Research ArticleRecognition of Reconstructed Frontal Face Images Using FFT-PCA/SVD Algorithm

Louis Asiedu , Felix O. Mettle, and Joseph A. Mensah

Department of Statistics & Actuarial Science, School of Physical and Mathematical Sciences, University of Ghana, Legon,Accra, Ghana

Correspondence should be addressed to Louis Asiedu; lasiedu@ug.edu.gh

Received 15 July 2020; Accepted 19 September 2020; Published 5 October 2020

Academic Editor: Jong Hae Kim

Copyright © 2020 Louis Asiedu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Face recognition has gained prominence among the various biometric-based methods (such as fingerprint and iris) due to itsnoninvasive characteristics. Modern face recognition modules/algorithms have been successful in many application areas (accesscontrol, entertainment/leisure, security system based on biometric data, and user-friendly human-machine interfaces). In spiteof these achievements, the performance of current face recognition algorithms/modules is still inhibited by varyingenvironmental constraints such as occlusions, expressions, varying poses, illumination, and ageing. This study assessed theperformance of Principal Component Analysis with singular value decomposition using Fast Fourier Transform (FFT-PCA/SVD) for preprocessing the face recognition algorithm on left and right reconstructed face images. The study found thataverage recognition rates for the FFT-PCA/SVD algorithm were 95% and 90% when the left and right reconstructed face imagesare used as test images, respectively. The result of the paired sample t-test revealed that the average recognition distances for theleft and right reconstructed face images are not significantly different when FFT-PCA/SVD is used for recognition. FFT-PCA/SVD is recommended as a viable algorithm for recognition of left and right reconstructed face images.

1. Introduction

Recognizing people using face images has gained promi-nence among the various biometric-based methods (suchas fingerprint and iris) due to its comparative advantage ofbeing nonintrusive and less cooperative (of subjects understudy). This task is easily carried out by humans. The designof machine-based face recognition systems (that mimichumans’ recognition prowess) and their underlying algo-rithms that give optimal classification or recognition rates,however, have been and continue to be challenging, espe-cially when face images are obtained under uncontrolledenvironments (poor illumination conditions, varying poses,expressions, and occlusions) [1]. Thus, there is a growinginterest in this field of research.

In the case of partially occluded faces (resulting from thewearing of mask and sunglasses, blockage by external objects,captured angle images, etc.), occlusion insensitive, localmatching, and reconstruction methods have been used forrecognition [2]. Performing face recognition using half-face

images can be considered a special case of partially occludedfaces where either the left or right face is occluded and seg-mented and the remaining half (nonoccluded) face is usedfor recognition [3]. Bilateral symmetry is a property of manynatural objects including the human face [4]. Leveraging thisproperty, the performances of holistic face representation-based algorithms have been evaluated on the left, right, andaverage half faces based on symmetry scores [5, 6].

Singh and Nandi [6] applied PCA on the full and left andright half faces and measured the performance of their algo-rithm in terms of the recognition rate and accuracy. Theirresults showed no difference in recognition rates betweenthe left and right half faces but achieved higher accuracy forthe left half face. They also reported no difference in accuracyrates between the full face and half faces. However, the recog-nition rate for half faces was half that for the full faces.

Harguess and Aggarwal [5] evaluated the recognition rateof full faces and average half faces using eigenfaces and thenearest neighbor as a classifier. They reported a significantlyhigher recognition rate using the average half face for both

HindawiJournal of Applied MathematicsVolume 2020, Article ID 9127465, 8 pageshttps://doi.org/10.1155/2020/9127465

https://orcid.org/0000-0002-2859-1215https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/9127465

men and women compared to the full face. Asiedu et al. [7]applied the PCA/SVD algorithm on full faces under varyingfacial expressions. They concluded that the algorithm wasmost consistent and efficient under varying expressions andachieved appreciable performance with an average recogni-tion rate of 92.86%.

Avuglah et al. [8] also applied the FFT-PCA/SVD algo-rithm on face images under angular constraints and statisti-cally evaluated the performance of the algorithm. Theyfound that the algorithm perfectly recognizes head tilts thatare 24° or less. They concluded that the algorithm has anappreciable performance with an average recognition rateof 92.5% in recognition of face images under angular con-straints. The question of whether this algorithm performswell on partial and reconstructed frontal face images, how-ever, has not been explored. This paper, therefore, intendsto assess the performance of the FFT-PCA/SVD algorithmon partial and reconstructed frontal face images based onbilateral symmetry.

2. Materials and Methods

2.1. Source of Data. Frontal face images from the Massachu-setts Institute of Technology (MIT) (20032005) and Japanese

Female Facial Expressions (JAFFE) database were used tobenchmark the face recognition algorithm. The train-imagedatabase contains twenty frontal face images. Ten of theseimages were 0° straight pose from the MIT (2003-2005) data-base, and ten were neutral pose face images from JAFFE. Theimages captured into the train-image database are denoted astrain images and are used to train the algorithm.

Twenty images reconstructed from the half-face images(created through vertical segmentation) of the train imageswere captured into the test-image database. The images cap-tured into the test-image database are called the test imagesand are used for testing the recognition algorithm.

To keep the data uniform, captured images were digitizedinto gray-scale precision and resized into 200 × 200 dimen-sions, and the data types were changed into double precisionfor preprocessing. This made the images (matrices) conform-able and enhanced easy computations. Figures 1 and 2 showsubjects in the train image database.

2.2. Image Reconstruction. The left segmented half-faceimages were reconstructed using the following steps:

(1) Rotate left segmented half-face image through 270°,and denote it as Nl1

Figure 1: Subjects in the MIT (2003-2005) database.

Figure 2: Subjects in the JAFFE database.

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(2) Rotate the left segmented half-face image through180°, and denote it as Nl2

(3) Concatenate Nl1 and Nl2 as

Tl = Nl1 ∣Nl2½ � ð1Þ

The right segmented half-face images were reconstructedusing the following steps:

(1) Rotate right segmented half-face image through 270°,and denote it as Nr1

(2) Rotate the right segmented half-face image through180°, and denote it as Nr2

(3) Concatenate Nr2 and Nr1 as

Tr = Nr2 ∣Nr1½ � ð2Þ

Figure 3 contains the original full images, left and righthalf-images, and their reconstructed images used in thetest-image database.

2.3. Research Design. Figure 4 shows a design of the recogni-tion module.

2.4. Preprocessing. Preprocessing is an important phase ofimage processing where the quality of the images isenhanced. The image acquisition process comes with variousforms of noise. The preprocessing techniques are used todenoise the images making them better conditioned forrecognition.

2.4.1. Fast Fourier Transform. Fast Fourier Transform (FFT)was adopted as a noise reduction mechanism. According toGlynn [9], the FFT algorithm reduces the computational bur-den to OðN log NÞ arithmetic operations.

Originalimage

Le� halfimage

Reconstructedimage

Right halfimage

Reconstructedimage

Figure 3: Reconstructed images from left and right segmented half images.

Classifier

Knowledge creation

Recognized imageTest image

Feature extractionPCA/SVD

Preprocessing(fast Fourier

transform & meancentering)

Train imagedatabase

Figure 4: Research design.

3Journal of Applied Mathematics

The DFT of a column vector ajk is represented mathe-matically as

a∗jk = DFT ajk� �

= 〠p−1

r=0ajke−i 2πsr/pð Þ, ð3Þ

where s = 0, 1,⋯, p − 1, j = 1, 2,⋯, n. ajk is the kth column ofthe image matrix, Aj [8].

Due to the Gaussian nature of illumination variations,the Gaussian filter is adopted for filtering the face imagesafter the Discrete Fourier transformation. After filtering, theinverse Discrete Fourier transformation (IDFT) was per-formed to reconstruct images into their original forms. Theinverse Discrete Fourier transformation (IDFT) is given by

ajk = IDFT a∗jkn o

=1p〠p−1

r=0a∗jkei 2πsr/pð Þ,

s = 0, 1,⋯, p − 1, j = 1, 2,⋯, n:

ð4Þ

The real components of the inverse transformations areextracted for the feature extraction stage whereas the imagi-nary component is discarded as noise. Figure 5 shows thestages in FFT preprocessing of an image (Avuglah et al. [8]).

2.5. Feature Extraction. Face image space is very large, and itsstorage requires reduction of the dimensions of originalimages using feature extraction methods. This study adoptedPrincipal Component Analysis (PCA) proposed by Kirbyand Sirovich [10] for feature extraction. PCA extracts themost significant components or those components which

are more informative and less redundant, from the originaldata. According to Shlens [11], PCA computes the mostmeaningful basis to reexpress a noisy garbled dataset. Therationale behind this new basis is to filter out the noise andreveal hidden dynamics in the dataset.

2.6. Implementation of FFT-PCA/SVD Algorithm. As notedearlier, used images were extracted from the MassachusettsInstitute of Technology (MIT) (2003-2005) database andJapanese Female Facial Expressions (JAFFE). The study con-sidered subjects captured under 0° pose from the MIT data-base and neutral facial expressions from JAFFE. Twentyface images, ten each from the MIT database and JAFFE,were trained, and their reconstructed images from left andright segmented half images were used for testing the recog-nition algorithm. To keep the data uniform, captured imageswere digitized into gray-scale precision and resized into 200× 200 dimensions and the data types changed into doubleprecision for preprocessing.

The FFT-PCA/SVD algorithm was used to train theimage database. Unique face features of the training set areextracted and stored in memory during the training phase.

Now, given the sample X = ðX1,X2,⋯,XnÞ, whose ele-ments are the vectorised form of the individual images inthe study. The mean centering preprocessing mechanism isperformed by subtracting the mean image from the individ-ual images under study. The mean is given by

�aj = E X j� �

, j = 1, 2,⋯, n,

�aj =1N〠N

i=1Xji =

1N〠p

i=1〠p

k=1ajik, j = 1, 2, 3,⋯, n,

ð5Þ

Original image Reconstructed image

Transformed image Filtered image

Discrete Fouriertransform

(DFT)

Inverse discreteFourier transform

(IDFT)

Gaussianfiltering

Figure 5: FFT preprocessing cycle.

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where N = ðp × pÞ, length (= rows of image × columns ofimage) of the image data, X j.

According to Avuglah et al. [8], the variance-covariancematrix C of the image space is given as

C = 1nWWT , ð6Þ

where the mean centered matrix is W = ðw1,w2,⋯,wnÞ.The eigenvalues and their corresponding eigenvectors are

extracted from singular value decomposition (SVD) of thematrix, C =UΣVT .

This decomposes the covariance matrix C into twoorthogonal matrices U and V and a diagonal matrix Σ:

zj = 〠n

j=1w juj, ð7Þ

where uj is the jth column vector of U.From the training set, the principal components are

extracted as

γj = zTj x j − �a� �

, ð8Þ

and ΓT = ½γ1, γ2,⋯, γn�.

When a new face (test image) is passed through the rec-ognition module, its unique features are extracted as

γ∗j = zTj xr − �að Þ, ð9Þ

and Γ∗Tr = ½γ∗1 , γ∗2 ,⋯, γ∗n �.The recognition distances (Euclidean distances) are com-

puted as

ψ = Γ − Γ∗rk k: ð10Þ

The minimum Euclidean distance dji =min ½ψ�, j = 1, 2,⋯, n, i = 1, 2, is chosen as the recognition distance.

3. Results and Discussion

Figures 6 and 7 present the recognition matches and dis-tances of the left and right reconstructed face images. It canbe seen in Figure 6 that there were two mismatches whenthe right reconstructed images are used as test images for rec-ognition. Also, there was one mismatch when the left recon-structed images are used as test images for recognition. Allrecognitions on the MIT (2003-2005) database gave correctmatches. This is evident from Figure 7.

Correctmatch

Wrongmatch

355.42

374.22

398.13

408.35

506.80

251.93

552.20

298.72

518.30

627.05

578.98

540.40

523.28

441.84

483.07

766.47

505.59

299.94

772.00

430.37

Originalimage

Le�reconstructed

image

Recognitiondistance

Recognitionstatus

Originalimage

Rightreconstructed

image

Recognitiondistance

Recognitionstatus

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Wrongmatch

Wrongmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Figure 6: Recognition of the reconstructed images (JAFFE).

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3.1. Numerical Evaluations. The average recognition rate wasthe numerical assessment criteria used in this study. Theaverage recognition rate, Ravg, of an algorithm is defined as

Ravg =∑truni=1nicrtrun × ntot

× 100, ð11Þ

where trun is the number of experimental runs, nicr is the

number of correct recognitions in the ith run, and ntot isthe total number of faces being tested in each run [7]. Theaverage error rate, Eavg, is defined as 100 − Ravg.

The total number of correct recognitions ∑10i−1nicr for the

study algorithm is 19.The total number of experimental runs trun = 10, and the

total number of images in a single experimental run ntot = 2.Now, using the left reconstructed face images as test

images, the average recognition rate of the study algorithm(FFT-PCA/SVD) is

Ravg =19

10ð Þ 2ð Þ × 100 = 95%: ð12Þ

The average error rate

Eavg = 100 − Ravg = 100 − 95 = 5%: ð13Þ

Using the right reconstructed half-face images as testimages, the average recognition rate of the study algorithm(FFT-PCA/SVD) is

Ravg =18

10ð Þ 2ð Þ × 100 = 90%: ð14Þ

The average error rate

Eavg = 100 − Ravg = 100 − 90 = 10%: ð15Þ

3.2. Statistical Evaluation. The paired sample t-test is usuallyused to evaluate measurements of the same individuals/unitsrecorded under different environmental conditions. Thepaired responses may then be analysed by computing theirdifferences, thereby eliminating much of the influence ofthe extraneous unit to unit variation (Johnson and Wichern[12]). Let dj1 denote the recognition distance recorded usingthe left reconstructed images as test images and dj2 denotethe recognition distance using the right reconstructed half-

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

Correctmatch

182.78

196.77

161.75

76.714

142.63

285.82

110.62

188.30

126.01

237.59

115.94

105.75

70.474

196.87

167.42

209.96

162.28

96.63

78.251

135.60

Originalimage

Originalimage

Le�reconstructed

image

Recognitiondistance

Recognitionstatus

Rightreconstructed

image

Recognitiondistance

Recognitionstatus

Figure 7: Recognition of the reconstructed images (MIT 2003-2005 database).Ex

pect

ed n

orm

al

Observed value

–3

–500 –250 0 250 500

–2

–1

0

1

2

3

Figure 8: Normal Q‐Q plot of observed difference.

6 Journal of Applied Mathematics

face images as test images for the jth individual, then thepaired differences

Dj = dj1 − dj2, j = 1, 2,⋯, n, ð16Þ

should reflect the differential effects of the treatments. Giventhat the difference Dj, j = 1, 2,⋯, n, are independent obser-vations from Nðδ, σ2dÞ distribution, the statistic

t =�D − δsd/

ffiffiffin

p , ð17Þ

where

�D =1n〠n

j=1Dj,

s2d =1

n − 1〠n

j=1Dj − �D� �2

ð18Þ

has a t-distribution with n − 1 degrees of freedom. Conse-quently, an α-level test of the hypothesis

H0:δ = 0 (mean difference of recognition distances fromleft and right reconstructed images is zero) against

H1:δ ≠ 0is conducted by comparing ∣t ∣ with tn−1ðα/2Þ, the

upper 100ðα/2Þ percentile of the t-distribution with n − 1degree of freedom. A 100ð1 − αÞ confidence interval forthe mean difference in recognition distance δ = Eðdj1 − dj2Þis constructed as

�d ± tn−1α

2

� � sdffiffiffin

p : ð19Þ

This test is sensitive to the assumption that the paireddifference should come from a univariate normal popula-tion. The Shapiro-Wilk test on the observed differencegave a value of 0.980 with a p value of 0.933. This showsthat the distribution of the observed difference is the sameas the expected distribution (normal). TheQ‐Q plot shown inFigure 8 confirms that the observed difference is normallydistributed.

The paired sample correlations between the distance forthe left and right reconstructed face images are 0.858 with ap value of 0.000. This indicates that there exists a strong pos-itive linear relationship between the Euclidean distance forthe left and right reconstructed face images. The p value of0.000 shows that this relationship is significant. The resultsof the paired sample t-test are shown in Table 1.

From Table 1, the average difference between the left andright reconstructed face images (LRI and RRI, respectively) is42.5865. The test statistic value from the paired sample test is0.928 corresponding to a p value of 0.365. It can be inferredfrom the p value that there is no significant differencebetween the average recognition distance for the left andright reconstructed face images. This means that the recon-structed images have the same average recognition distancesat 5% level of significance when used as test images.

4. Conclusion and Recommendation

The average recognition rates for the FFT-PCA/SVD algo-rithm were 95% and 90% when left and right recon-structed face images are used as test images, respectively.It could be inferred from the above results that the recogni-tion algorithm has relatively higher performance when leftreconstructed images are used as test images. The statisticalassessment revealed that there is no significant differencebetween the average recognition distances for the left andright reconstructed images.

It can therefore be concluded that the average recognitiondistance for left reconstructed images is not significantly dif-ferent from the average recognition distance for right recon-structed images when FFT-PCA/SVD is used for recognition.These results are consistent with the findings of Singh andNandi [6]. It is worthy of note that apart from the numericalevaluations which are mostly adopted in literature, this studyused a more data-driven approach or a statistical approach toalso assess the performance of the recognition algorithm onleft and right reconstructed face image databases. The perfor-mances of FFT-PCA/SVD on both databases are viable. FFT-PCA/SVD is therefore recommended for recognition of leftand right reconstructed face images.

Data Availability

The image database supporting this research article is frompreviously reported studies and datasets, which have beencited. The processed data are available from the correspond-ing author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interest.

References

[1] M. Turk and A. Pentland, “Face recognition using eigenfaces,”Proceedings. 1991 IEEE computer society conference on com-puter vision and pattern recognition, pp. 586-587, 1991.

Table 1: Paired sample t-test.

RRI-LRIStd. Dev. SE

95% CIt df Sig.

Mean Lower Upper

42.58565 205.2544 45.89628 -53.47637 138.64767 0.928 19 0.365

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[2] X. Wei, Unconstrained face recognition with occlusions, Ph.D.thesis University of Warwick, 2014, http://go.warwick.ac.uk/wrap/66778.

[3] H. Jia and A. M. Martinez, “Support vector machines in facerecognition with occlusions,” in 2009 IEEE Conference onComputer Vision and Pattern Recognition, pp. 136–141,Miami, FL, USA, 2009.

[4] D. O’Mara and R. Owens, “Measuring bilateral symmetry indigital images,” in Proceedings of Digital Processing Applica-tions (TENCON’96), pp. 151–156, Perth, WA, Australia, Aus-tralia, 1996.

[5] J. Harguess and J. K. Aggarwal, “Is there a connection betweenface symmetry and face recognition?,” CVPR 2011 WORK-SHOPS, pp. 66–73, 2011.

[6] A. K. Singh and G. C. Nandi, “Face recognition using facialsymmetry,” Proceedings of the Second International Conferenceon Computational Science, Engineering and Information Tech-nology, pp. 550–554, 2012.

[7] L. Asiedu, A. Adebanji, F. Oduro, and F. O. Mettle, “Statisticalevaluation of face recognition techniques under variable envi-ronmental constraints,” International Journal of Statistics andProbability, vol. 4, pp. 93–111, 2015.

[8] R. K. Avuglah, L. Asiedu, E. N. Nortey, and F. N. Yirenkyi,“Recognition of face images under varying head-poses usingfft-pca/svd algorithm,” Far East Journal of Mathematical Sci-ences (FJMS), vol. 103, no. 11, pp. 1769–1788, 2018.

[9] E. Glynn, Fourier analysis and image processing’, scientific pro-grammer. Bioinformatics, S Towers Institute for medicalResearch, 2007.

[10] M. Kirby and L. Sirovich, “Application of the karhunen-loeveprocedure for the characterization of human faces,” IEEETransactions on Pattern Analysis and Machine Intelligence,vol. 12, no. 1, pp. 103–108, 1990.

[11] J. Shlens, “A tutorial on principal component analysis: deriva-tion, discussion and singular value decomposition,” Mar,vol. 25, p. 16, 2003.

[12] R. A. Johnson and D.W.Wichern, Applied Multivariate Statis-tical Analysis Volume 5, Prentice hall Upper Saddle River, NJ,2002.

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http://go.warwick.ac.uk/wrap/66778http://go.warwick.ac.uk/wrap/66778

Recognition of Reconstructed Frontal Face Images Using FFT-PCA/SVD Algorithm1. Introduction2. Materials and Methods2.1. Source of Data2.2. Image Reconstruction2.3. Research Design2.4. Preprocessing2.4.1. Fast Fourier Transform

2.5. Feature Extraction2.6. Implementation of FFT-PCA/SVD Algorithm

3. Results and Discussion3.1. Numerical Evaluations3.2. Statistical Evaluation

4. Conclusion and RecommendationData AvailabilityConflicts of Interest