expresses several distinctive outcomes one ofthe drawbacks is over a high complexity due to larger matrixoperation A large amount of memory is also required sincethe memory requirement grows with the image quality highresolution and the number of training images used for imagematching or classification [10 11]

With PCA derivations for years there are many attemptsto overcome the drawbacks of a traditional PCA to enhancethe performance of PCA recognition scheme for examplea symmetrical PCA and two-dimensional PCA [9] Sometechniques were involved in finding alternative approachesto lessen the computational complexity of PCA for exampleapplying QR decomposition instead of singular value decom-position (SVD) [11] Nevertheless a few of them focus onthe feasibility to absorb the high computational stage matrixmanipulation and to optimize the computational stagesespecially utilizing the parallelism to enhance the speed-upof PCA More details will be discussed in the related worksection

With a high computational task one of the probableapproaches to lessen the serial constraint is the usage of par-allelism concepts Recently with a rapid increase of computerchips and advances in integrated circuit technology resultsin affordable multicore CPUs and enhanced parallelizationtechniques [12] which have risen for a specific computa-tional task becoming a practical solution As a result thisresearch has also investigated a possibility to improve the facerecognition system and so our contribution lies in two foldsfirst using Expectation-Maximization (EM) instead of PCAcovariance matrix computation for face feature extractions

and second a novel parallel architecture for enhancingthe speed-up on matrix computational operations over ourfirst optimization called EM-PCA including the enhance-ment over face classification parallelizationThe contributionforms novel parallel face recognition architecture so-called aParallel Expectation-Maximization PCA architecture (PEM-PCA)

This research paper is organized as follows In Section 2background of face recognition systems is briefly revisitedincluding PCA and its limitation Then in Section 3 closelyrelated works of a traditional PCA PCA derivationsrsquo opti-mizations and a parallel system for PCA face recognitionsare comparatively surveyed Section 4 presents the parallelarchitecture proposal including the first PCA optimiza-tion (EM-PCA) and the second parallel face recognitionarchitecture in three substages (PEM-PCA) After that inSection 5 our comparative performance of various proposalsis discussed Finally the conclusions and future work aredrawn in Section 6

2 Background

In general a face recognition system consists of four com-ponents shown in Figure 1 These components are acqui-sition preprocessing feature extractor and classifier [13]Additionally in order to recognize the face identity thereare two main processes training and testing For trainingprocess given an image acquisition step from various inputsfor example captured photos scanned images and videoframes (acquisition) face images are fed as inputs intothe preprocessing step for example histogram equalizationbackground removal pose adjustment image size normal-ization gray-scale transformation median filtering high-pass filtering translation and rotational normalizations andillumination normalization in order to generate normalizedface imagesThen the system extracts themain features of theimages (feature extractor) resulting in feature vectors storedin a training set

The testing process is similar to the training process butwith fewer stepsThe testing image will be processed in orderto generate proper normalized face images which are readyfor face image classifier (classifier) in order to figure out theleast feature matching distance between testing and trainedfeatures Normally there are several techniques of featureextractor as well as classifier that is PCA ICA and LDA asface image matching for example Euclidian distance (ED)support vector machine (SVM) and K-nearest neighbor [1]It should be noted that PCA and ED are commonly used due


Mean estimation

Eigen selection

Mean subtraction



Mean subtraction




Eigen projection

Trained image








3 Related Work

















(EM-PCA)

(EM-PCA)



























Mean subtraction

EM stage derivation

Data projection


Trained image











Input Matrix x


119873

119894=1119909119894



119879119880)minus1

119880119879119883

(6) M-step 119880 = 119883119860119879(119860119860119879)minus1





1015840

(11) Rotate 119880 for 119880 1198801015840= 119880119880

1015840




119860 = (119880119879119880)minus1

119880119879119883 (1)

119880 = 119883119860119879(119860119860119879)minus1

(2)








1119895times 1198611198961) or sixteen

threads (119860119894119895times 119861119896119898




119888 [119894 119895] = 119886 [119894 119895] minus 119887 [119894 119895] (3)

















119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)



119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)


119888 [119894 119895] =119886 [119894 119895]

119887 (8)



[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)














119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Mean estimation

Eigen selection

Mean subtraction



Mean subtraction




Eigen projection

Trained image








3 Related Work

















(EM-PCA)

(EM-PCA)



























Mean subtraction

EM stage derivation

Data projection


Trained image











Input Matrix x


119873

119894=1119909119894



119879119880)minus1

119880119879119883

(6) M-step 119880 = 119883119860119879(119860119860119879)minus1





1015840

(11) Rotate 119880 for 119880 1198801015840= 119880119880

1015840




119860 = (119880119879119880)minus1

119880119879119883 (1)

119880 = 119883119860119879(119860119860119879)minus1

(2)








1119895times 1198611198961) or sixteen

threads (119860119894119895times 119861119896119898




119888 [119894 119895] = 119886 [119894 119895] minus 119887 [119894 119895] (3)

















119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)



119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)


119888 [119894 119895] =119886 [119894 119895]

119887 (8)



[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)














119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in


















(EM-PCA)

(EM-PCA)



























Mean subtraction

EM stage derivation

Data projection


Trained image











Input Matrix x


119873

119894=1119909119894



119879119880)minus1

119880119879119883

(6) M-step 119880 = 119883119860119879(119860119860119879)minus1





1015840

(11) Rotate 119880 for 119880 1198801015840= 119880119880

1015840




119860 = (119880119879119880)minus1

119880119879119883 (1)

119880 = 119883119860119879(119860119860119879)minus1

(2)








1119895times 1198611198961) or sixteen

threads (119860119894119895times 119861119896119898




119888 [119894 119895] = 119886 [119894 119895] minus 119887 [119894 119895] (3)

















119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)



119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)


119888 [119894 119895] =119886 [119894 119895]

119887 (8)



[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)














119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





(EM-PCA)

(EM-PCA)



























Mean subtraction

EM stage derivation

Data projection


Trained image











Input Matrix x


119873

119894=1119909119894



119879119880)minus1

119880119879119883

(6) M-step 119880 = 119883119860119879(119860119860119879)minus1





1015840

(11) Rotate 119880 for 119880 1198801015840= 119880119880

1015840




119860 = (119880119879119880)minus1

119880119879119883 (1)

119880 = 119883119860119879(119860119860119879)minus1

(2)








1119895times 1198611198961) or sixteen

threads (119860119894119895times 119861119896119898




119888 [119894 119895] = 119886 [119894 119895] minus 119887 [119894 119895] (3)

















119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)



119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)


119888 [119894 119895] =119886 [119894 119895]

119887 (8)



[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)














119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Mean subtraction

EM stage derivation

Data projection


Trained image











Input Matrix x


119873

119894=1119909119894



119879119880)minus1

119880119879119883

(6) M-step 119880 = 119883119860119879(119860119860119879)minus1





1015840

(11) Rotate 119880 for 119880 1198801015840= 119880119880

1015840




119860 = (119880119879119880)minus1

119880119879119883 (1)

119880 = 119883119860119879(119860119860119879)minus1

(2)








1119895times 1198611198961) or sixteen

threads (119860119894119895times 119861119896119898




119888 [119894 119895] = 119886 [119894 119895] minus 119887 [119894 119895] (3)

















119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)



119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)


119888 [119894 119895] =119886 [119894 119895]

119887 (8)



[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)














119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Input Matrix x


119873

119894=1119909119894



119879119880)minus1

119880119879119883

(6) M-step 119880 = 119883119860119879(119860119860119879)minus1





1015840

(11) Rotate 119880 for 119880 1198801015840= 119880119880

1015840




119860 = (119880119879119880)minus1

119880119879119883 (1)

119880 = 119883119860119879(119860119860119879)minus1

(2)








1119895times 1198611198961) or sixteen

threads (119860119894119895times 119861119896119898




119888 [119894 119895] = 119886 [119894 119895] minus 119887 [119894 119895] (3)

















119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)



119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)


119888 [119894 119895] =119886 [119894 119895]

119887 (8)



[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)














119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



















119888 [1 119895] = (119886 [1 1] times 119887 [1 119895]) + 119886 [1 2] times 119887 [2 119895]

+ 119886 [1 3] times 119887 [3 119895]

(4)

119888 [2 119895] = (119886 [2 1] times 119887 [1 119895]) + 119886 [2 2] times 119887 [2 119895]

+ 119886 [2 3] times 119887 [3 119895]

(5)

119888 [3 119895] = (119886 [3 1] times 119887 [1 119895]) + 119886 [3 2] times 119887 [2 119895]

+ 119886 [3 3] times 119887 [3 119895]

(6)



119888 [119894 119895] = 119886 [119894 119895] times 119887 (7)


119888 [119894 119895] =119886 [119894 119895]

119887 (8)



[

[

a11 a12 a1311988621

11988622

11988623

11988631

11988632

11988633

]

]

119879

= [

[

c11 11988812

11988813

c21 11988822

11988823

c31 11988832

11988833

]

]

= [

[

a11 11988621

11988631

a12 11988622

11988632

a13 11988623

11988633

]

]

(9)














119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in















119888 [119895 119894] = 119886 [119894 119895]119879

= 119886 [119895 119894] (10)




Diag [119894 119895] = 119860[119894 119895]2

(11)



sum

119894=0

119899

sum

119895=0

119886 [119894 119895] (12)


[

[

119886 [0 0]2+ 119886 [0 1]

2+ 119886 [0 2]

2119860 119862

119860 119886 [1 0]2+ 119886 [1 1]

2+ 119886 [1 2]

2119861

119862 119861 119886 [2 0]2+ 119886 [2 1]

2+ 119886 [2 2]

2

]

]

[

[

119886 [0 0] 119886 [0 1] 119886 [0 2]

119886 [1 0] 119886 [1 1] 119886 [1 2]

119886 [2 0] 119886 [2 1] 119886 [2 2]

]

]

times [

[

119886 [0 0] 119886 [1 0] 119886 [2 0]

119886 [0 1] 119886 [1 1] 119886 [2 1]

119886 [0 2] 119886 [1 2] 119886 [2 2]

]

]

(13)


119862119894119895= (minus1)

119894+119895119860119894119895 (14)



cofactor of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

c11 11988812

11988813

11988821

11988822

11988823

11988831

11988832

11988833

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) + ((119886

21times 11988632) minus (119886

22times 11988631))


13times 11988632)) + ((119886

11times 11988633) minus (119886

13times 11988631)) minus ((119886

11times 11988632) minus (119886

12times 11988631))

+ ((11988612times 11988623) minus (119886

13times 11988622)) minus ((119886

11times 11988623) minus (119886

13times 11988621)) + ((119886

11times 11988622) minus (119886

12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(15)


Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Input Matrix input








(10) b++


(12) c++

(13) b=0



(18) endfor(19) c=0

(20) b=0






119899119895119862119899119895

=

119899

sum

119894=1

119886119894119895119862119894119895

(16)



119894119899119862119894119899

=

119899

sum

119895=1

119886119894119895119862119894119895

(17)



determinant of [

[

11988611

11988612

11988613

11988621

11988622

11988623

11988631

11988632

11988633

]

]

= [

[

d11 11988912

11988913

11988921

11988922

11988923

11988931

11988932

11988933

]

]

=

100381610038161003816100381610038161003816100381610038161003816100381610038161003816


23times 11988631)) +119886


22times 11988631))


13times 11988632)) +119886


13times 11988631)) minus119886


12times 11988631))


13times 11988622)) minus119886


13times 11988621)) +119886


12times 11988621))

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(18)




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Input Matrix input



119894+119895times 119889119890119905119890119903119898119894119899119886119899119905 (119888119903119890119886119905119890119878119906119887119872119886119905119903119894119909 (119903119890119904119906119897119905 [119894 119895]))





(9) result [119894 119895] = (minus1)119894+119895


(13) endif


1 middot 4 times 1

1 2 3 4


Normalize w5



u

u

u

u

u

u

u

u

w






11 11988912 11988913 and 119889

21will be






Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Input Matrix a

Output Matrix q


(2) V119895= 119886119895


(4) 119903119894119895= 119902119894times V119895



119895119895=10038171003817100381710038171003817V119895

100381710038171003817100381710038172


(9) endfor





(2) sum = 00


(4) 119904119906119898 += (119860 [119903119900119908 119888119900119897] minus 119861 [119888119900119897])2












A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




A BDEuclidean of

a11 a12 a13

a21 a22 a23

a31 a32 a33

a41 a42 a43


2



d11 d12 d13

d21 d22 d23

d31 d32 d33

d41 d42 d43


001003005

007009

19520

20521

21522

225


Com

puta

tiona

l tim

e (s)

e1e3

e5e7

e9


60

70

80

90

100


Accu

racy

()

001003005

007009

e1e3

e5e7

e9




100 200 300 400 5000

100

200

300

400



Com

puta

tiona

l tim

e (s)







0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

()

100 200 300 400 500



100

200

300

P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

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(c)

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Advances in

FuzzySystems


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0

20

40

60

80

100

PCAP-PCA

EM-PCAPEM-PCA


Accu

racy

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100 200 300 400 500



100

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P-PCAEM-PCAPEM-PCA

Number of CPU cores

Com

puta

tiona

l tim

e (s)




0

20

40

60

80

100

P-PCAEM-PCAPEM-PCA

Accu

racy

()



(a)

(b)

(c)

(d)







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







References


























































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Research Article PEM-PCA: A Parallel Expectation ...downloads.hindawi.com/journals/tswj/2014/468176.pdf · and/or verify a human face is facial recognition systems. Facial recognition