Face Recognition Using Gabor Wavelet Features with PCA and KPCA - A Comparative Study

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Procedia Computer Science 57 ( 2015 ) 650 – 659

Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015) doi: 10.1016/j.procs.2015.07.434

ScienceDirect

3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015)

Face Recognition using Gabor Wavelet Features with PCA and

KPCA - A Comparative Study

Vinay.A, Vinay.S.Shekhar, K.N.Balasubramanya Murthy, S.Natarajan

PES University, 100 Feet Ring Road, BSK III Stage, Bangalore 560085, Karanataka, India

Abstract

Face recognition (FR) is one of the most prominent forms of biometric recognition that proffers a myriad of cross-domain applications and augmenting its proficiency has been on the forefront of research efforts for the past two decades. The efficacy of FR systems is dictated by the choice of the feature extractor, and to that end, GABOR Wavelet Transform has emerged as one of the most successful methodologies for feature extraction of faces in digital images. Many variants have been proposed to improve performance of conventional GABOR, and therefore in this paper, we conduct a comprehensive study to provide an in-depth comparison of the efficacy of the GABOR-PCA (linear) and GABOR-KPCA (non-linear) techniques. Our experimentations have been conducted on the publicly available ORL database. We will demonstrate using pertinent mathematical arguments and extensive experimentations, the difference in performance between linear and non-linear choices, and show that the GABOR-PCA variant is better suited for FR tasks when GABOR is employed as the feature extractor. The results of this study, coupled with our other works, form a series that is intended to assist developers in making prudent choices in designing proficient FR systems.

Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015).

Keywords:Gabor; PCA; KPCA; Face Recognition.

1.Introduction and Related Work

Face recognition (FR) has evolved to be a widely adopted form of Biometric, which stands resolutely as one of the most effervescent and challenging areas in computer vision. Apart from access control and surveillance, it is extensively used in authentication tasks to identify an individual’s identity based on the geometric or statistical features that are extracted from the face images [1]. FR based authentication systems are witnessing a lot of recent innovations in terms of selective market applications involving Human Machine Interface such as the upcoming ATM user identity verification systems [26] and Driving License identity authentication [25] and so on. Although FR systems have achieved significant strides and have been ubiquitously implemented in a wide number of applications, they are riddled with a plethora of issues and are only effective in certain constrained scenarios [23]. Their performance tends to decline in the presence of strong variations in terms of © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015)

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illumination, expression, viewpoints and so on [1]. A number of renowned and effective mechanisms have been proposed to mitigate the reduction in performance, yet their proficiency is still to reach a stage where it can achieve impeccable accuracy. Hence there is a dire need for more dexterous mechanisms to effectively handle the drawbacks of contemporary FR systems. One of the main tasks in any FR system is Facial Feature Extraction, and many approaches have been proposed over the years to carry it out (see [37] for an excellent survey) and since the recognition performance of FR systems rely heavily on the choice of the feature detector, a prudent choice regarding this is paramount. [1] states that the principal criteria to be considered in designing an FR system is the choice of (a) the features to be extracted from the face and (b) ensuring that those selected features aptly represent the discriminating properties between different faces.

Among the prevalent extraction techniques, Gabor Wavelet Transform has been demonstrated to be remarkably effective due to its biological relevance and standout computational properties [24]. Gabor feature extraction is modeled after the human visual sense principles and exhibits exemplary characteristics such as the capability of capturing salient visual properties such as spatial localization, spatial frequency and orientation selectivity [6][32]. Even though Gabor is highly effective, the face images rendered by it are convolved with a bank of Gabor filters and hence the dimensionality of the Gabor feature space is overwhelmingly large, yielding high complexity [1][24]. Although many down sampling techniques have been proposed in order to reduce the dimensions by utilizing select feature points [19], their output still consists of a large number of high dimensions of feature matrix leading to the possibility of partial loss of feature discriminative information and marked reduction of accuracy in the classification stage. Hence to deal with this, PCA was subsequently opted and has been reliably effective in reducing the size of the dimensions of the Gabor feature space [20].

PCA [22] is a widely employed method for De-noising and Dimensionality Reduction. In certain scenarios, it has been suggested that PCA may not be effective enough, as it is a linear approach and face images have been shown to be of a nonlinear nature, when projected under uncontrolled laboratory conditions [27]. The higher order dependencies in an image consist of nonlinear relations among the pixel intensity values, such as the relationships among three or more pixels in an edge or a curve and include crucial information necessary for accurate recognition. It has been shown that the higher order statistics may be more adept at representing complex patterns [27], and these revelations led to the emergence of non-linear techniques. One of the most effective non-linear approaches is the Kernel PCA [12], a non-linear extension of conventional PCA. It is able to extract nonlinear features and in some cases provide better performance and achieve lower error rate and compared to other techniques for nonlinear feature extraction, kernel methods have the advantage of not requiring nonlinear optimization [12][28]. Furthermore, for the face authentication scenarios, we make an intelligent assumption (demonstrated to be viable in [31]) that at least one large face is present in the given complex background and skip the face detection step.

In our proceedings, we apply the PCA and KPCA techniques independently as a post-processing step after feature extraction to reduce the dimensionality of the filtered images and compare their efficacy. We hope to clearly elucidate the difference in performance in terms of choosing linear and non-linear techniques and thus assist developers in making vigilant decisions in designing effective FR systems.

The rest of the paper is organized as follows: Section 2 proffers background information; Section 3 describes the methodology; Section 4 details the evaluation criteria and the experimental setup; Section 5 elucidates the experimentation results and Section 6 proffers a discussion of the proposed approach and outlines future work.

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2.Background

2.1 Gabor Wavelet Transform

The characteristics of the Gabor wavelets [6], particularly the representations of frequency and orientation are based on the human visual system and have been proven to be suitable for representation and discrimination of texture information. A Two dimensional Gabor wavelet is essentially a Gaussian kernel function modulated by a sinusoidal plane wave [1] and can be represented as follows:

(-'786/$0+,07!7"05"0$0,0887#8 7/8

where x"_{= x cos}_Θ_{+ y sin}_Θ_{and y}"_{= -xsin}_Θ_{+ ycos}_Θ

In the above equation, (x, y) denotes the position of the pixel in the spatial domain and - signifies the central angular frequency of the sinusoidal plane wave. _Θdenotes the anti-clockwise rotation of the Gaussian function i.e. the orientation of the Gabor filter and σ signifies the Gaussian function’s sharpness along the x and y directions. An ideal value for σ in order to appropriately define the relationship between σ and - has been shown to be σ≈+$-9/:. The Gabor filter bank that is constituted of the various Gabor filters along with the various frequencies and the orientations are then utilized to extract facial features from the images. The choice of 5 frequencies and 8 orientations has been shown to be effective in most scenarios [1] [8]. The choice of 5 frequencies (m = 1,..5) and 8 orientations (n = 1,…8) for the Gabor filter bank yields the following equations:

-67+$0840&7!/8 708

Θn = 7+$387!/8 718

The Gabor feature representation G m,n can be obtained by convolving the input image I(x,y) with the Gabor

filter (-'78 Gm,n = I (x,y)

⊗

(-'78728 The change in the phase of the Gabor feature representation i.e. G m,n (x,y) is linear with small displacement

of the sinusoid direction, but its magnitude change with respect to displacement is slow [9]. Therefore, we utilize the magnitude of the convolution outputs as it showed promise in [1]. The Gabor feature representation of an image I (x,y) is the convolution of the image with the Gabor filter bank ψ (x,y, -Θn) and is represented

as follows: Omn(x,y)=I(x,y)

⊗

ψ(x,y,-Θn) (5)

2.2 PCA

PCA [22] is a popular technique that reduces high dimensionality by projects high dimensional data onto lower dimensional space in a manner that best represents the original data in a least-squares sense [36]. Let us suppose that a set of N sample images {x1,x2,x3……xN} is represented by t-dimensional feature vector (Gabor

in our case). Then PCA finds a linear transformation mapping the original t-dimensional feature space onto an

f-dimensional feature subspace such that f is considerably smaller than t. The resultant feature vector yi €

Rf_{is as follows:}

_y

i= xi (i=1,2,……N) (6)

where WPCA signifies the linear transformation matrix and i is the number of sample images. The columns of

WPCA are the f eigenvectors associated with the f largest eigenvalues of the scatter matrix SΓ, which can be

defined as follows [36]: SΓ = (7) where μ € Rt _{is the mean image of all the samples.}

2.3 Kernel PCA

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on the statistics of face images in a given input. The necessity for higher order statistics has led to a nonlinear extension that operates by mapping a pattern in the original input space to a higher dimensional feature vector in the feature space in a non-linear fashion [12]. Kernel-PCA is a renowned non-linear dimensionality reduction technique whose emergence was motivated by the need to carry out PCA in the feature space, which was not feasible due to the high computational expense of the dot product computation in the high dimensional feature space [1]. The kernel approach overcomes this by carrying out the implementation in the input space by employing various kernel tricks and does not require the mapping to be performed explicitly [13].

Suppose that the input space has a set of M samples( x1 ,x2,…,xM € RN) and there exists a nonlinear mapping Φ between the input and the feature space [1] such that Φ:RN_→_{F. F is the non-linear mapping which defines}

the kernel function. Let us consider that the data matrix in the feature space is represented by D i.e. D = [Φ(x1) Φ(x2) … Φ(xM)]. Furthermore, if we suppose that K € RMxM represents a kernel matrix that is obtained by

carrying out the dot product in the feature space [1] as shown in Eqn 8.

Kij = Φ (xi) . Φ (xj) (8)

It has been demonstrated [1][14] that in KPCA, we can derive the eigenvalues λ1, λ2 ,λ3… λM and the

eigenvectors V1 ,V2 , V3…VM by solving the following eigenvalue equation

KA= MAΛ (9)

Where A= [α1, α2,α3 …..αM] and Λ = diag {λ1, λ2,λ3,….λM}

Furthermore in Eqn 9, A € RMxM _{is an orthogonal vector matrix and}_Λ_{€ R}MxM _{is a diagonal eigenvalue}

matrix that contains diagonal elements in a decreasing order (λ1≥λ2≥ …≥λM) and M is the number of training

samples and is a constant. The derivation of KPCA’s eigenvector matrix V=[V1, V2,V3 ……VM ] requires the

normalization of A such that *%%)%%061 where i=1,2,3,…..M and is carried out as follows [1]:

6 7/.8

Φ(x) the

Kernel-PCA features of x can be computed as follows [1]:

F=Vt _Φ_{(x) = A}t _7//8

69Φ(x1).Φ(x) Φ(x2).Φ(x)…….Φ(xM).Φ(x)]t

2.4 Similarity Measure using MAHCOS (Cosine Mahalanobis Distance)

The similarity measure between the probe image and the gallery image can be defined as the Cosine Mahalanobis Distance between the projections of the probe and the gallery images. The effectiveness of MAHCOS has been demonstrated in [34] and has also shown promising results in [35]. Let us consider that

Γ1,Γ2.….ΓN are the reduced vectors (after feature extraction and dimensionality reduction) from the gallery

image and Θ =

is the average face. Then we calculate Φi = Γi– Θ as the mean subtracted faces. We

define the data matrix A as: A= [Φ1Φ2 ...ΦN]. The faces are essentially the eigenvectors of AAT. Then we

compute the eigenvectors [35] of AT_{A as A}T_Av i = μivi..

We then multiply both sides by A to obtain AAT_Av

i = μiAvi. Here, Avi are the eigenvectors of AAT.

Subsequently, in order to calculate the projection of the face image onto the above space, we subtract the average face from the probe image. Suppose we consider that wi is the projection of the mean subtracted face

onto the ith _{eigen face, we have the projection coefficients of the face as u=[w}

1, w2,...wN]. We now use

Mahalanobis Cosine Distance to measure the similarity between the aforementioned projection coefficients. The image space is spanned by the eigenfaces and the eigenvalues that correspond to the variance along each

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Eigenface. It is necessary to know the transformation between the image space and the Mahalanobis space before computing MAHCOS and furthermore, the Mahalanobis space consists of unit variance along each dimension. Consider that the Eigenspace has two vectors u and v and let μi = σ2i be the variance along the ith dimension. Furthermore, if m and n are the vectors that correspond in the Mahalanobis space, then the relationship between them can be defined as follows [35]:

m

i

=

and n

i

=

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Mahalanobis cosine (MAHCOS) is generally defined as the cosine of the angle between the projections of the images onto the Mahalanobis space. Hence the cosine Mahalanobis distance between u and v can be calculated in terms of m and n as follows:

D

MahCosine(u,v)

= cos (Θ

mn

) =

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3.Methodology

Fig.1.Framework for the Applied Methodology

The sequence of steps followed in our approach is illustrated in Fig .1. Initially, we load the images from the ORL database and construct a Gabor filter bank of 40 filters (8 orientation x 5 scales) to extract the Gabor magnitude features. Subsequently we partition the data into training, evaluation and verification sets i.e. the first three images of each subject serve as the training/gallery set, the next three images serve as the evaluation set and the remaining form the verification set. The next step involves computation of the training, evaluation and verification feature vectors using the PCA and KPCA methods for dimensionality reduction. For both PCA and KPCA, we compute the subspace using the training data, which comprised of 120 samples (images) with 10240 variables (pixels). After the subspace construction, we perform evaluation and test image projection. Finally, we compute the matching scores between the training feature vectors and evaluation feature vectors using the Mahalanobis Similarity Measure (MAHCOS) [34]. Subsequently, we perform the same for the verification features and based on the results, we determine whether there is a match. The two input structures were used solely for producing verification rates at certain thresholds. This produced verification rates at certain operating points, as well as Expected Performance Curve (EPC) [39] data. In terms of computation, the first input structure is used for threshold calculation and the second for verification rate calculation.

4.Experimental Design

The various evaluation metrics that are employed to quantify our results along with the particulars of the experimental setup are outlined in this section.

4.1 Evaluation Metrics

The efficacy of the proposed technique is demonstrated in terms of metrics such as: ROC, CMC, EPC and DET curves. Receiver Operating Curves (ROC) is a widely employed metric that emphasizes the need to increase the number of correct positives, while minimizing the number of false positives. ROC is well suited to evaluate

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classifiers, as the false detection rate is well defined [15] [16]. Additionally, the performance of a closed-set identification system can be summarized using the Cumulative Match Characteristic (CMC) curve [16] obtained by the ranking of match scores on a per-identity basis and is used as a reliability measure for FR systems. The Expected Performance Curves (EPC) [39] is another popularly adopted metric, as it is known to provide unbiased estimates of performance at various operating points. Finally, the Detection Error Trade-off (DET) Curves [30] are similarly a widely adopted means of representing performance on detection tasks that involve a tradeoff of error types and are preferred due to its ability to produce approximately linear curves. The feature matching has been carried out using different distance classifiers such as Euclidean [39], Cosine [40], City-Block [39] and Mahalanobis Cosine (MAHCOS) [34] [35] in order to establish the validity of our results.

4.2 Experimental Setup

The approaches has been tested and compared exhaustively on the ORL database using three sets: Training set, Evaluation set and the Verification set. The first three images of a subject were used for the training set. Subsequently, the next three images were used for the Evaluation set, which was used to gauge how well the system’s original intended goals have been achieved i.e. to determine that for a given input, the expected outcome achieved. Finally, the remaining images are used for the verification of the algorithm by the user set i.e. to compare the probe set against the gallery set. The experimentations were conducted on the publicly available benchmark ORL [33] database to compare the efficacy of Gabor-PCA and Gabor-KPCA techniques. 5.Results and Comparison

5.1GABOR-PCA Results 5.1.1 Using Euclidean Distance

In the identification experiments, the rank one recognition rate of the GABOR-PCA experiments equalled 87.50%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 5.00%, the minimal HTER (half total error rate) was 3.55% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 92.50%, 83.33% and 0.83% respectively. Finally the verification experiments on the test data (pre-set thresholds on the evaluation data) yielded verification rates at 1% and 0.1% FAR on the test set as 81.25% and 71.88% respectively.

5.1.2 Using Cosine Distance

In the identification experiments, the rank one recognition rate of the GABOR-PCA experiments equalled 87.50%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 4.13%, the minimal HTER (half total error rate) was 3.44% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 90.83%, 82.50% and 0.83% respectively. Finally the verification experiments on the test data (pre-set thresholds on the evaluation data) yielded verification rates at 1% and 0.1% FAR on the test set as 85.63% and 75.00% respectively.

5.1.3 Using City-Block Distance

In the identification experiments, the rank one recognition rate of the GABOR-PCA experiments equalled 80.83%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 5.84%, the minimal HTER (half total error rate) was 4.03% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 88.33%, 75.83% and 0.83% respectively. Finally the verification

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experiments on the test data (pre-set thresholds on the evaluation data) yielded verification rates at 1% and 0.1% FAR on the test set as 75.63% and 56.25% respectively.

5.1.4 Using Mahalanobis Cosine (MAHCOS) Distance

In the identification experiments, the rank one recognition rate of the GABOR-PCA experiments equalled 75.83%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 2.50%, the minimal HTER (half total error rate) was 1.57% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 97.50%, 92.50% and 0.83% respectively. Finally the verification experiments on the test data (preset thresholds on evaluation data) yielded verification rates at 1% and 0.1% FAR on the test set as 78.75% and 65.63% respectively.

The results are corroborated by the corresponding sample ROC (Fig.2), CMC (Fig.3), EPC (Fig.4) and DET (Fig.5) curves on the ORL dataset.

Fig.2 ROC Curve for Gabor-PCA on the ORL Dataset. Fig.3 CMC Curve for Gabor-PCA on the ORL Dataset.

Fig.4 EPC Curve for Gabor-PCA on the ORL Dataset Fig.5 DET Curve for Gabor-PCA on the ORL Dataset.

5.2 GABOR-KPCA Results 5.2.1 Using Euclidean Distance

In the identification experiments, the rank one recognition rate of the GABOR-KPCA experiments equalled 80.83%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 5.00%, the minimal HTER (half total error rate) was 3.75% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 91.67%, 78.33% and 0.83% respectively. Finally the verification experiments on the test data (pre-set thresholds on evaluation data) yielded verification rates at 1% and 0.1% FAR on the test set as 76.25% and 57.50% respectively.

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5.2.2 Using Cosine Distance

5.2.3 Using City Block Distance

5.2.4 Using Mahalanobis Cosine Distance

In the identification experiments, the rank one recognition rate of the GABOR-KPCA experiments equalled 80.00%. The verification experiments on the evaluation data yielded the following: the EER (equal error rate) on the evaluation set was 4.06%, the minimal HTER (half total error rate) was 2.87% and the verification rates at 1%, 0.1% and 0.01% FAR was found to be 95.00%, 92.50% and 0.83% respectively. Finally the verification experiments on the test data (preset thresholds on evaluation data) yielded verification rates at 1% and 0.1% FAR on the test set as 75.63% and 58.13% respectively.

The results are corroborated by the corresponding sample ROC (Fig.6), CMC (Fig.7), EPC (Fig.8) and DET (Fig.9) curves on the ORL dataset.

Fig.6 ROC Curve for Gabor-KPCA on the ORL Dataset. Fig.7 CMC Curve for Gabor-KPCA on the ORL Dataset.

Fig.8 EPC Curve for Gabor-KPCA on the ORL Dataset. Fig.9 DET Curve for Gabor-KPCA on the ORL Dataset.

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5.3 Comparison

The results of our experiments elucidated that the linear GABOR-PCA variant outperformed the non-linear GABOR-KPCA by 6.67% on the Euclidean distance measure, 0.83% on the cosine measure, 12.00% using the City-Block distance measure and 4.17% using MAHCOS. The choice of the distance measure also factors in and in the case of the cosine measure, the variation was minimal and with the city-block measure, it was maximal. Our results suggest that the PCA variant is more effective than the KPCA variant when the GABOR feature extractor is employed.

6.Discussion and Future Work

This study aimed at proffering an in-depth comparison of the GABOR-PCA and GABOR-KPCA variants in order to ascertain the performance variation. The experimentations were conducted on the publicly available ORL database and the results were corroborated with the corresponding sample ROC, CMC, EPC and DET curves. The results demonstrated that the GABOR-PCA method outperformed GABOR-KPCA by 6.67% (Euclidean), 0.83% (Cosine), 12.00% (City Block) and 4.17% (MAHCOS). This upsurge is contrary to the general assumption that KPCA is by default a better choice, and suggests that developers should consider both PCA and KPCA approaches based on the given scenario, instead of directly opting for KPCA. The performance of an FR system varies based on the choice of feature extractor and for GABOR in particular, our findings convey that the PCA approach is more effective.

Future work is currently being steered towards performing several similar studies (to be conducted in the forthcoming academic year) on the performance variation of other closely related techniques. Our immediate focus area is the investigation of the difference between the GABOR-LPP (GABOR using Locality Preserving Projection) and GABOR-KLPP (GABOR using Kernel Locality Preserving Projection) approaches.

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